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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,
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171:(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
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Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, quadratic and cosine.
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2896:{\displaystyle K(u)={\frac {1}{2}}e^{-{\frac {|u|}{\sqrt {2}}}}\cdot \sin \left({\frac {|u|}{\sqrt {2}}}+{\frac {\pi }{4}}\right)}
515:
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785:. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.
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337:{\displaystyle p(x|\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
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Note that the factor in front of the exponential has been omitted, even though it contains the parameter
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For most applications, it is desirable to define the function to satisfy two additional requirements:
199:
For many distributions, the kernel can be written in closed form, but not the normalization constant.
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3130:
Epanechnikov, V. A. (1969). "Non-Parametric
Estimation of a Multivariate Probability Density".
2320:
1981:
1788:
1605:
1420:
1239:
1063:
100:
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573:
188:
3378:
Comaniciu, D; Meer, P (2002). "Mean shift: A robust approach toward feature space analysis".
3397:
3213:
3204:(1988). "Locally weighted regression: An approach to regression analysis by local fitting".
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Commonly, kernel widths must also be specified when running a non-parametric estimation.
812:), where λ > 0. This can be used to select a scale that is appropriate for the data.
781:
The first requirement ensures that the method of kernel density estimation results in a
2915:
834:
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489:
151:. The term "kernel" has several distinct meanings in different branches of statistics.
449:{\displaystyle p(x|\mu ,\sigma ^{2})\propto e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
3412:
3014:
597:
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3173:
3160:(1992). "An introduction to kernel and nearest neighbor nonparametric regression".
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187:, most sampling algorithms ignore the normalization factor. In addition, in
2018:
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17:
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596:. An additional use is in the estimation of a time-varying intensity for a
2309:{\displaystyle K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)}
3401:
2161:{\displaystyle K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}}
3182:
600:
where window functions (kernels) are convolved with time-series data.
3143:
820:
534:
on data in an implicit space. This usage is particularly common in
3098:"Estimation of a probability density function and its derivatives"
1970:{\displaystyle K(u)={\frac {70}{81}}(1-{\left|u\right|}^{3})^{3}}
3359:"APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation"
2676:{\displaystyle K(u)={\frac {2}{\pi }}{\frac {1}{e^{u}+e^{-u}}}}
3380:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
3019:
36:
3305:{\displaystyle {\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du}
183:, and are unnecessary in many situations. For example, in
771:{\displaystyle K(-u)=K(u){\mbox{ for all values of }}u\,.}
702:{\displaystyle \int _{-\infty }^{+\infty }K(u)\,du=1\,;}
824:
All of the kernels below in a common coordinate system.
968:
917:
755:
486:, because it is not a function of the domain variable
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if the distribution only needs to be sampled from).
3324:
Density
Estimation for Statistics and Data Analysis
175:of the pdf or pmf. These factors form part of the
67:. Unsourced material may be challenged and removed.
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1777:{\displaystyle K(u)={\frac {35}{32}}(1-u^{2})^{3}}
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1594:{\displaystyle K(u)={\frac {15}{16}}(1-u^{2})^{2}}
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336:
2516:{\displaystyle K(u)={\frac {1}{e^{u}+2+e^{-u}}}}
3338:Nonparametric Econometrics: Theory and Practice
3206:Journal of the American Statistical Association
1010:Efficiency relative to the Epanechnikov kernel
580:of a random variable. Kernels are also used in
8:
2229:{\displaystyle {\frac {1}{2{\sqrt {\pi }}}}}
1409:{\displaystyle K(u)={\frac {3}{4}}(1-u^{2})}
518:is used in the suite of techniques known as
953:{\displaystyle \textstyle \int u^{2}K(u)du}
560:estimation techniques. Kernels are used in
556:, a kernel is a weighting function used in
2963:{\displaystyle {\frac {3{\sqrt {2}}}{16}}}
1001:{\displaystyle \textstyle \int K(u)^{2}du}
896:
3391:
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3070:Learn how and when to remove this message
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127:Learn how and when to remove this message
3033:This article includes a list of general
819:
3088:
2396:{\displaystyle 1-{\frac {8}{\pi ^{2}}}}
3010:Multivariate kernel density estimation
2432:{\displaystyle {\frac {\pi ^{2}}{16}}}
3102:The Annals of Mathematical Statistics
2755:{\displaystyle {\frac {2}{\pi ^{2}}}}
2719:{\displaystyle {\frac {\pi ^{2}}{4}}}
2559:{\displaystyle {\frac {\pi ^{2}}{3}}}
792:is a kernel, then so is the function
7:
65:adding citations to reliable sources
3336:Li, Qi; Racine, Jeffrey S. (2007).
