[{"data":1,"prerenderedAt":15588},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere":3},{"id":4,"title":5,"body":6,"description":15574,"draft":15575,"enableComment":15576,"extension":15577,"image":15549,"meta":15578,"navigation":15576,"onday":15579,"path":15580,"seo":15581,"stem":15582,"summary":15583,"tags":15584,"__hash__":15587},"blog\u002Fblog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere.md","Introduction of vMF Distribution on Hypersphere",{"type":7,"value":8,"toc":15548},"minimark",[9,144,149,227,462,465,638,641,898,901,1424,1503,1507,1777,1780,1884,1888,1893,1896,1899,2096,2099,2159,2162,2173,2176,2245,2249,2383,2385,2715,2804,2808,3275,3279,3312,3315,3404,3407,3502,3505,3509,3512,3515,3804,3807,3811,3814,4145,4177,4181,4184,4425,4499,4503,4539,4764,4865,4868,4968,4999,5076,5079,5578,5667,5918,6020,6075,6078,6199,6237,6413,6416,7101,7104,7107,7168,7349,7352,7591,7594,7757,7789,8068,8142,8145,8257,8260,8474,8506,8861,8863,9185,9188,9634,9637,9803,9805,10080,10158,10161,10395,10398,10582,10614,10750,10753,10785,10788,10911,11031,11035,11067,11099,11384,11387,11686,11689,11922,11954,12096,12098,12420,12423,12427,12433,12487,12816,12864,12867,12871,12877,12880,13246,13249,14027,14030,14033,14037,14040,14043,14518,14521,14554,14585,14805,14808,15010,15013,15046,15077,15252,15255,15346,15349],[10,11,12,13,61,62,138,139,143],"p",{},"The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere. In Euclidean space, a normal distribution is typically parameterized by a mean ",[14,15,19,42],"span",{"className":16,"translate":18},[17],"katex","no",[14,20,23],{"className":21},[22],"katex-mathml",[24,25,27],"math",{"xmlns":26},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[28,29,30,37],"semantics",{},[31,32,33],"mrow",{},[34,35,36],"mi",{},"μ",[38,39,41],"annotation",{"encoding":40},"application\u002Fx-tex","\\mu",[14,43,47],{"className":44,"ariaHidden":46},[45],"katex-html","true",[14,48,51,56],{"className":49},[50],"base",[14,52],{"className":53,"style":55},[54],"strut","height:0.625em;vertical-align:-0.1944em;",[14,57,36],{"className":58},[59,60],"mord","mathnormal"," and variance ",[14,63,65,87],{"className":64,"translate":18},[17],[14,66,68],{"className":67},[22],[24,69,70],{"xmlns":26},[28,71,72,84],{},[31,73,74],{},[75,76,77,80],"msup",{},[34,78,79],{},"σ",[81,82,83],"mn",{},"2",[38,85,86],{"encoding":40},"\\sigma^2",[14,88,90],{"className":89,"ariaHidden":46},[45],[14,91,93,97],{"className":92},[50],[14,94],{"className":95,"style":96},[54],"height:0.8141em;",[14,98,100,104],{"className":99},[59],[14,101,79],{"className":102,"style":103},[59,60],"margin-right:0.03588em;",[14,105,108],{"className":106},[107],"msupsub",[14,109,112],{"className":110},[111],"vlist-t",[14,113,116],{"className":114},[115],"vlist-r",[14,117,120],{"className":118,"style":96},[119],"vlist",[14,121,123,128],{"style":122},"top:-3.063em;margin-right:0.05em;",[14,124],{"className":125,"style":127},[126],"pstrut","height:2.7em;",[14,129,135],{"className":130},[131,132,133,134],"sizing","reset-size6","size3","mtight",[14,136,83],{"className":137},[59,134],", and in one dimension its density forms the familiar bell-shaped curve. Just as the normal distribution is the distribution with the maximum entropy given the mean and variance constraints, the vMF distribution is the distribution with the maximum entropy on a sphere given the average direction constraint. According to the ",[140,141,142],"em",{},"Central Limit Theorem",": the limiting distribution of a random walk or accumulated direction vector on a sphere will also tend towards the vMF distribution.",[145,146,148],"h2",{"id":147},"basic-concepts","Basic Concepts",[10,150,151,152,226],{},"Domain: 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sphere):",[14,642,644],{"className":643,"translate":18},[230],[14,645,647,715],{"className":646,"translate":18},[17],[14,648,650],{"className":649},[22],[24,651,652],{"xmlns":26,"display":239},[28,653,654,712],{},[31,655,656,659,661,663,666,668,671,673,675,677,685,687,689,691,694,697,699,701,708,710],{},[34,657,658],{},"f",[165,660,168],{"stretchy":167},[34,662,267],{"mathvariant":266},[165,664,665],{},"∣",[34,667,36],{"mathvariant":486},[165,669,670],{"separator":46},",",[34,672,596],{},[165,674,180],{"stretchy":167},[165,676,260],{},[678,679,680,683],"msub",{},[34,681,682],{},"C",[34,684,171],{},[165,686,168],{"stretchy":167},[34,688,596],{},[165,690,180],{"stretchy":167},[34,692,693],{},"exp",[165,695,696],{},"⁡",[165,698,168],{"stretchy":167},[34,700,596],{},[75,702,703,705],{},[34,704,36],{"mathvariant":486},[34,706,707],{"mathvariant":283},"⊤",[34,709,267],{"mathvariant":266},[165,711,180],{"stretchy":167},[38,713,714],{"encoding":40},"f(\\mathbf{x} \\mid \\boldsymbol{\\mu}, 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\\mathbf{x})",[14,716,718,743,781],{"className":717,"ariaHidden":46},[45],[14,719,721,724,728,731,734,737,740],{"className":720},[50],[14,722],{"className":723,"style":193},[54],[14,725,658],{"className":726,"style":727},[59,60],"margin-right:0.10764em;",[14,729,168],{"className":730},[197],[14,732,267],{"className":733},[59,372],[14,735],{"className":736,"style":352},[204],[14,738,665],{"className":739},[356],[14,741],{"className":742,"style":352},[204],[14,744,746,749,758,762,766,769,772,775,778],{"className":745},[50],[14,747],{"className":748,"style":193},[54],[14,750,752],{"className":751},[59],[14,753,755],{"className":754},[59],[14,756,36],{"className":757},[59,523],[14,759,670],{"className":760},[761],"mpunct",[14,763],{"className":764,"style":765},[204],"margin-right:0.1667em;",[14,767,596],{"className":768},[59,60],[14,770,180],{"className":771},[225],[14,773],{"className":774,"style":352},[204],[14,776,260],{"className":777},[356],[14,779],{"className":780,"style":352},[204],[14,782,784,788,835,838,841,844,847,851,854,857,892,895],{"className":783},[50],[14,785],{"className":786,"style":787},[54],"height:1.1491em;vertical-align:-0.25em;",[14,789,791,795],{"className":790},[59],[14,792,682],{"className":793,"style":794},[59,60],"margin-right:0.07153em;",[14,796,798],{"className":797},[107],[14,799,802,826],{"className":800},[111,801],"vlist-t2",[14,803,805,821],{"className":804},[115],[14,806,809],{"className":807,"style":808},[119],"height:0.3361em;",[14,810,812,815],{"style":811},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[14,813],{"className":814,"style":127},[126],[14,816,818],{"className":817},[131,132,133,134],[14,819,171],{"className":820},[59,60,134],[14,822,825],{"className":823},[824],"vlist-s","​",[14,827,829],{"className":828},[115],[14,830,833],{"className":831,"style":832},[119],"height:0.15em;",[14,834],{},[14,836,168],{"className":837},[197],[14,839,596],{"className":840},[59,60],[14,842,180],{"className":843},[225],[14,845],{"className":846,"style":765},[204],[14,848,693],{"className":849},[850],"mop",[14,852,168],{"className":853},[197],[14,855,596],{"className":856},[59,60],[14,858,860,869],{"className":859},[59],[14,861,863],{"className":862},[59],[14,864,866],{"className":865},[59],[14,867,36],{"className":868},[59,523],[14,870,872],{"className":871},[107],[14,873,875],{"className":874},[111],[14,876,878],{"className":877},[115],[14,879,881],{"className":880,"style":308},[119],[14,882,883,886],{"style":330},[14,884],{"className":885,"style":127},[126],[14,887,889],{"className":888},[131,132,133,134],[14,890,707],{"className":891},[59,134],[14,893,267],{"className":894},[59,372],[14,896,180],{"className":897},[225],[10,899,900],{},"where",[14,902,904],{"className":903,"translate":18},[230],[14,905,907,997],{"className":906,"translate":18},[17],[14,908,910],{"className":909},[22],[24,911,912],{"xmlns":26,"display":239},[28,913,914,994],{},[31,915,916,922,924,926,928,930],{},[678,917,918,920],{},[34,919,682],{},[34,921,171],{},[165,923,168],{"stretchy":167},[34,925,596],{},[165,927,180],{"stretchy":167},[165,929,260],{},[931,932,933,949],"mfrac",{},[75,934,935,937],{},[34,936,596],{},[31,938,939,945,947],{},[931,940,941,943],{},[34,942,171],{},[81,944,83],{},[165,946,174],{},[81,948,177],{},[31,950,951,953,955,958,971,988,990,992],{},[165,952,168],{"stretchy":167},[81,954,83],{},[34,956,957],{},"π",[75,959,960,962],{},[165,961,180],{"stretchy":167},[31,963,964,966,969],{},[34,965,171],{},[34,967,968],{"mathvariant":283},"\u002F",[81,970,83],{},[678,972,973,976],{},[34,974,975],{},"I",[31,977,978,984,986],{},[931,979,980,982],{},[34,981,171],{},[81,983,83],{},[165,985,174],{},[81,987,177],{},[165,989,168],{"stretchy":167},[34,991,596],{},[165,993,180],{"stretchy":167},[38,995,996],{"encoding":40},"C_d(\\kappa) = \\frac{\\kappa^{\\frac{d}{2} - 1}}{(2\\pi)^{d\u002F2} I_{\\frac{d}{2} - 1}(\\kappa)}",[14,998,1000,1064],{"className":999,"ariaHidden":46},[45],[14,1001,1003,1006,1046,1049,1052,1055,1058,1061],{"className":1002},[50],[14,1004],{"className":1005,"style":193},[54],[14,1007,1009,1012],{"className":1008},[59],[14,1010,682],{"className":1011,"style":794},[59,60],[14,1013,1015],{"className":1014},[107],[14,1016,1018,1038],{"className":1017},[111,801],[14,1019,1021,1035],{"className":1020},[115],[14,1022,1024],{"className":1023,"style":808},[119],[14,1025,1026,1029],{"style":811},[14,1027],{"className":1028,"style":127},[126],[14,1030,1032],{"className":1031},[131,132,133,134],[14,1033,171],{"className":1034},[59,60,134],[14,1036,825],{"className":1037},[824],[14,1039,1041],{"className":1040},[115],[14,1042,1044],{"className":1043,"style":832},[119],[14,1045],{},[14,1047,168],{"className":1048},[197],[14,1050,596],{"className":1051},[59,60],[14,1053,180],{"className":1054},[225],[14,1056],{"className":1057,"style":352},[204],[14,1059,260],{"className":1060},[356],[14,1062],{"className":1063,"style":352},[204],[14,1065,1067,1071],{"className":1066},[50],[14,1068],{"className":1069,"style":1070},[54],"height:2.872em;vertical-align:-1.216em;",[14,1072,1074,1078,1421],{"className":1073},[59],[14,1075],{"className":1076},[197,1077],"nulldelimiter",[14,1079,1081],{"className":1080},[931],[14,1082,1084,1412],{"className":1083},[111,801],[14,1085,1087,1409],{"className":1086},[115],[14,1088,1091,1285,1295],{"className":1089,"style":1090},[119],"height:1.656em;",[14,1092,1094,1098],{"style":1093},"top:-2.296em;",[14,1095],{"className":1096,"style":1097},[126],"height:3em;",[14,1099,1101,1104,1107,1110,1148,1276,1279,1282],{"className":1100},[59],[14,1102,168],{"className":1103},[197],[14,1105,83],{"className":1106},[59],[14,1108,957],{"className":1109,"style":103},[59,60],[14,1111,1113,1116],{"className":1112},[225],[14,1114,180],{"className":1115},[225],[14,1117,1119],{"className":1118},[107],[14,1120,1122],{"className":1121},[111],[14,1123,1125],{"className":1124},[115],[14,1126,1129],{"className":1127,"style":1128},[119],"height:0.814em;",[14,1130,1132,1135],{"style":1131},"top:-2.989em;margin-right:0.05em;",[14,1133],{"className":1134,"style":127},[126],[14,1136,1138],{"className":1137},[131,132,133,134],[14,1139,1141,1144],{"className":1140},[59,134],[14,1142,171],{"className":1143},[59,60,134],[14,1145,1147],{"className":1146},[59,134],"\u002F2",[14,1149,1151,1155],{"className":1150},[59],[14,1152,975],{"className":1153,"style":1154},[59,60],"margin-right:0.07847em;",[14,1156,1158],{"className":1157},[107],[14,1159,1161,1267],{"className":1160},[111,801],[14,1162,1164,1264],{"className":1163},[115],[14,1165,1168],{"className":1166,"style":1167},[119],"height:0.3448em;",[14,1169,1171,1174],{"style":1170},"top:-2.7288em;margin-left:-0.0785em;margin-right:0.05em;",[14,1172],{"className":1173,"style":1097},[126],[14,1175,1177],{"className":1176},[131,132,133,134],[14,1178,1180,1258,1261],{"className":1179},[59,134],[14,1181,1183,1188,1255],{"className":1182},[59,134],[14,1184],{"className":1185},[197,1077,131,1186,1187],"reset-size3","size6",[14,1189,1191],{"className":1190},[931],[14,1192,1194,1246],{"className":1193},[111,801],[14,1195,1197,1243],{"className":1196},[115],[14,1198,1201,1217,1228],{"className":1199,"style":1200},[119],"height:0.88em;",[14,1202,1204,1207],{"style":1203},"top:-2.656em;",[14,1205],{"className":1206,"style":1097},[126],[14,1208,1211],{"className":1209},[131,1186,1210,134],"size1",[14,1212,1214],{"className":1213},[59,134],[14,1215,83],{"className":1216},[59,134],[14,1218,1220,1223],{"style":1219},"top:-3.2255em;",[14,1221],{"className":1222,"style":1097},[126],[14,1224],{"className":1225,"style":1227},[1226,134],"frac-line","border-bottom-width:0.049em;",[14,1229,1231,1234],{"style":1230},"top:-3.384em;",[14,1232],{"className":1233,"style":1097},[126],[14,1235,1237],{"className":1236},[131,1186,1210,134],[14,1238,1240],{"className":1239},[59,134],[14,1241,171],{"className":1242},[59,60,134],[14,1244,825],{"className":1245},[824],[14,1247,1249],{"className":1248},[115],[14,1250,1253],{"className":1251,"style":1252},[119],"height:0.344em;",[14,1254],{},[14,1256],{"className":1257},[225,1077,131,1186,1187],[14,1259,174],{"className":1260},[209,134],[14,1262,177],{"className":1263},[59,134],[14,1265,825],{"className":1266},[824],[14,1268,1270],{"className":1269},[115],[14,1271,1274],{"className":1272,"style":1273},[119],"height:0.512em;",[14,1275],{},[14,1277,168],{"className":1278},[197],[14,1280,596],{"className":1281},[59,60],[14,1283,180],{"className":1284},[225],[14,1286,1288,1291],{"style":1287},"top:-3.23em;",[14,1289],{"className":1290,"style":1097},[126],[14,1292],{"className":1293,"style":1294},[1226],"border-bottom-width:0.04em;",[14,1296,1298,1301],{"style":1297},"top:-3.677em;",[14,1299],{"className":1300,"style":1097},[126],[14,1302,1304],{"className":1303},[59],[14,1305,1307,1310],{"className":1306},[59],[14,1308,596],{"className":1309},[59,60],[14,1311,1313],{"className":1312},[107],[14,1314,1316],{"className":1315},[111],[14,1317,1319],{"className":1318},[115],[14,1320,1323],{"className":1321,"style":1322},[119],"height:0.979em;",[14,1324,1326,1329],{"style":1325},"top:-3.363em;margin-right:0.05em;",[14,1327],{"className":1328,"style":1097},[126],[14,1330,1332],{"className":1331},[131,132,133,134],[14,1333,1335,1403,1406],{"className":1334},[59,134],[14,1336,1338,1341,1400],{"className":1337},[59,134],[14,1339],{"className":1340},[197,1077,131,1186,1187],[14,1342,1344],{"className":1343},[931],[14,1345,1347,1392],{"className":1346},[111,801],[14,1348,1350,1389],{"className":1349},[115],[14,1351,1353,1367,1375],{"className":1352,"style":1200},[119],[14,1354,1355,1358],{"style":1203},[14,1356],{"className":1357,"style":1097},[126],[14,1359,1361],{"className":1360},[131,1186,1210,134],[14,1362,1364],{"className":1363},[59,134],[14,1365,83],{"className":1366},[59,134],[14,1368,1369,1372],{"style":1219},[14,1370],{"className":1371,"style":1097},[126],[14,1373],{"className":1374,"style":1227},[1226,134],[14,1376,1377,1380],{"style":1230},[14,1378],{"className":1379,"style":1097},[126],[14,1381,1383],{"className":1382},[131,1186,1210,134],[14,1384,1386],{"className":1385},[59,134],[14,1387,171],{"className":1388},[59,60,134],[14,1390,825],{"className":1391},[824],[14,1393,1395],{"className":1394},[115],[14,1396,1398],{"className":1397,"style":1252},[119],[14,1399],{},[14,1401],{"className":1402},[225,1077,131,1186,1187],[14,1404,174],{"className":1405},[209,134],[14,1407,177],{"className":1408},[59,134],[14,1410,825],{"className":1411},[824],[14,1413,1415],{"className":1414},[115],[14,1416,1419],{"className":1417,"style":1418},[119],"height:1.216em;",[14,1420],{},[14,1422],{"className":1423},[225,1077],[10,1425,1426,1427,1502],{},"and ",[14,1428,1430,1449],{"className":1429,"translate":18},[17],[14,1431,1433],{"className":1432},[22],[24,1434,1435],{"xmlns":26},[28,1436,1437,1446],{},[31,1438,1439],{},[678,1440,1441,1443],{},[34,1442,975],{},[34,1444,1445],{},"ν",[38,1447,1448],{"encoding":40},"I_\\nu",[14,1450,1452],{"className":1451,"ariaHidden":46},[45],[14,1453,1455,1459],{"className":1454},[50],[14,1456],{"className":1457,"style":1458},[54],"height:0.8333em;vertical-align:-0.15em;",[14,1460,1462,1465],{"className":1461},[59],[14,1463,975],{"className":1464,"style":1154},[59,60],[14,1466,1468],{"className":1467},[107],[14,1469,1471,1494],{"className":1470},[111,801],[14,1472,1474,1491],{"className":1473},[115],[14,1475,1478],{"className":1476,"style":1477},[119],"height:0.1514em;",[14,1479,1481,1484],{"style":1480},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[14,1482],{"className":1483,"style":127},[126],[14,1485,1487],{"className":1486},[131,132,133,134],[14,1488,1445],{"className":1489,"style":1490},[59,60,134],"margin-right:0.06366em;",[14,1492,825],{"className":1493},[824],[14,1495,1497],{"className":1496},[115],[14,1498,1500],{"className":1499,"style":832},[119],[14,1501],{}," is the modified Bessel function of the first kind.",[145,1504,1506],{"id":1505},"intuitive-understanding","Intuitive Understanding",[466,1508,1509,1564,1687],{},[469,1510,1511,1563],{},[14,1512,1514,1532],{"className":1513,"translate":18},[17],[14,1515,1517],{"className":1516},[22],[24,1518,1519],{"xmlns":26},[28,1520,1521,1529],{},[31,1522,1523,1525,1527],{},[34,1524,596],{},[165,1526,260],{},[81,1528,602],{},[38,1530,1531],{"encoding":40},"\\kappa = 0",[14,1533,1535,1554],{"className":1534,"ariaHidden":46},[45],[14,1536,1538,1542,1545,1548,1551],{"className":1537},[50],[14,1539],{"className":1540,"style":1541},[54],"height:0.4306em;",[14,1543,596],{"className":1544},[59,60],[14,1546],{"className":1547,"style":352},[204],[14,1549,260],{"className":1550},[356],[14,1552],{"className":1553,"style":352},[204],[14,1555,1557,1560],{"className":1556},[50],[14,1558],{"className":1559,"style":634},[54],[14,1561,602],{"className":1562},[59],": uniform distribution on the sphere (no directional preference)",[469,1565,1566,1619,1620,1656,1657,1686],{},[14,1567,1569,1588],{"className":1568,"translate":18},[17],[14,1570,1572],{"className":1571},[22],[24,1573,1574],{"xmlns":26},[28,1575,1576,1585],{},[31,1577,1578,1580,1583],{},[34,1579,596],{},[165,1581,1582],{},">",[81,1584,602],{},[38,1586,1587],{"encoding":40},"\\kappa > 0",[14,1589,1591,1610],{"className":1590,"ariaHidden":46},[45],[14,1592,1594,1598,1601,1604,1607],{"className":1593},[50],[14,1595],{"className":1596,"style":1597},[54],"height:0.5782em;vertical-align:-0.0391em;",[14,1599,596],{"className":1600},[59,60],[14,1602],{"className":1603,"style":352},[204],[14,1605,1582],{"className":1606},[356],[14,1608],{"className":1609,"style":352},[204],[14,1611,1613,1616],{"className":1612},[50],[14,1614],{"className":1615,"style":634},[54],[14,1617,602],{"className":1618},[59],": distribution concentrated around ",[14,1621,1623,1637],{"className":1622,"translate":18},[17],[14,1624,1626],{"className":1625},[22],[24,1627,1628],{"xmlns":26},[28,1629,1630,1634],{},[31,1631,1632],{},[34,1633,36],{"mathvariant":486},[38,1635,1636],{"encoding":40},"\\boldsymbol{\\mu}",[14,1638,1640],{"className":1639,"ariaHidden":46},[45],[14,1641,1643,1647],{"className":1642},[50],[14,1644],{"className":1645,"style":1646},[54],"height:0.6389em;vertical-align:-0.1944em;",[14,1648,1650],{"className":1649},[59],[14,1651,1653],{"className":1652},[59],[14,1654,36],{"className":1655},[59,523],", with larger ",[14,1658,1660,1674],{"className":1659,"translate":18},[17],[14,1661,1663],{"className":1662},[22],[24,1664,1665],{"xmlns":26},[28,1666,1667,1671],{},[31,1668,1669],{},[34,1670,596],{},[38,1672,1673],{"encoding":40},"\\kappa",[14,1675,1677],{"className":1676,"ariaHidden":46},[45],[14,1678,1680,1683],{"className":1679},[50],[14,1681],{"className":1682,"style":1541},[54],[14,1684,596],{"className":1685},[59,60]," implying higher concentration",[469,1688,1689,1742,1743],{},[14,1690,1692,1712],{"className":1691,"translate":18},[17],[14,1693,1695],{"className":1694},[22],[24,1696,1697],{"xmlns":26},[28,1698,1699,1709],{},[31,1700,1701,1703,1706],{},[34,1702,596],{},[165,1704,1705],{},"→",[34,1707,1708],{"mathvariant":283},"∞",[38,1710,1711],{"encoding":40},"\\kappa \\to \\infty",[14,1713,1715,1733],{"className":1714,"ariaHidden":46},[45],[14,1716,1718,1721,1724,1727,1730],{"className":1717},[50],[14,1719],{"className":1720,"style":1541},[54],[14,1722,596],{"className":1723},[59,60],[14,1725],{"className":1726,"style":352},[204],[14,1728,1705],{"className":1729},[356],[14,1731],{"className":1732,"style":352},[204],[14,1734,1736,1739],{"className":1735},[50],[14,1737],{"className":1738,"style":1541},[54],[14,1740,1708],{"className":1741},[59],": tends to a point mass at ",[14,1744,1746,1759],{"className":1745,"translate":18},[17],[14,1747,1749],{"className":1748},[22],[24,1750,1751],{"xmlns":26},[28,1752,1753,1757],{},[31,1754,1755],{},[34,1756,36],{"mathvariant":486},[38,1758,1636],{"encoding":40},[14,1760,1762],{"className":1761,"ariaHidden":46},[45],[14,1763,1765,1768],{"className":1764},[50],[14,1766],{"className":1767,"style":1646},[54],[14,1769,1771],{"className":1770},[59],[14,1772,1774],{"className":1773},[59],[14,1775,36],{"className":1776},[59,523],[10,1778,1779],{},"Thus, the vMF distribution provides an analogy:",[466,1781,1782,1847],{},[469,1783,1784,1785,1846],{},"In ",[14,1786,1788,1806],{"className":1787,"translate":18},[17],[14,1789,1791],{"className":1790},[22],[24,1792,1793],{"xmlns":26},[28,1794,1795,1803],{},[31,1796,1797],{},[75,1798,1799,1801],{},[34,1800,275],{"mathvariant":248},[34,1802,10],{},[38,1804,1805],{"encoding":40},"\\mathbb{R}^p",[14,1807,1809],{"className":1808,"ariaHidden":46},[45],[14,1810,1812,1816],{"className":1811},[50],[14,1813],{"className":1814,"style":1815},[54],"height:0.6889em;",[14,1817,1819,1822],{"className":1818},[59],[14,1820,275],{"className":1821},[59,315],[14,1823,1825],{"className":1824},[107],[14,1826,1828],{"className":1827},[111],[14,1829,1831],{"className":1830},[115],[14,1832,1835],{"className":1833,"style":1834},[119],"height:0.6644em;",[14,1836,1837,1840],{"style":122},[14,1838],{"className":1839,"style":127},[126],[14,1841,1843],{"className":1842},[131,132,133,134],[14,1844,10],{"className":1845},[59,60,134],", the normal distribution controls concentration via the precision matrix",[469,1848,1849,1850],{},"On the sphere, the vMF distribution controls concentration around ",[14,1851,1853,1866],{"className":1852,"translate":18},[17],[14,1854,1856],{"className":1855},[22],[24,1857,1858],{"xmlns":26},[28,1859,1860,1864],{},[31,1861,1862],{},[34,1863,36],{"mathvariant":486},[38,1865,1636],{"encoding":40},[14,1867,1869],{"className":1868,"ariaHidden":46},[45],[14,1870,1872,1875],{"className":1871},[50],[14,1873],{"className":1874,"style":1646},[54],[14,1876,1878],{"className":1877},[59],[14,1879,1881],{"className":1880},[59],[14,1882,36],{"className":1883},[59,523],[145,1885,1887],{"id":1886},"properties","Properties",[1889,1890,1892],"h3",{"id":1891},"exponential-family-property","Exponential Family Property",[10,1894,1895],{},"The vMF distribution belongs to the exponential family.",[10,1897,1898],{},"Its form is:",[14,1900,1902],{"className":1901,"translate":18},[230],[14,1903,1905,1969],{"className":1904,"translate":18},[17],[14,1906,1908],{"className":1907},[22],[24,1909,1910],{"xmlns":26,"display":239},[28,1911,1912,1966],{},[31,1913,1914,1916,1918,1920,1922,1925,1927,1929,1932,1934,1936,1938,1940,1942,1944,1951,1953,1955,1958,1960,1962,1964],{},[34,1915,658],{},[165,1917,168],{"stretchy":167},[34,1919,267],{},[34,1921,665],{"mathvariant":283},[34,1923,1924],{},"θ",[165,1926,180],{"stretchy":167},[165,1928,260],{},[34,1930,1931],{},"h",[165,1933,168],{"stretchy":167},[34,1935,267],{},[165,1937,180],{"stretchy":167},[34,1939,693],{},[165,1941,696],{},[165,1943,168],{"stretchy":167},[75,1945,1946,1948],{},[34,1947,1924],{},[34,1949,1950],{},"T",[34,1952,267],{},[165,1954,174],{},[34,1956,1957],{},"A",[165,1959,168],{"stretchy":167},[34,1961,1924],{},[165,1963,180],{"stretchy":167},[165,1965,180],{"stretchy":167},[38,1967,1968],{"encoding":40},"f(x|\\theta) = h(x)\\exp(\\theta^T x - A(\\theta))",[14,1970,1972,2006,2077],{"className":1971,"ariaHidden":46},[45],[14,1973,1975,1978,1981,1984,1987,1990,1994,1997,2000,2003],{"className":1974},[50],[14,1976],{"className":1977,"style":193},[54],[14,1979,658],{"className":1980,"style":727},[59,60],[14,1982,168],{"className":1983},[197],[14,1985,267],{"className":1986},[59,60],[14,1988,665],{"className":1989},[59],[14,1991,1924],{"className":1992,"style":1993},[59,60],"margin-right:0.02778em;",[14,1995,180],{"className":1996},[225],[14,1998],{"className":1999,"style":352},[204],[14,2001,260],{"className":2002},[356],[14,2004],{"className":2005,"style":352},[204],[14,2007,2009,2013,2016,2019,2022,2025,2028,2031,2034,2065,2068,2071,2074],{"className":2008},[50],[14,2010],{"className":2011,"style":2012},[54],"height:1.1413em;vertical-align:-0.25em;",[14,2014,1931],{"className":2015},[59,60],[14,2017,168],{"className":2018},[197],[14,2020,267],{"className":2021},[59,60],[14,2023,180],{"className":2024},[225],[14,2026],{"className":2027,"style":765},[204],[14,2029,693],{"className":2030},[850],[14,2032,168],{"className":2033},[197],[14,2035,2037,2040],{"className":2036},[59],[14,2038,1924],{"className":2039,"style":1993},[59,60],[14,2041,2043],{"className":2042},[107],[14,2044,2046],{"className":2045},[111],[14,2047,2049],{"className":2048},[115],[14,2050,2053],{"className":2051,"style":2052},[119],"height:0.8913em;",[14,2054,2055,2058],{"style":330},[14,2056],{"className":2057,"style":127},[126],[14,2059,2061],{"className":2060},[131,132,133,134],[14,2062,1950],{"className":2063,"style":2064},[59,60,134],"margin-right:0.13889em;",[14,2066,267],{"className":2067},[59,60],[14,2069],{"className":2070,"style":205},[204],[14,2072,174],{"className":2073},[209],[14,2075],{"className":2076,"style":205},[204],[14,2078,2080,2083,2086,2089,2092],{"className":2079},[50],[14,2081],{"className":2082,"style":193},[54],[14,2084,1957],{"className":2085},[59,60],[14,2087,168],{"className":2088},[197],[14,2090,1924],{"className":2091,"style":1993},[59,60],[14,2093,2095],{"className":2094},[225],"))",[10,2097,2098],{},"with",[14,2100,2102],{"className":2101,"translate":18},[230],[14,2103,2105,2125],{"className":2104,"translate":18},[17],[14,2106,2108],{"className":2107},[22],[24,2109,2110],{"xmlns":26,"display":239},[28,2111,2112,2122],{},[31,2113,2114,2116,2118,2120],{},[34,2115,1924],{},[165,2117,260],{},[34,2119,596],{},[34,2121,36],{},[38,2123,2124],{"encoding":40},"\\theta = \\kappa \\mu",[14,2126,2128,2147],{"className":2127,"ariaHidden":46},[45],[14,2129,2131,2135,2138,2141,2144],{"className":2130},[50],[14,2132],{"className":2133,"style":2134},[54],"height:0.6944em;",[14,2136,1924],{"className":2137,"style":1993},[59,60],[14,2139],{"className":2140,"style":352},[204],[14,2142,260],{"className":2143},[356],[14,2145],{"className":2146,"style":352},[204],[14,2148,2150,2153,2156],{"className":2149},[50],[14,2151],{"className":2152,"style":55},[54],[14,2154,596],{"className":2155},[59,60],[14,2157,36],{"className":2158},[59,60],[10,2160,2161],{},"Hence, vMF enjoys standard exponential family properties:",[466,2163,2164,2167,2170],{},[469,2165,2166],{},"Existence of a sufficient statistic",[469,2168,2169],{},"Simple maximum likelihood estimation",[469,2171,2172],{},"Convenient for use in EM algorithms",[10,2174,2175],{},"Sufficient statistic:",[14,2177,2179],{"className":2178,"translate":18},[230],[14,2180,2182,2206],{"className":2181,"translate":18},[17],[14,2183,2185],{"className":2184},[22],[24,2186,2187],{"xmlns":26,"display":239},[28,2188,2189,2203],{},[31,2190,2191,2193,2195,2197,2199,2201],{},[34,2192,1950],{},[165,2194,168],{"stretchy":167},[34,2196,267],{},[165,2198,180],{"stretchy":167},[165,2200,260],{},[34,2202,267],{},[38,2204,2205],{"encoding":40},"T(x) = x",[14,2207,2209,2236],{"className":2208,"ariaHidden":46},[45],[14,2210,2212,2215,2218,2221,2224,2227,2230,2233],{"className":2211},[50],[14,2213],{"className":2214,"style":193},[54],[14,2216,1950],{"className":2217,"style":2064},[59,60],[14,2219,168],{"className":2220},[197],[14,2222,267],{"className":2223},[59,60],[14,2225,180],{"className":2226},[225],[14,2228],{"className":2229,"style":352},[204],[14,2231,260],{"className":2232},[356],[14,2234],{"className":2235,"style":352},[204],[14,2237,2239,2242],{"className":2238},[50],[14,2240],{"className":2241,"style":1541},[54],[14,2243,267],{"className":2244},[59,60],[1889,2246,2248],{"id":2247},"first-moment-mean","First Moment (Mean)",[14,2250,2252],{"className":2251,"translate":18},[230],[14,2253,2255,2294],{"className":2254,"translate":18},[17],[14,2256,2258],{"className":2257},[22],[24,2259,2260],{"xmlns":26,"display":239},[28,2261,2262,2291],{},[31,2263,2264,2267,2270,2272,2275,2277,2283,2285,2287,2289],{},[34,2265,2266],{},"E",[165,2268,2269],{"stretchy":167},"[",[34,2271,267],{},[165,2273,2274],{"stretchy":167},"]",[165,2276,260],{},[678,2278,2279,2281],{},[34,2280,1957],{},[34,2282,171],{},[165,2284,168],{"stretchy":167},[34,2286,596],{},[165,2288,180],{"stretchy":167},[34,2290,36],{},[38,2292,2293],{"encoding":40},"E[x] = A_d(\\kappa)\\mu",[14,2295,2297,2324],{"className":2296,"ariaHidden":46},[45],[14,2298,2300,2303,2306,2309,2312,2315,2318,2321],{"className":2299},[50],[14,2301],{"className":2302,"style":193},[54],[14,2304,2266],{"className":2305,"style":546},[59,60],[14,2307,2269],{"className":2308},[197],[14,2310,267],{"className":2311},[59,60],[14,2313,2274],{"className":2314},[225],[14,2316],{"className":2317,"style":352},[204],[14,2319,260],{"className":2320},[356],[14,2322],{"className":2323,"style":352},[204],[14,2325,2327,2330,2371,2374,2377,2380],{"className":2326},[50],[14,2328],{"className":2329,"style":193},[54],[14,2331,2333,2336],{"className":2332},[59],[14,2334,1957],{"className":2335},[59,60],[14,2337,2339],{"className":2338},[107],[14,2340,2342,2363],{"className":2341},[111,801],[14,2343,2345,2360],{"className":2344},[115],[14,2346,2348],{"className":2347,"style":808},[119],[14,2349,2351,2354],{"style":2350},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[14,2352],{"className":2353,"style":127},[126],[14,2355,2357],{"className":2356},[131,132,133,134],[14,2358,171],{"className":2359},[59,60,134],[14,2361,825],{"className":2362},[824],[14,2364,2366],{"className":2365},[115],[14,2367,2369],{"className":2368,"style":832},[119],[14,2370],{},[14,2372,168],{"className":2373},[197],[14,2375,596],{"className":2376},[59,60],[14,2378,180],{"className":2379},[225],[14,2381,36],{"className":2382},[59,60],[10,2384,900],{},[14,2386,2388],{"className":2387,"translate":18},[230],[14,2389,2391,2463],{"className":2390,"translate":18},[17],[14,2392,2394],{"className":2393},[22],[24,2395,2396],{"xmlns":26,"display":239},[28,2397,2398,2460],{},[31,2399,2400,2406,2408,2410,2412,2414],{},[678,2401,2402,2404],{},[34,2403,1957],{},[34,2405,171],{},[165,2407,168],{"stretchy":167},[34,2409,596],{},[165,2411,180],{"stretchy":167},[165,2413,260],{},[931,2415,2416,2436],{},[31,2417,2418,2430,2432,2434],{},[678,2419,2420,2422],{},[34,2421,975],{},[31,2423,2424,2426,2428],{},[34,2425,171],{},[34,2427,968],{"mathvariant":283},[81,2429,83],{},[165,2431,168],{"stretchy":167},[34,2433,596],{},[165,2435,180],{"stretchy":167},[31,2437,2438,2454,2456,2458],{},[678,2439,2440,2442],{},[34,2441,975],{},[31,2443,2444,2446,2448,2450,2452],{},[34,2445,171],{},[34,2447,968],{"mathvariant":283},[81,2449,83],{},[165,2451,174],{},[81,2453,177],{},[165,2455,168],{"stretchy":167},[34,2457,596],{},[165,2459,180],{"stretchy":167},[38,2461,2462],{"encoding":40},"A_d(\\kappa) = \\frac{I_{d\u002F2}(\\kappa)}{I_{d\u002F2-1}(\\kappa)}",[14,2464,2466,2530],{"className":2465,"ariaHidden":46},[45],[14,2467,2469,2472,2512,2515,2518,2521,2524,2527],{"className":2468},[50],[14,2470],{"className":2471,"style":193},[54],[14,2473,2475,2478],{"className":2474},[59],[14,2476,1957],{"className":2477},[59,60],[14,2479,2481],{"className":2480},[107],[14,2482,2484,2504],{"className":2483},[111,801],[14,2485,2487,2501],{"className":2486},[115],[14,2488,2490],{"className":2489,"style":808},[119],[14,2491,2492,2495],{"style":2350},[14,2493],{"className":2494,"style":127},[126],[14,2496,2498],{"className":2497},[131,132,133,134],[14,2499,171],{"className":2500},[59,60,134],[14,2502,825],{"className":2503},[824],[14,2505,2507],{"className":2506},[115],[14,2508,2510],{"className":2509,"style":832},[119],[14,2511],{},[14,2513,168],{"className":2514},[197],[14,2516,596],{"className":2517},[59,60],[14,2519,180],{"className":2520},[225],[14,2522],{"className":2523,"style":352},[204],[14,2525,260],{"className":2526},[356],[14,2528],{"className":2529,"style":352},[204],[14,2531,2533,2537],{"className":2532},[50],[14,2534],{"className":2535,"style":2536},[54],"height:2.5364em;vertical-align:-1.0412em;",[14,2538,2540,2543,2712],{"className":2539},[59],[14,2541],{"className":2542},[197,1077],[14,2544,2546],{"className":2545},[931],[14,2547,2549,2703],{"className":2548},[111,801],[14,2550,2552,2700],{"className":2551},[115],[14,2553,2556,2628,2636],{"className":2554,"style":2555},[119],"height:1.4952em;",[14,2557,2559,2562],{"style":2558},"top:-2.314em;",[14,2560],{"className":2561,"style":1097},[126],[14,2563,2565,2619,2622,2625],{"className":2564},[59],[14,2566,2568,2571],{"className":2567},[59],[14,2569,975],{"className":2570,"style":1154},[59,60],[14,2572,2574],{"className":2573},[107],[14,2575,2577,2610],{"className":2576},[111,801],[14,2578,2580,2607],{"className":2579},[115],[14,2581,2583],{"className":2582,"style":1167},[119],[14,2584,2586,2589],{"style":2585},"top:-2.5198em;margin-left:-0.0785em;margin-right:0.05em;",[14,2587],{"className":2588,"style":127},[126],[14,2590,2592],{"className":2591},[131,132,133,134],[14,2593,2595,2598,2601,2604],{"className":2594},[59,134],[14,2596,171],{"className":2597},[59,60,134],[14,2599,1147],{"className":2600},[59,134],[14,2602,174],{"className":2603},[209,134],[14,2605,177],{"className":2606},[59,134],[14,2608,825],{"className":2609},[824],[14,2611,2613],{"className":2612},[115],[14,2614,2617],{"className":2615,"style":2616},[119],"height:0.3552em;",[14,2618],{},[14,2620,168],{"className":2621},[197],[14,2623,596],{"className":2624},[59,60],[14,2626,180],{"className":2627},[225],[14,2629,2630,2633],{"style":1287},[14,2631],{"className":2632,"style":1097},[126],[14,2634],{"className":2635,"style":1294},[1226],[14,2637,2639,2642],{"style":2638},"top:-3.7452em;",[14,2640],{"className":2641,"style":1097},[126],[14,2643,2645,2691,2694,2697],{"className":2644},[59],[14,2646,2648,2651],{"className":2647},[59],[14,2649,975],{"className":2650,"style":1154},[59,60],[14,2652,2654],{"className":2653},[107],[14,2655,2657,2683],{"className":2656},[111,801],[14,2658,2660,2680],{"className":2659},[115],[14,2661,2663],{"className":2662,"style":1167},[119],[14,2664,2665,2668],{"style":2585},[14,2666],{"className":2667,"style":127},[126],[14,2669,2671],{"className":2670},[131,132,133,134],[14,2672,2674,2677],{"className":2673},[59,134],[14,2675,171],{"className":2676},[59,60,134],[14,2678,1147],{"className":2679},[59,134],[14,2681,825],{"className":2682},[824],[14,2684,2686],{"className":2685},[115],[14,2687,2689],{"className":2688,"style":2616},[119],[14,2690],{},[14,2692,168],{"className":2693},[197],[14,2695,596],{"className":2696},[59,60],[14,2698,180],{"className":2699},[225],[14,2701,825],{"className":2702},[824],[14,2704,2706],{"className":2705},[115],[14,2707,2710],{"className":2708,"style":2709},[119],"height:1.0412em;",[14,2711],{},[14,2713],{"className":2714},[225,1077],[10,2716,2717,2718,2803],{},"The function ",[14,2719,2721,2745],{"className":2720,"translate":18},[17],[14,2722,2724],{"className":2723},[22],[24,2725,2726],{"xmlns":26},[28,2727,2728,2742],{},[31,2729,2730,2736,2738,2740],{},[678,2731,2732,2734],{},[34,2733,1957],{},[34,2735,171],{},[165,2737,168],{"stretchy":167},[34,2739,596],{},[165,2741,180],{"stretchy":167},[38,2743,2744],{"encoding":40},"A_d(\\kappa)",[14,2746,2748],{"className":2747,"ariaHidden":46},[45],[14,2749,2751,2754,2794,2797,2800],{"className":2750},[50],[14,2752],{"className":2753,"style":193},[54],[14,2755,2757,2760],{"className":2756},[59],[14,2758,1957],{"className":2759},[59,60],[14,2761,2763],{"className":2762},[107],[14,2764,2766,2786],{"className":2765},[111,801],[14,2767,2769,2783],{"className":2768},[115],[14,2770,2772],{"className":2771,"style":808},[119],[14,2773,2774,2777],{"style":2350},[14,2775],{"className":2776,"style":127},[126],[14,2778,2780],{"className":2779},[131,132,133,134],[14,2781,171],{"className":2782},[59,60,134],[14,2784,825],{"className":2785},[824],[14,2787,2789],{"className":2788},[115],[14,2790,2792],{"className":2791,"style":832},[119],[14,2793],{},[14,2795,168],{"className":2796},[197],[14,2798,596],{"className":2799},[59,60],[14,2801,180],{"className":2802},[225]," is called the mean resultant length function.",[1889,2805,2807],{"id":2806},"second-moment","Second Moment",[14,2809,2811],{"className":2810,"translate":18},[230],[14,2812,2814,2903],{"className":2813,"translate":18},[17],[14,2815,2817],{"className":2816},[22],[24,2818,2819],{"xmlns":26,"display":239},[28,2820,2821,2900],{},[31,2822,2823,2825,2827,2829,2835,2837,2839,2857,2859,2862,2892,2894],{},[34,2824,2266],{},[165,2826,2269],{"stretchy":167},[34,2828,267],{},[75,2830,2831,2833],{},[34,2832,267],{},[34,2834,1950],{},[165,2836,2274],{"stretchy":167},[165,2838,260],{},[931,2840,2841,2855],{},[31,2842,2843,2849,2851,2853],{},[678,2844,2845,2847],{},[34,2846,1957],{},[34,2848,171],{},[165,2850,168],{"stretchy":167},[34,2852,596],{},[165,2854,180],{"stretchy":167},[34,2856,596],{},[34,2858,975],{},[165,2860,2861],{},"+",[31,2863,2864,2866,2868,2870,2890],{},[165,2865,168],{"fence":46},[81,2867,177],{},[165,2869,174],{},[931,2871,2872,2888],{},[31,2873,2874,2876,2882,2884,2886],{},[34,2875,171],{},[678,2877,2878,2880],{},[34,2879,1957],{},[34,2881,171],{},[165,2883,168],{"stretchy":167},[34,2885,596],{},[165,2887,180],{"stretchy":167},[34,2889,596],{},[165,2891,180],{"fence":46},[34,2893,36],{},[75,2895,2896,2898],{},[34,2897,36],{},[34,2899,1950],{},[38,2901,2902],{"encoding":40},"E[xx^T] = \\frac{A_d(\\kappa)}{\\kappa} I + \\left(1 - \\frac{dA_d(\\kappa)}{\\kappa}\\right)\\mu\\mu^T",[14,2904,2906,2962,3091],{"className":2905,"ariaHidden":46},[45],[14,2907,2909,2912,2915,2918,2921,2950,2953,2956,2959],{"className":2908},[50],[14,2910],{"className":2911,"style":2012},[54],[14,2913,2266],{"className":2914,"style":546},[59,60],[14,2916,2269],{"className":2917},[197],[14,2919,267],{"className":2920},[59,60],[14,2922,2924,2927],{"className":2923},[59],[14,2925,267],{"className":2926},[59,60],[14,2928,2930],{"className":2929},[107],[14,2931,2933],{"className":2932},[111],[14,2934,2936],{"className":2935},[115],[14,2937,2939],{"className":2938,"style":2052},[119],[14,2940,2941,2944],{"style":330},[14,2942],{"className":2943,"style":127},[126],[14,2945,2947],{"className":2946},[131,132,133,134],[14,2948,1950],{"className":2949,"style":2064},[59,60,134],[14,2951,2274],{"className":2952},[225],[14,2954],{"className":2955,"style":352},[204],[14,2957,260],{"className":2958},[356],[14,2960],{"className":2961,"style":352},[204],[14,2963,2965,2969,3079,3082,3085,3088],{"className":2964},[50],[14,2966],{"className":2967,"style":2968},[54],"height:2.113em;vertical-align:-0.686em;",[14,2970,2972,2975,3076],{"className":2971},[59],[14,2973],{"className":2974},[197,1077],[14,2976,2978],{"className":2977},[931],[14,2979,2981,3067],{"className":2980},[111,801],[14,2982,2984,3064],{"className":2983},[115],[14,2985,2988,2999,3007],{"className":2986,"style":2987},[119],"height:1.427em;",[14,2989,2990,2993],{"style":2558},[14,2991],{"className":2992,"style":1097},[126],[14,2994,2996],{"className":2995},[59],[14,2997,596],{"className":2998},[59,60],[14,3000,3001,3004],{"style":1287},[14,3002],{"className":3003,"style":1097},[126],[14,3005],{"className":3006,"style":1294},[1226],[14,3008,3009,3012],{"style":1297},[14,3010],{"className":3011,"style":1097},[126],[14,3013,3015,3055,3058,3061],{"className":3014},[59],[14,3016,3018,3021],{"className":3017},[59],[14,3019,1957],{"className":3020},[59,60],[14,3022,3024],{"className":3023},[107],[14,3025,3027,3047],{"className":3026},[111,801],[14,3028,3030,3044],{"className":3029},[115],[14,3031,3033],{"className":3032,"style":808},[119],[14,3034,3035,3038],{"style":2350},[14,3036],{"className":3037,"style":127},[126],[14,3039,3041],{"className":3040},[131,132,133,134],[14,3042,171],{"className":3043},[59,60,134],[14,3045,825],{"className":3046},[824],[14,3048,3050],{"className":3049},[115],[14,3051,3053],{"className":3052,"style":832},[119],[14,3054],{},[14,3056,168],{"className":3057},[197],[14,3059,596],{"className":3060},[59,60],[14,3062,180],{"className":3063},[225],[14,3065,825],{"className":3066},[824],[14,3068,3070],{"className":3069},[115],[14,3071,3074],{"className":3072,"style":3073},[119],"height:0.686em;",[14,3075],{},[14,3077],{"className":3078},[225,1077],[14,3080,975],{"className":3081,"style":1154},[59,60],[14,3083],{"className":3084,"style":205},[204],[14,3086,2861],{"className":3087},[209],[14,3089],{"className":3090,"style":205},[204],[14,3092,3094,3098,3240,3243,3246],{"className":3093},[50],[14,3095],{"className":3096,"style":3097},[54],"height:2.4em;vertical-align:-0.95em;",[14,3099,3102,3111,3114,3117,3120,3123,3234],{"className":3100},[3101],"minner",[14,3103,3107],{"className":3104,"style":3106},[197,3105],"delimcenter","top:0em;",[14,3108,168],{"className":3109},[3110,133],"delimsizing",[14,3112,177],{"className":3113},[59],[14,3115],{"className":3116,"style":205},[204],[14,3118,174],{"className":3119},[209],[14,3121],{"className":3122,"style":205},[204],[14,3124,3126,3129,3231],{"className":3125},[59],[14,3127],{"className":3128},[197,1077],[14,3130,3132],{"className":3131},[931],[14,3133,3135,3223],{"className":3134},[111,801],[14,3136,3138,3220],{"className":3137},[115],[14,3139,3141,3152,3160],{"className":3140,"style":2987},[119],[14,3142,3143,3146],{"style":2558},[14,3144],{"className":3145,"style":1097},[126],[14,3147,3149],{"className":3148},[59],[14,3150,596],{"className":3151},[59,60],[14,3153,3154,3157],{"style":1287},[14,3155],{"className":3156,"style":1097},[126],[14,3158],{"className":3159,"style":1294},[1226],[14,3161,3162,3165],{"style":1297},[14,3163],{"className":3164,"style":1097},[126],[14,3166,3168,3171,3211,3214,3217],{"className":3167},[59],[14,3169,171],{"className":3170},[59,60],[14,3172,3174,3177],{"className":3173},[59],[14,3175,1957],{"className":3176},[59,60],[14,3178,3180],{"className":3179},[107],[14,3181,3183,3203],{"className":3182},[111,801],[14,3184,3186,3200],{"className":3185},[115],[14,3187,3189],{"className":3188,"style":808},[119],[14,3190,3191,3194],{"style":2350},[14,3192],{"className":3193,"style":127},[126],[14,3195,3197],{"className":3196},[131,132,133,134],[14,3198,171],{"className":3199},[59,60,134],[14,3201,825],{"className":3202},[824],[14,3204,3206],{"className":3205},[115],[14,3207,3209],{"className":3208,"style":832},[119],[14,3210],{},[14,3212,168],{"className":3213},[197],[14,3215,596],{"className":3216},[59,60],[14,3218,180],{"className":3219},[225],[14,3221,825],{"className":3222},[824],[14,3224,3226],{"className":3225},[115],[14,3227,3229],{"className":3228,"style":3073},[119],[14,3230],{},[14,3232],{"className":3233},[225,1077],[14,3235,3237],{"className":3236,"style":3106},[225,3105],[14,3238,180],{"className":3239},[3110,133],[14,3241],{"className":3242,"style":765},[204],[14,3244,36],{"className":3245},[59,60],[14,3247,3249,3252],{"className":3248},[59],[14,3250,36],{"className":3251},[59,60],[14,3253,3255],{"className":3254},[107],[14,3256,3258],{"className":3257},[111],[14,3259,3261],{"className":3260},[115],[14,3262,3264],{"className":3263,"style":2052},[119],[14,3265,3266,3269],{"style":330},[14,3267],{"className":3268,"style":127},[126],[14,3270,3272],{"className":3271},[131,132,133,134],[14,3273,1950],{"className":3274,"style":2064},[59,60,134],[1889,3276,3278],{"id":3277},"rotation-invariance","Rotation Invariance",[10,3280,3281,3282,670],{},"For any orthogonal matrix ",[14,3283,3285,3298],{"className":3284,"translate":18},[17],[14,3286,3288],{"className":3287},[22],[24,3289,3290],{"xmlns":26},[28,3291,3292,3296],{},[31,3293,3294],{},[34,3295,275],{},[38,3297,275],{"encoding":40},[14,3299,3301],{"className":3300,"ariaHidden":46},[45],[14,3302,3304,3308],{"className":3303},[50],[14,3305],{"className":3306,"style":3307},[54],"height:0.6833em;",[14,3309,275],{"className":3310,"style":3311},[59,60],"margin-right:0.00773em;",[10,3313,3314],{},"if",[14,3316,3318],{"className":3317,"translate":18},[230],[14,3319,3321,3352],{"className":3320,"translate":18},[17],[14,3322,3324],{"className":3323},[22],[24,3325,3326],{"xmlns":26,"display":239},[28,3327,3328,3349],{},[31,3329,3330,3332,3335,3339,3341,3343,3345,3347],{},[34,3331,267],{},[165,3333,3334],{},"∼",[3336,3337,3338],"mtext",{},"vMF",[165,3340,168],{"stretchy":167},[34,3342,36],{},[165,3344,670],{"separator":46},[34,3346,596],{},[165,3348,180],{"stretchy":167},[38,3350,3351],{"encoding":40},"x \\sim \\text{vMF}(\\mu,\\kappa)",[14,3353,3355,3373],{"className":3354,"ariaHidden":46},[45],[14,3356,3358,3361,3364,3367,3370],{"className":3357},[50],[14,3359],{"className":3360,"style":1541},[54],[14,3362,267],{"className":3363},[59,60],[14,3365],{"className":3366,"style":352},[204],[14,3368,3334],{"className":3369},[356],[14,3371],{"className":3372,"style":352},[204],[14,3374,3376,3379,3386,3389,3392,3395,3398,3401],{"className":3375},[50],[14,3377],{"className":3378,"style":193},[54],[14,3380,3383],{"className":3381},[59,3382],"text",[14,3384,3338],{"className":3385},[59],[14,3387,168],{"className":3388},[197],[14,3390,36],{"className":3391},[59,60],[14,3393,670],{"className":3394},[761],[14,3396],{"className":3397,"style":765},[204],[14,3399,596],{"className":3400},[59,60],[14,3402,180],{"className":3403},[225],[10,3405,3406],{},"then",[14,3408,3410],{"className":3409,"translate":18},[230],[14,3411,3413,3445],{"className":3412,"translate":18},[17],[14,3414,3416],{"className":3415},[22],[24,3417,3418],{"xmlns":26,"display":239},[28,3419,3420,3442],{},[31,3421,3422,3424,3426,3428,3430,3432,3434,3436,3438,3440],{},[34,3423,275],{},[34,3425,267],{},[165,3427,3334],{},[3336,3429,3338],{},[165,3431,168],{"stretchy":167},[34,3433,275],{},[34,3435,36],{},[165,3437,670],{"separator":46},[34,3439,596],{},[165,3441,180],{"stretchy":167},[38,3443,3444],{"encoding":40},"Rx \\sim \\text{vMF}(R\\mu,\\kappa)",[14,3446,3448,3469],{"className":3447,"ariaHidden":46},[45],[14,3449,3451,3454,3457,3460,3463,3466],{"className":3450},[50],[14,3452],{"className":3453,"style":3307},[54],[14,3455,275],{"className":3456,"style":3311},[59,60],[14,3458,267],{"className":3459},[59,60],[14,3461],{"className":3462,"style":352},[204],[14,3464,3334],{"className":3465},[356],[14,3467],{"className":3468,"style":352},[204],[14,3470,3472,3475,3481,3484,3487,3490,3493,3496,3499],{"className":3471},[50],[14,3473],{"className":3474,"style":193},[54],[14,3476,3478],{"className":3477},[59,3382],[14,3479,3338],{"className":3480},[59],[14,3482,168],{"className":3483},[197],[14,3485,275],{"className":3486,"style":3311},[59,60],[14,3488,36],{"className":3489},[59,60],[14,3491,670],{"className":3492},[761],[14,3494],{"className":3495,"style":765},[204],[14,3497,596],{"className":3498},[59,60],[14,3500,180],{"className":3501},[225],[10,3503,3504],{},"This shows that the vMF distribution is invariant under the rotation group.",[1889,3506,3508],{"id":3507},"information-geometry-properties","Information Geometry Properties",[10,3510,3511],{},"The family of vMF distributions forms an information-geometric manifold.",[10,3513,3514],{},"Fisher information:",[14,3516,3518],{"className":3517,"translate":18},[230],[14,3519,3521,3581],{"className":3520,"translate":18},[17],[14,3522,3524],{"className":3523},[22],[24,3525,3526],{"xmlns":26,"display":239},[28,3527,3528,3578],{},[31,3529,3530,3532,3534,3536,3538,3540,3542,3561,3564,3566,3572,3574,3576],{},[34,3531,975],{},[165,3533,168],{"stretchy":167},[34,3535,596],{},[165,3537,180],{"stretchy":167},[165,3539,260],{},[165,3541,174],{},[931,3543,3544,3551],{},[75,3545,3546,3549],{},[34,3547,3548],{"mathvariant":283},"∂",[81,3550,83],{},[31,3552,3553,3555],{},[34,3554,3548],{"mathvariant":283},[75,3556,3557,3559],{},[34,3558,596],{},[81,3560,83],{},[34,3562,3563],{},"log",[165,3565,696],{},[678,3567,3568,3570],{},[34,3569,682],{},[34,3571,171],{},[165,3573,168],{"stretchy":167},[34,3575,596],{},[165,3577,180],{"stretchy":167},[38,3579,3580],{"encoding":40},"I(\\kappa) = -\\frac{\\partial^2}{\\partial \\kappa^2} \\log