3015:
1447:
2161:
2896:
1815:
2676:
2516:
1266:
32:
2008:
1632:
1090:
2347:
810:
184:
distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated. Otherwise, it may be unnecessary (for example,
2890:
331:
443:
2303:
2155:
1964:
2670:
3299:
765:
2763:
696:
1771:
1588:
2510:
2223:
1403:
947:
205:
2957:
995:
160:(pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the
2390:
2426:
2749:
2713:
2553:
1046:
2067:
1874:
2038:
2582:
1845:
1691:
1662:
1506:
1477:
1325:
1296:
1156:
1127:
1222:
473:
1334:
2340:
2001:
1808:
1625:
1440:
1259:
1083:
877:
2185:
2919:
838:
493:
817:
Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, quadratic and cosine.
342:
2237:
2083:
1888:
2598:
3222:
2885:{\displaystyle K(u)={\frac {1}{2}}e^{-{\frac {|u|}{\sqrt {2}}}}\cdot \sin \left({\frac {|u|}{\sqrt {2}}}+{\frac {\pi }{4}}\right)}
709:
96:
2998:
634:
68:
3334:
49:
774:. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.
75:
1705:
1522:
2442:
3407:
3058:
3036:
115:
3029:
326:{\displaystyle p(x|\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
504:
82:
2192:
1344:
902:
2926:
953:
53:
64:
2355:
173:
2397:
3422:
2720:
2684:
2524:
771:
196:
153:
1009:
448:
Note that the factor in front of the exponential has been omitted, even though it contains the parameter
2045:
1852:
2016:
512:
620:
For most applications, it is desirable to define the function to satisfy two additional requirements:
188:
For many distributions, the kernel can be written in closed form, but not the normalization constant.
2973:
550:
157:
2560:
1823:
1669:
1640:
1484:
1455:
1303:
1274:
1134:
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3417:
3381:
3023:
2988:
542:
169:
1170:
566:
451:
42:
20:
3376:
3119:
Epanechnikov, V. A. (1969). "Non-Parametric
Estimation of a Multivariate Probability Density".
3040:
2309:
1970:
1777:
1594:
1409:
1228:
1052:
89:
3367:
Comaniciu, D; Meer, P (2002). "Mean shift: A robust approach toward feature space analysis".
2435:
841:
3193:(1988). "Locally weighted regression: An approach to regression analysis by local fitting".
847:
3412:
3347:
3186:
624:
610:
165:
8:
2169:
2076:
592:
Commonly, kernel widths must also be specified when running a non-parametric estimation.
516:
192:
149:
801:), where λ > 0. This can be used to select a scale that is appropriate for the data.
770:
The first requirement ensures that the method of kernel density estimation results in a
2993:
2904:
823:
478:
140:. The term "kernel" has several distinct meanings in different branches of statistics.
438:{\displaystyle p(x|\mu ,\sigma ^{2})\propto e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
3330:
2983:
562:
177:
3386:
3206:
3202:
3166:
3162:
3158:
3149:(1992). "An introduction to kernel and nearest neighbor nonparametric regression".
3128:
3098:
2591:
582:
578:
558:
536:
524:
520:
3190:
2978:
1095:
601:
554:
181:
137:
546:
508:
3103:
3086:
3401:
3003:
586:
176:, most sampling algorithms ignore the normalization factor. In addition, in
3146:
607:
585:. An additional use is in the estimation of a time-varying intensity for a
2298:{\displaystyle K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)}
574:
570:
3390:
2150:{\displaystyle K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}}
613:
133:
3171:
1446:
589:
where window functions (kernels) are convolved with time-series data.
2160:
161:
3132:
31:
2895:
1814:
523:
on data in an implicit space. This usage is particularly common in
3087:"Estimation of a probability density function and its derivatives"
2675:
2515:
1265:
2007:
1959:{\displaystyle K(u)={\frac {70}{81}}(1-{\left|u\right|}^{3})^{3}}
1631:
1089:
3348:"APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation"
2346:
2665:{\displaystyle K(u)={\frac {2}{\pi }}{\frac {1}{e^{u}+e^{-u}}}}
3369:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
3294:{\displaystyle {\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du}
3185:
172:, and are unnecessary in many situations. For example, in
809:
760:{\displaystyle K(-u)=K(u){\mbox{ for all values of }}u\,.}
691:{\displaystyle \int _{-\infty }^{+\infty }K(u)\,du=1\,;}
813:
All of the kernels below in a common coordinate system.