1052:{\displaystyle K(u)={\frac {1}{2}}}
3039:it lacks sufficient corresponding
2073:{\displaystyle {\frac {175}{247}}}
1880:{\displaystyle {\frac {350}{429}}}
665:
657:
25:
2044:{\displaystyle {\frac {35}{243}}}
3096:Schuster, Eugene (August 1969).
3024:
2905:
2685:
2525:
2356:
2170:
2017:
1824:
1641:
1456:
1275:
1099:
516:reproducing kernel Hilbert space
41:
1015:Uniform ("rectangular window")
52:needs additional citations for
3340:. Princeton University Press.
3286:
3279:
3261:
3255:
3218:10.1080/01621459.1988.10478639
3174:10.1080/00031305.1992.10475879
2863:
2855:
2823:
2815:
2787:
2781:
2622:
2616:
2588:{\displaystyle {\frac {1}{6}}}
2466:
2460:
2333:
2325:
2261:
2255:
2107:
2101:
1994:
1986:
1958:
1928:
1912:
1906:
1851:{\displaystyle {\frac {1}{9}}}
1801:
1793:
1765:
1745:
1729:
1723:
1697:{\displaystyle {\frac {5}{7}}}
1668:{\displaystyle {\frac {1}{7}}}
1618:
1610:
1582:
1562:
1546:
1540:
1512:{\displaystyle {\frac {3}{5}}}
1483:{\displaystyle {\frac {1}{5}}}
1433:
1425:
1403:
1384:
1368:
1362:
1331:{\displaystyle {\frac {2}{3}}}
1302:{\displaystyle {\frac {1}{6}}}
1252:
1244:
1222:
1218:
1210:
1200:
1194:
1188:
1162:{\displaystyle {\frac {1}{2}}}
1133:{\displaystyle {\frac {1}{3}}}
1076:
1068:
1033:
1027:
982:
975:
940:
934:
871:
865:
816:Kernel functions in common use
751:
745:
736:
727:
679:
673:
417:
404:
387:
367:
360:
305:
292:
250:
230:
223:
1:
757: for all values of
347:and the associated kernel is
185:pseudo-random number sampling
159:In statistics, especially in
1228:{\displaystyle K(u)=(1-|u|)}
783:probability density function
208:probability density function
165:probability density function
3326:. Chapman and Hall, London.
894:lying outside the support.
479:{\displaystyle \sigma ^{2}}
3450:
610:
545:
524:statistical classification
29:
3322:Silverman, B. W. (1986).
3230:Efficiency is defined as
3162:The American Statistician
2985:Kernel density estimation
2346:{\displaystyle |u|\leq 1}
2007:{\displaystyle |u|\leq 1}
1814:{\displaystyle |u|\leq 1}
1631:{\displaystyle |u|\leq 1}
1446:{\displaystyle |u|\leq 1}
1265:{\displaystyle |u|\leq 1}
1089:{\displaystyle |u|\leq 1}
899:
562:kernel density estimation
522:to perform tasks such as
169:probability mass function
3419:Nonparametric statistics
3000:Positive-definite kernel
851:is given with a bounded
592:where they are known as
554:nonparametric statistics
542:Nonparametric statistics
181:probability distribution
76:"Kernel" statistics
3115:10.1214/aoms/1177697495
3054:more precise citations.