C_d(\\kappa)",[14,3582,3584,3611],{"className":3583,"ariaHidden":46},[45],[14,3585,3587,3590,3593,3596,3599,3602,3605,3608],{"className":3586},[50],[14,3588],{"className":3589,"style":193},[54],[14,3591,975],{"className":3592,"style":1154},[59,60],[14,3594,168],{"className":3595},[197],[14,3597,596],{"className":3598},[59,60],[14,3600,180],{"className":3601},[225],[14,3603],{"className":3604,"style":352},[204],[14,3606,260],{"className":3607},[356],[14,3609],{"className":3610,"style":352},[204],[14,3612,3614,3618,3621,3741,3744,3752,3755,3795,3798,3801],{"className":3613},[50],[14,3615],{"className":3616,"style":3617},[54],"height:2.1771em;vertical-align:-0.686em;",[14,3619,174],{"className":3620},[59],[14,3622,3624,3627,3738],{"className":3623},[59],[14,3625],{"className":3626},[197,1077],[14,3628,3630],{"className":3629},[931],[14,3631,3633,3730],{"className":3632},[111,801],[14,3634,3636,3727],{"className":3635},[115],[14,3637,3640,3682,3690],{"className":3638,"style":3639},[119],"height:1.4911em;",[14,3641,3642,3645],{"style":2558},[14,3643],{"className":3644,"style":1097},[126],[14,3646,3648,3652],{"className":3647},[59],[14,3649,3548],{"className":3650,"style":3651},[59],"margin-right:0.05556em;",[14,3653,3655,3658],{"className":3654},[59],[14,3656,596],{"className":3657},[59,60],[14,3659,3661],{"className":3660},[107],[14,3662,3664],{"className":3663},[111],[14,3665,3667],{"className":3666},[115],[14,3668,3671],{"className":3669,"style":3670},[119],"height:0.7401em;",[14,3672,3673,3676],{"style":1131},[14,3674],{"className":3675,"style":127},[126],[14,3677,3679],{"className":3678},[131,132,133,134],[14,3680,83],{"className":3681},[59,134],[14,3683,3684,3687],{"style":1287},[14,3685],{"className":3686,"style":1097},[126],[14,3688],{"className":3689,"style":1294},[1226],[14,3691,3692,3695],{"style":1297},[14,3693],{"className":3694,"style":1097},[126],[14,3696,3698],{"className":3697},[59],[14,3699,3701,3704],{"className":3700},[59],[14,3702,3548],{"className":3703,"style":3651},[59],[14,3705,3707],{"className":3706},[107],[14,3708,3710],{"className":3709},[111],[14,3711,3713],{"className":3712},[115],[14,3714,3716],{"className":3715,"style":96},[119],[14,3717,3718,3721],{"style":122},[14,3719],{"className":3720,"style":127},[126],[14,3722,3724],{"className":3723},[131,132,133,134],[14,3725,83],{"className":3726},[59,134],[14,3728,825],{"className":3729},[824],[14,3731,3733],{"className":3732},[115],[14,3734,3736],{"className":3735,"style":3073},[119],[14,3737],{},[14,3739],{"className":3740},[225,1077],[14,3742],{"className":3743,"style":765},[204],[14,3745,3747,3748],{"className":3746},[850],"lo",[14,3749,3751],{"style":3750},"margin-right:0.01389em;","g",[14,3753],{"className":3754,"style":765},[204],[14,3756,3758,3761],{"className":3757},[59],[14,3759,682],{"className":3760,"style":794},[59,60],[14,3762,3764],{"className":3763},[107],[14,3765,3767,3787],{"className":3766},[111,801],[14,3768,3770,3784],{"className":3769},[115],[14,3771,3773],{"className":3772,"style":808},[119],[14,3774,3775,3778],{"style":811},[14,3776],{"className":3777,"style":127},[126],[14,3779,3781],{"className":3780},[131,132,133,134],[14,3782,171],{"className":3783},[59,60,134],[14,3785,825],{"className":3786},[824],[14,3788,3790],{"className":3789},[115],[14,3791,3793],{"className":3792,"style":832},[119],[14,3794],{},[14,3796,168],{"className":3797},[197],[14,3799,596],{"className":3800},[59,60],[14,3802,180],{"className":3803},[225],[10,3805,3806],{},"This allows the use of natural gradient and information-geometric optimization.",[1889,3808,3810],{"id":3809},"entropy","Entropy",[10,3812,3813],{},"The entropy of the vMF distribution is:",[14,3815,3817],{"className":3816,"translate":18},[230],[14,3818,3820,3891],{"className":3819,"translate":18},[17],[14,3821,3823],{"className":3822},[22],[24,3824,3825],{"xmlns":26,"display":239},[28,3826,3827,3888],{},[31,3828,3829,3832,3834,3836,3838,3844,3846,3848,3850,3852,3854,3856,3862,3864,3866,3868,3870,3876,3878,3880,3882,3884,3886],{},[34,3830,3831],{},"H",[165,3833,260],{},[165,3835,174],{},[34,3837,596],{},[678,3839,3840,3842],{},[34,3841,1957],{},[34,3843,171],{},[165,3845,168],{"stretchy":167},[34,3847,596],{},[165,3849,180],{"stretchy":167},[165,3851,2861],{},[34,3853,3563],{},[165,3855,696],{},[678,3857,3858,3860],{},[34,3859,682],{},[34,3861,171],{},[165,3863,168],{"stretchy":167},[34,3865,596],{},[165,3867,180],{"stretchy":167},[165,3869,2861],{},[931,3871,3872,3874],{},[34,3873,171],{},[81,3875,83],{},[34,3877,3563],{},[165,3879,696],{},[165,3881,168],{"stretchy":167},[81,3883,83],{},[34,3885,957],{},[165,3887,180],{"stretchy":167},[38,3889,3890],{"encoding":40},"H = -\\kappa A_d(\\kappa) + \\log C_d(\\kappa) + \\frac{d}{2}\\log(2\\pi)",[14,3892,3894,3913,3983,4055],{"className":3893,"ariaHidden":46},[45],[14,3895,3897,3900,3904,3907,3910],{"className":3896},[50],[14,3898],{"className":3899,"style":3307},[54],[14,3901,3831],{"className":3902,"style":3903},[59,60],"margin-right:0.08125em;",[14,3905],{"className":3906,"style":352},[204],[14,3908,260],{"className":3909},[356],[14,3911],{"className":3912,"style":352},[204],[14,3914,3916,3919,3922,3925,3965,3968,3971,3974,3977,3980],{"className":3915},[50],[14,3917],{"className":3918,"style":193},[54],[14,3920,174],{"className":3921},[59],[14,3923,596],{"className":3924},[59,60],[14,3926,3928,3931],{"className":3927},[59],[14,3929,1957],{"className":3930},[59,60],[14,3932,3934],{"className":3933},[107],[14,3935,3937,3957],{"className":3936},[111,801],[14,3938,3940,3954],{"className":3939},[115],[14,3941,3943],{"className":3942,"style":808},[119],[14,3944,3945,3948],{"style":2350},[14,3946],{"className":3947,"style":127},[126],[14,3949,3951],{"className":3950},[131,132,133,134],[14,3952,171],{"className":3953},[59,60,134],[14,3955,825],{"className":3956},[824],[14,3958,3960],{"className":3959},[115],[14,3961,3963],{"className":3962,"style":832},[119],[14,3964],{},[14,3966,168],{"className":3967},[197],[14,3969,596],{"className":3970},[59,60],[14,3972,180],{"className":3973},[225],[14,3975],{"className":3976,"style":205},[204],[14,3978,2861],{"className":3979},[209],[14,3981],{"className":3982,"style":205},[204],[14,3984,3986,3989,3994,3997,4037,4040,4043,4046,4049,4052],{"className":3985},[50],[14,3987],{"className":3988,"style":193},[54],[14,3990,3747,3992],{"className":3991},[850],[14,3993,3751],{"style":3750},[14,3995],{"className":3996,"style":765},[204],[14,3998,4000,4003],{"className":3999},[59],[14,4001,682],{"className":4002,"style":794},[59,60],[14,4004,4006],{"className":4005},[107],[14,4007,4009,4029],{"className":4008},[111,801],[14,4010,4012,4026],{"className":4011},[115],[14,4013,4015],{"className":4014,"style":808},[119],[14,4016,4017,4020],{"style":811},[14,4018],{"className":4019,"style":127},[126],[14,4021,4023],{"className":4022},[131,132,133,134],[14,4024,171],{"className":4025},[59,60,134],[14,4027,825],{"className":4028},[824],[14,4030,4032],{"className":4031},[115],[14,4033,4035],{"className":4034,"style":832},[119],[14,4036],{},[14,4038,168],{"className":4039},[197],[14,4041,596],{"className":4042},[59,60],[14,4044,180],{"className":4045},[225],[14,4047],{"className":4048,"style":205},[204],[14,4050,2861],{"className":4051},[209],[14,4053],{"className":4054,"style":205},[204],[14,4056,4058,4062,4125,4128,4133,4136,4139,4142],{"className":4057},[50],[14,4059],{"className":4060,"style":4061},[54],"height:2.0574em;vertical-align:-0.686em;",[14,4063,4065,4068,4122],{"className":4064},[59],[14,4066],{"className":4067},[197,1077],[14,4069,4071],{"className":4070},[931],[14,4072,4074,4114],{"className":4073},[111,801],[14,4075,4077,4111],{"className":4076},[115],[14,4078,4081,4092,4100],{"className":4079,"style":4080},[119],"height:1.3714em;",[14,4082,4083,4086],{"style":2558},[14,4084],{"className":4085,"style":1097},[126],[14,4087,4089],{"className":4088},[59],[14,4090,83],{"className":4091},[59],[14,4093,4094,4097],{"style":1287},[14,4095],{"className":4096,"style":1097},[126],[14,4098],{"className":4099,"style":1294},[1226],[14,4101,4102,4105],{"style":1297},[14,4103],{"className":4104,"style":1097},[126],[14,4106,4108],{"className":4107},[59],[14,4109,171],{"className":4110},[59,60],[14,4112,825],{"className":4113},[824],[14,4115,4117],{"className":4116},[115],[14,4118,4120],{"className":4119,"style":3073},[119],[14,4121],{},[14,4123],{"className":4124},[225,1077],[14,4126],{"className":4127,"style":765},[204],[14,4129,3747,4131],{"className":4130},[850],[14,4132,3751],{"style":3750},[14,4134,168],{"className":4135},[197],[14,4137,83],{"className":4138},[59],[14,4140,957],{"className":4141,"style":103},[59,60],[14,4143,180],{"className":4144},[225],[10,4146,4147,4148,4176],{},"Property: larger ",[14,4149,4151,4164],{"className":4150,"translate":18},[17],[14,4152,4154],{"className":4153},[22],[24,4155,4156],{"xmlns":26},[28,4157,4158,4162],{},[31,4159,4160],{},[34,4161,596],{},[38,4163,1673],{"encoding":40},[14,4165,4167],{"className":4166,"ariaHidden":46},[45],[14,4168,4170,4173],{"className":4169},[50],[14,4171],{"className":4172,"style":1541},[54],[14,4174,596],{"className":4175},[59,60]," gives smaller entropy.",[1889,4178,4180],{"id":4179},"spherical-harmonic-expansion","Spherical Harmonic Expansion",[10,4182,4183],{},"The vMF distribution can be expanded as:",[14,4185,4187],{"className":4186,"translate":18},[230],[14,4188,4190,4241],{"className":4189,"translate":18},[17],[14,4191,4193],{"className":4192},[22],[24,4194,4195],{"xmlns":26,"display":239},[28,4196,4197,4238],{},[31,4198,4199,4201,4203,4205,4207,4209,4218,4225,4232,4234,4236],{},[34,4200,658],{},[165,4202,168],{"stretchy":167},[34,4204,267],{},[165,4206,180],{"stretchy":167},[165,4208,260],{},[4210,4211,4212,4215],"munder",{},[165,4213,4214],{},"∑",[34,4216,4217],{},"l",[678,4219,4220,4223],{},[34,4221,4222],{},"a",[34,4224,4217],{},[678,4226,4227,4230],{},[34,4228,4229],{},"Y",[34,4231,4217],{},[165,4233,168],{"stretchy":167},[34,4235,267],{},[165,4237,180],{"stretchy":167},[38,4239,4240],{"encoding":40},"f(x) = \\sum_l a_l Y_l(x)",[14,4242,4244,4271],{"className":4243,"ariaHidden":46},[45],[14,4245,4247,4250,4253,4256,4259,4262,4265,4268],{"className":4246},[50],[14,4248],{"className":4249,"style":193},[54],[14,4251,658],{"className":4252,"style":727},[59,60],[14,4254,168],{"className":4255},[197],[14,4257,267],{"className":4258},[59,60],[14,4260,180],{"className":4261},[225],[14,4263],{"className":4264,"style":352},[204],[14,4266,260],{"className":4267},[356],[14,4269],{"className":4270,"style":352},[204],[14,4272,4274,4278,4331,4334,4374,4416,4419,4422],{"className":4273},[50],[14,4275],{"className":4276,"style":4277},[54],"height:2.3521em;vertical-align:-1.3021em;",[14,4279,4282],{"className":4280},[850,4281],"op-limits",[14,4283,4285,4322],{"className":4284},[111,801],[14,4286,4288,4319],{"className":4287},[115],[14,4289,4292,4306],{"className":4290,"style":4291},[119],"height:1.05em;",[14,4293,4295,4299],{"style":4294},"top:-1.8479em;margin-left:0em;",[14,4296],{"className":4297,"style":4298},[126],"height:3.05em;",[14,4300,4302],{"className":4301},[131,132,133,134],[14,4303,4217],{"className":4304,"style":4305},[59,60,134],"margin-right:0.01968em;",[14,4307,4309,4312],{"style":4308},"top:-3.05em;",[14,4310],{"className":4311,"style":4298},[126],[14,4313,4314],{},[14,4315,4214],{"className":4316},[850,4317,4318],"op-symbol","large-op",[14,4320,825],{"className":4321},[824],[14,4323,4325],{"className":4324},[115],[14,4326,4329],{"className":4327,"style":4328},[119],"height:1.3021em;",[14,4330],{},[14,4332],{"className":4333,"style":765},[204],[14,4335,4337,4340],{"className":4336},[59],[14,4338,4222],{"className":4339},[59,60],[14,4341,4343],{"className":4342},[107],[14,4344,4346,4366],{"className":4345},[111,801],[14,4347,4349,4363],{"className":4348},[115],[14,4350,4352],{"className":4351,"style":808},[119],[14,4353,4354,4357],{"style":2350},[14,4355],{"className":4356,"style":127},[126],[14,4358,4360],{"className":4359},[131,132,133,134],[14,4361,4217],{"className":4362,"style":4305},[59,60,134],[14,4364,825],{"className":4365},[824],[14,4367,4369],{"className":4368},[115],[14,4370,4372],{"className":4371,"style":832},[119],[14,4373],{},[14,4375,4377,4381],{"className":4376},[59],[14,4378,4229],{"className":4379,"style":4380},[59,60],"margin-right:0.22222em;",[14,4382,4384],{"className":4383},[107],[14,4385,4387,4408],{"className":4386},[111,801],[14,4388,4390,4405],{"className":4389},[115],[14,4391,4393],{"className":4392,"style":808},[119],[14,4394,4396,4399],{"style":4395},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[14,4397],{"className":4398,"style":127},[126],[14,4400,4402],{"className":4401},[131,132,133,134],[14,4403,4217],{"className":4404,"style":4305},[59,60,134],[14,4406,825],{"className":4407},[824],[14,4409,4411],{"className":4410},[115],[14,4412,4414],{"className":4413,"style":832},[119],[14,4415],{},[14,4417,168],{"className":4418},[197],[14,4420,267],{"className":4421},[59,60],[14,4423,180],{"className":4424},[225],[10,4426,4427,4428,4498],{},"where ",[14,4429,4431,4449],{"className":4430,"translate":18},[17],[14,4432,4434],{"className":4433},[22],[24,4435,4436],{"xmlns":26},[28,4437,4438,4446],{},[31,4439,4440],{},[678,4441,4442,4444],{},[34,4443,4229],{},[34,4445,4217],{},[38,4447,4448],{"encoding":40},"Y_l",[14,4450,4452],{"className":4451,"ariaHidden":46},[45],[14,4453,4455,4458],{"className":4454},[50],[14,4456],{"className":4457,"style":1458},[54],[14,4459,4461,4464],{"className":4460},[59],[14,4462,4229],{"className":4463,"style":4380},[59,60],[14,4465,4467],{"className":4466},[107],[14,4468,4470,4490],{"className":4469},[111,801],[14,4471,4473,4487],{"className":4472},[115],[14,4474,4476],{"className":4475,"style":808},[119],[14,4477,4478,4481],{"style":4395},[14,4479],{"className":4480,"style":127},[126],[14,4482,4484],{"className":4483},[131,132,133,134],[14,4485,4217],{"className":4486,"style":4305},[59,60,134],[14,4488,825],{"className":4489},[824],[14,4491,4493],{"className":4492},[115],[14,4494,4496],{"className":4495,"style":832},[119],[14,4497],{}," are spherical harmonics.",[1889,4500,4502],{"id":4501},"spherical-cap","Spherical Cap",[10,4504,4505,4506,4534,4535,4538],{},"In the von Mises–Fisher distribution, probability mass is mainly concentrated around the mean direction ",[14,4507,4509,4522],{"className":4508,"translate":18},[17],[14,4510,4512],{"className":4511},[22],[24,4513,4514],{"xmlns":26},[28,4515,4516,4520],{},[31,4517,4518],{},[34,4519,36],{},[38,4521,41],{"encoding":40},[14,4523,4525],{"className":4524,"ariaHidden":46},[45],[14,4526,4528,4531],{"className":4527},[50],[14,4529],{"className":4530,"style":55},[54],[14,4532,36],{"className":4533},[59,60],".",[4536,4537],"br",{},"\nHence, a spherical cap is commonly used to describe confidence regions.",[10,4540,4541,4542,4623,4624,4733,4734,4763],{},"Let ",[14,4543,4545,4572],{"className":4544,"translate":18},[17],[14,4546,4548],{"className":4547},[22],[24,4549,4550],{"xmlns":26},[28,4551,4552,4570],{},[31,4553,4554,4556,4558,4560,4562,4564,4566,4568],{},[34,4555,267],{},[165,4557,3334],{},[3336,4559,3338],{},[165,4561,168],{"stretchy":167},[34,4563,36],{},[165,4565,670],{"separator":46},[34,4567,596],{},[165,4569,180],{"stretchy":167},[38,4571,3351],{"encoding":40},[14,4573,4575,4593],{"className":4574,"ariaHidden":46},[45],[14,4576,4578,4581,4584,4587,4590],{"className":4577},[50],[14,4579],{"className":4580,"style":1541},[54],[14,4582,267],{"className":4583},[59,60],[14,4585],{"className":4586,"style":352},[204],[14,4588,3334],{"className":4589},[356],[14,4591],{"className":4592,"style":352},[204],[14,4594,4596,4599,4605,4608,4611,4614,4617,4620],{"className":4595},[50],[14,4597],{"className":4598,"style":193},[54],[14,4600,4602],{"className":4601},[59,3382],[14,4603,3338],{"className":4604},[59],[14,4606,168],{"className":4607},[197],[14,4609,36],{"className":4610},[59,60],[14,4612,670],{"className":4613},[761],[14,4615],{"className":4616,"style":765},[204],[14,4618,596],{"className":4619},[59,60],[14,4621,180],{"className":4622},[225],", ",[14,4625,4627,4659],{"className":4626,"translate":18},[17],[14,4628,4630],{"className":4629},[22],[24,4631,4632],{"xmlns":26},[28,4633,4634,4656],{},[31,4635,4636,4638,4640,4642,4644],{},[34,4637,267],{},[165,4639,670],{"separator":46},[34,4641,36],{},[165,4643,270],{},[75,4645,4646,4648],{},[34,4647,249],{},[31,4649,4650,4652,4654],{},[34,4651,171],{},[165,4653,174],{},[81,4655,177],{},[38,4657,4658],{"encoding":40},"x,\\mu \\in S^{d-1}",[14,4660,4662,4689],{"className":4661,"ariaHidden":46},[45],[14,4663,4665,4668,4671,4674,4677,4680,4683,4686],{"className":4664},[50],[14,4666],{"className":4667,"style":513},[54],[14,4669,267],{"className":4670},[59,60],[14,4672,670],{"className":4673},[761],[14,4675],{"className":4676,"style":765},[204],[14,4678,36],{"className":4679},[59,60],[14,4681],{"className":4682,"style":352},[204],[14,4684,270],{"className":4685},[356],[14,4687],{"className":4688,"style":352},[204],[14,4690,4692,4695],{"className":4691},[50],[14,4693],{"className":4694,"style":539},[54],[14,4696,4698,4701],{"className":4697},[59],[14,4699,249],{"className":4700,"style":546},[59,60],[14,4702,4704],{"className":4703},[107],[14,4705,4707],{"className":4706},[111],[14,4708,4710],{"className":4709},[115],[14,4711,4713],{"className":4712,"style":539},[119],[14,4714,4715,4718],{"style":122},[14,4716],{"className":4717,"style":127},[126],[14,4719,4721],{"className":4720},[131,132,133,134],[14,4722,4724,4727,4730],{"className":4723},[59,134],[14,4725,171],{"className":4726},[59,60,134],[14,4728,174],{"className":4729},[209,134],[14,4731,177],{"className":4732},[59,134],", and let ",[14,4735,4737,4751],{"className":4736,"translate":18},[17],[14,4738,4740],{"className":4739},[22],[24,4741,4742],{"xmlns":26},[28,4743,4744,4748],{},[31,4745,4746],{},[34,4747,1924],{},[38,4749,4750],{"encoding":40},"\\theta",[14,4752,4754],{"className":4753,"ariaHidden":46},[45],[14,4755,4757,4760],{"className":4756},[50],[14,4758],{"className":4759,"style":2134},[54],[14,4761,1924],{"className":4762,"style":1993},[59,60]," be the angle between the vectors:",[14,4765,4767],{"className":4766,"translate":18},[230],[14,4768,4770,4799],{"className":4769,"translate":18},[17],[14,4771,4773],{"className":4772},[22],[24,4774,4775],{"xmlns":26,"display":239},[28,4776,4777,4796],{},[31,4778,4779,4782,4784,4786,4788,4794],{},[34,4780,4781],{},"cos",[165,4783,696],{},[34,4785,1924],{},[165,4787,260],{},[75,4789,4790,4792],{},[34,4791,36],{},[34,4793,1950],{},[34,4795,267],{},[38,4797,4798],{"encoding":40},"\\cos\\theta = \\mu^T x",[14,4800,4802,4826],{"className":4801,"ariaHidden":46},[45],[14,4803,4805,4808,4811,4814,4817,4820,4823],{"className":4804},[50],[14,4806],{"className":4807,"style":2134},[54],[14,4809,4781],{"className":4810},[850],[14,4812],{"className":4813,"style":765},[204],[14,4815,1924],{"className":4816,"style":1993},[59,60],[14,4818],{"className":4819,"style":352},[204],[14,4821,260],{"className":4822},[356],[14,4824],{"className":4825,"style":352},[204],[14,4827,4829,4833,4862],{"className":4828},[50],[14,4830],{"className":4831,"style":4832},[54],"height:1.0858em;vertical-align:-0.1944em;",[14,4834,4836,4839],{"className":4835},[59],[14,4837,36],{"className":4838},[59,60],[14,4840,4842],{"className":4841},[107],[14,4843,4845],{"className":4844},[111],[14,4846,4848],{"className":4847},[115],[14,4849,4851],{"className":4850,"style":2052},[119],[14,4852,4853,4856],{"style":330},[14,4854],{"className":4855,"style":127},[126],[14,4857,4859],{"className":4858},[131,132,133,134],[14,4860,1950],{"className":4861,"style":2064},[59,60,134],[14,4863,267],{"className":4864},[59,60],[10,4866,4867],{},"A confidence region can be written as",[14,4869,4871],{"className":4870,"translate":18},[230],[14,4872,4874,4898],{"className":4873,"translate":18},[17],[14,4875,4877],{"className":4876},[22],[24,4878,4879],{"xmlns":26,"display":239},[28,4880,4881,4895],{},[31,4882,4883,4885,4888],{},[34,4884,1924],{},[165,4886,4887],{},"≤",[678,4889,4890,4892],{},[34,4891,1924],{},[34,4893,4894],{},"c",[38,4896,4897],{"encoding":40},"\\theta \\le \\theta_c",[14,4899,4901,4920],{"className":4900,"ariaHidden":46},[45],[14,4902,4904,4908,4911,4914,4917],{"className":4903},[50],[14,4905],{"className":4906,"style":4907},[54],"height:0.8304em;vertical-align:-0.136em;",[14,4909,1924],{"className":4910,"style":1993},[59,60],[14,4912],{"className":4913,"style":352},[204],[14,4915,4887],{"className":4916},[356],[14,4918],{"className":4919,"style":352},[204],[14,4921,4923,4927],{"className":4922},[50],[14,4924],{"className":4925,"style":4926},[54],"height:0.8444em;vertical-align:-0.15em;",[14,4928,4930,4933],{"className":4929},[59],[14,4931,1924],{"className":4932,"style":1993},[59,60],[14,4934,4936],{"className":4935},[107],[14,4937,4939,4960],{"className":4938},[111,801],[14,4940,4942,4957],{"className":4941},[115],[14,4943,4945],{"className":4944,"style":1477},[119],[14,4946,4948,4951],{"style":4947},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[14,4949],{"className":4950,"style":127},[126],[14,4952,4954],{"className":4953},[131,132,133,134],[14,4955,4894],{"className":4956},[59,60,134],[14,4958,825],{"className":4959},[824],[14,4961,4963],{"className":4962},[115],[14,4964,4966],{"className":4965,"style":832},[119],[14,4967],{},[10,4969,4970,4971,4534],{},"i.e., a spherical cap with axis ",[14,4972,4974,4987],{"className":4973,"translate":18},[17],[14,4975,4977],{"className":4976},[22],[24,4978,4979],{"xmlns":26},[28,4980,4981,4985],{},[31,4982,4983],{},[34,4984,36],{},[38,4986,41],{"encoding":40},[14,4988,4990],{"className":4989,"ariaHidden":46},[45],[14,4991,4993,4996],{"className":4992},[50],[14,4994],{"className":4995,"style":55},[54],[14,4997,36],{"className":4998},[59,60],[10,5000,5001,5002,670],{},"On the unit sphere ",[14,5003,5005,5029],{"className":5004,"translate":18},[17],[14,5006,5008],{"className":5007},[22],[24,5009,5010],{"xmlns":26},[28,5011,5012,5026],{},[31,5013,5014],{},[75,5015,5016,5018],{},[34,5017,249],{"mathvariant":248},[31,5019,5020,5022,5024],{},[34,5021,171],{},[165,5023,174],{},[81,5025,177],{},[38,5027,5028],{"encoding":40},"\\mathbb{S}^{d-1}",[14,5030,5032],{"className":5031,"ariaHidden":46},[45],[14,5033,5035,5038],{"className":5034},[50],[14,5036],{"className":5037,"style":539},[54],[14,5039,5041,5044],{"className":5040},[59],[14,5042,249],{"className":5043},[59,315],[14,5045,5047],{"className":5046},[107],[14,5048,5050],{"className":5049},[111],[14,5051,5053],{"className":5052},[115],[14,5054,5056],{"className":5055,"style":539},[119],[14,5057,5058,5061],{"style":122},[14,5059],{"className":5060,"style":127},[126],[14,5062,5064],{"className":5063},[131,132,133,134],[14,5065,5067,5070,5073],{"className":5066},[59,134],[14,5068,171],{"className":5069},[59,60,134],[14,5071,174],{"className":5072},[209,134],[14,5074,177],{"className":5075},[59,134],[10,5077,5078],{},"the area of a spherical cap