957:
906:
744:
475:, because it is not a function of the domain variable
3225:
2929:
2907:
2766:
2723:
2687:
2601:
2563:
2527:
2445:
2400:
2358:
2312:
2240:
2195:
2172:
2086:
2048:
2019:
1973:
1891:
1855:
1826:
1780:
1708:
1672:
1643:
1597:
1525:
1487:
1458:
1412:
1347:
1306:
1277:
1231:
1173:
1137:
1108:
1055:
1012:
956:
905:
850:
826:
712:
637:
481:
454:
345:
208:
3324:
185:
if the distribution only needs to be sampled from).
3313:
Density
Estimation for Statistics and Data Analysis
2922:
2716:
2556:
2188:
1636:
1270:
164:of the pdf or pmf. These factors form part of the
56:. Unsourced material may be challenged and removed.
3293:
2951:
2913:
2884:
2743:
2707:
2680:
2664:
2576:
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2504:
2420:
2393:
2384:
2351:
2334:
2297:
2217:
2179:
2149:
2061:
2032:
2012:
1995:
1958:
1868:
1848:
1839:
1819:
1802:
1766:{\displaystyle K(u)={\frac {35}{32}}(1-u^{2})^{3}}
1765:
1685:
1665:
1656:
1619:
1583:{\displaystyle K(u)={\frac {15}{16}}(1-u^{2})^{2}}
1582:
1500:
1471:
1434:
1397:
1319:
1299:
1290:
1253:
1216:
1150:
1121:
1101:
1077:
1040:
989:
941:
871:
832:
759:
690:
487:
467:
437:
325:
2520:
2041:
804:
3399:
2505:{\displaystyle K(u)={\frac {1}{e^{u}+2+e^{-u}}}}
3327:Nonparametric Econometrics: Theory and Practice
3195:Journal of the American Statistical Association
999:Efficiency relative to the Epanechnikov kernel
569:of a random variable. Kernels are also used in
3366:
3145:
3118:
2218:{\displaystyle {\frac {1}{2{\sqrt {\pi }}}}}
1398:{\displaystyle K(u)={\frac {3}{4}}(1-u^{2})}
507:is used in the suite of techniques known as
942:{\displaystyle \textstyle \int u^{2}K(u)du}
549:estimation techniques. Kernels are used in
545:, a kernel is a weighting function used in
530:
2952:{\displaystyle {\frac {3{\sqrt {2}}}{16}}}
990:{\displaystyle \textstyle \int K(u)^{2}du}
3380:
3310:
3284:
3253:
3170:
3102:
3059:Learn how and when to remove this message
2176:
753:
684:
671:
116:Learn how and when to remove this message
3084:
3022:This article includes a list of general
808:
2385:{\displaystyle 1-{\frac {8}{\pi ^{2}}}}
3400:
2999:Multivariate kernel density estimation
2421:{\displaystyle {\frac {\pi ^{2}}{16}}}
143:
3091:The Annals of Mathematical Statistics
2744:{\displaystyle {\frac {2}{\pi ^{2}}}}
2708:{\displaystyle {\frac {\pi ^{2}}{4}}}
2548:{\displaystyle {\frac {\pi ^{2}}{3}}}
781:is a kernel, then so is the function
3345:
3008:
54:adding citations to reliable sources
25:
3325:Li, Qi; Racine, Jeffrey S. (2007).
1480:
1451:
1130:
1041:{\displaystyle K(u)={\frac {1}{2}}}
498:
13:
3028:it lacks sufficient corresponding
2062:{\displaystyle {\frac {175}{247}}}
1869:{\displaystyle {\frac {350}{429}}}
654:
646:
14:
3434:
2900:
2165:
2033:{\displaystyle {\frac {35}{243}}}
3085:Schuster, Eugene (August 1969).
3013:
2894:
2674:
2514:
2345:
2159:
2006:
1813:
1630:
1445:
1264:
1088:
505:reproducing kernel Hilbert space
30:
1004:Uniform ("rectangular window")
41:needs additional citations for
3329:. Princeton University Press.