831:In the table below, if
578:conditional expectation
32:Kernel (disambiguation)
3306:
2964:
2926:
2897:
2756:
2720:
2677:
2589:
2560:
2517:
2433:
2397:
2347:
2310:
2230:
2192:
2162:
2074:
2045:
2008:
1971:
1881:
1852:
1815:
1778:
1698:
1669:
1632:
1595:
1513:
1484:
1447:
1410:
1332:
1303:
1266:
1229:
1163:
1134:
1090:
1053:
1002:
954:
884:
883:{\displaystyle K(u)=0}
845:
825:
772:
703:
500:
480:
450:
338:
3307:
2965:
2927:
2898:
2757:
2721:
2678:
2590:
2561:
2518:
2434:
2398:
2348:
2311:
2231:
2193:
2163:
2075:
2046:
2009:
1972:
1882:
1853:
1816:
1779:
1699:
1670:
1633:
1596:
1514:
1485:
1448:
1411:
1333:
1304:
1267:
1230:
1164:
1135:
1091:
1054:
1003:
955:
885:
846:
823:
773:
704:
611:Further information:
546:Further information:
501:
481:
451:
339:
3234:
2938:
2916:
2775:
2732:
2696:
2610:
2572:
2536:
2454:
2409:
2367:
2321:
2249:
2204:
2181:
2095:
2057:
2028:
1982:
1900:
1864:
1835:
1789:
1717:
1681:
1652:
1606:
1534:
1496:
1467:
1421:
1356:
1315:
1286:
1240:
1182:
1146:
1117:
1064:
1021:
965:
914:
859:
835:
721:
646:
584:, in the use of the
490:
463:
354:
217:
177:normalization factor
145:statistical analysis
61:improve this article
30:For other uses, see
3434:Bayesian statistics
3132:Theory Probab. Appl
2191:{\displaystyle 1\,}
669:
528:regression analysis
204:normal distribution
161:Bayesian statistics
155:Bayesian statistics
3402:10.1109/34.1000236
3357:Zucchini, Walter.
3302:
3005:Density estimation
2960:
2922:
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2752:
2716:
2673:
2585:
2556:
2513:
2429:
2393:
2343:
2306:
2226:
2188:
2158:
2070:
2041:
2004:
1967:
1877:
1848:
1811:
1774:
1694:
1665:
1628:
1591:
1509:
1480:
1443:
1406:
1328:
1299:
1262:
1225:
1159:
1130:
1086:
1049:
998:
997:
950:
949:
900:Kernel Functions,
880:
841:
826:
768:
759:
699:
649:
496:
476:
446:
334:
202:An example is the
163:, the kernel of a
3347:978-0-691-12161-1
3271:
3080:
3079:
3072:
2995:Stochastic kernel
2976:
2975:
2958:
2952:
2925:{\displaystyle 0}
2886:
2873:
2872:
2833:
2832:
2801:
2769:Silverman kernel
2750:
2714:
2671:
2636:
2583:
2554:
2511:
2427:
2391:
2296:
2275:
2224:
2221:
2144:
2126:
2125:
2068:
2039:
1926:
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1846:
1743:
1692:
1663:
1560:
1507:
1478:
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1326:
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1128:
1047:
844:{\displaystyle K}
758:
574:kernel regression
570:density functions
499:{\displaystyle x}
442:
330:
279:
278:
189:Bayesian analysis
137:
136:
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111:
16:(Redirected from
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3221:
3212:(403): 596–610.
3198:Cleveland, W. S.
3194:
3188:
3187:
3185:
3154:
3148:
3147:
3126:
3120:
3119:
3117:
3108:(4): 1187-1195.