is:",[14,5080,5082],{"className":5081,"translate":18},[230],[14,5083,5085,5173],{"className":5084,"translate":18},[17],[14,5086,5088],{"className":5087},[22],[24,5089,5090],{"xmlns":26,"display":239},[28,5091,5092,5170],{},[31,5093,5094,5096,5098,5100,5102,5104,5110,5122,5125,5144],{},[34,5095,1957],{},[165,5097,168],{"stretchy":167},[34,5099,1924],{},[165,5101,180],{"stretchy":167},[165,5103,260],{},[931,5105,5106,5108],{},[81,5107,177],{},[81,5109,83],{},[678,5111,5112,5114],{},[34,5113,1957],{},[31,5115,5116,5118,5120],{},[34,5117,171],{},[165,5119,174],{},[81,5121,177],{},[3336,5123,5124],{}," ",[678,5126,5127,5129],{},[34,5128,975],{},[31,5130,5131,5142],{},[75,5132,5133,5140],{},[31,5134,5135,5138],{},[34,5136,5137],{},"sin",[165,5139,696],{},[81,5141,83],{},[34,5143,1924],{},[31,5145,5146,5148,5160,5162,5168],{},[165,5147,168],{"fence":46},[931,5149,5150,5158],{},[31,5151,5152,5154,5156],{},[34,5153,171],{},[165,5155,174],{},[81,5157,177],{},[81,5159,83],{},[165,5161,670],{"separator":46},[931,5163,5164,5166],{},[81,5165,177],{},[81,5167,83],{},[165,5169,180],{"fence":46},[38,5171,5172],{"encoding":40},"A(\\theta) = \\frac{1}{2} A_{d-1} \\, I_{\\sin^2\\theta}\\left(\\frac{d-1}{2},\\frac12\\right)",[14,5174,5176,5203],{"className":5175,"ariaHidden":46},[45],[14,5177,5179,5182,5185,5188,5191,5194,5197,5200],{"className":5178},[50],[14,5180],{"className":5181,"style":193},[54],[14,5183,1957],{"className":5184},[59,60],[14,5186,168],{"className":5187},[197],[14,5189,1924],{"className":5190,"style":1993},[59,60],[14,5192,180],{"className":5193},[225],[14,5195],{"className":5196,"style":352},[204],[14,5198,260],{"className":5199},[356],[14,5201],{"className":5202,"style":352},[204],[14,5204,5206,5209,5272,5322,5325,5418,5421],{"className":5205},[50],[14,5207],{"className":5208,"style":3097},[54],[14,5210,5212,5215,5269],{"className":5211},[59],[14,5213],{"className":5214},[197,1077],[14,5216,5218],{"className":5217},[931],[14,5219,5221,5261],{"className":5220},[111,801],[14,5222,5224,5258],{"className":5223},[115],[14,5225,5228,5239,5247],{"className":5226,"style":5227},[119],"height:1.3214em;",[14,5229,5230,5233],{"style":2558},[14,5231],{"className":5232,"style":1097},[126],[14,5234,5236],{"className":5235},[59],[14,5237,83],{"className":5238},[59],[14,5240,5241,5244],{"style":1287},[14,5242],{"className":5243,"style":1097},[126],[14,5245],{"className":5246,"style":1294},[1226],[14,5248,5249,5252],{"style":1297},[14,5250],{"className":5251,"style":1097},[126],[14,5253,5255],{"className":5254},[59],[14,5256,177],{"className":5257},[59],[14,5259,825],{"className":5260},[824],[14,5262,5264],{"className":5263},[115],[14,5265,5267],{"className":5266,"style":3073},[119],[14,5268],{},[14,5270],{"className":5271},[225,1077],[14,5273,5275,5278],{"className":5274},[59],[14,5276,1957],{"className":5277},[59,60],[14,5279,5281],{"className":5280},[107],[14,5282,5284,5313],{"className":5283},[111,801],[14,5285,5287,5310],{"className":5286},[115],[14,5288,5290],{"className":5289,"style":808},[119],[14,5291,5292,5295],{"style":2350},[14,5293],{"className":5294,"style":127},[126],[14,5296,5298],{"className":5297},[131,132,133,134],[14,5299,5301,5304,5307],{"className":5300},[59,134],[14,5302,171],{"className":5303},[59,60,134],[14,5305,174],{"className":5306},[209,134],[14,5308,177],{"className":5309},[59,134],[14,5311,825],{"className":5312},[824],[14,5314,5316],{"className":5315},[115],[14,5317,5320],{"className":5318,"style":5319},[119],"height:0.2083em;",[14,5321],{},[14,5323],{"className":5324,"style":765},[204],[14,5326,5328,5331],{"className":5327},[59],[14,5329,975],{"className":5330,"style":1154},[59,60],[14,5332,5334],{"className":5333},[107],[14,5335,5337,5409],{"className":5336},[111,801],[14,5338,5340,5406],{"className":5339},[115],[14,5341,5343],{"className":5342,"style":1167},[119],[14,5344,5346,5349],{"style":5345},"top:-2.5021em;margin-left:-0.0785em;margin-right:0.05em;",[14,5347],{"className":5348,"style":127},[126],[14,5350,5352],{"className":5351},[131,132,133,134],[14,5353,5355,5399,5403],{"className":5354},[59,134],[14,5356,5358,5373],{"className":5357},[850,134],[14,5359,5361,5365,5369],{"className":5360},[850,134],[14,5362,5364],{"className":5363},[134],"s",[14,5366,5368],{"className":5367},[134],"i",[14,5370,5372],{"className":5371},[134],"n",[14,5374,5376],{"className":5375},[107],[14,5377,5379],{"className":5378},[111],[14,5380,5382],{"className":5381},[115],[14,5383,5386],{"className":5384,"style":5385},[119],"height:0.7753em;",[14,5387,5389,5393],{"style":5388},"top:-2.815em;margin-right:0.0714em;",[14,5390],{"className":5391,"style":5392},[126],"height:2.5em;",[14,5394,5396],{"className":5395},[131,1186,1210,134],[14,5397,83],{"className":5398},[59,134],[14,5400],{"className":5401,"style":5402},[204,134],"margin-right:0.1952em;",[14,5404,1924],{"className":5405,"style":1993},[59,60,134],[14,5407,825],{"className":5408},[824],[14,5410,5412],{"className":5411},[115],[14,5413,5416],{"className":5414,"style":5415},[119],"height:0.1979em;",[14,5417],{},[14,5419],{"className":5420,"style":765},[204],[14,5422,5424,5430,5504,5507,5510,5572],{"className":5423},[3101],[14,5425,5427],{"className":5426,"style":3106},[197,3105],[14,5428,168],{"className":5429},[3110,133],[14,5431,5433,5436,5501],{"className":5432},[59],[14,5434],{"className":5435},[197,1077],[14,5437,5439],{"className":5438},[931],[14,5440,5442,5493],{"className":5441},[111,801],[14,5443,5445,5490],{"className":5444},[115],[14,5446,5448,5459,5467],{"className":5447,"style":4080},[119],[14,5449,5450,5453],{"style":2558},[14,5451],{"className":5452,"style":1097},[126],[14,5454,5456],{"className":5455},[59],[14,5457,83],{"className":5458},[59],[14,5460,5461,5464],{"style":1287},[14,5462],{"className":5463,"style":1097},[126],[14,5465],{"className":5466,"style":1294},[1226],[14,5468,5469,5472],{"style":1297},[14,5470],{"className":5471,"style":1097},[126],[14,5473,5475,5478,5481,5484,5487],{"className":5474},[59],[14,5476,171],{"className":5477},[59,60],[14,5479],{"className":5480,"style":205},[204],[14,5482,174],{"className":5483},[209],[14,5485],{"className":5486,"style":205},[204],[14,5488,177],{"className":5489},[59],[14,5491,825],{"className":5492},[824],[14,5494,5496],{"className":5495},[115],[14,5497,5499],{"className":5498,"style":3073},[119],[14,5500],{},[14,5502],{"className":5503},[225,1077],[14,5505,670],{"className":5506},[761],[14,5508],{"className":5509,"style":765},[204],[14,5511,5513,5516,5569],{"className":5512},[59],[14,5514],{"className":5515},[197,1077],[14,5517,5519],{"className":5518},[931],[14,5520,5522,5561],{"className":5521},[111,801],[14,5523,5525,5558],{"className":5524},[115],[14,5526,5528,5539,5547],{"className":5527,"style":5227},[119],[14,5529,5530,5533],{"style":2558},[14,5531],{"className":5532,"style":1097},[126],[14,5534,5536],{"className":5535},[59],[14,5537,83],{"className":5538},[59],[14,5540,5541,5544],{"style":1287},[14,5542],{"className":5543,"style":1097},[126],[14,5545],{"className":5546,"style":1294},[1226],[14,5548,5549,5552],{"style":1297},[14,5550],{"className":5551,"style":1097},[126],[14,5553,5555],{"className":5554},[59],[14,5556,177],{"className":5557},[59],[14,5559,825],{"className":5560},[824],[14,5562,5564],{"className":5563},[115],[14,5565,5567],{"className":5566,"style":3073},[119],[14,5568],{},[14,5570],{"className":5571},[225,1077],[14,5573,5575],{"className":5574,"style":3106},[225,3105],[14,5576,180],{"className":5577},[3110,133],[10,5579,4427,5580,5666],{},[14,5581,5583,5607],{"className":5582,"translate":18},[17],[14,5584,5586],{"className":5585},[22],[24,5587,5588],{"xmlns":26},[28,5589,5590,5604],{},[31,5591,5592],{},[678,5593,5594,5596],{},[34,5595,1957],{},[31,5597,5598,5600,5602],{},[34,5599,171],{},[165,5601,174],{},[81,5603,177],{},[38,5605,5606],{"encoding":40},"A_{d-1}",[14,5608,5610],{"className":5609,"ariaHidden":46},[45],[14,5611,5613,5617],{"className":5612},[50],[14,5614],{"className":5615,"style":5616},[54],"height:0.8917em;vertical-align:-0.2083em;",[14,5618,5620,5623],{"className":5619},[59],[14,5621,1957],{"className":5622},[59,60],[14,5624,5626],{"className":5625},[107],[14,5627,5629,5658],{"className":5628},[111,801],[14,5630,5632,5655],{"className":5631},[115],[14,5633,5635],{"className":5634,"style":808},[119],[14,5636,5637,5640],{"style":2350},[14,5638],{"className":5639,"style":127},[126],[14,5641,5643],{"className":5642},[131,132,133,134],[14,5644,5646,5649,5652],{"className":5645},[59,134],[14,5647,171],{"className":5648},[59,60,134],[14,5650,174],{"className":5651},[209,134],[14,5653,177],{"className":5654},[59,134],[14,5656,825],{"className":5657},[824],[14,5659,5661],{"className":5660},[115],[14,5662,5664],{"className":5663,"style":5319},[119],[14,5665],{}," is the total surface area of the sphere, given by:",[14,5668,5670],{"className":5669,"translate":18},[230],[14,5671,5673,5732],{"className":5672,"translate":18},[17],[14,5674,5676],{"className":5675},[22],[24,5677,5678],{"xmlns":26,"display":239},[28,5679,5680,5729],{},[31,5681,5682,5694,5696],{},[678,5683,5684,5686],{},[34,5685,1957],{},[31,5687,5688,5690,5692],{},[34,5689,171],{},[165,5691,174],{},[81,5693,177],{},[165,5695,260],{},[931,5697,5698,5714],{},[31,5699,5700,5702],{},[81,5701,83],{},[75,5703,5704,5706],{},[34,5705,957],{},[31,5707,5708,5710,5712],{},[34,5709,171],{},[34,5711,968],{"mathvariant":283},[81,5713,83],{},[31,5715,5716,5719,5721,5723,5725,5727],{},[34,5717,5718],{"mathvariant":283},"Γ",[165,5720,168],{"stretchy":167},[34,5722,171],{},[34,5724,968],{"mathvariant":283},[81,5726,83],{},[165,5728,180],{"stretchy":167},[38,5730,5731],{"encoding":40},"A_{d-1} = 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is the regularized incomplete beta function.",[10,6021,6022,6023,280],{},"For ",[14,6024,6026,6045],{"className":6025,"translate":18},[17],[14,6027,6029],{"className":6028},[22],[24,6030,6031],{"xmlns":26},[28,6032,6033,6042],{},[31,6034,6035,6037,6039],{},[34,6036,171],{},[165,6038,260],{},[81,6040,6041],{},"3",[38,6043,6044],{"encoding":40},"d=3",[14,6046,6048,6066],{"className":6047,"ariaHidden":46},[45],[14,6049,6051,6054,6057,6060,6063],{"className":6050},[50],[14,6052],{"className":6053,"style":2134},[54],[14,6055,171],{"className":6056},[59,60],[14,6058],{"className":6059,"style":352},[204],[14,6061,260],{"className":6062},[356],[14,6064],{"className":6065,"style":352},[204],[14,6067,6069,6072],{"className":6068},[50],[14,6070],{"className":6071,"style":634},[54],[14,6073,6041],{"className":6074},[59],[10,6076,6077],{},"Cap 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\\le \\theta_c) = \\frac{ \\int_0^{\\theta_c} e^{\\kappa\\cos\\theta} (\\sin\\theta)^{d-2} \\,d\\theta }{ \\int_0^{\\pi} e^{\\kappa\\cos\\theta} (\\sin\\theta)^{d-2} \\,d\\theta 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294},[1226],[14,6868,6870,6873],{"style":6869},"top:-3.7898em;",[14,6871],{"className":6872,"style":6678},[126],[14,6874,6876,6971,6974,7027,7030,7033,7036,7039,7077,7080,7083],{"className":6875},[59],[14,6877,6879,6882],{"className":6878},[850],[14,6880,6458],{"className":6881,"style":6689},[850,4317,6688],[14,6883,6885],{"className":6884},[107],[14,6886,6888,6963],{"className":6887},[111,801],[14,6889,6891,6960],{"className":6890},[115],[14,6892,6895,6906],{"className":6893,"style":6894},[119],"height:1.044em;",[14,6896,6897,6900],{"style":6705},[14,6898],{"className":6899,"style":127},[126],[14,6901,6903],{"className":6902},[131,132,133,134],[14,6904,602],{"className":6905},[59,134],[14,6907,6908,6911],{"style":6717},[14,6909],{"className":6910,"style":127},[126],[14,6912,6914],{"className":6913},[131,132,133,134],[14,6915,6917],{"className":6916},[59,134],[14,6918,6920,6923],{"className":6919},[59,134],[14,6921,1924],{"className":6922,"style":1993},[59,60,134],[14,6924,6926],{"className":6925},[107],[14,6927,6929,6951],{"className":6928},[111,801],[14,6930,6932,6948],{"className":6931},[115],[14,6933,6936],{"className":6934,"style":6935},[119],"height:0.1645em;",[14,6937,6939,6942],{"style":6938},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[14,6940],{"className":6941,"style":5392},[126],[14,6943,6945],{"className":6944},[131,1186,1210,134],[14,6946,4894],{"className":6947},[59,60,134],[14,6949,825],{"className":6950},[824],[14,6952,6954],{"className":6953},[115],[14,6955,6958],{"className":6956,"style":6957},[119],"height:0.143em;",[14,6959],{},[14,6961,825],{"className":6962},[824],[14,6964,6966],{"className":6965},[115],[14,6967,6969],{"className":6968,"style":6739},[119],[14,6970],{},[14,6972],{"className":6973,"style":765},[204],[14,6975,6977,6980],{"className":6976},[59],[14,6978,6471],{"className":6979},[59,60],[14,6981,6983],{"className":6982},[107],[14,6984,6986],{"className":6985},[111],[14,6987,6989],{"className":6988},[115],[14,6990,6992],{"className":6991,"style":539},[119],[14,6993,6994,6997],{"style":122},[14,6995],{"className":6996,"style":127},[126],[14,6998,7000],{"className":6999},[131,132,133,134],[14,7001,7003,7006,7009,7021,7024],{"className":7002},[59,134],[14,7004,596],{"className":7005},[59,60,134],[14,7007],{"className":7008,"style":5402},[204,134],[14,7010,7012,7015,7018],{"className":7011},[850,134],[14,7013,4894],{"className":7014},[134],[14,7016,6388],{"className":7017},[134],[14,7019,5364],{"className":7020},[134],[14,7022],{"className":7023,"style":5402},[204,134],[14,7025,1924],{"className":7026,"style":1993},[59,60,134],[14,7028,168],{"className":7029},[197],[14,7031,5137],{"className":7032},[850],[14,7034],{"className":7035,"style":765},[204],[14,7037,1924],{"className":7038,"style":1993},[59,60],[14,7040,7042,7045],{"className":7041},[225],[14,7043,180],{"className":7044},[225],[14,7046,7048],{"className":7047},[107],[14,7049,7051],{"className":7050},[111],[14,7052,7054],{"className":7053},[115],[14,7055,7057],{"className":7056,"style":539},[119],[14,7058,7059,7062],{"style":122},[14,7060],{"className":7061,"style":127},[126],[14,7063,7065],{"className":7064},[131,132,133,134],[14,7066,7068,7071,7074],{"className":7067},[59,134],[14,7069,171],{"className":7070},[59,60,134],[14,7072,174],{"className":7073},[209,134],[14,7075,83],{"className":7076},[59,134],[14,7078],{"className":7079,"style":765},[204],[14,7081,171],{"className":7082},[59,60],[14,7084,1924],{"className":7085,"style":1993},[59,60],[14,7087,825],{"className":7088},[824],[14,7090,7092],{"className":7091},[115],[14,7093,7096],{"className":7094,"style":7095},[119],"height:1.1051em;",[14,7097],{},[14,7099],{"className":7100},[225,1077],[10,7102,7103],{},"There is no simple closed-form solution.",[10,7105,7106],{},"Approximations are commonly used.",[10,7108,7109,7110,7138,7139,7167],{},"When ",[14,7111,7113,7126],{"className":7112,"translate":18},[17],[14,7114,7116],{"className":7115},[22],[24,7117,7118],{"xmlns":26},[28,7119,7120,7124],{},[31,7121,7122],{},[34,7123,596],{},[38,7125,1673],{"encoding":40},[14,7127,7129],{"className":7128,"ariaHidden":46},[45],[14,7130,7132,7135],{"className":7131},[50],[14,7133],{"className":7134,"style":1541},[54],[14,7136,596],{"className":7137},[59,60]," is large, the distribution is concentrated around ",[14,7140,7142,7155],{"className":7141,"translate":18},[17],[14,7143,7145],{"className":7144},[22],[24,7146,7147],{"xmlns":26},[28,7148,7149,7153],{},[31,7150,7151],{},[34,7152,36],{},[38,7154,41],{"encoding":40},[14,7156,7158],{"className":7157,"ariaHidden":46},[45],[14,7159,7161,7164],{"className":7160},[50],[14,7162],{"className":7163,"style":55},[54],[14,7165,36],{"className":7166},[59,60],". Using",[14,7169,7171],{"className":7170,"translate":18},[230],[14,7172,7174,7209],{"className":7173,"translate":18},[17],[14,7175,7177],{"className":7176},[22],[24,7178,7179],{"xmlns":26,"display":239},[28,7180,7181,7206],{},[31,7182,7183,7185,7187,7189,7192,7194,7196],{},[34,7184,4781],{},[165,7186,696],{},[34,7188,1924],{},[165,7190,7191],{},"≈",[81,7193,177],{},[165,7195,174],{},[931,7197,7198,7204],{},[75,7199,7200,7202],{},[34,7201,1924],{},[81,7203,83],{},[81,7205,83],{},[38,7207,7208],{"encoding":40},"\\cos\\theta \\approx 1 - \\frac{\\theta^2}{2}",[14,7210,7212,7236,7255],{"className":7211,"ariaHidden":46},[45],[14,7213,7215,7218,7221,7224,7227,7230,7233],{"className":7214},[50],[14,7216],{"className":7217,"style":2134},[54],[14,7219,4781],{"className":7220},[850],[14,7222],{"className":7223,"style":765},[204],[14,7225,1924],{"className":7226,"style":1993},[59,60],[14,7228],{"className":7229,"style":352},[204],[14,7231,7191],{"className":7232},[356],[14,7234],{"className":7235,"style":352},[204],[14,7237,7239,7243,7246,7249,7252],{"className":7238},[50],[14,7240],{"className":7241,"style":7242},[54],"height:0.7278em;vertical-align:-0.0833em;",[14,7244,177],{"className":7245},[59],[14,7247],{"className":7248,"style":205},[204],[14,7250,174],{"className":7251},[209],[14,7253],{"className":7254,"style":205},[204],[14,7256,7258,7261],{"className":7257},[50],[14,7259],{"className":7260,"style":3617},[54],[14,7262,7264,7267,7346],{"className":7263},[59],[14,7265],{"className":7266},[197,1077],[14,7268,7270],{"className":7269},[931],[14,7271,7273,7338],{"className":7272},[111,801],[14,7274,7276,7335],{"className":7275},[115],[14,7277,7279,7290,7298],{"className":7278,"style":3639},[119],[14,7280,7281,7284],{"style":2558},[14,7282],{"className":7283,"style":1097},[126],[14,7285,7287],{"className":7286},[59],[14,7288,83],{"className":7289},[59],[14,7291,7292,7295],{"style":1287},[14,7293],{"className":7294,"style":1097},[126],[14,7296],{"className":7297,"style":1294},[1226],[14,7299,7300,7303],{"style":1297},[14,7301],{"className":7302,"style":1097},[126],[14,7304,7306],{"className":7305},[59],[14,7307,7309,7312],{"className":7308},[59],[14,7310,1924],{"className":7311,"style":1993},[59,60],[14,7313,7315],{"className":7314},[107],[14,7316,7318],{"className":7317},[111],[14,7319,7321],{"className":7320},[115],[14,7322,7324],{"className":7323,"style":96},[119],[14,7325,7326,7329],{"style":122},[14,7327],{"className":7328,"style":127},[126],[14,7330,7332],{"className":7331},[131,132,133,134],[14,7333,83],{"className":7334},[59,134],[14,7336,825],{"className":7337},[824],[14,7339,7341],{"className":7340},[115],[14,7342,7344],{"className":7343,"style":3073},[119],[14,7345],{},[14,7347],{"className":7348},[225,1077],[10,7350,7351],{},"we obtain",[14,7353,7355],{"className":7354,"translate":18},[230],[14,7356,7358,7412],{"className":7357,"translate":18},[17],[14,7359,7361],{"className":7360},[22],[24,7362,7363],{"xmlns":26,"display":239},[28,7364,7365,7409],{},[31,7366,7367,7381,7383,7389],{},[75,7368,7369,7371],{},[34,7370,6471],{},[31,7372,7373,7375,7377,7379],{},[34,7374,596],{},[34,7376,4781],{},[165,7378,696],{},[34,7380,1924],{},[165,7382,7191],{},[75,7384,7385,7387],{},[34,7386,6471],{},[34,7388,596],{},[75,7390,7391,7393],{},[34,7392,6471],{},[31,7394,7395,7397,7399,7405,7407],{},[165,7396,174],{},[34,7398,596],{},[75,7400,7401,7403],{},[34,7402,1924],{},[81,7404,83],{},[34,7406,968],{"mathvariant":283},[81,7408,83],{},[38,7410,7411],{"encoding":40},"e^{\\kappa\\cos\\theta} \\approx e^{\\kappa} e^{-\\kappa\\theta^2\u002F2}",[14,7413,7415,7483],{"className":7414,"ariaHidden":46},[45],[14,7416,7418,7421,7474,7477,7480],{"className":7417},[50],[14,7419],{"className":7420,"style":308},[54],[14,7422,7424,7427],{"className":7423},[59],[14,7425,6471],{"className":7426},[59,60],[14,7428,7430],{"className":7429},[107],[14,7431,7433],{"className":7432},[111],[14,7434,7436],{"className":7435},[115],[14,7437,7439],{"className":7438,"style":308},[119],[14,7440,7441,7444],{"style":330},[14,7442],{"className":7443,"style":127},[126],[14,7445,7447],{"className":7446},[131,132,133,134],[14,7448,7450,7453,7456,7468,7471],{"className":7449},[59,134],[14,7451,596],{"className":7452},[59,60,134],[14,7454],{"className":7455,"style":5402},[204,134],[14,7457,7459,7462,7465],{"className":7458},[850,134],[14,7460,4894],{"className":7461},[134],[14,7463,6388],{"className":7464},[134],[14,7466,5364],{"className":7467},[134],[14,7469],{"className":7470,"style":5402},[204,134],[14,7472,1924],{"className":7473,"style":1993},[59,60,134],[14,7475],{"className":7476,"style":352},[204],[14,7478,7191],{"className":7479},[356],[14,7481],{"className":7482,"style":352},[204],[14,7484,7486,7490,7523],{"className":7485},[50],[14,7487],{"className":7488,"style":7489},[54],"height:1.0369em;",[14,7491,7493,7496],{"className":7492},[59],[14,7494,6471],{"className":7495},[59,60],[14,7497,7499],{"className":7498},[107],[14,7500,7502],{"className":7501},[111],[14,7503,7505],{"className":7504},[115],[14,7506,7509],{"className":7507,"style":7508},[119],"height:0.7144em;",[14,7510,7511,7514],{"style":330},[14,7512],{"className":7513,"style":127},[126],[14,7515,7517],{"className":7516},[131,132,133,134],[14,7518,7520],{"className":7519},[59,134],[14,7521,596],{"className":7522},[59,60,134],[14,7524,7526,7529],{"className":7525},[59],[14,7527,6471],{"className":7528},[59,60],[14,7530,7532],{"className":7531},[107],[14,7533,7535],{"className":7534},[111],[14,7536,7538],{"className":7537},[115],[14,7539,7541],{"className":7540,"style":7489},[119],[14,7542,7543,7546],{"style":330},[14,7544],{"className":7545,"style":127},[126],[14,7547,7549],{"className":7548},[131,132,133,134],[14,7550,7552,7555,7558,7588],{"className":7551},[59,134],[14,7553,174],{"className":7554},[59,134],[14,7556,596],{"className":7557},[59,60,134],[14,7559,7561,7564],{"className":7560},[59,134],[14,7562,1924],{"className":7563,"style":1993},[59,60,134],[14,7565,7567],{"className":7566},[107],[14,7568,7570],{"className":7569},[111],[14,7571,7573],{"className":7572},[115],[14,7574,7576],{"className":7575,"style":2052},[119],[14,7577,7579,7582],{"style":7578},"top:-2.931em;margin-right:0.0714em;",[14,7580],{"className":7581,"style":5392},[126],[14,7583,7585],{"className":7584},[131,1186,1210,134],[14,7586,83],{"className":7587},[59,134],[14,7589,1147],{"className":7590},[59,134],[10,7592,7593],{},"Thus, locally, the vMF distribution approximates a Gaussian distribution:",[14,7595,7597],{"className":7596,"translate":18},[230],[14,7598,7600,7636],{"className":7599,"translate":18},[17],[14,7601,7603],{"className":7602},[22],[24,7604,7605],{"xmlns":26,"display":239},[28,7606,7607,7633],{},[31,7608,7609,7611,7613,7617],{},[34,7610,1924],{},[165,7612,3334],{},[34,7614,7616],{"mathvariant":7615},"script","N",[31,7618,7619,7621,7623,7625,7631],{},[165,7620,168],{"fence":46},[81,7622,602],{},[165,7624,670],{"separator":46},[931,7626,7627,7629],{},[81,7628,177],{},[34,7630,596],{},[165,7632,180],{"fence":46},[38,7634,7635],{"encoding":40},"\\theta \\sim 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the confidence probability is ",[14,7761,7763,7776],{"className":7762,"translate":18},[17],[14,7764,7766],{"className":7765},[22],[24,7767,7768],{"xmlns":26},[28,7769,7770,7774],{},[31,7771,7772],{},[34,7773,10],{},[38,7775,10],{"encoding":40},[14,7777,7779],{"className":7778,"ariaHidden":46},[45],[14,7780,7782,7785],{"className":7781},[50],[14,7783],{"className":7784,"style":55},[54],[14,7786,10],{"className":7787},[59,60],", then",[14,7790,7792],{"className":7791,"translate":18},[230],[14,7793,7795,7833],{"className":7794,"translate":18},[17],[14,7796,7798],{"className":7797},[22],[24,7799,7800],{"xmlns":26,"display":239},[28,7801,7802,7830],{},[31,7803,7804,7810,7812,7821,7823],{},[678,7805,7806,7808],{},[34,7807,1924],{},[34,7809,4894],{},[165,7811,7191],{},[7813,7814,7815],"msqrt",{},[931,7816,7817,7819],{},[81,7818,83],{},[34,7820,596],{},[3336,7822,5124],{},[678,7824,7825,7828],{},[34,7826,7827],{},"z",[34,7829,10],{},[38,7831,7832],{"encoding":40},"\\theta_c \\approx 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z_p",[14,7834,7836,7891],{"className":7835,"ariaHidden":46},[45],[14,7837,7839,7842,7882,7885,7888],{"className":7838},[50],[14,7840],{"className":7841,"style":4926},[54],[14,7843,7845,7848],{"className":7844},[59],[14,7846,1924],{"className":7847,"style":1993},[59,60],[14,7849,7851],{"className":7850},[107],[14,7852,7854,7874],{"className":7853},[111,801],[14,7855,7857,7871],{"className":7856},[115],[14,7858,7860],{"className":7859,"style":1477},[119],[14,7861,7862,7865],{"style":4947},[14,7863],{"className":7864,"style":127},[126],[14,7866,7868],{"className":7867},[131,132,133,134],[14,7869,4894],{"className":7870},[59,60,134],[14,7872,825],{"className":7873},[824],[14,7875,7877],{"className":7876},[115],[14,7878,7880],{"className":7879,"style":832},[119],[14,7881],{},[14,7883],{"className":7884,"style":352},[204],[14,7886,7191],{"className":7887},[356],[14,7889],{"className":7890,"style":352},[204],[14,7892,7894,7898,8022,8025],{"className":7893},[50],[14,7895],{"className":7896,"style":7897},[54],"height:2.44em;vertical-align:-0.7884em;",[14,7899,7902],{"className":7900},[59,7901],"sqrt",[14,7903,7905,8013],{"className":7904},[111,801],[14,7906,7908,8010],{"className":7907},[115],[14,7909,7912,7987],{"className":7910,"style":7911},[119],"height:1.6516em;",[14,7913,7917,7921],{"className":7914,"style":7916},[7915],"svg-align","top:-4.4em;",[14,7918],{"className":7919,"style":7920},[126],"height:4.4em;",[14,7922,7925],{"className":7923,"style":7924},[59],"padding-left:1em;",[14,7926,7928,7931,7984],{"className":7927},[59],[14,7929],{"className":7930},[197,1077],[14,7932,7934],{"className":7933},[931],[14,7935,7937,7976],{"className":7936},[111,801],[14,7938,7940,7973],{"className":7939},[115],[14,7941,7943,7954,7962],{"className":7942,"style":5227},[119],[14,7944,7945,7948],{"style":2558},[14,7946],{"className":7947,"style":1097},[126],[14,7949,7951],{"className":7950},[59],[14,7952,596],{"className":7953},[59,60],[14,7955,7956,7959],{"style":1287},[14,7957],{"className":7958,"style":1097},[126],[14,7960],{"className":7961,"style":1294},[1226],[14,7963,7964,7967],{"style":1297},[14,7965],{"className":7966,"style":1097},[126],[14,7968,7970],{"className":7969},[59],[14,7971,83],{"className":7972},[59],[14,7974,825],{"className":7975},[824],[14,7977,7979],{"className":7978},[115],[14,7980,7982],{"className":7981,"style":3073},[119],[14,7983],{},[14,7985],{"className":7986},[225,1077],[14,7988,7990,7993],{"style":7989},"top:-3.6116em;",[14,7991],{"className":7992,"style":7920},[126],[14,7994,7998],{"className":7995,"style":7997},[7996],"hide-tail","min-width:1.02em;height:2.48em;",[7999,8000,8006],"svg",{"xmlns":8001,"width":8002,"height":8003,"viewBox":8004,"preserveAspectRatio":8005},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","2.48em","0 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is the quantile of the normal distribution.",[10,8143,8144],{},"Examples:",[8146,8147,8148,8229],"table",{},[8149,8150,8151],"thead",{},[8152,8153,8154,8158],"tr",{},[8155,8156,8157],"th",{},"Confidence",[8155,8159,8160],{},[14,8161,8163,8180],{"className":8162,"translate":18},[17],[14,8164,8166],{"className":8165},[22],[24,8167,8168],{"xmlns":26},[28,8169,8170,8178],{},[31,8171,8172],{},[678,8173,8174,8176],{},[34,8175,7827],{},[34,8177,10],{},[38,8179,8090],{"encoding":40},[14,8181,8183],{"className":8182,"ariaHidden":46},[45],[14,8184,8186,8189],{"className":8185},[50],[14,8187],{"className":8188,"style":8100},[54],[14,8190,8192,8195],{"className":8191},[59],[14,8193,7827],{"className":8194,"style":8031},[59,60],[14,8196,8198],{"className":8197},[107],[14,8199,8201,8221],{"className":8200},[111,801],[14,8202,8204,8218],{"className":8203},[115],[14,8205,8207],{"className":8206,"style":1477},[119],[14,8208,8209,8212],{"style":8046},[14,8210],{"className":8211,"style":127},[126],[14,8213,8215],{"className":8214},[131,132,133,134],[14,8216,10],{"className":8217},[59,60,134],[14,8219,825],{"className":8220},[824],[14,8222,8224],{"className":8223},[115],[14,8225,8227],{"className":8226,"style":8065},[119],[14,8228],{},[8230,8231,8232,8241,8249],"tbody",{},[8152,8233,8234,8238],{},[8235,8236,8237],"td",{},"90%",[8235,8239,8240],{},"1.64",[8152,8242,8243,8246],{},[8235,8244,8245],{},"95%",[8235,8247,8248],{},"1.96",[8152,8250,8251,8254],{},[8235,8252,8253],{},"99%",[8235,8255,8256],{},"2.58",[10,8258,8259],{},"Thus, the 