3304:
3275:
3268:
3250:
3244:
3213:
3207:10.1080/01621459.1988.10478639
3179:
3163:10.1080/00031305.1992.10475879
3139:
3111:
3078:
2852:
2844:
2812:
2804:
2776:
2770:
2611:
2605:
2577:{\displaystyle {\frac {1}{6}}}
2455:
2449:
2322:
2314:
2250:
2244:
2096:
2090:
1983:
1975:
1947:
1917:
1901:
1895:
1840:{\displaystyle {\frac {1}{9}}}
1790:
1782:
1754:
1734:
1718:
1712:
1686:{\displaystyle {\frac {5}{7}}}
1657:{\displaystyle {\frac {1}{7}}}
1607:
1599:
1571:
1551:
1535:
1529:
1501:{\displaystyle {\frac {3}{5}}}
1472:{\displaystyle {\frac {1}{5}}}
1422:
1414:
1392:
1373:
1357:
1351:
1320:{\displaystyle {\frac {2}{3}}}
1291:{\displaystyle {\frac {1}{6}}}
1241:
1233:
1211:
1207:
1199:
1189:
1183:
1177:
1151:{\displaystyle {\frac {1}{2}}}
1122:{\displaystyle {\frac {1}{3}}}
1065:
1057:
1022:
1016:
971:
964:
929:
923:
860:
854:
805:Kernel functions in common use
740:
734:
725:
716:
668:
662:
406:
393:
376:
356:
349:
294:
281:
239:
219:
212:
1:
3071:
746: for all values of
595:
336:and the associated kernel is
174:pseudo-random number sampling
148:In statistics, especially in
1217:{\displaystyle K(u)=(1-|u|)}
772:probability density function
197:probability density function
154:probability density function
7:
3315:. Chapman and Hall, London.
2967:
883:lying outside the support.
468:{\displaystyle \sigma ^{2}}
10:
3439:
599:
534:
513:statistical classification
18:
3311:Silverman, B. W. (1986).
3219:Efficiency is defined as
3151:The American Statistician
2974:Kernel density estimation
2335:{\displaystyle |u|\leq 1}
1996:{\displaystyle |u|\leq 1}
1803:{\displaystyle |u|\leq 1}
1620:{\displaystyle |u|\leq 1}
1435:{\displaystyle |u|\leq 1}
1254:{\displaystyle |u|\leq 1}
1078:{\displaystyle |u|\leq 1}
888:
551:kernel density estimation
511:to perform tasks such as
158:probability mass function
3408:Nonparametric statistics
2989:Positive-definite kernel
840:is given with a bounded
581:where they are known as
543:nonparametric statistics
531:Nonparametric statistics
170:probability distribution
65:"Kernel" statistics
3104:10.1214/aoms/1177697495
3043:more precise citations.
820:In the table below, if
567:conditional expectation
21:Kernel (disambiguation)
3295:
2953:
2915:
2886:
2745:
2709:
2666:
2578:
2549:
2506:
2422:
2386:
2336:
2299:
2219:
2181:
2151:
2063:
2034:
1997:
1960:
1870:
1841:
1804:
1767:
1687:
1658:
1621:
1584:
1502:
1473:
1436:
1399:
1321:
1292:
1255:
1218:
1152:
1123:
1079:
1042:
991:
943:
873:
872:{\displaystyle K(u)=0}
834:
814:
761:
692:
489:
469:
439:
327:
3296:
2954:
2916:
2887:
2746:
2710:
2667:
2579:
2550:
2507:
2423:
2387:
2337:
2300:
2220:
2182:
2152:
2064:
2035:
1998:
1961:
1871:
1842:
1805:
1768:
1688:
1659:
1622:
1585:
1503:
1474:
1437:
1400:
1322:
1293:
1256:
1219:
1153:
1124:
1080:
1043:
992:
944:
874:
835:
812:
762:
693:
600:Further information:
535:Further information:
490:
470:
440:
328:
3223:
2927:
2905:
2764:
2721:
2685:
2599:
2561:
2525:
2443:
2398:
2356:
2310:
2238:
2193:
2170:
2084:
2046:
2017:
1971:
1889:
1853:
1824:
1778:
1706:
1670:
1641:
1595:
1523:
1485:
1456:
1410:
1345:
1304:
1275:
1229:
1171:
1135:
1106:
1053:
1010:
954:
903:
848:
824:
710:
635:
573:, in the use of the
479:
452:
343:
206:
166:normalization factor
134:statistical analysis
50:improve this article
19:For other uses, see
3423:Bayesian statistics
3121:Theory Probab. Appl
2180:{\displaystyle 1\,}
658:
517:regression analysis
193:normal distribution
150:Bayesian statistics
144:Bayesian statistics
3391:10.1109/34.1000236
3346:Zucchini, Walter.
