3621:
3629:
2823:
3636:
Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional
3708:
between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100,
3610:
3007:
3637:
data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with
41:(FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in
2343:
1071:
1464:
1723:
2652:
2051:
820:
613:
20:
are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of
3437:
2873:
432:
682:
282:
898:
2123:
1939:
2490:
1833:
353:
3712:
Compared to female mortality data, modes of variation of male mortality data shows higher mortality after around age 20, possibly related to the fact that life expectancy for women is higher than that for men.
1519:
150:
3308:
1571:
1203:
2203:
3124:
971:
3231:
1309:
1317:
214:
2625:
3175:
1586:
179:
3676:
2574:
2406:
505:
2596:
527:
483:
379:
1128:
731:
2818:{\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})}
1253:
2865:
2647:
2552:
943:
457:
3341:
2416:
The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance.
2366:
2195:
963:
3376:
3699:
3429:
2175:
1229:
1970:
739:
3400:
3038:
2843:
2530:
2510:
2146:
1962:
1094:
921:
702:
539:
3883:
Kirkpatrick, Mark; Heckman, Nancy (August 1989). "A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters".
3605:{\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in .}
3002:{\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in .}
387:
621:
222:
825:
2056:
1844:
3782:
2430:
1738:
293:
3730:
Castro, P. E.; Lawton, W. H.; Sylvestre, E. A. (November 1986). "Principal Modes of
Variation for Processes with Continuous Sample Curves".
3704:
The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log
33:. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in
1472:
75:
3311:
3014:
1135:
58:
38:
3236:
2338:{\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in }
1524:
1150:
1066:{\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in ,}
3043:
1459:{\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),}
3180:
1258:
3934:
Jones, M. C.; Rice, John A. (May 1992). "Displaying the
Important Features of Large Collections of Similar Curves".
3819:
Kleffe, JĂĽrgen (January 1973). "Principal components of random variables with values in a seperable hilbert space".
184:
1718:{\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.}
2601:
3985:
356:
3135:
155:
1729:
530:
3980:
3975:
2421:
1578:
288:
217:
66:
54:
34:
22:
3640:
2557:
2371:
488:
2579:
510:
466:
362:
3127:
1142:
1099:
707:
42:
1234:
2848:
2630:
2535:
926:
440:
3796:
3317:
2351:
2180:
948:
3346:
3681:
2046:{\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},}
815:{\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},}
3624:
The first and second modes of variation of female mortality data from 41 countries in 2003
3405:
2151:
608:{\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},}
8:
1208:
3632:
The first and second modes of variation of male mortality data from 41 countries in 2003
3916:
3755:
3385:
3130:
3023:
2828:
2515:
2495:
2131:
1947:
1079:
906:
687:
3951:
3908:
3900:
3836:
3801:
3747:
460:
3920:
3947:
3943:
3892:
3828:
3791:
3739:
427:{\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},}
533:
for random vectors, one can express the centered random vector in the eigenbasis
1145:
677:{\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})}
277:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0}
1574:
285:
3832:
893:{\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.}
3969:
3955:
3904:
3840:
3805:
3751:
3379:
30:
2118:{\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.}
1934:{\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt}
3705:
3912:
2485:{\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}}
1828:{\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),}
348:{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}}
3896:
3759:
26:
3743:
3777:
3776:
Wang, Jane-Ling; Chiou, Jeng-Min; MĂĽller, Hans-Georg (June 2016).
1514:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0}
3709:
mortality is related to congenital disease or natural death.