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3061:
3055:
3050:this article by
3041:inline citations
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2603:Sigmoid function
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1271:
1269:
1268:
1263:
1255:
1247:
1234:
1232:
1231:
1226:
1221:
1213:
1168:
1166:
1165:
1160:
1158:
1150:
1139:
1137:
1136:
1131:
1129:
1121:
1103:
1095:
1093:
1092:
1087:
1079:
1071:
1058:
1056:
1055:
1050:
1048:
1040:
1007:
1005:
1004:
999:
990:
989:
959:
957:
956:
951:
930:
929:
897:
889:
887:
886:
881:
850:
848:
847:
842:
777:
775:
774:
769:
760:
756:
708:
706:
705:
700:
668:
660:
594:window functions
590:spectral density
588:to estimate the
576:to estimate the
566:random variables
548:Kernel smoothing
536:machine learning
532:cluster analysis
514:The kernel of a
510:Pattern analysis
505:
503:
502:
497:
485:
483:
482:
477:
475:
474:
455:
453:
452:
447:
445:
444:
443:
441:
440:
439:
426:
425:
424:
402:
386:
385:
370:
343:
341:
340:
335:
333:
332:
331:
329:
328:
327:
314:
313:
312:
290:
280:
277:
276:
261:
257:
249:
248:
233:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
3449:
3448:
3444:
3443:
3442:
3440:
3439:
3438:
3429:Point processes
3409:
3408:
3377:
3366:
3364:
3361:
3356:
3348:
3335:
3332:
3331:
3321:
3320:
3316:
3285:
3242:
3232:
3231:
3229:
3225:
3196:
3195:
3191:
3156:
3155:
3151:
3144:10.1137/1114019
3129:
3127:
3123:
3095:
3094:
3090:
3085:
3076:
3065:
3059:
3056:
3046:Please help to
3045:
3029:
3025:
2990:Kernel smoother
2981:
2972:not applicable
2943:
2936:
2935:
2914:
2913:
2853:
2850:
2846:
2813:
2803:
2773:
2772:
2740:
2730:
2729:
2701:
2694:
2693:
2657:
2644:
2643:
2608:
2607:
2570:
2569:
2541:
2534:
2533:
2497:
2478:
2477:
2452:
2451:
2414:
2407:
2406:
2381:
2365:
2364:
2319:
2318:
2287:
2283:
2247:
2246:
2212:
2202:
2201:
2179:
2178:
2146:
2128:
2093:
2092:
2055:
2054:
2026:
2025:
1980:
1979:
1957:
1939:
1937:
1898:
1897:
1862:
1861:
1833:
1832:
1787:
1786:
1764:
1754:
1715:
1714:
1679:
1678:
1650:
1649:
1604:
1603:
1581:
1571:
1532:
1531:
1527:
1494:
1493:
1465:
1464:
1419:
1418:
1393:
1354:
1353:
1313:
1312:
1284:
1283:
1238:
1237:
1180:
1179:
1144:
1143:
1115:
1114:
1107:Boxcar function
1062:
1061:
1019:
1018:
981:
963:
962:
921:
912:
911:
857:
856:
833:
832:
818:
719:
718:
644:
643:
615:
613:Integral kernel
609:
550:
544:
512:
488:
487:
466:
461:
460:
431:
427:
416:
403:
393:
377:
352:
351:
319:
315:
304:
291:
281:
268:
240:
215:
214:
193:conjugate prior
157:
149:window function
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
27:Window function
23:
22:
15:
12:
11:
5:
3447:
3445:
3437:
3436:
3431:
3426:
3421:
3411:
3410:
3407:
3406:
3393:10.1.1.76.8968
3386:(5): 603–619.
3374:
3373:
3353:
3352:
3346:
3330:
3329:
3314:
3301:
3298:
3292:
3288:
3284:
3281:
3278:
3275:
3270:
3267:
3263:
3260:
3257:
3254:
3249:
3245:
3241:
3223:
3189:
3168:(3): 175–185.
3149:
3138:(1): 153–158.