95% confidence cone angle is:",[14,8261,8263],{"className":8262,"translate":18},[230],[14,8264,8266,8297],{"className":8265,"translate":18},[17],[14,8267,8269],{"className":8268},[22],[24,8270,8271],{"xmlns":26,"display":239},[28,8272,8273,8294],{},[31,8274,8275,8282,8284,8286],{},[678,8276,8277,8279],{},[34,8278,1924],{},[81,8280,8281],{},"95",[165,8283,7191],{},[81,8285,8248],{},[7813,8287,8288],{},[931,8289,8290,8292],{},[81,8291,83],{},[34,8293,596],{},[38,8295,8296],{"encoding":40},"\\theta_{95} \\approx 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A_d(\\kappa) - z_p 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small angles:",[14,9638,9640],{"className":9639,"translate":18},[230],[14,9641,9643,9685],{"className":9642,"translate":18},[17],[14,9644,9646],{"className":9645},[22],[24,9647,9648],{"xmlns":26,"display":239},[28,9649,9650,9682],{},[31,9651,9652,9654,9656,9658,9660,9662,9668,9670],{},[34,9653,1957],{},[165,9655,168],{"stretchy":167},[34,9657,1924],{},[165,9659,180],{"stretchy":167},[165,9661,7191],{},[678,9663,9664,9666],{},[34,9665,682],{},[34,9667,171],{},[3336,9669,5124],{},[75,9671,9672,9674],{},[34,9673,1924],{},[31,9675,9676,9678,9680],{},[34,9677,171],{},[165,9679,174],{},[81,9681,177],{},[38,9683,9684],{"encoding":40},"A(\\theta) \\approx C_d \\, \\theta^{d-1}",[14,9686,9688,9715],{"className":9687,"ariaHidden":46},[45],[14,9689,9691,9694,9697,9700,9703,9706,9709,9712],{"className":9690},[50],[14,9692],{"className":9693,"style":193},[54],[14,9695,1957],{"className":9696},[59,60],[14,9698,168],{"className":9699},[197],[14,9701,1924],{"className":9702,"style":1993},[59,60],[14,9704,180],{"className":9705},[225],[14,9707],{"className":9708,"style":352},[204],[14,9710,7191],{"className":9711},[356],[14,9713],{"className":9714,"style":352},[204],[14,9716,9718,9722,9762,9765],{"className":9717},[50],[14,9719],{"className":9720,"style":9721},[54],"height:1.0491em;vertical-align:-0.15em;",[14,9723,9725,9728],{"className":9724},[59],[14,9726,682],{"className":9727,"style":794},[59,60],[14,9729,9731],{"className":9730},[107],[14,9732,9734,9754],{"className":9733},[111,801],[14,9735,9737,9751],{"className":9736},[115],[14,9738,9740],{"className":9739,"style":808},[119],[14,9741,9742,9745],{"style":811},[14,9743],{"className":9744,"style":127},[126],[14,9746,9748],{"className":9747},[131,132,133,134],[14,9749,171],{"className":9750},[59,60,134],[14,9752,825],{"className":9753},[824],[14,9755,9757],{"className":9756},[115],[14,9758,9760],{"className":9759,"style":832},[119],[14,9761],{},[14,9763],{"className":9764,"style":765},[204],[14,9766,9768,9771],{"className":9767},[59],[14,9769,1924],{"className":9770,"style":1993},[59,60],[14,9772,9774],{"className":9773},[107],[14,9775,9777],{"className":9776},[111],[14,9778,9780],{"className":9779},[115],[14,9781,9783],{"className":9782,"style":308},[119],[14,9784,9785,9788],{"style":330},[14,9786],{"className":9787,"style":127},[126],[14,9789,9791],{"className":9790},[131,132,133,134],[14,9792,9794,9797,9800],{"className":9793},[59,134],[14,9795,171],{"className":9796},[59,60,134],[14,9798,174],{"className":9799},[209,134],[14,9801,177],{"className":9802},[59,134],[10,9804,900],{},[14,9806,9808],{"className":9807,"translate":18},[230],[14,9809,9811,9879],{"className":9810,"translate":18},[17],[14,9812,9814],{"className":9813},[22],[24,9815,9816],{"xmlns":26,"display":239},[28,9817,9818,9876],{},[31,9819,9820,9826,9828],{},[678,9821,9822,9824],{},[34,9823,682],{},[34,9825,171],{},[165,9827,260],{},[931,9829,9830,9854],{},[31,9831,9832,9834],{},[81,9833,83],{},[75,9835,9836,9838],{},[34,9837,957],{},[31,9839,9840,9842,9844,9846,9848,9850,9852],{},[165,9841,168],{"stretchy":167},[34,9843,171],{},[165,9845,174],{},[81,9847,177],{},[165,9849,180],{"stretchy":167},[34,9851,968],{"mathvariant":283},[81,9853,83],{},[31,9855,9856,9858,9860,9862,9864,9866,9868,9870,9872,9874],{},[34,9857,5718],{"mathvariant":283},[165,9859,168],{"stretchy":167},[165,9861,168],{"stretchy":167},[34,9863,171],{},[165,9865,174],{},[81,9867,177],{},[165,9869,180],{"stretchy":167},[34,9871,968],{"mathvariant":283},[81,9873,83],{},[165,9875,180],{"stretchy":167},[38,9877,9878],{"encoding":40},"C_d = \\frac{2\\pi^{(d-1)\u002F2}}{\\Gamma((d-1)\u002F2)}",[14,9880,9882,9937],{"className":9881,"ariaHidden":46},[45],[14,9883,9885,9888,9928,9931,9934],{"className":9884},[50],[14,9886],{"className":9887,"style":1458},[54],[14,9889,9891,9894],{"className":9890},[59],[14,9892,682],{"className":9893,"style":794},[59,60],[14,9895,9897],{"className":9896},[107],[14,9898,9900,9920],{"className":9899},[111,801],[14,9901,9903,9917],{"className":9902},[115],[14,9904,9906],{"className":9905,"style":808},[119],[14,9907,9908,9911],{"style":811},[14,9909],{"className":9910,"style":127},[126],[14,9912,9914],{"className":9913},[131,132,133,134],[14,9915,171],{"className":9916},[59,60,134],[14,9918,825],{"className":9919},[824],[14,9921,9923],{"className":9922},[115],[14,9924,9926],{"className":9925,"style":832},[119],[14,9927],{},[14,9929],{"className":9930,"style":352},[204],[14,9932,260],{"className":9933},[356],[14,9935],{"className":9936,"style":352},[204],[14,9938,9940,9943],{"className":9939},[50],[14,9941],{"className":9942,"style":5805},[54],[14,9944,9946,9949,10077],{"className":9945},[59],[14,9947],{"className":9948},[197,1077],[14,9950,9952],{"className":9951},[931],[14,9953,9955,10069],{"className":9954},[111,801],[14,9956,9958,10066],{"className":9957},[115],[14,9959,9961,10000,10008],{"className":9960,"style":5824},[119],[14,9962,9963,9966],{"style":2558},[14,9964],{"className":9965,"style":1097},[126],[14,9967,9969,9972,9976,9979,9982,9985,9988,9991,9994,9997],{"className":9968},[59],[14,9970,5718],{"className":9971},[59],[14,9973,9975],{"className":9974},[197],"((",[14,9977,171],{"className":9978},[59,60],[14,9980],{"className":9981,"style":205},[204],[14,9983,174],{"className":9984},[209],[14,9986],{"className":9987,"style":205},[204],[14,9989,177],{"className":9990},[59],[14,9992,180],{"className":9993},[225],[14,9995,1147],{"className":9996},[59],[14,9998,180],{"className":9999},[225],[14,10001,10002,10005],{"style":1287},[14,10003],{"className":10004,"style":1097},[126],[14,10006],{"className":10007,"style":1294},[1226],[14,10009,10010,10013],{"style":1297},[14,10011],{"className":10012,"style":1097},[126],[14,10014,10016,10019],{"className":10015},[59],[14,10017,83],{"className":10018},[59],[14,10020,10022,10025],{"className":10021},[59],[14,10023,957],{"className":10024,"style":103},[59,60],[14,10026,10028],{"className":10027},[107],[14,10029,10031],{"className":10030},[111],[14,10032,10034],{"className":10033},[115],[14,10035,10037],{"className":10036,"style":5885},[119],[14,10038,10039,10042],{"style":122},[14,10040],{"className":10041,"style":127},[126],[14,10043,10045],{"className":10044},[131,132,133,134],[14,10046,10048,10051,10054,10057,10060,10063],{"className":10047},[59,134],[14,10049,168],{"className":10050},[197,134],[14,10052,171],{"className":10053},[59,60,134],[14,10055,174],{"className":10056},[209,134],[14,10058,177],{"className":10059},[59,134],[14,10061,180],{"className":10062},[225,134],[14,10064,1147],{"className":10065},[59,134],[14,10067,825],{"className":10068},[824],[14,10070,10072],{"className":10071},[115],[14,10073,10075],{"className":10074,"style":5912},[119],[14,10076],{},[14,10078],{"className":10079},[225,1077],[10,10081,10082,10083,10157],{},"Hence, the spherical cap area grows as ",[14,10084,10086,10110],{"className":10085,"translate":18},[17],[14,10087,10089],{"className":10088},[22],[24,10090,10091],{"xmlns":26},[28,10092,10093,10107],{},[31,10094,10095],{},[75,10096,10097,10099],{},[34,10098,1924],{},[31,10100,10101,10103,10105],{},[34,10102,171],{},[165,10104,174],{},[81,10106,177],{},[38,10108,10109],{"encoding":40},"\\theta^{d-1}",[14,10111,10113],{"className":10112,"ariaHidden":46},[45],[14,10114,10116,10119],{"className":10115},[50],[14,10117],{"className":10118,"style":539},[54],[14,10120,10122,10125],{"className":10121},[59],[14,10123,1924],{"className":10124,"style":1993},[59,60],[14,10126,10128],{"className":10127},[107],[14,10129,10131],{"className":10130},[111],[14,10132,10134],{"className":10133},[115],[14,10135,10137],{"className":10136,"style":539},[119],[14,10138,10139,10142],{"style":122},[14,10140],{"className":10141,"style":127},[126],[14,10143,10145],{"className":10144},[131,132,133,134],[14,10146,10148,10151,10154],{"className":10147},[59,134],[14,10149,171],{"className":10150},[59,60,134],[14,10152,174],{"className":10153},[209,134],[14,10155,177],{"className":10156},[59,134],", which is a key reason why high-dimensional sphere volume concentrates near the equator.",[10,10159,10160],{},"In many embedding papers, the following is used:",[14,10162,10164],{"className":10163,"translate":18},[230],[14,10165,10167,10202],{"className":10166,"translate":18},[17],[14,10168,10170],{"className":10169},[22],[24,10171,10172],{"xmlns":26,"display":239},[28,10173,10174,10199],{},[31,10175,10176,10183,10185],{},[678,10177,10178,10180],{},[34,10179,1924],{},[3336,10181,10182],{},"typical",[165,10184,7191],{},[7813,10186,10187],{},[931,10188,10189,10197],{},[31,10190,10191,10193,10195],{},[34,10192,171],{},[165,10194,174],{},[81,10196,177],{},[34,10198,596],{},[38,10200,10201],{"encoding":40},"\\theta_{\\text{typical}} \\approx \\sqrt{\\frac{d-1}{\\kappa}}",[14,10203,10205,10267],{"className":10204,"ariaHidden":46},[45],[14,10206,10208,10212,10258,10261,10264],{"className":10207},[50],[14,10209],{"className":10210,"style":10211},[54],"height:0.9805em;vertical-align:-0.2861em;",[14,10213,10215,10218],{"className":10214},[59],[14,10216,1924],{"className":10217,"style":1993},[59,60],[14,10219,10221],{"className":10220},[107],[14,10222,10224,10250],{"className":10223},[111,801],[14,10225,10227,10247],{"className":10226},[115],[14,10228,10230],{"className":10229,"style":808},[119],[14,10231,10232,10235],{"style":4947},[14,10233],{"className":10234,"style":127},[126],[14,10236,10238],{"className":10237},[131,132,133,134],[14,10239,10241],{"className":10240},[59,134],[14,10242,10244],{"className":10243},[59,3382,134],[14,10245,10182],{"className":10246},[59,134],[14,10248,825],{"className":10249},[824],[14,10251,10253],{"className":10252},[115],[14,10254,10256],{"className":10255,"style":8065},[119],[14,10257],{},[14,10259],{"className":10260,"style":352},[204],[14,10262,7191],{"className":10263},[356],[14,10265],{"className":10266,"style":352},[204],[14,10268,10270,10274],{"className":10269},[50],[14,10271],{"className":10272,"style":10273},[54],"height:2.44em;vertical-align:-0.7634em;",[14,10275,10277],{"className":10276},[59,7901],[14,10278,10280,10386],{"className":10279},[111,801],[14,10281,10283,10383],{"className":10282},[115],[14,10284,10287,10370],{"className":10285,"style":10286},[119],"height:1.6766em;",[14,10288,10290,10293],{"className":10289,"style":7916},[7915],[14,10291],{"className":10292,"style":7920},[126],[14,10294,10296],{"className":10295,"style":7924},[59],[14,10297,10299,10302,10367],{"className":10298},[59],[14,10300],{"className":10301},[197,1077],[14,10303,10305],{"className":10304},[931],[14,10306,10308,10359],{"className":10307},[111,801],[14,10309,10311,10356],{"className":10310},[115],[14,10312,10314,10325,10333],{"className":10313,"style":4080},[119],[14,10315,10316,10319],{"style":2558},[14,10317],{"className":10318,"style":1097},[126],[14,10320,10322],{"className":10321},[59],[14,10323,596],{"className":10324},[59,60],[14,10326,10327,10330],{"style":1287},[14,10328],{"className":10329,"style":1097},[126],[14,10331],{"className":10332,"style":1294},[1226],[14,10334,10335,10338],{"style":1297},[14,10336],{"className":10337,"style":1097},[126],[14,10339,10341,10344,10347,10350,10353],{"className":10340},[59],[14,10342,171],{"className":10343},[59,60],[14,10345],{"className":10346,"style":205},[204],[14,10348,174],{"className":10349},[209],[14,10351],{"className":10352,"style":205},[204],[14,10354,177],{"className":10355},[59],[14,10357,825],{"className":10358},[824],[14,10360,10362],{"className":10361},[115],[14,10363,10365],{"className":10364,"style":3073},[119],[14,10366],{},[14,10368],{"className":10369},[225,1077],[14,10371,10373,10376],{"style":10372},"top:-3.6366em;",[14,10374],{"className":10375,"style":7920},[126],[14,10377,10379],{"className":10378,"style":7997},[7996],[7999,10380,10381],{"xmlns":8001,"width":8002,"height":8003,"viewBox":8004,"preserveAspectRatio":8005},[8007,10382],{"d":8009},[14,10384,825],{"className":10385},[824],[14,10387,10389],{"className":10388},[115],[14,10390,10393],{"className":10391,"style":10392},[119],"height:0.7634em;",[14,10394],{},[10,10396,10397],{},"Interpretation: most probability mass of the vMF distribution lies in",[14,10399,10401],{"className":10400,"translate":18},[230],[14,10402,10404,10435],{"className":10403,"translate":18},[17],[14,10405,10407],{"className":10406},[22],[24,10408,10409],{"xmlns":26,"display":239},[28,10410,10411,10432],{},[31,10412,10413,10415,10418],{},[34,10414,1924],{},[165,10416,10417],{},"≲",[7813,10419,10420],{},[931,10421,10422,10430],{},[31,10423,10424,10426,10428],{},[34,10425,171],{},[165,10427,174],{},[81,10429,177],{},[34,10431,596],{},[38,10433,10434],{"encoding":40},"\\theta \\lesssim \\sqrt{\\frac{d-1}{\\kappa}}",[14,10436,10438,10458],{"className":10437,"ariaHidden":46},[45],[14,10439,10441,10445,10448,10451,10455],{"className":10440},[50],[14,10442],{"className":10443,"style":10444},[54],"height:0.9592em;vertical-align:-0.2296em;",[14,10446,1924],{"className":10447,"style":1993},[59,60],[14,10449],{"className":10450,"style":352},[204],[14,10452,10417],{"className":10453},[356,10454],"amsrm",[14,10456],{"className":10457,"style":352},[204],[14,10459,10461,10464],{"className":10460},[50],[14,10462],{"className":10463,"style":10273},[54],[14,10465,10467],{"className":10466},[59,7901],[14,10468,10470,10574],{"className":10469},[111,801],[14,10471,10473,10571],{"className":10472},[115],[14,10474,10476,10559],{"className":10475,"style":10286},[119],[14,10477,10479,10482],{"className":10478,"style":7916},[7915],[14,10480],{"className":10481,"style":7920},[126],[14,10483,10485],{"className":10484,"style":7924},[59],[14,10486,10488,10491,10556],{"className":10487},[59],[14,10489],{"className":10490},[197,1077],[14,10492,10494],{"className":10493},[931],[14,10495,10497,10548],{"className":10496},[111,801],[14,10498,10500,10545],{"className":10499},[115],[14,10501,10503,10514,10522],{"className":10502,"style":4080},[119],[14,10504,10505,10508],{"style":2558},[14,10506],{"className":10507,"style":1097},[126],[14,10509,10511],{"className":10510},[59],[14,10512,596],{"className":10513},[59,60],[14,10515,10516,10519],{"style":1287},[14,10517],{"className":10518,"style":1097},[126],[14,10520],{"className":10521,"style":1294},[1226],[14,10523,10524,10527],{"style":1297},[14,10525],{"className":10526,"style":1097},[126],[14,10528,10530,10533,10536,10539,10542],{"className":10529},[59],[14,10531,171],{"className":10532},[59,60],[14,10534],{"className":10535,"style":205},[204],[14,10537,174],{"className":10538},[209],[14,10540],{"className":10541,"style":205},[204],[14,10543,177],{"className":10544},[59],[14,10546,825],{"className":10547},[824],[14,10549,10551],{"className":10550},[115],[14,10552,10554],{"className":10553,"style":3073},[119],[14,10555],{},[14,10557],{"className":10558},[225,1077],[14,10560,10561,10564],{"style":10372},[14,10562],{"className":10563,"style":7920},[126],[14,10565,10567],{"className":10566,"style":7997},[7996],[7999,10568,10569],{"xmlns":8001,"width":8002,"height":8003,"viewBox":8004,"preserveAspectRatio":8005},[8007,10570],{"d":8009},[14,10572,825],{"className":10573},[824],[14,10575,10577],{"className":10576},[115],[14,10578,10580],{"className":10579,"style":10392},[119],[14,10581],{},[10,10583,10584,10585,10613],{},"Intuition: ",[14,10586,10588,10601],{"className":10587,"translate":18},[17],[14,10589,10591],{"className":10590},[22],[24,10592,10593],{"xmlns":26},[28,10594,10595,10599],{},[31,10596,10597],{},[34,10598,596],{},[38,10600,1673],{"encoding":40},[14,10602,10604],{"className":10603,"ariaHidden":46},[45],[14,10605,10607,10610],{"className":10606},[50],[14,10608],{"className":10609,"style":1541},[54],[14,10611,596],{"className":10612},[59,60]," determines the directional noise magnitude:",[14,10615,10617],{"className":10616,"translate":18},[230],[14,10618,10620,10653],{"className":10619,"translate":18},[17],[14,10621,10623],{"className":10622},[22],[24,10624,10625],{"xmlns":26,"display":239},[28,10626,10627,10650],{},[31,10628,10629,10631,10633,10636,10638,10648],{},[34,10630,1924],{},[165,10632,3334],{},[34,10634,10635],{},"O",[165,10637,168],{"stretchy":167},[7813,10639,10640],{},[31,10641,10642,10644,10646],{},[81,10643,177],{},[34,10645,968],{"mathvariant":283},[34,10647,596],{},[165,10649,180],{"stretchy":167},[38,10651,10652],{"encoding":40},"\\theta \\sim O(\\sqrt{1\u002F\\kappa})",[14,10654,10656,10674],{"className":10655,"ariaHidden":46},[45],[14,10657,10659,10662,10665,10668,10671],{"className":10658},[50],[14,10660],{"className":10661,"style":2134},[54],[14,10663,1924],{"className":10664,"style":1993},[59,60],[14,10666],{"className":10667,"style":352},[204],[14,10669,3334],{"className":10670},[356],[14,10672],{"className":10673,"style":352},[204],[14,10675,10677,10681,10684,10687,10747],{"className":10676},[50],[14,10678],{"className":10679,"style":10680},[54],"height:1.24em;vertical-align:-0.2561em;",[14,10682,10635],{"className":10683,"style":1993},[59,60],[14,10685,168],{"className":10686},[197],[14,10688,10690],{"className":10689},[59,7901],[14,10691,10693,10738],{"className":10692},[111,801],[14,10694,10696,10735],{"className":10695},[115],[14,10697,10700,10718],{"className":10698,"style":10699},[119],"height:0.9839em;",[14,10701,10704,10708],{"className":10702,"style":10703},[7915],"top:-3.2em;",[14,10705],{"className":10706,"style":10707},[126],"height:3.2em;",[14,10709,10711,10715],{"className":10710,"style":7924},[59],[14,10712,10714],{"className":10713},[59],"1\u002F",[14,10716,596],{"className":10717},[59,60],[14,10719,10721,10724],{"style":10720},"top:-2.9439em;",[14,10722],{"className":10723,"style":10707},[126],[14,10725,10728],{"className":10726,"style":10727},[7996],"min-width:1.02em;height:1.28em;",[7999,10729,10732],{"xmlns":8001,"width":8002,"height":10730,"viewBox":10731,"preserveAspectRatio":8005},"1.28em","0 0 400000 1296",[8007,10733],{"d":10734},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[14,10736,825],{"className":10737},[824],[14,10739,10741],{"className":10740},[115],[14,10742,10745],{"className":10743,"style":10744},[119],"height:0.2561em;",[14,10746],{},[14,10748,180],{"className":10749},[225],[10,10751,10752],{},"These formulas are frequently used in sphere embedding, CLIP embedding, contrastive learning, spherical clustering, and directional statistics.",[10,10754,10755,10756,10784],{},"For example, ",[14,10757,10759,10772],{"className":10758,"translate":18},[17],[14,10760,10762],{"className":10761},[22],[24,10763,10764],{"xmlns":26},[28,10765,10766,10770],{},[31,10767,10768],{},[34,10769,596],{},[38,10771,1673],{"encoding":40},[14,10773,10775],{"className":10774,"ariaHidden":46},[45],[14,10776,10778,10781],{"className":10777},[50],[14,10779],{"className":10780,"style":1541},[54],[14,10782,596],{"className":10783},[59,60]," can be used to estimate: angular noise in embeddings, angular radius of clusters, confidence cone of prototypes.",[10,10786,10787],{},"The probability mass of the vMF distribution is concentrated at",[14,10789,10791],{"className":10790,"translate":18},[230],[14,10792,10794,10825],{"className":10793,"translate":18},[17],[14,10795,10797],{"className":10796},[22],[24,10798,10799],{"xmlns":26,"display":239},[28,10800,10801,10823],{},[31,10802,10803,10805,10807,10809,10811,10821],{},[34,10804,1924],{},[165,10806,3334],{},[34,10808,10635],{},[165,10810,168],{"stretchy":167},[7813,10812,10813],{},[31,10814,10815,10817,10819],{},[81,10816,177],{},[34,10818,968],{"mathvariant":283},[34,10820,596],{},[165,10822,180],{"stretchy":167},[38,10824,10652],{"encoding":40},[14,10826,10828,10846],{"className":10827,"ariaHidden":46},[45],[14,10829,10831,10834,10837,10840,10843],{"className":10830},[50],[14,10832],{"className":10833,"style":2134},[54],[14,10835,1924],{"className":10836,"style":1993},[59,60],[14,10838],{"className":10839,"style":352},[204],[14,10841,3334],{"className":10842},[356],[14,10844],{"className":10845,"style":352},[204],[14,10847,10849,10852,10855,10858,10908],{"className":10848},[50],[14,10850],{"className":10851,"style":10680},[54],[14,10853,10635],{"className":10854,"style":1993},[59,60],[14,10856,168],{"className":10857},[197],[14,10859,10861],{"className":10860},[59,7901],[14,10862,10864,10900],{"className":10863},[111,801],[14,10865,10867,10897],{"className":10866},[115],[14,10868,10870,10885],{"className":10869,"style":10699},[119],[14,10871,10873,10876],{"className":10872,"style":10703},[7915],[14,10874],{"className":10875,"style":10707},[126],[14,10877,10879,10882],{"className":10878,"style":7924},[59],[14,10880,10714],{"className":10881},[59],[14,10883,596],{"className":10884},[59,60],[14,10886,10887,10890],{"style":10720},[14,10888],{"className":10889,"style":10707},[126],[14,10891,10893],{"className":10892,"style":10727},[7996],[7999,10894,10895],{"xmlns":8001,"width":8002,"height":10730,"viewBox":10731,"preserveAspectRatio":8005},[8007,10896],{"d":10734},[14,10898,825],{"className":10899},[824],[14,10901,10903],{"className":10902},[115],[14,10904,10906],{"className":10905,"style":10744},[119],[14,10907],{},[14,10909,180],{"className":10910},[225],[10,10912,10913,10914,10942,10943,4534],{},"The corresponding confidence region is a spherical cap with axis ",[14,10915,10917,10930],{"className":10916,"translate":18},[17],[14,10918,10920],{"className":10919},[22],[24,10921,10922],{"xmlns":26},[28,10923,10924,10928],{},[31,10925,10926],{},[34,10927,36],{},[38,10929,41],{"encoding":40},[14,10931,10933],{"className":10932,"ariaHidden":46},[45],[14,10934,10936,10939],{"className":10935},[50],[14,10937],{"className":10938,"style":55},[54],[14,10940,36],{"className":10941},[59,60]," and angular radius roughly ",[14,10944,10946,10968],{"className":10945,"translate":18},[17],[14,10947,10949],{"className":10948},[22],[24,10950,10951],{"xmlns":26},[28,10952,10953,10965],{},[31,10954,10955],{},[7813,10956,10957],{},[31,10958,10959,10961,10963],{},[81,10960,177],{},[34,10962,968],{"mathvariant":283},[34,10964,596],{},[38,10966,10967],{"encoding":40},"\\sqrt{1\u002F\\kappa}",[14,10969,10971],{"className":10970,"ariaHidden":46},[45],[14,10972,10974,10978],{"className":10973},[50],[14,10975],{"className":10976,"style":10977},[54],"height:1.24em;vertical-align:-0.305em;",[14,10979,10981],{"className":10980},[59,7901],[14,10982,10984,11022],{"className":10983},[111,801],[14,10985,10987,11019],{"className":10986},[115],[14,10988,10991,11006],{"className":10989,"style":10990},[119],"height:0.935em;",[14,10992,10994,10997],{"className":10993,"style":10703},[7915],[14,10995],{"className":10996,"style":10707},[126],[14,10998,11000,11003],{"className":10999,"style":7924},[59],[14,11001,10714],{"className":11002},[59],[14,11004,596],{"className":11005},[59,60],[14,11007,11009,11012],{"style":11008},"top:-2.895em;",[14,11010],{"className":11011,"style":10707},[126],[14,11013,11015],{"className":11014,"style":10727},[7996],[7999,11016,11017],{"xmlns":8001,"width":8002,"height":10730,"viewBox":10731,"preserveAspectRatio":8005},[8007,11018],{"d":10734},[14,11020,825],{"className":11021},[824],[14,11023,11025],{"className":11024},[115],[14,11026,11029],{"className":11027,"style":11028},[119],"height:0.305em;",[14,11030],{},[145,11032,11034],{"id":11033},"parameter-estimation","Parameter Estimation",[10,11036,11037,11038,11066],{},"Estimating ",[14,11039,11041,11054],{"className":11040,"translate":18},[17],[14,11042,11044],{"className":11043},[22],[24,11045,11046],{"xmlns":26},[28,11047,11048,11052],{},[31,11049,11050],{},[34,11051,596],{},[38,11053,1673],{"encoding":40},[14,11055,11057],{"className":11056,"ariaHidden":46},[45],[14,11058,11060,11063],{"className":11059},[50],[14,11061],{"className":11062,"style":1541},[54],[14,11064,596],{"className":11065},[59,60]," is the most challenging part, as it requires solving an equation involving modified Bessel functions.",[10,11068,11069,11070,11098],{},"Below