3291:
2994:Density estimation
2949:
2911:
2882:
2741:
2705:
2662:
2574:
2545:
2502:
2418:
2382:
2332:
2295:
2215:
2177:
2147:
2059:
2030:
1993:
1956:
1866:
1837:
1800:
1763:
1683:
1654:
1617:
1580:
1498:
1469:
1432:
1395:
1317:
1288:
1251:
1214:
1148:
1119:
1075:
1038:
987:
986:
939:
938:
889:Kernel Functions,
869:
830:
815:
757:
748:
688:
638:
485:
465:
435:
323:
191:An example is the
152:, the kernel of a
3336:978-0-691-12161-1
3260:
3069:
3068:
3061:
2984:Stochastic kernel
2965:
2964:
2947:
2941:
2914:{\displaystyle 0}
2875:
2862:
2861:
2822:
2821:
2790:
2758:Silverman kernel
2739:
2703:
2660:
2625:
2572:
2543:
2500:
2416:
2380:
2285:
2264:
2213:
2210:
2133:
2115:
2114:
2057:
2028:
1915:
1864:
1835:
1732:
1681:
1652:
1549:
1496:
1467:
1371:
1315:
1286:
1146:
1117:
1036:
833:{\displaystyle K}
747:
563:kernel regression
559:density functions
488:{\displaystyle x}
431:
319:
268:
267:
178:Bayesian analysis
126:
125:
118:
100:
3430:
3394:
3384:
3361:
3359:
3357:
3352:
3340:
3317:
3316:
3308:
3302:
3300:
3298:
3297:
3292:
3283:
3282:
3261:
3240:
3239:
3227:
3217:
3211:
3210:
3201:(403): 596–610.
3187:Cleveland, W. S.
3183:
3177:
3176:
3174:
3143:
3137:
3136:
3115:
3109:
3108:
3106:
3097:(4): 1187-1195.
3082:
3064:
3057:
3053:
3050:
3044:
3039:this article by
3030:inline citations
3017:
3016:
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2668:
2663:
2661:
2659:
2658:
2657:
2642:
2641:
2628:
2626:
2618:
2592:Sigmoid function
2583:
2581:
2580:
2575:
2573:
2565:
2554:
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2224:
2222:
2221:
2216:
2214:
2212:
2211:
2206:
2197:
2186:
2184:
2183:
2178:
2163:
2156:
2154:
2153:
2148:
2146:
2145:
2144:
2143:
2134:
2126:
2116:
2107:
2103:
2068:
2066:
2065:
2060:
2058:
2050:
2039:
2037:
2036:
2031:
2029:
2021:
2010:
2002:
2000:
1999:
1994:
1986:
1978:
1965:
1963:
1962:
1957:
1955:
1954:
1945:
1944:
1939:
1938:
1916:
1908:
1875:
1873:
1872:
1867:
1865:
1857:
1846:
1844:
1843:
1838:
1836:
1828:
1817:
1809:
1807:
1806:
1801:
1793:
1785:
1772:
1770:
1769:
1764:
1762:
1761:
1752:
1751:
1733:
1725:
1692:
1690:
1689:
1684:
1682:
1674:
1663:
1661:
1660:
1655:
1653:
1645:
1634:
1626:
1624:
1623:
1618:
1610:
1602:
1589:
1587:
1586:
1581:
1579:
1578:
1569:
1568:
1550:
1542:
1507:
1505:
1504:
1499:
1497:
1489:
1478:
1476:
1475:
1470:
1468:
1460:
1449:
1441:
1439:
1438:
1433:
1425:
1417:
1404:
1402:
1401:
1396:
1391:
1390:
1372:
1364:
1326:
1324:
1323:
1318:
1316:
1308:
1297:
1295:
1294:
1289:
1287:
1279:
1268:
1260:
1258:
1257:
1252:
1244:
1236:
1223:
1221:
1220:
1215:
1210:
1202:
1157:
1155:
1154:
1149:
1147:
1139:
1128:
1126:
1125:
1120:
1118:
1110:
1092:
1084:
1082:
1081:
1076:
1068:
1060:
1047:
1045:
1044:
1039:
1037:
1029:
996:
994:
993:
988:
979:
978:
948:
946:
945:
940:
919:
918:
886:
885:
878:
876:
875:
870:
839:
837:
836:
831:
766:
764:
763:
758:
749:
745:
697:
695:
694:
689:
657:
649:
583:window functions
579:spectral density
577:to estimate the
565:to estimate the
555:random variables
537:Kernel smoothing
525:machine learning
521:cluster analysis
503:The kernel of a
499:Pattern analysis
494:
492:
491:
486:
474:
472:
471:
466:
464:
463:
444:
442:
441:
436:
434:
433:
432:
430:
429:
428:
415:
414:
413:
391:
375:
374:
359:
332:
330:
329:
324:
322:
321:
320:
318:
317:
316:
303:
302:
301:
279:
269:
266:
265:
250:
246:
238:
237:
222:
121:
114:
110:
107:
101:
99:
58:
34:
26:
3438:
3437:
3433:
3432:
3431:
3429:
3428:
3427:
3418:Point processes
3398:
3397:
3355:
3353:
3350:
3337:
3321:
3320:
3309:
3305:
3278:
3274:
3235:
3231:
3226:
3224:
3221:
3220:
3218:
3214:
3184:
3180:
3144:
3140:
3133:10.1137/1114019
3116:
3112:
3083:
3079:
3074:
3065:
3054:
3048:
3045:
3035:Please help to
3034:
3018:
3014:
2979:Kernel smoother
2970:
2961:not applicable
2936:
2932:
2930:
2928:
2925:
2924:
2906:
2903:
2902:
2867:
2851:
2843:
2842:
2840:
2839:
2835:
2811:
2803:
2802:
2800:
2796:
2792:
2782:
2765:
2762:
2761:
2733:
2729:
2724:
2722:
2719:
2718:
2694:
2690:
2688:
2686:
2683:
2682:
2650:
2646:
2637:
2633:
2632:
2627:
2617:
2600:
2597:
2596:
2564:
2562:
2559:
2558:
2534:
2530:
2528:
2526:
2523:
2522:
2490:
2486:
2471:
2467:
2466:
2461:
2444:
2441:
2440:
2407:
2403:
2401:
2399:
2396:
2395:
2374:
2370:
2365:
2357:
2354:
2353:
2321:
2313:
2311:
2308:
2307:
2277:
2276:
2272:
2256:
2239:
2236:
2235:
2205:
2201:
2196:
2194:
2191:
2190:
2171:
2168:
2167:
2139:
2135:
2125:
2121:
2117:
2102:
2085:
2082:
2081:
2049:
2047:
2044:
2043:
2020:
2018:
2015:
2014:
1982:
1974:
1972:
1969:
1968:
1950:
1946:
1940:
1928:
1927:
1926:
1907:
1890:
1887:
1886:
1856:
1854:
1851:
1850:
1827:
1825:
1822:
1821:
1789:
1781:
1779:
1776:
1775:
1757:
1753:
1747:
1743:
1724:
1707:
1704:
1703:
1673:
1671:
1668:
1667:
1644:
1642:
1639:
1638:
1606:
1598:
1596:
1593:
1592:
1574:
1570:
1564:
1560:
1541:
1524:
1521:
1520:
1516:
1488:
1486:
1483:
1482:
1459:
1457:
1454:
1453:
1421:
1413:
1411:
1408:
1407:
1386:
1382:
1363:
1346:
1343:
1342:
1307:
1305:
1302:
1301:
1278:
1276:
1273:
1272:
1240:
1232:
1230:
1227:
1226:
1206:
1198:
1172:
1169:
1168:
1138:
1136:
1133:
1132:
1109:
1107:
1104:
1103:
1096:Boxcar function
1064:
1056:
1054:
1051:
1050:
1028:
1011:
1008:
1007:
974:
970:
955:
952:
951:
914:
910:
904:
901:
900:
849:
846:
845:
825:
822:
821:
807:
743:
711:
708:
707:
650:
642:
636:
633:
632:
604:
602:Integral kernel
598:
539:
533:
501:
480:
477:
476:
459:
455:
453:
450:
449:
424:
420:
416:
409:
405:
392:
390:
386:
382:
370:
366:
355:
344:
341:
340:
312:
308:
304:
297:
293:
280:
278:
274:
270:
261:
257:
245:
233:
229:
218:
207:
204:
203:
182:conjugate prior
146:
138:window function
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
16:Window function
12:
11:
5:
3436:
3426:
3425:
3420:
3415:
3410:
3396:
3395:
3382:10.1.1.76.8968
3375:(5): 603–619.