3121:
3087:
3086:
3084:
3081:
3078:
3077:
3032:
3030:
3023:
3018:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2980:
2977:
2974:
2973:
2970:
2957:
2951:
2946:
2932:
2921:
2910:
2903:
2891:
2885:
2882:
2877:
2871:
2865:
2861:
2857:
2849:
2845:
2842:
2839:
2831:
2825:
2821:
2817:
2810:
2806:
2800:
2797:
2792:
2789:
2786:
2783:
2780:
2770:
2766:
2765:
2762:
2747:
2743:
2739:
2726:
2713:
2708:
2704:
2690:
2683:
2667:
2664:
2660:
2656:
2651:
2647:
2642:
2635:
2632:
2627:
2624:
2621:
2618:
2615:
2605:
2599:
2598:
2595:
2582:
2579:
2566:
2553:
2548:
2544:
2530:
2523:
2507:
2504:
2500:
2496:
2493:
2490:
2485:
2481:
2476:
2471:
2468:
2465:
2462:
2459:
2449:
2443:
2442:
2439:
2426:
2421:
2417:
2403:
2388:
2384:
2380:
2375:
2372:
2361:
2354:
2342:
2339:
2335:
2331:
2327:
2304:
2300:
2295:
2292:
2286:
2282:
2279:
2274:
2271:
2266:
2263:
2260:
2257:
2254:
2244:
2240:
2239:
2236:
2220:
2215:
2211:
2198:
2186:
2175:
2168:
2153:
2149:
2143:
2140:
2135:
2131:
2124:
2121:
2117:
2112:
2109:
2106:
2103:
2100:
2090:
2084:
2083:
2080:
2067:
2064:
2051:
2038:
2035:
2022:
2015:
2003:
2000:
1996:
1992:
1988:
1964:
1960:
1954:
1948:
1945:
1942:
1936:
1933:
1930:
1925:
1922:
1917:
1914:
1911:
1908:
1905:
1895:
1891:
1890:
1887:
1874:
1871:
1858:
1845:
1842:
1829:
1822:
1810:
1807:
1803:
1799:
1795:
1771:
1767:
1761:
1757:
1753:
1750:
1747:
1742:
1739:
1734:
1731:
1728:
1725:
1722:
1712:
1708:
1707:
1704:
1691:
1688:
1675:
1662:
1659:
1646:
1639:
1627:
1624:
1620:
1616:
1612:
1588:
1584:
1578:
1574:
1570:
1567:
1564:
1559:
1556:
1551:
1548:
1545:
1542:
1539:
1529:
1523:
1522:
1519:
1506:
1503:
1490:
1477:
1474:
1461:
1454:
1442:
1439:
1435:
1431:
1427:
1405:
1400:
1396:
1392:
1389:
1386:
1381:
1378:
1373:
1370:
1367:
1364:
1361:
1351:
1342:
1341:
1338:
1325:
1322:
1309:
1296:
1293:
1280:
1273:
1261:
1258:
1254:
1250:
1246:
1224:
1220:
1216:
1212:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1177:
1173:
1172:
1169:
1156:
1153:
1140:
1127:
1124:
1111:
1097:
1085:
1082:
1078:
1074:
1070:
1046:
1043:
1038:
1035:
1032:
1029:
1026:
1016:
1012:
1011:
1008:
996:
993:
988:
984:
980:
977:
974:
971:
960:
948:
945:
942:
939:
936:
933:
928:
924:
920:
909:
890:for values of
879:
876:
873:
870:
867:
864:
840:
817:
814:
779:
778:
767:
763:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
715:
714:
710:
709:
698:
694:
691:
688:
685:
681:
678:
675:
672:
667:
664:
659:
656:
652:
640:
639:
617:A kernel is a
608:
605:
558:non-parametric
543:
540:
520:kernel methods
511:
508:
495:
473:
469:
457:
456:
438:
434:
430:
423:
419:
415:
412:
409:
406:
400:
396:
392:
389:
384:
380:
376:
373:
369:
365:
362:
359:
345:
344:
326:
322:
318:
311:
307:
303:
300:
297:
294:
288:
284:
275:
271:
267:
264:
260:
255:
252:
247:
243:
239:
236:
232:
228:
225:
222:
156:
153:
147:to refer to a
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3446:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3416:
3414:
3403:
3399:
3394:
3389:
3385:
3381:
3376:
3375:
3360:
3355:
3354:
3349:
3343:
3339:
3334:
3333:
3325:
3318:
3315:
3299:
3296:
3290:
3282:
3276:
3273:
3268:
3265:
3258:
3252:
3247:
3243:
3239:
3227:
3224:
3219:
3215:
3211:
3207:
3203:
3202:Devlin, S. J.