are several practical approximation algorithms\u002Fformulas for estimating the vMF ",[14,11071,11073,11086],{"className":11072,"translate":18},[17],[14,11074,11076],{"className":11075},[22],[24,11077,11078],{"xmlns":26},[28,11079,11080,11084],{},[31,11081,11082],{},[34,11083,596],{},[38,11085,1673],{"encoding":40},[14,11087,11089],{"className":11088,"ariaHidden":46},[45],[14,11090,11092,11095],{"className":11091},[50],[14,11093],{"className":11094,"style":1541},[54],[14,11096,596],{"className":11097},[59,60]," parameter, listed in approximate order of common usage.",[10,11100,11101,11102,11244,11245,4534],{},"Given samples ",[14,11103,11105,11139],{"className":11104,"translate":18},[17],[14,11106,11108],{"className":11107},[22],[24,11109,11110],{"xmlns":26},[28,11111,11112,11136],{},[31,11113,11114,11120,11122,11124,11126,11128,11130],{},[678,11115,11116,11118],{},[34,11117,267],{},[81,11119,177],{},[165,11121,670],{"separator":46},[34,11123,4534],{"mathvariant":283},[34,11125,4534],{"mathvariant":283},[34,11127,4534],{"mathvariant":283},[165,11129,670],{"separator":46},[678,11131,11132,11134],{},[34,11133,267],{},[34,11135,5372],{},[38,11137,11138],{"encoding":40},"x_1,...,x_n",[14,11140,11142],{"className":11141,"ariaHidden":46},[45],[14,11143,11145,11148,11188,11191,11194,11198,11201,11204],{"className":11144},[50],[14,11146],{"className":11147,"style":55},[54],[14,11149,11151,11154],{"className":11150},[59],[14,11152,267],{"className":11153},[59,60],[14,11155,11157],{"className":11156},[107],[14,11158,11160,11180],{"className":11159},[111,801],[14,11161,11163,11177],{"className":11162},[115],[14,11164,11166],{"className":11165,"style":8324},[119],[14,11167,11168,11171],{"style":2350},[14,11169],{"className":11170,"style":127},[126],[14,11172,11174],{"className":11173},[131,132,133,134],[14,11175,177],{"className":11176},[59,134],[14,11178,825],{"className":11179},[824],[14,11181,11183],{"className":11182},[115],[14,11184,11186],{"className":11185,"style":832},[119],[14,11187],{},[14,11189,670],{"className":11190},[761],[14,11192],{"className":11193,"style":765},[204],[14,11195,11197],{"className":11196},[59],"...",[14,11199,670],{"className":11200},[761],[14,11202],{"className":11203,"style":765},[204],[14,11205,11207,11210],{"className":11206},[59],[14,11208,267],{"className":11209},[59,60],[14,11211,11213],{"className":11212},[107],[14,11214,11216,11236],{"className":11215},[111,801],[14,11217,11219,11233],{"className":11218},[115],[14,11220,11222],{"className":11221,"style":1477},[119],[14,11223,11224,11227],{"style":2350},[14,11225],{"className":11226,"style":127},[126],[14,11228,11230],{"className":11229},[131,132,133,134],[14,11231,5372],{"className":11232},[59,60,134],[14,11234,825],{"className":11235},[824],[14,11237,11239],{"className":11238},[115],[14,11240,11242],{"className":11241,"style":832},[119],[14,11243],{},", each ",[14,11246,11248,11280],{"className":11247,"translate":18},[17],[14,11249,11251],{"className":11250},[22],[24,11252,11253],{"xmlns":26},[28,11254,11255,11277],{},[31,11256,11257,11263,11265],{},[678,11258,11259,11261],{},[34,11260,267],{},[34,11262,5368],{},[165,11264,270],{},[75,11266,11267,11269],{},[34,11268,249],{"mathvariant":248},[31,11270,11271,11273,11275],{},[34,11272,171],{},[165,11274,174],{},[81,11276,177],{},[38,11278,11279],{"encoding":40},"x_i \\in \\mathbb{S}^{d-1}",[14,11281,11283,11340],{"className":11282,"ariaHidden":46},[45],[14,11284,11286,11290,11331,11334,11337],{"className":11285},[50],[14,11287],{"className":11288,"style":11289},[54],"height:0.6891em;vertical-align:-0.15em;",[14,11291,11293,11296],{"className":11292},[59],[14,11294,267],{"className":11295},[59,60],[14,11297,11299],{"className":11298},[107],[14,11300,11302,11323],{"className":11301},[111,801],[14,11303,11305,11320],{"className":11304},[115],[14,11306,11309],{"className":11307,"style":11308},[119],"height:0.3117em;",[14,11310,11311,11314],{"style":2350},[14,11312],{"className":11313,"style":127},[126],[14,11315,11317],{"className":11316},[131,132,133,134],[14,11318,5368],{"className":11319},[59,60,134],[14,11321,825],{"className":11322},[824],[14,11324,11326],{"className":11325},[115],[14,11327,11329],{"className":11328,"style":832},[119],[14,11330],{},[14,11332],{"className":11333,"style":352},[204],[14,11335,270],{"className":11336},[356],[14,11338],{"className":11339,"style":352},[204],[14,11341,11343,11346],{"className":11342},[50],[14,11344],{"className":11345,"style":539},[54],[14,11347,11349,11352],{"className":11348},[59],[14,11350,249],{"className":11351},[59,315],[14,11353,11355],{"className":11354},[107],[14,11356,11358],{"className":11357},[111],[14,11359,11361],{"className":11360},[115],[14,11362,11364],{"className":11363,"style":539},[119],[14,11365,11366,11369],{"style":122},[14,11367],{"className":11368,"style":127},[126],[14,11370,11372],{"className":11371},[131,132,133,134],[14,11373,11375,11378,11381],{"className":11374},[59,134],[14,11376,171],{"className":11377},[59,60,134],[14,11379,174],{"className":11380},[209,134],[14,11382,177],{"className":11383},[59,134],[10,11385,11386],{},"First compute:",[14,11388,11390],{"className":11389,"translate":18},[230],[14,11391,11393,11446],{"className":11392,"translate":18},[17],[14,11394,11396],{"className":11395},[22],[24,11397,11398],{"xmlns":26,"display":239},[28,11399,11400,11443],{},[31,11401,11402,11410,11412],{},[11403,11404,11405,11407],"mover",{"accent":46},[34,11406,275],{},[165,11408,11409],{},"ˉ",[165,11411,260],{},[931,11413,11414,11441],{},[31,11415,11416,11418,11433,11439],{},[34,11417,665],{"mathvariant":283},[11419,11420,11421,11423,11431],"munderover",{},[165,11422,4214],{},[31,11424,11425,11427,11429],{},[34,11426,5368],{},[165,11428,260],{},[81,11430,177],{},[34,11432,5372],{},[678,11434,11435,11437],{},[34,11436,267],{},[34,11438,5368],{},[34,11440,665],{"mathvariant":283},[34,11442,5372],{},[38,11444,11445],{"encoding":40},"\\bar{R} = \\frac{|\\sum_{i=1}^n x_i|}{n}",[14,11447,11449,11501],{"className":11448,"ariaHidden":46},[45],[14,11450,11452,11456,11492,11495,11498],{"className":11451},[50],[14,11453],{"className":11454,"style":11455},[54],"height:0.8201em;",[14,11457,11460],{"className":11458},[59,11459],"accent",[14,11461,11463],{"className":11462},[111],[14,11464,11466],{"className":11465},[115],[14,11467,11469,11478],{"className":11468,"style":11455},[119],[14,11470,11472,11475],{"style":11471},"top:-3em;",[14,11473],{"className":11474,"style":1097},[126],[14,11476,275],{"className":11477,"style":3311},[59,60],[14,11479,11481,11484],{"style":11480},"top:-3.2523em;",[14,11482],{"className":11483,"style":1097},[126],[14,11485,11489],{"className":11486,"style":11488},[11487],"accent-body","left:-0.1667em;",[14,11490,11409],{"className":11491},[59],[14,11493],{"className":11494,"style":352},[204],[14,11496,260],{"className":11497},[356],[14,11499],{"className":11500,"style":352},[204],[14,11502,11504,11508],{"className":11503},[50],[14,11505],{"className":11506,"style":11507},[54],"height:2.18em;vertical-align:-0.686em;",[14,11509,11511,11514,11683],{"className":11510},[59],[14,11512],{"className":11513},[197,1077],[14,11515,11517],{"className":11516},[931],[14,11518,11520,11675],{"className":11519},[111,801],[14,11521,11523,11672],{"className":11522},[115],[14,11524,11527,11538,11546],{"className":11525,"style":11526},[119],"height:1.494em;",[14,11528,11529,11532],{"style":2558},[14,11530],{"className":11531,"style":1097},[126],[14,11533,11535],{"className":11534},[59],[14,11536,5372],{"className":11537},[59,60],[14,11539,11540,11543],{"style":1287},[14,11541],{"className":11542,"style":1097},[126],[14,11544],{"className":11545,"style":1294},[1226],[14,11547,11549,11552],{"style":11548},"top:-3.6897em;",[14,11550],{"className":11551,"style":1097},[126],[14,11553,11555,11558,11561,11626,11629,11669],{"className":11554},[59],[14,11556,665],{"className":11557},[59],[14,11559],{"className":11560,"style":765},[204],[14,11562,11564,11568],{"className":11563},[850],[14,11565,4214],{"className":11566,"style":11567},[850,4317,6688],"position:relative;top:0em;",[14,11569,11571],{"className":11570},[107],[14,11572,11574,11617],{"className":11573},[111,801],[14,11575,11577,11614],{"className":11576},[115],[14,11578,11581,11602],{"className":11579,"style":11580},[119],"height:0.8043em;",[14,11582,11584,11587],{"style":11583},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[14,11585],{"className":11586,"style":127},[126],[14,11588,11590],{"className":11589},[131,132,133,134],[14,11591,11593,11596,11599],{"className":11592},[59,134],[14,11594,5368],{"className":11595},[59,60,134],[14,11597,260],{"className":11598},[356,134],[14,11600,177],{"className":11601},[59,134],[14,11603,11605,11608],{"style":11604},"top:-3.2029em;margin-right:0.05em;",[14,11606],{"className":11607,"style":127},[126],[14,11609,11611],{"className":11610},[131,132,133,134],[14,11612,5372],{"className":11613},[59,60,134],[14,11615,825],{"className":11616},[824],[14,11618,11620],{"className":11619},[115],[14,11621,11624],{"className":11622,"style":11623},[119],"height:0.2997em;",[14,11625],{},[14,11627],{"className":11628,"style":765},[204],[14,11630,11632,11635],{"className":11631},[59],[14,11633,267],{"className":11634},[59,60],[14,11636,11638],{"className":11637},[107],[14,11639,11641,11661],{"className":11640},[111,801],[14,11642,11644,11658],{"className":11643},[115],[14,11645,11647],{"className":11646,"style":11308},[119],[14,11648,11649,11652],{"style":2350},[14,11650],{"className":11651,"style":127},[126],[14,11653,11655],{"className":11654},[131,132,133,134],[14,11656,5368],{"className":11657},[59,60,134],[14,11659,825],{"className":11660},[824],[14,11662,11664],{"className":11663},[115],[14,11665,11667],{"className":11666,"style":832},[119],[14,11668],{},[14,11670,665],{"className":11671},[59],[14,11673,825],{"className":11674},[824],[14,11676,11678],{"className":11677},[115],[14,11679,11681],{"className":11680,"style":3073},[119],[14,11682],{},[14,11684],{"className":11685},[225,1077],[10,11687,11688],{},"Mean direction:",[14,11690,11692],{"className":11691,"translate":18},[230],[14,11693,11695,11737],{"className":11694,"translate":18},[17],[14,11696,11698],{"className":11697},[22],[24,11699,11700],{"xmlns":26,"display":239},[28,11701,11702,11734],{},[31,11703,11704,11706,11708],{},[34,11705,36],{},[165,11707,260],{},[931,11709,11710,11720],{},[31,11711,11712,11714],{},[165,11713,4214],{},[678,11715,11716,11718],{},[34,11717,267],{},[34,11719,5368],{},[31,11721,11722,11724,11726,11732],{},[34,11723,665],{"mathvariant":283},[165,11725,4214],{},[678,11727,11728,11730],{},[34,11729,267],{},[34,11731,5368],{},[34,11733,665],{"mathvariant":283},[38,11735,11736],{"encoding":40},"\\mu = \\frac{\\sum x_i}{|\\sum x_i|}",[14,11738,11740,11758],{"className":11739,"ariaHidden":46},[45],[14,11741,11743,11746,11749,11752,11755],{"className":11742},[50],[14,11744],{"className":11745,"style":55},[54],[14,11747,36],{"className":11748},[59,60],[14,11750],{"className":11751,"style":352},[204],[14,11753,260],{"className":11754},[356],[14,11756],{"className":11757,"style":352},[204],[14,11759,11761,11765],{"className":11760},[50],[14,11762],{"className":11763,"style":11764},[54],"height:2.363em;vertical-align:-0.936em;",[14,11766,11768,11771,11919],{"className":11767},[59],[14,11769],{"className":11770},[197,1077],[14,11772,11774],{"className":11773},[931],[14,11775,11777,11911],{"className":11776},[111,801],[14,11778,11780,11908],{"className":11779},[115],[14,11781,11783,11846,11854],{"className":11782,"style":2987},[119],[14,11784,11785,11788],{"style":2558},[14,11786],{"className":11787,"style":1097},[126],[14,11789,11791,11794,11797,11800,11803,11843],{"className":11790},[59],[14,11792,665],{"className":11793},[59],[14,11795],{"className":11796,"style":765},[204],[14,11798,4214],{"className":11799,"style":11567},[850,4317,6688],[14,11801],{"className":11802,"style":765},[204],[14,11804,11806,11809],{"className":11805},[59],[14,11807,267],{"className":11808},[59,60],[14,11810,11812],{"className":11811},[107],[14,11813,11815,11835],{"className":11814},[111,801],[14,11816,11818,11832],{"className":11817},[115],[14,11819,11821],{"className":11820,"style":11308},[119],[14,11822,11823,11826],{"style":2350},[14,11824],{"className":11825,"style":127},[126],[14,11827,11829],{"className":11828},[131,132,133,134],[14,11830,5368],{"className":11831},[59,60,134],[14,11833,825],{"className":11834},[824],[14,11836,11838],{"className":11837},[115],[14,11839,11841],{"className":11840,"style":832},[119],[14,11842],{},[14,11844,665],{"className":11845},[59],[14,11847,11848,11851],{"style":1287},[14,11849],{"className":11850,"style":1097},[126],[14,11852],{"className":11853,"style":1294},[1226],[14,11855,11856,11859],{"style":1297},[14,11857],{"className":11858,"style":1097},[126],[14,11860,11862,11865,11868],{"className":11861},[59],[14,11863,4214],{"className":11864,"style":11567},[850,4317,6688],[14,11866],{"className":11867,"style":765},[204],[14,11869,11871,11874],{"className":11870},[59],[14,11872,267],{"className":11873},[59,60],[14,11875,11877],{"className":11876},[107],[14,11878,11880,11900],{"className":11879},[111,801],[14,11881,11883,11897],{"className":11882},[115],[14,11884,11886],{"className":11885,"style":11308},[119],[14,11887,11888,11891],{"style":2350},[14,11889],{"className":11890,"style":127},[126],[14,11892,11894],{"className":11893},[131,132,133,134],[14,11895,5368],{"className":11896},[59,60,134],[14,11898,825],{"className":11899},[824],[14,11901,11903],{"className":11902},[115],[14,11904,11906],{"className":11905,"style":832},[119],[14,11907],{},[14,11909,825],{"className":11910},[824],[14,11912,11914],{"className":11913},[115],[14,11915,11917],{"className":11916,"style":5912},[119],[14,11918],{},[14,11920],{"className":11921},[225,1077],[10,11923,11924,11925,11953],{},"The exact equation for ",[14,11926,11928,11941],{"className":11927,"translate":18},[17],[14,11929,11931],{"className":11930},[22],[24,11932,11933],{"xmlns":26},[28,11934,11935,11939],{},[31,11936,11937],{},[34,11938,596],{},[38,11940,1673],{"encoding":40},[14,11942,11944],{"className":11943,"ariaHidden":46},[45],[14,11945,11947,11950],{"className":11946},[50],[14,11948],{"className":11949,"style":1541},[54],[14,11951,596],{"className":11952},[59,60]," is:",[14,11955,11957],{"className":11956,"translate":18},[230],[14,11958,11960,11992],{"className":11959,"translate":18},[17],[14,11961,11963],{"className":11962},[22],[24,11964,11965],{"xmlns":26,"display":239},[28,11966,11967,11989],{},[31,11968,11969,11975,11977,11979,11981,11983],{},[678,11970,11971,11973],{},[34,11972,1957],{},[34,11974,171],{},[165,11976,168],{"stretchy":167},[34,11978,596],{},[165,11980,180],{"stretchy":167},[165,11982,260],{},[11403,11984,11985,11987],{"accent":46},[34,11986,275],{},[165,11988,11409],{},[38,11990,11991],{"encoding":40},"A_d(\\kappa) = \\bar{R}",[14,11993,11995,12059],{"className":11994,"ariaHidden":46},[45],[14,11996,11998,12001,12041,12044,12047,12050,12053,12056],{"className":11997},[50],[14,11999],{"className":12000,"style":193},[54],[14,12002,12004,12007],{"className":12003},[59],[14,12005,1957],{"className":12006},[59,60],[14,12008,12010],{"className":12009},[107],[14,12011,12013,12033],{"className":12012},[111,801],[14,12014,12016,12030],{"className":12015},[115],[14,12017,12019],{"className":12018,"style":808},[119],[14,12020,12021,12024],{"style":2350},[14,12022],{"className":12023,"style":127},[126],[14,12025,12027],{"className":12026},[131,132,133,134],[14,12028,171],{"className":12029},[59,60,134],[14,12031,825],{"className":12032},[824],[14,12034,12036],{"className":12035},[115],[14,12037,12039],{"className":12038,"style":832},[119],[14,12040],{},[14,12042,168],{"className":12043},[197],[14,12045,596],{"className":12046},[59,60],[14,12048,180],{"className":12049},[225],[14,12051],{"className":12052,"style":352},[204],[14,12054,260],{"className":12055},[356],[14,12057],{"className":12058,"style":352},[204],[14,12060,12062,12065],{"className":12061},[50],[14,12063],{"className":12064,"style":11455},[54],[14,12066,12068],{"className":12067},[59,11459],[14,12069,12071],{"className":12070},[111],[14,12072,12074],{"className":12073},[115],[14,12075,12077,12085],{"className":12076,"style":11455},[119],[14,12078,12079,12082],{"style":11471},[14,12080],{"className":12081,"style":1097},[126],[14,12083,275],{"className":12084,"style":3311},[59,60],[14,12086,12087,12090],{"style":11480},[14,12088],{"className":12089,"style":1097},[126],[14,12091,12093],{"className":12092,"style":11488},[11487],[14,12094,11409],{"className":12095},[59],[10,12097,900],{},[14,12099,12101],{"className":12100,"translate":18},[230],[14,12102,12104,12175],{"className":12103,"translate":18},[17],[14,12105,12107],{"className":12106},[22],[24,12108,12109],{"xmlns":26,"display":239},[28,12110,12111,12173],{},[31,12112,12113,12119,12121,12123,12125,12127],{},[678,12114,12115,12117],{},[34,12116,1957],{},[34,12118,171],{},[165,12120,168],{"stretchy":167},[34,12122,596],{},[165,12124,180],{"stretchy":167},[165,12126,260],{},[931,12128,12129,12149],{},[31,12130,12131,12143,12145,12147],{},[678,12132,12133,12135],{},[34,12134,975],{},[31,12136,12137,12139,12141],{},[34,12138,171],{},[34,12140,968],{"mathvariant":283},[81,12142,83],{},[165,12144,168],{"stretchy":167},[34,12146,596],{},[165,12148,180],{"stretchy":167},[31,12150,12151,12167,12169,12171],{},[678,12152,12153,12155],{},[34,12154,975],{},[31,12156,12157,12159,12161,12163,12165],{},[34,12158,171],{},[34,12160,968],{"mathvariant":283},[81,12162,83],{},[165,12164,174],{},[81,12166,177],{},[165,12168,168],{"stretchy":167},[34,12170,596],{},[165,12172,180],{"stretchy":167},[38,12174,2462],{"encoding":40},[14,12176,12178,12242],{"className":12177,"ariaHidden":46},[45],[14,12179,12181,12184,12224,12227,12230,12233,12236,12239],{"className":12180},[50],[14,12182],{"className":12183,"style":193},[54],[14,12185,12187,12190],{"className":12186},[59],[14,12188,1957],{"className":12189},[59,60],[14,12191,12193],{"className":12192},[107],[14,12194,12196,12216],{"className":12195},[111,801],[14,12197,12199,12213],{"className":12198},[115],[14,12200,12202],{"className":12201,"style":808},[119],[14,12203,12204,12207],{"style":2350},[14,12205],{"className":12206,"style":127},[126],[14,12208,12210],{"className":12209},[131,132,133,134],[14,12211,171],{"className":12212},[59,60,134],[14,12214,825],{"className":12215},[824],[14,12217,12219],{"className":12218},[115],[14,12220,12222],{"className":12221,"style":832},[119],[14,12223],{},[14,12225,168],{"className":12226},[197],[14,12228,596],{"className":12229},[59,60],[14,12231,180],{"className":12232},[225],[14,12234],{"className":12235,"style":352},[204],[14,12237,260],{"className":12238},[356],[14,12240],{"className":12241,"style":352},[204],[14,12243,12245,12248],{"className":12244},[50],[14,12246],{"className":12247,"style":2536},[54],[14,12249,12251,12254,12417],{"className":12250},[59],[14,12252],{"className":12253},[197,1077],[14,12255,12257],{"className":12256},[931],[14,12258,12260,12409],{"className":12259},[111,801],[14,12261,12263,12406],{"className":12262},[115],[14,12264,12266,12335,12343],{"className":12265,"style":2555},[119],[14,12267,12268,12271],{"style":2558},[14,12269],{"className":12270,"style":1097},[126],[14,12272,12274,12326,12329,12332],{"className":12273},[59],[14,12275,12277,12280],{"className":12276},[59],[14,12278,975],{"className":12279,"style":1154},[59,60],[14,12281,12283],{"className":12282},[107],[14,12284,12286,12318],{"className":12285},[111,801],[14,12287,12289,12315],{"className":12288},[115],[14,12290,12292],{"className":12291,"style":1167},[119],[14,12293,12294,12297],{"style":2585},[14,12295],{"className":12296,"style":127},[126],[14,12298,12300],{"className":12299},[131,132,133,134],[14,12301,12303,12306,12309,12312],{"className":12302},[59,134],[14,12304,171],{"className":12305},[59,60,134],[14,12307,1147],{"className":12308},[59,134],[14,12310,174],{"className":12311},[209,134],[14,12313,177],{"className":12314},[59,134],[14,12316,825],{"className":12317},[824],[14,12319,12321],{"className":12320},[115],[14,12322,12324],{"className":12323,"style":2616},[119],[14,12325],{},[14,12327,168],{"className":12328},[197],[14,12330,596],{"className":12331},[59,60],[14,12333,180],{"className":12334},[225],[14,12336,12337,12340],{"style":1287},[14,12338],{"className":12339,"style":1097},[126],[14,12341],{"className":12342,"style":1294},[1226],[14,12344,12345,12348],{"style":2638},[14,12346],{"className":12347,"style":1097},[126],[14,12349,12351,12397,12400,12403],{"className":12350},[59],[14,12352,12354,12357],{"className":12353},[59],[14,12355,975],{"className":12356,"style":1154},[59,60],[14,12358,12360],{"className":12359},[107],[14,12361,12363,12389],{"className":12362},[111,801],[14,12364,12366,12386],{"className":12365},[115],[14,12367,12369],{"className":12368,"style":1167},[119],[14,12370,12371,12374],{"style":2585},[14,12372],{"className":12373,"style":127},[126],[14,12375,12377],{"className":12376},[131,132,133,134],[14,12378,12380,12383],{"className":12379},[59,134],[14,12381,171],{"className":12382},[59,60,134],[14,12384,1147],{"className":12385},[59,134],[14,12387,825],{"className":12388},[824],[14,12390,12392],{"className":12391},[115],[14,12393,12395],{"className":12394,"style":2616},[119],[14,12396],{},[14,12398,168],{"className":12399},[197],[14,12401,596],{"className":12402},[59,60],[14,12404,180],{"className":12405},[225],[14,12407,825],{"className":12408},[824],[14,12410,12412],{"className":12411},[115],[14,12413,12415],{"className":12414,"style":2709},[119],[14,12416],{},[14,12418],{"className":12419},[225,1077],[10,12421,12422],{},"This involves the modified Bessel function of the first kind, so approximations or numerical solutions are necessary.",[1889,12424,12426],{"id":12425},"banerjee-approximation-most-commonly-used","Banerjee Approximation (most commonly used)",[10,12428,12429,12430],{},"Source: Arindam Banerjee, 2005: ",[140,12431,12432],{},"Clustering on the Unit Hypersphere using von Mises-Fisher",[10,12434,12435,12436],{},"Applicable for high dimensions ",[14,12437,12439,12457],{"className":12438,"translate":18},[17],[14,12440,12442],{"className":12441},[22],[24,12443,12444],{"xmlns":26},[28,12445,12446,12454],{},[31,12447,12448,12450,12452],{},[34,12449,171],{},[165,12451,599],{},[81,12453,6041],{},[38,12455,12456],{"encoding":40},"d \\ge 3",[14,12458,12460,12478],{"className":12459,"ariaHidden":46},[45],[14,12461,12463,12466,12469,12472,12475],{"className":12462},[50],[14,12464],{"className":12465,"style":4907},[54],[14,12467,171],{"className":12468},[59,60],[14,12470],{"className":12471,"style":352},[204],[14,12473,599],{"className":12474},[356],[14,12476],{"className":12477,"style":352},[204],[14,12479,12481,12484],{"className":12480},[50],[14,12482],{"className":12483,"style":634},[54],[14,12485,6041],{"className":12486},[59],[14,12488,12490],{"className":12489,"translate":18},[230],[14,12491,12493,12553],{"className":12492,"translate":18},[17],[14,12494,12496],{"className":12495},[22],[24,12497,12498],{"xmlns":26,"display":239},[28,12499,12500,12550],{},[31,12501,12502,12504,12506],{},[34,12503,596],{},[165,12505,7191],{},[931,12507,12508,12534],{},[31,12509,12510,12516,12518,12520,12522,12532],{},[11403,12511,12512,12514],{"accent":46},[34,12513,275],{},[165,12515,11409],{},[165,12517,168],{"stretchy":167},[34,12519,171],{},[165,12521,174],{},[75,12523,12524,12530],{},[11403,12525,12526,12528],{"accent":46},[34,12527,275],{},[165,12529,11409],{},[81,12531,83],{},[165,12533,180],{"stretchy":167},[31,12535,12536,12538,12540],{},[81,12537,177],{},[165,12539,174],{},[75,12541,12542,12548],{},[11403,12543,12544,12546],{"accent":46},[34,12545,275],{},[165,12547,11409],{},[81,12549,83],{},[38,12551,12552],{"encoding":40},"\\kappa \\approx \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2}",[14,12554,12556,12575],{"className":12555,"ariaHidden":46},[45],[14,12557,12559,12563,12566,12569,12572],{"className":12558},[50],[14,12560],{"className":12561,"style":12562},[54],"height:0.4831em;",[14,12564,596],{"className":12565},[59,60],[14,12567],{"className":12568,"style":352},[204],[14,12570,7191],{"className":12571},[356],[14,12573],{"className":12574,"style":352},[204],[14,12576,12578,12582],{"className":12577},[50],[14,12579],{"className":12580,"style":12581},[54],"height:2.2905em;vertical-align:-0.7934em;",[14,12583,12585,12588,12813],{"className":12584},[59],[14,12586],{"className":12587},[197,1077],[14,12589,12591],{"className":12590},[931],[14,12592,12594,12804],{"className":12593},[111,801],[14,12595,12597,12801],{"className":12596},[115],[14,12598,12601,12679,12687],{"className":12599,"style":12600},[119],"height:1.4971em;",[14,12602,12604,12607],{"style":12603},"top:-2.2899em;",[14,12605],{"className":12606,"style":1097},[126],[14,12608,12610,12613,12616,12619,12622],{"className":12609},[59],[14,12611,177],{"className":12612},[59],[14,12614],{"className":12615,"style":205},[204],[14,12617,174],{"className":12618},[209],[14,12620],{"className":12621,"style":205},[204],[14,12623,12625,12656],{"className":12624},[59],[14,12626,12628],{"className":12627},[59,11459],[14,12629,12631],{"className":12630},[111],[14,12632,12634],{"className":12633},[115],[14,12635,12637,12645],{"className":12636,"style":11455},[119],[14,12638,12639,12642],{"style":11471},[14,12640],{"className":12641,"style":1097},[126],[14,12643,275],{"className":12644,"style":3311},[59,60],[14,12646,12647,12650],{"style":11480},[14,12648],{"className":12649,"style":1097},[126],[14,12651,12653],{"className":12652,"style":11488},[11487],[14,12654,11409],{"className":12655},[59],[14,12657,12659],{"className":12658},[107],[14,12660,12662],{"className":12661},[111],[14,12663,12665],{"className":12664},[115],[14,12666,12668],{"className":12667,"style":3670},[119],[14,12669,12670,12673],{"style":1131},[14,12671],{"className":12672,"style":127},[126],[14,12674,12676],{"className":12675},[131,132,133,134],[14,12677,83],{"className":12678},[59,134],[14,12680,12681,12684],{"style":1287},[14,12682],{"className":12683,"style":1097},[126],[14,12685],{"className":12686,"style":1294},[1226],[14,12688,12689,12692],{"style":1297},[14,12690],{"className":12691,"style":1097},[126],[14,12693,12695,12726,12729,12732,12735,12738,12741,12798],{"className":12694},[59],[14,12696,12698],{"className":12697},[59,11459],[14,12699,12701],{"className":12700},[111],[14,12702,12704],{"className":12703},[115],[14,12705,12707,12715],{"className":12706,"style":11455},[119],[14,12708,12709,12712],{"style":11471},[14,12710],{"className":12711,"style":1097},[126],[14,12713,275],{"className":12714,"style":3311},[59,60],[14,12716,12717,12720],{"style":11480},[14,12718],{"className":12719,"style":1097},[126],[14,12721,12723],{"className":12722,"style":11488},[11487],[14,12724,11409],{"className":12725},[59],[14,12727,168],{"className":12728},[197],[14,12730,171],{"className":12731},[59,60],[14,12733],{"className":12734,"style":205},[204],[14,12736,174],{"className":12737},[209],[14,12739],{"className":12740,"style":205},[204],[14,12742,12744,12775],{"className":12743},[59],[14,12745,12747],{"className":12746},[59,11459],[14,12748,12750],{"className":12749},[111],[14,12751,12753],{"className":12752},[115],[14,12754,12756,12764],{"className":12755,"style":11455},[119],[14,12757,12758,12761],{"style":11471},[14,12759],{"className":12760,"style":1097},[126],[14,12762,275],{"className":12763,"style":3311},[59,60],[14,12765,12766,12769],{"style":11480},[14,12767],{"className":12768,"style":1097},[126],[14,12770,12772],{"className":12771,"style":11488},[11487],[14,12773,11409],{"className":12774},[59],[14,12776,12778],{"className":12777},[107],[14,12779,12781],{"className":12780},[111],[14,12782,12784],{"className":12783},[115],[14,12785,12787],{"className":12786,"style":96},[119],[14,12788,12789,12792],{"style":122},[14,12790],{"className":12791,"style":127},[126],[14,12793,12795],{"className":12794},[131,132,133,134],[14,12796,83],{"className":12797},[59,134],[14,12799,180],{"className":12800},[225],[14,12802,825],{"className":12803},[824],[14,12805,12807],{"className":12806},[115],[14,12808,12811],{"className":12809,"style":12810},[119],"height:0.7934em;",[14,12812],{},[14,12814],{"className":12815},[225,1077],[10,12817,12818,12819,12863],{},"Advantages: extremely simple, ",[14,12820,12822,12842],{"className":12821,"translate":18},[17],[14,12823,12825],{"className":12824},[22],[24,12826,12827],{"xmlns":26},[28,12828,12829,12839],{},[31,12830,12831,12833,12835,12837],{},[34,12832,10635],{},[165,12834,168],{"stretchy":167},[81,12836,177],{},[165,12838,180],{"stretchy":167},[38,12840,12841],{"encoding":40},"O(1)",[14,12843,12845],{"className":12844,"ariaHidden":46},[45],[14,12846,12848,12851,12854,12857,12860],{"className":12847},[50],[14,12849],{"className":12850,"style":193},[54],[14,12852,10635],{"className":12853,"style":1993},[59,60],[14,12855,168],{"className":12856},[197],[14,12858,177],{"className":12859},[59],[14,12861,180],{"className":12862},[225]," computation, works well in high dimensions. Disadvantages: larger bias in low dimensions.",[10,12865,12866],{},"This is the most common initialization formula for spherical k-means \u002F vMF mixtures.",[1889,12868,12870],{"id":12869},"sra-approximation-newton-refinement-recommended","Sra Approximation + Newton Refinement (recommended)",[10,12872,12873,12874],{},"Source: Suvrit Sra, 2012: ",[140,12875,12876],{},"A Short Note on Parameter Approximation for vMF",[10,12878,12879],{},"Initial value:",[14,12881,12883],{"className":12882,"translate":18},[230],[14,12884,12886,12950],{"className":12885,"translate":18},[17],[14,12887,12889],{"className":12888},[22],[24,12890,12891],{"xmlns":26,"display":239},[28,12892,12893,12947],{},[31,12894,12895,12901,12903],{},[678,12896,12897,12899],{},[34,12898,596],{},[81,12900,602],{},[165,12902,260],{},[931,12904,12905,12931],{},[31,12906,12907,12913,12915,12917,12919,12929],{},[11403,12908,12909,12911],{"accent":46},[34,12910,275],{},[165,12912,11409],{},[165,12914,168],{"stretchy":167},[34,12916,171],{},[165,12918,174],{},[75,12920,12921,12927],{},[11403,12922,12923,12925],{"accent":46},[34,12924,275],{},[165,12926,11409],{},[81,12928,83],{},[165,12930,180],{"stretchy":167},[31,12932,12933,12935,12937],{},[81,12934,177],{},[165,12936,174],{},[75,12938,12939,12945],{},[11403,12940,12941,12943],{"accent":46},[34,12942,275],{},[165,12944,11409],{},[81,12946,83],{},[38,12948,12949],{"encoding":40},"\\kappa_0 = \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2}",[14,12951,12953,13009],{"className":12952,"ariaHidden":46},[45],[14,12954,12956,12960,13000,13003,13006],{"className":12955},[50],[14,12957],{"className":12958,"style":12959},[54],"height:0.5806em;vertical-align:-0.15em;",[14,12961,12963,12966],{"className":12962},[59],[14,12964,596],{"className":12965},[59,60],[14,12967,12969],{"className":12968},[107],[14,12970,12972,12992],{"className":12971},[111,801],[14,12973,12975,12989],{"className":12974},[115],[14,12976,12978],{"className":12977,"style":8324},[119],[14,12979,12980,12983],{"style":2350},[14,12981],{"className":12982,"style":127},[126],[14,12984,12986],{"className":12985},[131,132,133,134],[14,12987,602],{"className":12988},[59,134],[14,12990,825],{"className":12991},[824],[14,12993,12995],{"className":12994},[115],[14,12996,12998],{"className":12997,"style":832},[119],[14,12999],{},[14,13001],{"className":13002,"style":352},[204],[14,13004,260],{"className":13005},[356],[14,13007],{"className":13008,"style":352},[204],[14,13010,13012,13015],{"className":13011},[50],[14,13013],{"className":13014,"style":12581},[54],[14,13016,13018,13021,13243],{"className":13017},[59],[14,13019],{"className":13020},[197,1077],[14,13022,13024],{"className":13023},[931],[14,13025,13027,13235],{"className":13026},[111,801],[14,13028,13030,13232],{"className":13029},[115],[14,13031,13033,13110,13118],{"className":13032,"style":12600},[119],[14,13034,13035,13038],{"style":12603},[14,13036],{"className":13037,"style":1097},[126],[14,13039,13041,13044,13047,13050,13053],{"className":13040},[59],[14,13042,177],{"className":13043},[59],[14,13045],{"className":13046,"style":205},[204],[14,13048,174],{"className":13049},[209],[14,13051],{"className":13052,"style":205},[204],[14,13054,13056,13087],{"className":13055},[59],[14,13057,13059],{"className":13058},[59,11459],[14,13060,13062],{"className":13061},[111],[14,13063,13065],{"className":13064},[115],[14,13066,13068,13076],{"className":13067,"style":11455},[119],[14,13069,13070,13073],{"style":11471},[14,13071],{"className":13072,"style":1097},[126],[14,13074,275],{"className":13075,"style":3311},[59,60],[14,13077,13078,13081],{"style":11480},[14,13079],{"className":13080,"style":1097},[126],[14,13082,13084],{"className":13083,"style":11488},[11487],[14,13085,11409],{"className":13086},[59],[14,13088,13090],{"className":13089},[107],[14,13091,13093],{"className":13092},[111],[14,13094,13096],{"className":13095},[115],[14,13097,13099],{"className":13098,"style":3670},[119],[14,13100,13101,13104],{"style":1131},[14,13102],{"className":13103,"style":127},[126],[14,13105,13107],{"className":13106},[131,132,133,134],[14,13108,83],{"className":13109},[59,134],[14,13111,13112,13115],{"style":1287},[14,13113],{"className":13114,"style":1097},[126],[14,13116],{"className":13117,"style":1294},[1226],[14,13119,13120,13123],{"style":1297},[14,13121],{"className":13122,"style":1097},[126],[14,13124,13126,13157,13160,13163,13166,13169,13172,13229],{"className":13125},[59],[14,13127,13129],{"className":13128},[59,11459],[14,13130,13132],{"className":13131},[111],[14,13133,13135],{"className":13134},[115],[14,13136,13138,13146],{"className":13137,"style":11455},[119],[14,13139,13140,13143],{"style":11471},[14,13141],{"className":13142,"style":1097},[126],[14,13144,275],{"className":13145,"style":3311},[59,60],[14,13147,13148,13151],{"style":11480},[14,13149],{"className":13150,"style":1097},[126],[14,13152,13154],{"className":13153,"style":11488},[11487],[14,13155,11409],{"className":13156},[59],[14,13158,168],{"className":13159},[197],[14,13161,171],{"className":13162},[59,60],[14,13164],{"className":13165,"style":205},[204],[14,13167,174],{"className":13168},[209],[14,13170],{"className":13171,"style":205},[204],[14,13173,13175,13206],{"className":13174},[59],[14,13176,13178],{"className":13177},[59,11459],[14,13179,13181],{"className":13180},[111],[14,13182,13184],{"className":13183},[115],[14,13185,13187,13195],{"className":13186,"style":11455},[119],[14,13188,13189,13192],{"style":11471},[14,13190],{"className":13191,"style":1097},[126],[14,13193,275],{"className":13194,"style":3311},[59,60],[14,13196,13197,13200],{"style":11480},[14,13198],{"className":13199,"style":1097},[126],[14,13201,13203],{"className":13202,"style":11488},[11487],[14,13204,11409],{"className":13205},[59],[14,13207,13209],{"className":13208},[107],[14,13210,13212],{"className":13211},[111],[14,13213,13215],{"className":13214},[115],[14,13216,13218],{"className":13217,"style":96},[119],[14,13219,13220,13223],{"style":122},[14,13221],{"className":13222,"style":127},[126],[14,13224,13226],{"className":13225},[131,132,133,134],[14,13227,83],{"className":13228},[59,134],[14,13230,180],{"className":13231},[225],[14,13233,825],{"className":13234},[824],[14,13236,13238],{"className":13237},[115],[14,13239,13241],{"className":13240,"style":12810},[119],[14,13242],{},[14,13244],{"className":13245},[225,1077],[10,13247,13248],{},"Then apply Newton iteration:",[14,13250,13252],{"className":13251,"translate":18},[230],[14,13253,13255,13378],{"className":13254,"translate":18},[17],[14,13256,13258],{"className":13257},[22],[24,13259,13260],{"xmlns":26,"display":239},[28,13261,13262,13375],{},[31,13263,13264,13277,13279,13285,13287],{},[678,13265,13266,13268],{},[34,13267,596],{},[31,13269,13270,13273,13275],{},[34,13271,13272],{},"t",[165,13274,2861],{},[81,13276,177],{},[165,13278,260],{},[678,13280,13281,13283],{},[34,13282,596],{},[34,13284,13272],{},[165,13286,174],{},[931,13288,13289,13315],{},[31,13290,13291,13297,13299,13305,13307,13309],{},[678,13292,13293,13295],{},[34,13294,1957],{},[34,13296,171],{},[165,13298,168],{"stretchy":167},[678,13300,13301,13303],{},[34,13302,596],{},[34,13304,13272],{},[165,13306,180],{"stretchy":167},[165,13308,174],{},[11403,13310,13311,13313],{"accent":46},[34,13312,275],{},[165,13314,11409],{},[31,13316,13317,13319,13321,13327,13329,13335,13341,13343,13359,13365,13367,13373],{},[81,13318,177],{},[165,13320,174],{},[678,13322,13323,13325],{},[34,13324,1957],{},[34,13326,171],{},[165,13328,168],{"stretchy":167},[678,13330,13331,13333],{},[34,13332,596],{},[34,13334,13272],{},[75,13336,13337,13339],{},[165,13338,180],{"stretchy":167},[81,13340,83],{},[165,13342,174],{},[931,13344,13345,13353],{},[31,13346,13347,13349,13351],{},[34,13348,171],{},[165,13350,174],{},[81,13352,177],{},[678,13354,13355,13357],{},[34,13356,596],{},[34,13358,13272],{},[678,13360,13361,13363],{},[34,13362,1957],{},[34,13364,171],{},[165,13366,168],{"stretchy":167},[678,13368,13369,13371],{},[34,13370,596],{},[34,13372,13272],{},[165,13374,180],{"stretchy":167},[38,13376,13377],{"encoding":40},"\\kappa_{t+1} = \\kappa_t - \\frac{A_d(\\kappa_t)-\\bar{R}}{1-A_d(\\kappa_t)^2-\\frac{d-1}{\\kappa_t}A_d(\\kappa_t)}",[14,13379,13381,13446,13503],{"className":13380,"ariaHidden":46},[45],[14,13382,13384,13388,13437,13440,13443],{"className":13383},[50],[14,13385],{"className":13386,"style":13387},[54],"height:0.6389em;vertical-align:-0.2083em;",[14,13389,13391,13394],{"className":13390},[59],[14,13392,596],{"className":13393},[59,60],[14,13395,13397],{"className":13396},[107],[14,13398,13400,13429],{"className":13399},[111,801],[14,13401,13403,13426],{"className":13402},[115],[14,13404,13406],{"className":13405,"style":8324},[119],[14,13407,13408,13411],{"style":2350},[14,13409],{"className":13410,"style":127},[126],[14,13412,13414],{"className":13413},[131,132,133,134],[14,13415,13417,13420,13423],{"className":13416},[59,134],[14,13418,13272],{"className":13419},[59,60,134],[14,13421,2861],{"className":13422},[209,134],[14,13424,177],{"className":13425},[59,134],[14,13427,825],{"className":13428},[824],[14,13430,13432],{"className":13431},[115],[14,13433,13435],{"className":13434,"style":5319},[119],[14,13436],{},[14,13438],{"className":13439,"style":352},[204],[14,13441,260],{"className":13442},[356],[14,13444],{"className":13445,"style":352},[204],[14,13447,13449,13453,13494,13497,13500],{"className":13448},[50],[14,13450],{"className":13451,"style":13452},[54],"height:0.7333em;vertical-align:-0.15em;",[14,13454,13456,13459],{"className":13455},[59],[14,13457,596],{"className":13458},[59,60],[14,13460,13462],{"className":13461},[107],[14,13463,13465,13486],{"className":13464},[111,801],[14,13466,13468,13483],{"className":13467},[115],[14,13469,13472],{"className":13470,"style":13471},[119],"height:0.2806em;",[14,13473,13474,13477],{"style":2350},[14,13475],{"className":13476,"style":127},[126],[14,13478,13480],{"className":13479},[131,132,133,134],[14,13481,13272],{"className":13482},[59,60,134],[14,13484,825],{"className":13485},[824],[14,13487,13489],{"className":13488},[115],[14,13490,13492],{"className":13491,"style":832},[119],[14,13493],{},[14,13495],{"className":13496,"style":205},[204],[14,13498,174],{"className":13499},[209],[14,13501],{"className":13502,"style":205},[204],[14,13504,13506,13510],{"className":13505},[50],[14,13507],{"className":13508,"style":13509},[54],"height:2.7123em;vertical-align:-1.2152em;",[14,13511,13513,13516,14024],{"className":13512},[59],[14,13514],{"className":13515},[197,1077],[14,13517,13519],{"className":13518},[931],[14,13520,13522,14015],{"className":13521},[111,801],[14,13523,13525,14012],{"className":13524},[115],[14,13526,13528,13870,13878],{"className":13527,"style":12600},[119],[14,13529,13531,13534],{"style":13530},"top:-2.2299em;",[14,13532],{"className":13533,"style":1097},[126],[14,13535,13537,13540,13543,13546,13549,13589,13592,13632,13661,13664,13667,13670,13784,13824,13827,13867],{"className":13536},[59],[14,13538,177],{"className":13539},[59],[14,13541],{"className":13542,"style":205},[204],[14,13544,174],{"className":13545},[209],[14,13547],{"className":13548,"style":205},[204],[14,13550,13552,13555],{"className":13551},[59],[14,13553,1957],{"className":13554},[59,60],[14,13556,13558],{"className":13557},[107],[14,13559,13561,13581],{"className":13560},[111,801],[14,13562,13564,13578],{"className":13563},[115],[14,13565,13567],{"className":13566,"style":808},[119],[14,13568,13569,13572],{"style":2350},[14,13570],{"className":13571,"style":127},[126],[14,13573,13575],{"className":13574},[131,132,133,134],[14,13576,171],{"className":13577},[59,60,134],[14,13579,825],{"className":13580},[824],[14,13582,13584],{"className":13583},[115],[14,13585,13587],{"className":13586,"style":832},[119],[14,13588],{},[14,13590,168],{"className":13591},[197],[14,13593,13595,13598],{"className":13594},[59],[14,13596,596],{"className":13597},[59,60],[14,13599,13601],{"className":13600},[107],[14,13602,13604,13624],{"className":13603},[111,801],[14,13605,13607,13621],{"className":13606},[115],[14,13608,13610],{"className":13609,"style":13471},[119],[14,13611,13612,13615],{"style":2350},[14,13613],{"className":13614,"style":127},[126],[14,13616,13618],{"className":13617},[131,132,133,134],[14,13619,13272],{"className":13620},[59,60,134],[14,13622,825],{"className":13623},[824],[14,13625,13627],{"className":13626},[115],[14,13628,13630],{"className":13629,"style":832},[119],[14,13631],{},[14,13633,13635,13638],{"className":13634},[225],[14,13636,180],{"className":13637},[225],[14,13639,13641],{"className":13640},[107],[14,13642,13644],{"className":13643},[111],[14,13645,13647],{"className":13646},[115],[14,13648,13650],{"className":13649,"style":3670},[119],[14,13651,13652,13655],{"style":1131},[14,13653],{"className":13654,"style":127},[126],[14,13656,13658],{"className":13657},[131,132,133,134],[14,13659,83],{"className":13660},[59,134],[14,13662],{"className":13663,"style":205},[204],[14,13665,174],{"className":13666},[209],[14,13668],{"className":13669,"style":205},[204],[14,13671,13673,13676,13781],{"className":13672},[59],[14,13674],{"className":13675},[197,1077],[14,13677,13679],{"className":13678},[931],[14,13680,13682,13772],{"className":13681},[111,801],[14,13683,13685,13769],{"className":13684},[115],[14,13686,13688,13741,13749],{"className":13687,"style":6337},[119],[14,13689,13690,13693],{"style":6340},[14,13691],{"className":13692,"style":1097},[126],[14,13694,13696],{"className":13695},[131,132,133,134],[14,13697,13699],{"className":13698},[59,134],[14,13700,13702,13705],{"className":13701},[59,134],[14,13703,596],{"className":13704},[59,60,134],[14,13706,13708],{"className":13707},[107],[14,13709,13711,13733],{"className":13710},[111,801],[14,13712,13714,13730],{"className":13713},[115],[14,13715,13718],{"className":13716,"style":13717},[119],"height:0.2963em;",[14,13719,13721,13724],{"style":13720},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[14,13722],{"className":13723,"style":5392},[126],[14,13725,13727],{"className":13726},[131,1186,1210,134],[14,13728,13272],{"className":13729},[59,60,134],[14,13731,825],{"className":13732},[824],[14,13734,13736],{"className":13735},[115],[14,13737,13739],{"className":13738,"style":6957},[119],[14,13740],{},[14,13742,13743,13746],{"style":1287},[14,13744],{"className":13745,"style":1097},[126],[14,13747],{"className":13748,"style":1294},[1226],[14,13750,13751,13754],{"style":6363},[14,13752],{"className":13753,"style":1097},[126],[14,13755,13757],{"className":13756},[131,132,133,134],[14,13758,13760,13763,13766],{"className":13759},[59,134],[14,13761,171],{"className":13762},[59,60,134],[14,13764,174],{"className":13765},[209,134],[14,13767,177],{"className":13768},[59,134],[14,13770,825],{"className":13771},[824],[14,13773,13775],{"className":13774},[115],[14,13776,13779],{"className":13777,"style":13778},[119],"height:0.4451em;",[14,13780],{},[14,13782],{"className":13783},[225,1077],[14,13785,13787,13790],{"className":13786},[59],[14,13788,1957],{"className":13789},[59,60],[14,13791,13793],{"className":13792},[107],[14,13794,13796,13816],{"className":13795},[111,801],[14,13797,13799,13813],{"className":13798},[115],[14,13800,13802],{"className":13801,"style":808},[119],[14,13803,13804,13807],{"style":2350},[14,13805],{"className":13806,"style":127},[126],[14,13808,13810],{"className":13809},[131,132,133,134],[14,13811,171],{"className":13812},[59,60,134],[14,13814,825],{"className":13815},[824],[14,13817,13819],{"className":13818},[115],[14,13820,13822],{"className":13821,"style":832},[119],[14,13823],{},[14,13825,168],{"className":13826},[197],[14,13828,13830,13833],{"className":13829},[59],[14,13831,596],{"className":13832},[59,60],[14,13834,13836],{"className":13835},[107],[14,13837,13839,13859],{"className":13838},[111,801],[14,13840,13842,13856],{"className":13841},[115],[14,13843,13845],{"className":13844,"style":13471},[119],[14,13846,13847,13850],{"style":2350},[14,13848],{"className":13849,"style":127},[126],[14,13851,13853],{"className":13852},[131,132,133,134],[14,13854,13272],{"className":13855},[59,60,134],[14,13857,825],{"className":13858},[824],[14,13860,13862],{"className":13861},[115],[14,13863,13865],{"className":13864,"style":832},[119],[14,13866],{},[14,13868,180],{"className":13869},[225],[14,13871,13872,13875],{"style":1287},[14,13873],{"className":13874,"style":1097},[126],[14,13876],{"className":13877,"style":1294},[1226],[14,13879,13880,13883],{"style":1297},[14,13881],{"className":13882,"style":1097},[126],[14,13884,13886,13926,13929,13969,13972,13975,13978,13981],{"className":13885},[59],[14,13887,13889,13892],{"className":13888},[59],[14,13890,1957],{"className":13891},[59,60],[14,13893,13895],{"className":13894},[107],[14,13896,13898,13918],{"className":13897},[111,801],[14,13899,13901,13915],{"className":13900},[115],[14,13902,13904],{"className":13903,"style":808},[119],[14,13905,13906,13909],{"style":2350},[14,13907],{"className":13908,"style":127},[126],[14,13910,13912],{"className":13911},[131,132,133,134],[14,13913,171],{"className":13914},[59,60,134],[14,13916,825],{"className":13917},[824],[14,13919,13921],{"className":13920},[115],[14,13922,13924],{"className":13923,"style":832},[119],[14,13925],{},[14,13927,168],{"className":13928},[197],[14,13930,13932,13935],{"className":13931},[59],[14,13933,596],{"className":13934},[59,60],[14,13936,13938],{"className":13937},[107],[14,13939,13941,13961],{"className":13940},[111,801],[14,13942,13944,13958],{"className":13943},[115],[14,13945,13947],{"className":13946,"style":13471},[119],[14,13948,13949,13952],{"style":2350},[14,13950],{"className":13951,"style":127},[126],[14,13953,13955],{"className":13954},[131,132,133,134],[14,13956,13272],{"className":13957},[59,60,134],[14,13959,825],{"className":13960},[824],[14,13962,13964],{"className":13963},[115],[14,13965,13967],{"className":13966,"style":832},[119],[14,13968],{},[14,13970,180],{"className":13971},[225],[14,13973],{"className":13974,"style":205},[204],[14,13976,174],{"className":13977},[209],[14,13979],{"className":13980,"style":205},[204],[14,13982,13984],{"className":13983},[59,11459],[14,13985,13987],{"className":13986},[111],[14,13988,13990],{"className":13989},[115],[14,13991,13993,14001],{"className":13992,"style":11455},[119],[14,13994,13995,13998],{"style":11471},[14,13996],{"className":13997,"style":1097},[126],[14,13999,275],{"className":14000,"style":3311},[59,60],[14,14002,14003,14006],{"style":11480},[14,14004],{"className":14005,"style":1097},[126],[14,14007,14009],{"className":14008,"style":11488},[11487],[14,14010,11409],{"className":14011},[59],[14,14013,825],{"className":14014},[824],[14,14016,14018],{"className":14017},[115],[14,14019,14022],{"className":14020,"style":14021},[119],"height:1.2152em;",[14,14023],{},[14,14025],{"className":14026},[225,1077],[10,14028,14029],{},"Advantages: highly accurate, only 2–3 iterations needed.",[10,14031,14032],{},"Many libraries use this method.",[1889,14034,14036],{"id":14035},"minka-approximation","Minka Approximation",[10,14038,14039],{},"Source: Thomas Minka",[10,14041,14042],{},"Formula:",[14,14044,14046],{"className":14045,"translate":18},[230],[14,14047,14049,14133],{"className":14048,"translate":18},[17],[14,14050,14052],{"className":14051},[22],[24,14053,14054],{"xmlns":26,"display":239},[28,14055,14056,14130],{},[31,14057,14058,14060,14062,14106,14108],{},[34,14059,596],{},[165,14061,7191],{},[931,14063,14064,14090],{},[31,14065,14066,14072,14074,14076,14078,14088],{},[11403,14067,14068,14070],{"accent":46},[34,14069,275],{},[165,14071,11409],{},[165,14073,168],{"stretchy":167},[34,14075,171],{},[165,14077,174],{},[75,14079,14080,14086],{},[11403,14081,14082,14084],{"accent":46},[34,14083,275],{},[165,14085,11409],{},[81,14087,83],{},[165,14089,180],{"stretchy":167},[31,14091,14092,14094,14096],{},[81,14093,177],{},[165,14095,174],{},[75,14097,14098,14104],{},[11403,14099,14100,14102],{"accent":46},[34,14101,275],{},[165,14103,11409],{},[81,14105,83],{},[165,14107,2861],{},[931,14109,14110,14116],{},[11403,14111,14112,14114],{"accent":46},[34,14113,275],{},[165,14115,11409],{},[31,14117,14118,14120,14122,14124,14126,14128],{},[81,14119,83],{},[165,14121,168],{"stretchy":167},[34,14123,171],{},[165,14125,174],{},[81,14127,177],{},[165,14129,180],{"stretchy":167},[38,14131,14132],{"encoding":40},"\\kappa \\approx \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2} + \\frac{\\bar{R}}{2(d-1)}",[14,14134,14136,14154,14400],{"className":14135,"ariaHidden":46},[45],[14,14137,14139,14142,14145,14148,14151],{"className":14138},[50],[14,14140],{"className":14141,"style":12562},[54],[14,14143,596],{"className":14144},[59,60],[14,14146],{"className":14147,"style":352},[204],[14,14149,7191],{"className":14150},[356],[14,14152],{"className":14153,"style":352},[204],[14,14155,14157,14160,14391,14394,14397],{"className":14156},[50],[14,14158],{"className":14159,"style":12581},[54],[14,14161,14163,14166,14388],{"className":14162},[59],[14,14164],{"className":14165},[197,1077],[14,14167,14169],{"className":14168},[931],[14,14170,14172,14380],{"className":14171},[111,801],[14,14173,14175,14377],{"className":14174},[115],[14,14176,14178,14255,14263],{"className":14177,"style":12600},[119],[14,14179,14180,14183],{"style":12603},[14,14181],{"className":14182,"style":1097},[126],[14,14184,14186,14189,14192,14195,14198],{"className":14185},[59],[14,14187,177],{"className":14188},[59],[14,14190],{"className":14191,"style":205},[204],[14,14193,174],{"className":14194},[209],[14,14196],{"className":14197,"style":205},[204],[14,14199,14201,14232],{"className":14200},[59],[14,14202,14204],{"className":14203},[59,11459],[14,14205,14207],{"className":14206},[111],[14,14208,14210],{"className":14209},[115],[14,14211,14213,14221],{"className":14212,"style":11455},[119],[14,14214,14215,14218],{"style":11471},[14,14216],{"className":14217,"style":1097},[126],[14,14219,275],{"className":14220,"style":3311},[59,60],[14,14222,14223,14226],{"style":11480},[14,14224],{"className":14225,"style":1097},[126],[14,14227,14229],{"className":14228,"style":11488},[11487],[14,14230,11409],{"className":14231},[59],[14,14233,14235],{"className":14234},[107],[14,14236,14238],{"className":14237},[111],[14,14239,14241],{"className":14240},[115],[14,14242,14244],{"className":14243,"style":3670},[119],[14,14245,14246,14249],{"style":1131},[14,14247],{"className":14248,"style":127},[126],[14,14250,14252],{"className":14251},[131,132,133,134],[14,14253,83],{"className":14254},[59,134],[14,14256,14257,14260],{"style":1287},[14,14258],{"className":14259,"style":1097},[126],[14,14261],{"className":14262,"style":1294},[1226],[14,14264,14265,14268],{"style":1297},[14,14266],{"className":14267,"style":1097},[126],[14,14269,14271,14302,14305,14308,14311,14314,14317,14374],{"className":14270},[59],[14,14272,14274],{"className":14273},[59,11459],[14,14275,14277],{"className":14276},[111],[14,14278,14280],{"className":14279},[115],[14,14281,14283,14291],{"className":14282,"style":11455},[119],[14,14284,14285,14288],{"style":11471},[14,14286],{"className":14287,"style":1097},[126],[14,14289,275],{"className":14290,"style":3311},[59,60],[14,14292,14293,14296],{"style":11480},[14,14294],{"className":14295,"style":1097},[126],[14,14297,14299],{"className":14298,"style":11488},[11487],[14,14300,11409],{"className":14301},[59],[14,14303,168],{"className":14304},[197],[14,14306,171],{"className":14307},[59,60],[14,14309],{"className":14310,"style":205},[204],[14,14312,174],{"className":14313},[209],[14,14315],{"className":14316,"style":205},[204],[14,14318,14320,14351],{"className":14319},[59],[14,14321,14323],{"className":14322},[59,11459],[14,14324,14326],{"className":14325},[111],[14,14327,14329],{"className":14328},[115],[14,14330,14332,14340],{"className":14331,"style":11455},[119],[14,14333,14334,14337],{"style":11471},[14,14335],{"className":14336,"style":1097},[126],[14,14338,275],{"className":14339,"style":3311},[59,60],[14,14341,14342,14345],{"style":11480},[14,14343],{"className":14344,"style":1097},[126],[14,14346,14348],{"className":14347,"style":11488},[11487],[14,14349,11409],{"className":14350},[59],[14,14352,14354],{"className":14353},[107],[14,14355,14357],{"className":14356},[111],[14,14358,14360],{"className":14359},[115],[14,14361,14363],{"className":14362,"style":96},[119],[14,14364,14365,14368],{"style":122},[14,14366],{"className":14367,"style":127},[126],[14,14369,14371],{"className":14370},[131,132,133,134],[14,14372,83],{"className":14373},[59,134],[14,14375,180],{"className":14376},[225],[14,14378,825],{"className":14379},[824],[14,14381,14383],{"className":14382},[115],[14,14384,14386],{"className":14385,"style":12810},[119],[14,14387],{},[14,14389],{"className":14390},[225,1077],[14,14392],{"className":14393,"style":205},[204],[14,14395,2861],{"className":14396},[209],[14,14398],{"className":14399,"style":205},[204],[14,14401,14403,14407],{"className":14402},[50],[14,14404],{"className":14405,"style":14406},[54],"height:2.4331em;vertical-align:-0.936em;",[14,14408,14410,14413,14515],{"className":14409},[59],[14,14411],{"className":14412},[197,1077],[14,14414,14416],{"className":14415},[931],[14,14417,14419,14507],{"className":14418},[111,801],[14,14420,14422,14504],{"className":14421},[115],[14,14423,14425,14457,14465],{"className":14424,"style":12600},[119],[14,14426,14427,14430],{"style":2558},[14,14428],{"className":14429,"style":1097},[126],[14,14431,14433,14436,14439,14442,14445,14448,14451,14454],{"className":14432},[59],[14,14434,83],{"className":14435},[59],[14,14437,168],{"className":14438},[197],[14,14440,171],{"className":14441},[59,60],[14,14443],{"className":14444,"style":205},[204],[14,14446,174],{"className":14447},[209],[14,14449],{"className":14450,"style":205},[204],[14,14452,177],{"className":14453},[59],[14,14455,180],{"className":14456},[225],[14,14458,14459,14462],{"style":1287},[14,14460],{"className":14461,"style":1097},[126],[14,14463],{"className":14464,"style":1294},[1226],[14,14466,14467,14470],{"style":1297},[14,14468],{"className":14469,"style":1097},[126],[14,14471,14473],{"className":14472},[59],[14,14474,14476],{"className":14475},[59,11459],[14,14477,14479],{"className":14478},[111],[14,14480,14482],{"className":14481},[115],[14,14483,14485,14493],{"className":14484,"style":11455},[119],[14,14486,14487,14490],{"style":11471},[14,14488],{"className":14489,"style":1097},[126],[14,14491,275],{"className":14492,"style":3311},[59,60],[14,14494,14495,14498],{"style":11480},[14,14496],{"className":14497,"style":1097},[126],[14,14499,14501],{"className":14500,"style":11488},[11487],[14,14502,11409],{"className":14503},[59],[14,14505,825],{"className":14506},[824],[14,14508,14510],{"className":14509},[115],[14,14511,14513],{"className":14512,"style":5912},[119],[14,14514],{},[14,14516],{"className":14517},[225,1077],[10,14519,14520],{},"Characteristics: slightly more accurate than Banerjee, still closed-form.",[1889,14522,14524,14525,14553],{"id":14523},"large-κkappaκ-asymptotic-approximation","Large-",[14,14526,14528,14541],{"className":14527,"translate":18},[17],[14,14529,14531],{"className":14530},[22],[24,14532,14533],{"xmlns":26},[28,14534,14535,14539],{},[31,14536,14537],{},[34,14538,596],{},[38,14540,1673],{"encoding":40},[14,14542,14544],{"className":14543,"ariaHidden":46},[45],[14,14545,14547,14550],{"className":14546},[50],[14,14548],{"className":14549,"style":1541},[54],[14,14551,596],{"className":14552},[59,60]," Asymptotic Approximation",[10,14555,7109,14556,14584],{},[14,14557,14559,14572],{"className":14558,"translate":18},[17],[14,14560,14562],{"className":14561},[22],[24,14563,14564],{"xmlns":26},[28,14565,14566,14570],{},[31,14567,14568],{},[34,14569,596],{},[38,14571,1673],{"encoding":40},[14,14573,14575],{"className":14574,"ariaHidden":46},[45],[14,14576,14578,14581],{"className":14577},[50],[14,14579],{"className":14580,"style":1541},[54],[14,14582,596],{"className":14583},[59,60]," is very large:",[14,14586,14588],{"className":14587,"translate":18},[230],[14,14589,14591,14637],{"className":14590,"translate":18},[17],[14,14592,14594],{"className":14593},[22],[24,14595,14596],{"xmlns":26,"display":239},[28,14597,14598,14634],{},[31,14599,14600,14606,14608,14610,14612,14614,14616,14618],{},[678,14601,14602,14604],{},[34,14603,1957],{},[34,14605,171],{},[165,14607,168],{"stretchy":167},[34,14609,596],{},[165,14611,180],{"stretchy":167},[165,14613,7191],{},[81,14615,177],{},[165,14617,174],{},[931,14619,14620,14628],{},[31,14621,14622,14624,14626],{},[34,14623,171],{},[165,14625,174],{},[81,14627,177],{},[31,14629,14630,14632],{},[81,14631,83],{},[34,14633,596],{},[38,14635,14636],{"encoding":40},"A_d(\\kappa) \\approx 1 - \\frac{d-1}{2\\kappa}",[14,14638,14640,14704,14722],{"className":14639,"ariaHidden":46},[45],[14,14641,14643,14646,14686,14689,14692,14695,14698,14701],{"className":14642},[50],[14,14644],{"className":14645,"style":193},[54],[14,14647,14649,14652],{"className":14648},[59],[14,14650,1957],{"className":14651},[59,60],[14,14653,14655],{"className":14654},[107],[14,14656,14658,14678],{"className":14657},[111,801],[14,14659,14661,14675],{"className":14660},[115],[14,14662,14664],{"className":14663,"style":808},[119],[14,14665,14666,14669],{"style":2350},[14,14667],{"className":14668,"style":127},[126],[14,14670,14672],{"className":14671},[131,132,133,134],[14,14673,171],{"className":14674},[59,60,134],[14,14676,825],{"className":14677},[824],[14,14679,14681],{"className":14680},[115],[14,14682,14684],{"className":14683,"style":832},[119],[14,14685],{},[14,14687,168],{"className":14688},[197],[14,14690,596],{"className":14691},[59,60],[14,14693,180],{"className":14694},[225],[14,14696],{"className":14697,"style":352},[204],[14,14699,7191],{"className":14700},[356],[14,14702],{"className":14703,"style":352},[204],[14,14705,14707,14710,14713,14716,14719],{"className":14706},[50],[14,14708],{"className":14709,"style":7242},[54],[14,14711,177],{"className":14712},[59],[14,14714],{"className":14715,"style":205},[204],[14,14717,174],{"className":14718},[209],[14,14720],{"className":14721,"style":205},[204],[14,14723,14725,14728],{"className":14724},[50],[14,14726],{"className":14727,"style":4061},[54],[14,14729,14731,14734,14802],{"className":14730},[59],[14,14732],{"className":14733},[197,1077],[14,14735,14737],{"className":14736},[931],[14,14738,14740,14794],{"className":14739},[111,801],[14,14741,14743,14791],{"className":14742},[115],[14,14744,14746,14760,14768],{"className":14745,"style":4080},[119],[14,14747,14748,14751],{"style":2558},[14,14749],{"className":14750,"style":1097},[126],[14,14752,14754,14757],{"className":14753},[59],[14,14755,83],{"className":14756},[59],[14,14758,596],{"className":14759},[59,60],[14,14761,14762,14765],{"style":1287},[14,14763],{"className":14764,"style":1097},[126],[14,14766],{"className":14767,"style":1294},[1226],[14,14769,14770,14773],{"style":1297},[14,14771],{"className":14772,"style":1097},[126],[14,14774,14776,14779,14782,14785,14788],{"className":14775},[59],[14,14777,171],{"className":14778},[59,60],[14,14780],{"className":14781,"style":205},[204],[14,14783,174],{"className":14784},[209],[14,14786],{"className":14787,"style":205},[204],[14,14789,177],{"className":14790},[59],[14,14792,825],{"className":14793},[824],[14,14795,14797],{"className":14796},[115],[14,14798,14800],{"className":14799,"style":3073},[119],[14,14801],{},[14,14803],{"className":14804},[225,1077],[10,14806,14807],{},"Inverting gives:",[14,14809,14811],{"className":14810,"translate":18},[230],[14,14812,14814,14858],{"className":14813,"translate":18},[17],[14,14815,14817],{"className":14816},[22],[24,14818,14819],{"xmlns":26,"display":239},[28,14820,14821,14855],{},[31,14822,14823,14825,14827],{},[34,14824,596],{},[165,14826,7191],{},[931,14828,14829,14837],{},[31,14830,14831,14833,14835],{},[34,14832,171],{},[165,14834,174],{},[81,14836,177],{},[31,14838,14839,14841,14843,14845,14847,14853],{},[81,14840,83],{},[165,14842,168],{"stretchy":167},[81,14844,177],{},[165,14846,174],{},[11403,14848,14849,14851],{"accent":46},[34,14850,275],{},[165,14852,11409],{},[165,14854,180],{"stretchy":167},[38,14856,14857],{"encoding":40},"\\kappa \\approx \\frac{d-1}{2(1-\\bar{R})}",[14,14859,14861,14879],{"className":14860,"ariaHidden":46},[45],[14,14862,14864,14867,14870,14873,14876],{"className":14863},[50],[14,14865],{"className":14866,"style":12562},[54],[14,14868,596],{"className":14869},[59,60],[14,14871],{"className":14872,"style":352},[204],[14,14874,7191],{"className":14875},[356],[14,14877],{"className":14878,"style":352},[204],[14,14880,14882,14886],{"className":14881},[50],[14,14883],{"className":14884,"style":14885},[54],"height:2.3316em;vertical-align:-0.9601em;",[14,14887,14889,14892,15007],{"className":14888},[59],[14,14890],{"className":14891},[197,1077],[14,14893,14895],{"className":14894},[931],[14,14896,14898,14998],{"className":14897},[111,801],[14,14899,14901,14995],{"className":14900},[115],[14,14902,14904,14964,14972],{"className":14903,"style":4080},[119],[14,14905,14906,14909],{"style":12603},[14,14907],{"className":14908,"style":1097},[126],[14,14910,14912,14915,14918,14921,14924,14927,14930,14961],{"className":14911},[59],[14,14913,83],{"className":14914},[59],[14,14916,168],{"className":14917},[197],[14,14919,177],{"className":14920},[59],[14,14922],{"className":14923,"style":205},[204],[14,14925,174],{"className":14926},[209],[14,14928],{"className":14929,"style":205},[204],[14,14931,14933],{"className":14932},[59,11459],[14,14934,14936],{"className":14935},[111],[14,14937,14939],{"className":14938},[115],[14,14940,14942,14950],{"className":14941,"style":11455},[119],[14,14943,14944,14947],{"style":11471},[14,14945],{"className":14946,"style":1097},[126],[14,14948,275],{"className":14949,"style":3311},[59,60],[14,14951,14952,14955],{"style":11480},[14,14953],{"className":14954,"style":1097},[126],[14,14956,14958],{"className":14957,"style":11488},[11487],[14,14959,11409],{"className":14960},[59],[14,14962,180],{"className":14963},[225],[14,14965,14966,14969],{"style":1287},[14,14967],{"className":14968,"style":1097},[126],[14,14970],{"className":14971,"style":1294},[1226],[14,14973,14974,14977],{"style":1297},[14,14975],{"className":14976,"style":1097},[126],[14,14978,14980,14983,14986,14989,14992],{"className":14979},[59],[14,14981,171],{"className":14982},[59,60],[14,14984],{"className":14985,"style":205},[204],[14,14987,174],{"className":14988},[209],[14,14990],{"className":14991,"style":205},[204],[14,14993,177],{"className":14994},[59],[14,14996,825],{"className":14997},[824],[14,14999,15001],{"className":15000},[115],[14,15002,15005],{"className":15003,"style":15004},[119],"height:0.9601em;",[14,15006],{},[14,15008],{"className":15009},[225,1077],[10,15011,15012],{},"Applicable when embeddings are highly concentrated, e.g., in contrastive learning.",[1889,15014,15016,15017,15045],{"id":15015},"small-κkappaκ-approximation","Small-",[14,15018,15020,15033],{"className":15019,"translate":18},[17],[14,15021,15023],{"className":15022},[22],[24,15024,15025],{"xmlns":26},[28,15026,15027,15031],{},[31,15028,15029],{},[34,15030,596],{},[38,15032,1673],{"encoding":40},[14,15034,15036],{"className":15035,"ariaHidden":46},[45],[14,15037,15039,15042],{"className":15038},[50],[14,15040],{"className":15041,"style":1541},[54],[14,15043,596],{"className":15044},[59,60]," Approximation",[10,15047,7109,15048,15076],{},[14,15049,15051,15064],{"className":15050,"translate":18},[17],[14,15052,15054],{"className":15053},[22],[24,15055,15056],{"xmlns":26},[28,15057,15058,15062],{},[31,15059,15060],{},[34,15061,596],{},[38,15063,1673],{"encoding":40},[14,15065,15067],{"className":15066,"ariaHidden":46},[45],[14,15068,15070,15073],{"className":15069},[50],[14,15071],{"className":15072,"style":1541},[54],[14,15074,596],{"className":15075},[59,60]," is very small:",[14,15078,15080],{"className":15079,"translate":18},[230],[14,15081,15083,15115],{"className":15082,"translate":18},[17],[14,15084,15086],{"className":15085},[22],[24,15087,15088],{"xmlns":26,"display":239},[28,15089,15090,15112],{},[31,15091,15092,15098,15100,15102,15104,15106],{},[678,15093,15094,15096],{},[34,15095,1957],{},[34,15097,171],{},[165,15099,168],{"stretchy":167},[34,15101,596],{},[165,15103,180],{"stretchy":167},[165,15105,7191],{},[931,15107,15108,15110],{},[34,15109,596],{},[34,15111,171],{},[38,15113,15114],{"encoding":40},"A_d(\\kappa) \\approx \\frac{\\kappa}{d}",[14,15116,15118,15182],{"className":15117,"ariaHidden":46},[45],[14,15119,15121,15124,15164,15167,15170,15173,15176,15179],{"className":15120},[50],[14,15122],{"className":15123,"style":193},[54],[14,15125,15127,15130],{"className":15126},[59],[14,15128,1957],{"className":15129},[59,60],[14,15131,15133],{"className":15132},[107],[14,15134,15136,15156],{"className":15135},[111,801],[14,15137,15139,15153],{"className":15138},[115],[14,15140,15142],{"className":15141,"style":808},[119],[14,15143,15144,15147],{"style":2350},[14,15145],{"className":15146,"style":127},[126],[14,15148,15150],{"className":15149},[131,132,133,134],[14,15151,171],{"className":15152},[59,60,134],[14,15154,825],{"className":15155},[824],[14,15157,15159],{"className":15158},[115],[14,15160,15162],{"className":15161,"style":832},[119],[14,15163],{},[14,15165,168],{"className":15166},[197],[14,15168,596],{"className":15169},[59,60],[14,15171,180],{"className":15172},[225],[14,15174],{"className":15175,"style":352},[204],[14,15177,7191],{"className":15178},[356],[14,15180],{"className":15181,"style":352},[204],[14,15183,15185,15189],{"className":15184},[50],[14,15186],{"className":15187,"style":15188},[54],"height:1.7936em;vertical-align:-0.686em;",[14,15190,15192,15195,15249],{"className":15191},[59],[14,15193],{"className":15194},[197,1077],[14,15196,15198],{"className":15197},[931],[14,15199,15201,15241],{"className":15200},[111,801],[14,15202,15204,15238],{"className":15203},[115],[14,15205,15208,15219,15227],{"className":15206,"style":15207},[119],"height:1.1076em;",[14,15209,15210,15213],{"style":2558},[14,15211],{"className":15212,"style":1097},[126],[14,15214,15216],{"className":15215},[59],[14,15217,171],{"className":15218},[59,60],[14,15220,15221,15224],{"style":1287},[14,15222],{"className":15223,"style":1097},[126],[14,15225],{"className":15226,"style":1294},[1226],[14,15228,15229,15232],{"style":1297},[14,15230],{"className":15231,"style":1097},[126],[14,15233,15235],{"className":15234},[59],[14,15236,596],{"className":15237},[59,60],[14,15239,825],{"className":15240},[824],[14,15242,15244],{"className":15243},[115],[14,15245,15247],{"className":15246,"style":3073},[119],[14,15248],{},[14,15250],{"className":15251},[225,1077],[10,15253,15254],{},"Thus,",[14,15256,15258],{"className":15257,"translate":18},[230],[14,15259,15261,15285],{"className":15260,"translate":18},[17],[14,15262,15264],{"className":15263},[22],[24,15265,15266],{"xmlns":26,"display":239},[28,15267,15268,15282],{},[31,15269,15270,15272,15274,15276],{},[34,15271,596],{},[165,15273,7191],{},[34,15275,171],{},[11403,15277,15278,15280],{"accent":46},[34,15279,275],{},[165,15281,11409],{},[38,15283,15284],{"encoding":40},"\\kappa \\approx d\\bar{R}",[14,15286,15288,15306],{"className":15287,"ariaHidden":46},[45],[14,15289,15291,15294,15297,15300,15303],{"className":15290},[50],[14,15292],{"className":15293,"style":12562},[54],[14,15295,596],{"className":15296},[59,60],[14,15298],{"className":15299,"style":352},[204],[14,15301,7191],{"className":15302},[356],[14,15304],{"className":15305,"style":352},[204],[14,15307,15309,15312,15315],{"className":15308},[50],[14,15310],{"className":15311,"style":11455},[54],[14,15313,171],{"className":15314},[59,60],[14,15316,15318],{"className":15317},[59,11459],[14,15319,15321],{"className":15320},[111],[14,15322,15324],{"className":15323},[115],[14,15325,15327,15335],{"className":15326,"style":11455},[119],[14,15328,15329,15332],{"style":11471},[14,15330],{"className":15331,"style":1097},[126],[14,15333,275],{"className":15334,"style":3311},[59,60],[14,15336,15337,15340],{"style":11480},[14,15338],{"className":15339,"style":1097},[126],[14,15341,15343],{"className":15342,"style":11488},[11487],[14,15344,11409],{"className":15345},[59],[10,15347,15348],{},"Applicable when data are nearly uniform.",[8146,15350,15351,15367],{},[8149,15352,15353],{},[8152,15354,15355,15358,15361,15364],{},[8155,15356,15357],{},"Estimation",[8155,15359,15360],{},"Complexity",[8155,15362,15363],{},"Accuracy",[8155,15365,15366],{},"Usage",[8230,15368,15369,15383,15396,15410,15479],{},[8152,15370,15371,15374,15377,15380],{},[8235,15372,15373],{},"Banerjee",[8235,15375,15376],{},"Very low",[8235,15378,15379],{},"Medium",[8235,15381,15382],{},"Initialization",[8152,15384,15385,15388,15390,15393],{},[8235,15386,15387],{},"Sra + Newton",[8235,15389,15379],{},[8235,15391,15392],{},"High",[8235,15394,15395],{},"Recommended",[8152,15397,15398,15401,15404,15407],{},[8235,15399,15400],{},"Minka",[8235,15402,15403],{},"Low",[8235,15405,15406],{},"Medium-High",[8235,15408,15409],{},"Fast estimation",[8152,15411,15412,15443,15445,15476],{},[8235,15413,15414,15415],{},"Large ",[14,15416,15418,15431],{"className":15417,"translate":18},[17],[14,15419,15421],{"className":15420},[22],[24,15422,15423],{"xmlns":26},[28,15424,15425,15429],{},[31,15426,15427],{},[34,15428,596],{},[38,15430,1673],{"encoding":40},[14,15432,15434],{"className":15433,"ariaHidden":46},[45],[14,15435,15437,15440],{"className":15436},[50],[14,15438],{"className":15439,"style":1541},[54],[14,15441,596],{"className":15442},[59,60],[8235,15444,15376],{},[8235,15446,15447,15448,180],{},"High (large ",[14,15449,15451,15464],{"className":15450,"translate":18},[17],[14,15452,15454],{"className":15453},[22],[24,15455,15456],{"xmlns":26},[28,15457,15458,15462],{},[31,15459,15460],{},[34,15461,596],{},[38,15463,1673],{"encoding":40},[14,15465,15467],{"className":15466,"ariaHidden":46},[45],[14,15468,15470,15473],{"className":15469},[50],[14,15471],{"className":15472,"style":1541},[54],[14,15474,596],{"className":15475},[59,60],[8235,15477,15478],{},"High concentration",[8152,15480,15481,15512,15514,15545],{},[8235,15482,15483,15484],{},"Small ",[14,15485,15487,15500],{"className":15486,"translate":18},[17],[14,15488,15490],{"className":15489},[22],[24,15491,15492],{"xmlns":26},[28,15493,15494,15498],{},[31,15495,15496],{},[34,15497,596],{},[38,15499,1673],{"encoding":40},[14,15501,15503],{"className":15502,"ariaHidden":46},[45],[14,15504,15506,15509],{"className":15505},[50],[14,15507],{"className":15508,"style":1541},[54],[14,15510,596],{"className":15511},[59,60],[8235,15513,15376],{},[8235,15515,15516,15517,180],{},"High (small ",[14,15518,15520,15533],{"className":15519,"translate":18},[17],[14,15521,15523],{"className":15522},[22],[24,15524,15525],{"xmlns":26},[28,15526,15527,15531],{},[31,15528,15529],{},[34,15530,596],{},[38,15532,1673],{"encoding":40},[14,15534,15536],{"className":15535,"ariaHidden":46},[45],[14,15537,15539,15542],{"className":15538},[50],[14,15540],{"className":15541,"style":1541},[54],[14,15543,596],{"className":15544},[59,60],[8235,15546,15547],{},"Near uniformity",{"title":15549,"searchDepth":15550,"depth":15550,"links":15551},"",2,[15552,15553,15554,15565],{"id":147,"depth":15550,"text":148},{"id":1505,"depth":15550,"text":1506},{"id":1886,"depth":15550,"text":1887,"children":15555},[15556,15558,15559,15560,15561,15562,15563,15564],{"id":1891,"depth":15557,"text":1892},3,{"id":2247,"depth":15557,"text":2248},{"id":2806,"depth":15557,"text":2807},{"id":3277,"depth":15557,"text":3278},{"id":3507,"depth":15557,"text":3508},{"id":3809,"depth":15557,"text":3810},{"id":4179,"depth":15557,"text":4180},{"id":4501,"depth":15557,"text":4502},{"id":11033,"depth":15550,"text":11034,"children":15566},[15567,15568,15569,15570,15572],{"id":12425,"depth":15557,"text":12426},{"id":12869,"depth":15557,"text":12870},{"id":14035,"depth":15557,"text":14036},{"id":14523,"depth":15557,"text":15571},"Large-κ\\kappaκ Asymptotic Approximation",{"id":15015,"depth":15557,"text":15573},"Small-κ\\kappaκ Approximation","The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere. In Euclidean space, a normal distribution is typically parameterized by a mean μ\\muμ and variance σ2\\sigma^2σ2, and in one dimension its density forms the familiar bell-shaped curve. Just as the normal distribution is the distribution with the maximum entropy given the mean and variance constraints, the vMF distribution is the distribution with the maximum entropy on a sphere given the average direction constraint. According to the Central Limit Theorem: the limiting distribution of a random walk or accumulated direction vector on a sphere will also tend towards the vMF distribution.",false,true,"md",{},"2026-04-09","\u002Fblog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere",{"title":5,"description":15574},"blog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere","The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere.",[15585,15586],"machine-learning","mathematics","yWV5oIebLE0_s02h0DOYkeIRVgCuycbL05Dnx2Zw5uk",1777090294952]