3363:
3362:
3342:
3341:
3335:
3319:
3318:
3303:
3290:
3287:
3281:
3277:
3273:
3270:
3267:
3264:
3259:
3256:
3252:
3249:
3246:
3243:
3238:
3234:
3230:
3212:
3178:
3157:(3): 175–185.
3138:
3127:(1): 153–158.
3110:
3076:
3075:
3073:
3070:
3067:
3066:
3021:
3019:
3012:
3007:
3006:
3001:
2996:
2991:
2986:
2981:
2976:
2969:
2966:
2963:
2962:
2959:
2946:
2940:
2935:
2921:
2910:
2899:
2892:
2880:
2874:
2871:
2866:
2860:
2854:
2850:
2846:
2838:
2834:
2831:
2828:
2820:
2814:
2810:
2806:
2799:
2795:
2789:
2786:
2781:
2778:
2775:
2772:
2769:
2759:
2755:
2754:
2751:
2736:
2732:
2728:
2715:
2702:
2697:
2693:
2679:
2672:
2656:
2653:
2649:
2645:
2640:
2636:
2631:
2624:
2621:
2616:
2613:
2610:
2607:
2604:
2594:
2588:
2587:
2584:
2571:
2568:
2555:
2542:
2537:
2533:
2519:
2512:
2496:
2493:
2489:
2485:
2482:
2479:
2474:
2470:
2465:
2460:
2457:
2454:
2451:
2448:
2438:
2432:
2431:
2428:
2415:
2410:
2406:
2392:
2377:
2373:
2369:
2364:
2361:
2350:
2343:
2331:
2328:
2324:
2320:
2316:
2293:
2289:
2284:
2281:
2275:
2271:
2268:
2263:
2260:
2255:
2252:
2249:
2246:
2243:
2233:
2229:
2228:
2225:
2209:
2204:
2200:
2187:
2175:
2164:
2157:
2142:
2138:
2132:
2129:
2124:
2120:
2113:
2110:
2106:
2101:
2098:
2095:
2092:
2089:
2079:
2073:
2072:
2069:
2056:
2053:
2040:
2027:
2024:
2011:
2004:
1992:
1989:
1985:
1981:
1977:
1953:
1949:
1943:
1937:
1934:
1931:
1925:
1922:
1919:
1914:
1911:
1906:
1903:
1900:
1897:
1894:
1884:
1880:
1879:
1876:
1863:
1860:
1847:
1834:
1831:
1818:
1811:
1799:
1796:
1792:
1788:
1784:
1760:
1756:
1750:
1746:
1742:
1739:
1736:
1731:
1728:
1723:
1720:
1717:
1714:
1711:
1701:
1697:
1696:
1693:
1680:
1677:
1664:
1651:
1648:
1635:
1628:
1616:
1613:
1609:
1605:
1601:
1577:
1573:
1567:
1563:
1559:
1556:
1553:
1548:
1545:
1540:
1537:
1534:
1531:
1528:
1518:
1512:
1511:
1508:
1495:
1492:
1479:
1466:
1463:
1450:
1443:
1431:
1428:
1424:
1420:
1416:
1394:
1389:
1385:
1381:
1378:
1375:
1370:
1367:
1362:
1359:
1356:
1353:
1350:
1340:
1331:
1330:
1327:
1314:
1311:
1298:
1285:
1282:
1269:
1262:
1250:
1247:
1243:
1239:
1235:
1213:
1209:
1205:
1201:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1166:
1162:
1161:
1158:
1145:
1142:
1129:
1116:
1113:
1100:
1086:
1074:
1071:
1067:
1063:
1059:
1035:
1032:
1027:
1024:
1021:
1018:
1015:
1005:
1001:
1000:
997:
985:
982:
977:
973:
969:
966:
963:
960:
949:
937:
934:
931:
928:
925:
922:
917:
913:
909:
898:
879:for values of
868:
865:
862:
859:
856:
853:
829:
806:
803:
768:
767:
756:
752:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
704:
703:
699:
698:
687:
683:
680:
677:
674:
670:
667:
664:
661:
656:
653:
648:
645:
641:
629:
628:
606:A kernel is a
597:
594:
547:non-parametric