3199:
3193:
3190:
3184:
3179:
3175:
3171:
3167:
3163:
3159:
3158:Altman, N. S.
3153:
3150:
3145:
3141:
3137:
3133:
3125:
3122:
3116:
3111:
3107:
3103:
3099:
3092:
3089:
3082:
3074:
3071:
3063:
3053:
3049:
3043:
3042:
3036:
3031:
3022:
3021:
3016:
3015:Kernel method
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2982:
2978:
2971:
2955:
2949:
2944:
2933:
2919:
2911:
2908:
2904:
2889:
2883:
2880:
2875:
2869:
2859:
2847:
2843:
2840:
2837:
2829:
2819:
2808:
2804:
2798:
2795:
2790:
2784:
2778:
2771:
2768:
2767:
2763:
2745:
2741:
2737:
2727:
2711:
2706:
2702:
2691:
2688:
2684:
2665:
2662:
2658:
2654:
2649:
2645:
2640:
2633:
2630:
2625:
2619:
2613:
2606:
2604:
2601:
2600:
2596:
2580:
2577:
2567:
2551:
2546:
2542:
2531:
2528:
2524:
2505:
2502:
2498:
2494:
2491:
2488:
2483:
2479:
2474:
2469:
2463:
2457:
2450:
2448:
2445:
2444:
2440:
2424:
2419:
2415:
2404:
2386:
2382:
2378:
2373:
2370:
2362:
2359:
2355:
2353:
2340:
2337:
2329:
2302:
2298:
2293:
2290:
2284:
2280:
2277:
2272:
2269:
2264:
2258:
2252:
2245:
2242:
2241:
2237:
2218:
2213:
2209:
2199:
2184:
2176:
2173:
2169:
2151:
2147:
2141:
2138:
2133:
2129:
2122:
2119:
2115:
2110:
2104:
2098:
2091:
2089:
2086:
2085:
2081:
2065:
2062:
2052:
2036:
2033:
2023:
2020:
2016:
2014:
2001:
1998:
1990:
1962:
1952:
1946:
1943:
1940:
1934:
1931:
1923:
1920:
1915:
1909:
1903:
1896:
1893:
1892:
1888:
1872:
1869:
1859:
1843:
1840:
1830:
1827:
1823:
1821:
1808:
1805:
1797:
1769:
1759:
1755:
1751:
1748:
1740:
1737:
1732:
1726:
1720:
1713:
1710:
1709:
1705:
1689:
1686:
1676:
1660:
1657:
1647:
1644:
1640:
1638:
1625:
1622:
1614:
1586:
1576:
1572:
1568:
1565:
1557:
1554:
1549:
1543:
1537:
1530:
1525:
1524:
1520:
1504:
1501:
1491:
1475:
1472:
1462:
1459:
1455:
1453:
1440:
1437:
1429:
1398:
1394:
1390:
1387:
1379:
1376:
1371:
1365:
1359:
1352:
1350:
1347:
1344:
1343:
1339:
1323:
1320:
1310:
1294:
1291:
1281:
1278:
1274:
1272:
1259:
1256:
1248:
1214:
1206:
1203:
1197:
1191:
1185:
1178:
1175:
1174:
1170:
1154:
1151:
1141:
1125:
1122:
1112:
1110:
1108:
1102:
1098:
1096:
1083:
1080:
1072:
1044:
1041:
1036:
1030:
1024:
1017:
1014:
1013:
1009:
994:
991:
986:
978:
972:
969:
961:
946:
943:
937:
931:
926:
922:
918:
910:
907:
903:
898:
895:
893:
877:
874:
868:
862:
854:
838:
829:
822:
815:
813:
811:
807:
803:
799:
796:* defined by
795:
791:
786:
784:
765:
761:
748:
742:
739:
733:
730:
724:
717:
716:
712:
711:
696:
692:
689:
686:
683:
676:
670:
662:
654:
650:
642:
641:
637:
636:Normalization
634:
633:
632:
630:
626:
623:
620:
614:
606:
604:
601:
599:
598:point process
595:
591:
587:
583:
579:
575:
571:
567:
563:
559:
555:
549:
541:
539:
537:
533:
529:
525:
521:
517:
509:
507:
493:
471:
467:
436:
432:
428:
421:
413:
410:
407:
398:
394:
390:
382:
378:
374:
371:
363:
357:
350:
349:
348:
324:
320:
316:
309:
301:
298:
295:
286:
282:
273:
269:
265:
262:
258:
253:
245:
241:
237:
234:
226:
220:
213:
212:
211:
209:
205:
200:
197:
194:
190:
186:
182:
178:
174:
170:
166:
162:
154:
152:
150:
146:
142:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
3383:
3379:
3365:. Retrieved
3337:
3323:
3317:
3226:
3209:
3205:
3192:
3165:
3161:
3152:
3135:
3131:
3124:
3105:
3101:
3091:
3066:
3057:
3038:
2316:
1977:
1784:
1601:
1416:
1349:(parabolic)
1348:
1346:Epanechnikov
1235:
1104:
1059:
905:
901:
891:
830:
827:
809:
805:
801:
797:
793:
789:
787:
780:
628:
619:non-negative
616:
602:
564:to estimate
551:
513:
458:
346:
201:
198:
158:
140:
138:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
18:Epanechnikov
3424:Time series
3367:6 September
3052:introducing
1528:(biweight)
1176:Triangular
622:real-valued
586:periodogram
582:time-series
143:is used in
3413:Categories
3183:1813/31637
3128:Named for
3083:References
3035:references
1711:Triweight
625:integrable
607:Definition
173:parameters
87:newspapers
3388:CiteSeerX
3274:∫
3240:∫
2881:π
2844:
2838:⋅
2809:−
2742:π
2703:π
2663:−
2634:π
2543:π
2503:−
2416:π
2383:π
2374:−
2338:≤
2317:Support:
2291:π
2281:
2270:π
2219:π
2134:−
2123:π
1999:≤
1978:Support:
1935:−
1806:≤
1785:Support:
1752:−
1623:≤
1602:Support:
1569:−
1438:≤
1417:Support:
1391:−
1257:≤
1236:Support:
1207:−
1081:≤
1060:Support:
970:∫
919:∫
731:−
713:Symmetry:
666:∞
658:∞
655:−
651:∫
627:function
468:σ
433:σ
414:μ
411:−
399:−
391:∝
379:σ
372:μ
321:σ
302:μ
299:−
287:−
270:σ
266:π
242:σ
235:μ
167:(pdf) or
139:The term
3060:May 2012
2979:See also
2447:Logistic
2088:Gaussian
1894:Tricube
1526:Quartic
572:, or in
117:May 2012
3048:improve
2934:
2912:
2728:
2692:
2568:
2532:
2405:
2363:
2243:Cosine
2200:
2177:
2053:
2024:
1860:
1831:
1677:
1648:
1492:
1463:
1311:
1282:
1142:
1113:
855:, then
853:support
206:. Its
179:of the
101:scholar
3390:
3344:
3037:, but
2764:84.3%
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