3859:
3620:
1732:, one can express the centered function in the eigenbasis,
145:{\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}}
3628:
507:
is a diagonal matrix whose entries are the eigenvalues of
3382:. Substituting estimates for the unknown quantities, the
3303:{\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))}
3729:
3684:
3643:
3440:
3408:
3388:
3349:
3320:
3239:
3183:
3138:
3046:
3026:
2876:
2851:
2831:
2655:
2633:
2604:
2582:
2560:
2538:
2518:
2498:
2433:
2374:
2354:
2206:
2183:
2154:
2134:
2059:
1973:
1950:
1847:
1741:
1589:
1566:{\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}}
1527:
1475:
1320:
1261:
1237:
1211:
1198:{\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}}
1153:
1102:
1082:
974:
951:
929:
909:
828:
742:
710:
690:
624:
542:
513:
491:
469:
443:
390:
365:
296:
225:
187:
158:
78:
25:
with the directions represented by the corresponding
3378:in detail, often involving point wise estimate and
3882:
3693:
3670:
3604:
3423:
3394:
3370:
3335:
3302:
3225:
3169:
3119:{\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)}
3118:
3032:
3001:
2859:
2837:
2817:
2641:
2619:
2590:
2568:
2546:
2524:
2504:
2484:
2400:
2360:
2337:
2189:
2169:
2140:
2117:
2045:
1956:
1933:
1827:
1717:
1565:
1513:
1458:
1303:
1247:
1223:
1197:
1122:
1088:
1065:
957:
937:
915:
892:
814:
725:
696:
676:
607:
521:
499:
477:
451:
426:
373:
347:
276:
208:
173:
144:
2348:that are viewed simultaneously over the range of
3967:
3226:{\displaystyle \mu (t)=\operatorname {E} (X(t))}
1304:{\displaystyle \mu (t)=\operatorname {E} (X(t))}
3821:Mathematische Operationsforschung und Statistik
3783:Annual Review of Statistics and Its Application
3775:
3012:
684:is the principal component associated with the
2419:
209:{\displaystyle \mathbf {\Sigma } _{p\times p}}
53:Modes of variation are a natural extension of
1133:
357:eigendecomposition of a real symmetric matrix
1964:-th principal component with the properties
1560:
1528:
1255:is an interval, denote the mean function by
64:
2620:{\displaystyle {\overline {\mathbf {x} }}}
2598:. These data yield the sample mean vector
3933:
3795:
1648:
3797:10.1146/annurev-statistics-041715-033624
3627:
3619:
3170:{\displaystyle X(t),t\in {\mathcal {T}}}
174:{\displaystyle {\boldsymbol {\mu }}_{p}}
3314:provides methods for the estimation of
3312:Functional principal component analysis
2562:
997:
667:
552:
161:
39:functional principal component analysis
3968:
3818:
463:whose columns are the eigenvectors of
3854:
3852:
3850:
3771:
3769:
3671:{\displaystyle \alpha =\pm 1,\pm 2,}
2569:{\displaystyle {\boldsymbol {\mu }}}
2177:is the set of functions, indexed by
2627:, and the sample covariance matrix
2401:{\displaystyle A=2\ {\text{or}}\ 3}
500:{\displaystyle \mathbf {\Lambda } }
13:
3567:
3199:
3162:
2649:with eigenvalue-eigenvector pairs
2591:{\displaystyle \mathbf {\Sigma } }
2300:
2060:
1974:
1867:
1788:
1670:
1637:
1611:
1400:
1311:, and the covariance function by
1277:
1240:
1177:
945:is the set of vectors, indexed by
829:
743:
522:{\displaystyle \mathbf {\Sigma } }
478:{\displaystyle \mathbf {\Sigma } }
374:{\displaystyle \mathbf {\Sigma } }
14:
3997:
3847:
3766:
1123:{\displaystyle 2\ {\text{or}}\ 3}
2953:
2910:
2882:
2853:
2796:
2738:
2686:
2635:
2608:
2584:
2540:
2472:
2451:
2436:
1023:
977:
931:
726:{\displaystyle \mathbf {e} _{k}}
713:
659:
640:
592:
544:
515:
493:
471:
445:
411:
405:
400:
392:
367:
335:
314:
299:
190:
80:
3885:Journal of Mathematical Biology
3615:
2512:independent drawings from some
3948:10.1080/00031305.1992.10475870
3927:
3876:
3812:
3723:
3596:
3581:
3553:
3547:
3535:
3514:
3496:
3490:
3484:
3472:
3466:
3448:
3418:
3412:
3365:
3353:
3330:
3324:
3297:
3294:
3288:
3279:
3273:
3267:
3255:
3243:
3220:
3217:
3211:
3205:
3193:
3187:
3148:
3142:
3113:
3107:
3085:
3079:
3063:
3057:
2993:
2978:
2957:
2934:
2886:
2812:
2800:
2776:
2766:
2754:
2742:
2718:
2708:
2702:
2690:
2666:
2656:
2332:
2317:
2283:
2277:
2244:
2238:
2229:
2223:
2164:
2158:
2089:
2066:
2024:
2011:
1993:
1980:
1922:
1916:
1903:
1900:
1894:
1885:
1879:
1873:
1819:
1813:
1766:
1760:
1751:
1745:
1703:
1697:
1691:
1679:
1658:
1652:
1642:
1632:
1619:
1616:
1606:
1450:
1444:
1431:
1425:
1378:
1375:
1369:
1360:
1354:
1348:
1336:
1324:
1298:
1295:
1289:
1283:
1271:
1265:
1248:{\displaystyle {\mathcal {T}}}
1163:
1157:
1057:
1042:
858:
835:
793:
780:
762:
749:
671:
655:
133:
87:
48:
1:
3716:
2411:
1577:eigenfunctions of the linear
3233:and the covariance function
2914:
2860:{\displaystyle \mathbf {X} }
2642:{\displaystyle \mathbf {S} }
2612:
2547:{\displaystyle \mathbf {X} }
938:{\displaystyle \mathbf {X} }
452:{\displaystyle \mathbf {Q} }
181:, and the covariance matrix
35:principal component analysis
7:
10:
4002:
3860:"Human Mortality Database"
3778:"Functional Data Analysis"
3936:The American Statistician
3833:10.1080/02331887308801137
3402:-th mode of variation of
2845:-th mode of variation of
2148:-th mode of variation of
1096:is typically selected as
923:-th mode of variation of
43:exploratory data analysis
2532:-dimensional population
1579:Hilbert–Schmidt operator
1521:are the eigenvalues and
531:Karhunen–Loève expansion
359:, the covariance matrix
3336:{\displaystyle \mu (t)}
3177:with the mean function
2361:{\displaystyle \alpha }
2190:{\displaystyle \alpha }
958:{\displaystyle \alpha }
3695:
3672:
3633:
3625:
3606:
3425:
3396:
3372:
3371:{\displaystyle G(s,t)}
3337:
3304:
3227:
3171:
3120:
3034:
3013:Modes of variation in
3003:
2861:
2839:
2819:
2643:
2621:
2592:
2576:and covariance matrix
2570:
2548:
2526:
2506:
2486:
2420:Modes of variation in
2402:
2362:
2339:
2191:
2171:
2142:
2119:
2047:
1958:
1935:
1829:
1792:
1730:Karhunen–Loève theorem
1719:
1567:
1515:
1460:
1404:
1305:
1249:
1225:
1199:
1134:Modes of variation in
1124:
1090:
1067:
959:
939:
917:
894:
816:
733:, with the properties
727:
698:
678:
609:
579:
523:
501:
479:
453:
428:
375:
349:
278:
210:
175:
146:
65:Modes of variation in
3986:Matrix decompositions
3696:
3694:{\displaystyle \pm 3}
3673:
3631:
3623:
3607:
3426:
3397:
3373:
3338:
3305:
3228:
3172:
3121:
3035:
3004:
2862:
2840:
2820:
2644:
2622:
2593:
2571:
2549:
2527:
2507:
2487:
2403:
2363:
2340:
2192:
2172:
2143:
2120:
2048:
1959:
1936:
1830:
1772:
1720:
1568:
1516:
1461:
1384:
1306:
1250:
1226:
1200:
1125:
1091:
1068:
960:
940:
918:
895:
817:
728:
699:
679:
610:
559:
524:
502:
480:
454:
429:
381:can be decomposed as
376:
350:
279:
211:
176:
147:
3682:
3641:
3438:
3431:can be estimated by
3424:{\displaystyle X(t)}
3406:
3386:
3347:
3318:
3237:
3181:
3136:
3044:
3024:
2874:
2867:can be estimated by
2849:
2829:
2653:
2631:
2602:
2580:
2558:
2536:
2516:
2496:
2431:
2372:
2352:
2204:
2181:
2170:{\displaystyle X(t)}
2152:
2132:
2057:
1971:
1948:
1845:
1739:
1587:
1525:
1473:
1318:
1259:
1235:
1209:
1151:
1100:
1080:
972:
949:
927:
907:
826:
740:
708:
688:
622:
540:
511:
489:
467:
441:
388:
363:
294:
223:
185:
156:
152:has the mean vector
76:
3981:Functional analysis
3976:Dimension reduction
1224:{\displaystyle p=1}
654:
72:If a random vector
3897:10.1007/bf00290638
3691:
3668:
3634:
3626:
3602:
3421:
3392:
3368:
3333:
3300:
3223:
3167:
3116:
3030:
2999:
2857:
2835:
2815:
2639:
2617:
2588:
2566:
2544:
2522:
2502:
2482:
2398:
2358:
2335:
2187:
2167:
2138:
2115:
2043:
1954:
1931:
1825:
1715:
1563:
1511:
1456:
1301:
1245:
1221:
1205:, where typically
1195:
1120:
1086:
1063:
955:
935:
913:
890:
812:
723:
694:
674:
638:
605:
519:
497:
475:
449:
424:
371:
345:
284:and corresponding
274:
206:
171:
142:
18:modes of variation
3864:www.mortality.org
3538:
3526:
3517:
3487:
3451:
3395:{\displaystyle k}
3128:square-integrable
3033:{\displaystyle n}
2960:
2946:
2937:
2917:
2889:
2838:{\displaystyle k}
2803:
2779:
2745:
2721:
2693:
2669:
2615:
2554:with mean vector
2525:{\displaystyle p}
2505:{\displaystyle n}
2427:Suppose the data
2394:
2390:
2386:
2310:
2291:
2265:
2141:{\displaystyle k}
2101:
1957:{\displaystyle k}
1143:square-integrable
1116:
1112:
1108:
1089:{\displaystyle A}
1019:
916:{\displaystyle k}
877:
873:
869:
697:{\displaystyle k}
461:orthogonal matrix
3993:
3960:
3959:
3931:
3925:
3924:
3880:
3874:
3873:
3871:
3870:
3856:
3845:
3844:
3816:
3810:
3809:
3799:
3773:
3764:
3763:
3727:
3701:, respectively.