532:
529:
509:kernel methods
500:
497:
484:
462:
458:
446:
445:
427:
423:
419:
412:
408:
404:
401:
398:
395:
389:
385:
381:
378:
373:
369:
365:
362:
358:
354:
351:
348:
334:
333:
315:
311:
307:
300:
296:
292:
289:
286:
283:
277:
273:
264:
260:
256:
253:
249:
244:
241:
236:
232:
228:
225:
221:
217:
214:
211:
145:
142:
136:to refer to a
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
3435:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3405:
3403:
3392:
3388:
3383:
3378:
3374:
3370:
3365:
3364:
3349:
3344:
3343:
3338:
3332:
3328:
3323:
3322:
3314:
3307:
3288:
3285:
3279:
3271:
3265:
3262:
3257:
3254:
3247:
3241:
3236:
3232:
3228:
3216:
3208:
3204:
3200:
3196:
3192:
3191:Devlin, S. J.
3188:
3182:
3173:
3168:
3164:
3160:
3156:
3152:
3148:
3147:Altman, N. S.
3142:
3134:
3130:
3126:
3122:
3114:
3105:
3100:
3096:
3092:
3088:
3081:
3077:
3063:
3060:
3052:
3042:
3038:
3032:
3031:
3025:
3020:
3011:
3010:
3005:
3004:Kernel method
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2971:
2960:
2944:
2938:
2933:
2908:
2897:
2893:
2878:
2872:
2869:
2864:
2858:
2848:
2836:
2832:
2829:
2826:
2818:
2808:
2797:
2793:
2787:
2784:
2779:
2773:
2767:
2760:
2757:
2756:
2752:
2734:
2730:
2726:
2700:
2695:
2691:
2677:
2673:
2654:
2651:
2647:
2643:
2638:
2634:
2629:
2622:
2619:
2614:
2608:
2602:
2595:
2593:
2590:
2589:
2585:
2569:
2566:
2540:
2535:
2531:
2517:
2513:
2494:
2491:
2487:
2483:
2480:
2477:
2472:
2468:
2463:
2458:
2452:
2446:
2439:
2437:
2434:
2433:
2429:
2413:
2408:
2404:
2375:
2371:
2367:
2362:
2359:
2348:
2344:
2342:
2329:
2326:
2318:
2291:
2287:
2282:
2279:
2273:
2269:
2266:
2261:
2258:
2253:
2247:
2241:
2234:
2231:
2230:
2226:
2207:
2202:
2198:
2173:
2162:
2158:
2140:
2136:
2130:
2127:
2122:
2118:
2111:
2108:
2104:
2099:
2093:
2087:
2080:
2078:
2075:
2074:
2070:
2054:
2051:
2025:
2022:
2009:
2005:
2003:
1990:
1987:
1979:
1951:
1941:
1935:
1932:
1929:
1923:
1920:
1912:
1909:
1904:
1898:
1892:
1885:
1882:
1881:
1877:
1861:
1858:
1832:
1829:
1816:
1812:
1810:
1797:
1794:
1786:
1758:
1748:
1744:
1740:
1737:
1729:
1726:
1721:
1715:
1709:
1702:
1699:
1698:
1694:
1678:
1675:
1649:
1646:
1633:
1629:
1627:
1614:
1611:
1603:
1575:
1565:
1561:
1557:
1554:
1546:
1543:
1538:
1532:
1526:
1519:
1514:
1513:
1509:
1493:
1490:
1464:
1461:
1448:
1444:
1442:
1429:
1426:
1418:
1387:
1383:
1379:
1376:
1368:
1365:
1360:
1354:
1348:
1341:
1339:
1336:
1333:
1332:
1328:
1312:
1309:
1283:
1280:
1267:
1263:
1261:
1248:
1245:
1237:
1203:
1195:
1192:
1186:
1180:
1174:
1167:
1164:
1163:
1159:
1143:
1140:
1114:
1111:
1099:
1097:
1091:
1087:
1085:
1072:
1069:
1061:
1033:
1030:
1025:
1019:
1013:
1006:
1003:
1002:
998:
983:
980:
975:
967:
961:
958:
950:
935:
932:
926:
920:
915:
911:
907:
899:
896:
892:
887:
884:
882:
866:
863:
857:
851:
843:
827:
818:
811:
802:
800:
796:
792:
788:
785:* defined by
784:
780:
775:
773:
754:
750:
737:
731:
728:
722:
719:
713:
706:
705:
701:
700:
685:
681:
678:
675:
672:
665:
659:
651:
643:
639:
631:
630:
626:
625:Normalization
623:
622:
621:
619:
615:
612:
609:
603:
593:
590:
588:
587:point process
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
538:
528:
526:
522:
518:
514:
510:
506:
496:
482:
460:
456:
425:
421:
417:
410:
402:
399:
396:
387:
383:
379:
371:
367:
363:
360:
352:
346:
339:
338:
337:
313:
309:
305:
298:
290:
287:
284:
275:
271:
262:
258:
254:
251:
247:
242:
234:
230:
226:
223:
215:
209:
202:
201:
200:
198:
194:
189:
186:
183:
179:
175:
171:
167:
163:
159:
155:
151:
141:
139:
135:
131:
120:
117:
109:
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
3372:
3368:
3354:. Retrieved
3326:
3312:
3306:
3215:
3198:
3194:
3181:
3154:
3150:
3141:
3124:
3120:
3113:
3094:
3090:
3080:
3055:
3046:
3027:
2305:
1966:
1773:
1590:
1405:
1338:(parabolic)
1337:
1335:Epanechnikov
1224:
1093:
1048:
894:
890:
880:
819:
816:
798:
794:
790:
786:
782:
778:
776:
769:
617:
608:non-negative
605:
591:
553:to estimate
540:
502:
447:
335:
190:
187:
147:
129:
127:
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
3413:Time series
3356:6 September
3041:introducing
1517:(biweight)
1165:Triangular
611:real-valued
575:periodogram
571:time-series
132:is used in
3402:Categories
3172:1813/31637
3117:Named for
3072:References
3024:references
1700:Triweight
614:integrable
596:Definition
162:parameters
76:newspapers
3377:CiteSeerX
3263:∫
3229:∫
2870:π
2833:
2827:⋅
2798:−
2731:π
2692:π
2652:−
2623:π
2532:π
2492:−
2405:π
2372:π
2363:−
2327:≤
2306:Support:
2280:π
2270:
2259:π
2208:π
2123:−
2112:π
1988:≤
1967:Support:
1924:−
1795:≤
1774:Support:
1741:−
1612:≤
1591:Support:
1558:−
1427:≤
1406:Support:
1380:−
1246:≤
1225:Support:
1196:−
1070:≤
1049:Support:
959:∫
908:∫
720:−
702:Symmetry:
655:∞
647:∞
644:−
640:∫
616:function
457:σ
422:σ
403:μ
400:−
388:−
380:∝
368:σ
361:μ
310:σ
291:μ
288:−
276:−
259:σ
255:π
231:σ
224:μ
156:(pdf) or
128:The term
3049:May 2012
2968:See also
2436:Logistic
2077:Gaussian
1883:Tricube
1515:Quartic
561:, or in
106:May 2012
3037:improve
2923:
2901:
2717:
2681:
2557:
2521:
2394:
2352:
2232:Cosine
2189:
2166:
2042:
2013:
1849:
1820:
1666:
1637:
1481:
1452:
1300:
1271:
1131:
1102:
844:, then
842:support
195:. Its
168:of the
90:scholar
3379:
3333:
3026:, but
2753:84.3%
2586:88.7%
2430:99.9%
2227:95.1%
2071:99.8%
1878:98.7%
1695:99.4%
1329:98.6%
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