3700:
3698:
3697:
3692:
3677:
3675:
3674:
3669:
3611:
3609:
3608:
3603:
3571:
3570:
3546:
3545:
3540:
3539:
3531:
3527:
3525:
3524:
3519:
3518:
3510:
3506:
3489:
3488:
3480:
3465:
3464:
3453:
3452:
3444:
3430:
3428:
3427:
3422:
3401:
3399:
3398:
3393:
3377:
3375:
3374:
3369:
3342:
3340:
3339:
3334:
3309:
3307:
3306:
3301:
3232:
3230:
3229:
3224:
3176:
3174:
3173:
3168:
3166:
3165:
3125:
3123:
3122:
3117:
3106:
3105:
3078:
3077:
3056:
3055:
3039:
3037:
3036:
3031:
3008:
3006:
3005:
3000:
2968:
2967:
2962:
2961:
2956:
2951:
2947:
2945:
2944:
2939:
2938:
2930:
2926:
2918:
2913:
2908:
2903:
2902:
2891:
2890:
2885:
2880:
2866:
2864:
2863:
2858:
2856:
2844:
2842:
2841:
2836:
2824:
2822:
2821:
2816:
2811:
2810:
2805:
2804:
2799:
2794:
2787:
2786:
2781:
2780:
2772:
2753:
2752:
2747:
2746:
2741:
2736:
2729:
2728:
2723:
2722:
2714:
2701:
2700:
2695:
2694:
2689:
2684:
2677:
2676:
2671:
2670:
2662:
2648:
2646:
2645:
2640:
2638:
2626:
2624:
2623:
2618:
2616:
2611:
2606:
2597:
2595:
2594:
2589:
2587:
2575:
2573:
2572:
2567:
2565:
2553:
2551:
2550:
2545:
2543:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2491:
2489:
2488:
2483:
2481:
2480:
2475:
2460:
2459:
2454:
2445:
2444:
2439:
2407:
2405:
2404:
2399:
2392:
2391:
2388:
2384:
2367:
2365:
2364:
2359:
2344:
2342:
2341:
2336:
2308:
2304:
2303:
2289:
2276:
2275:
2266:
2264:
2263:
2254:
2222:
2221:
2196:
2194:
2193:
2188:
2176:
2174:
2173:
2168:
2147:
2145:
2144:
2139:
2124:
2122:
2121:
2116:
2102:
2099:
2088:
2087:
2078:
2077:
2052:
2050:
2049:
2044:
2039:
2038:
2023:
2022:
1992:
1991:
1963:
1961:
1960:
1955:
1940:
1938:
1937:
1932:
1915:
1914:
1872:
1871:
1870:
1857:
1856:
1834:
1832:
1831:
1826:
1812:
1811:
1802:
1801:
1791:
1786:
1724:
1722:
1721:
1716:
1675:
1674:
1673:
1641:
1640:
1631:
1630:
1615:
1614:
1605:
1604:
1572:
1570:
1569:
1564:
1553:
1552:
1540:
1539:
1520:
1518:
1517:
1512:
1498:
1497:
1485:
1484:
1465:
1463:
1462:
1457:
1443:
1442:
1424:
1423:
1414:
1413:
1403:
1398:
1310:
1308:
1307:
1302:
1254:
1252:
1251:
1246:
1244:
1243:
1230:
1228:
1227:
1222:
1204:
1202:
1201:
1196:
1194:
1193:
1181:
1180:
1129:
1127:
1126:
1121:
1114:
1113:
1110:
1106:
1095:
1093:
1092:
1087:
1072:
1070:
1069:
1064:
1032:
1031:
1026:
1020:
1018:
1017:
1008:
1000:
992:
991:
980:
964:
962:
961:
956:
944:
942:
941:
936:
934:
922:
920:
919:
914:
899:
897:
896:
891:
875:
874:
871:
867:
857:
856:
847:
846:
821:
819:
818:
813:
808:
807:
792:
791:
761:
760:
732:
730:
729:
724:
722:
721:
716:
704:-th eigenvector
703:
701:
700:
695:
683:
681:
680:
675:
670:
662:
653:
648:
643:
634:
633:
614:
612:
611:
606:
601:
600:
595:
589:
588:
578:
573:
555:
547:
528:
526:
525:
520:
518:
506:
504:
503:
498:
496:
484:
482:
481:
476:
474:
458:
456:
455:
450:
448:
433:
431:
430:
425:
420:
419:
414:
408:
403:
395:
380:
378:
377:
372:
370:
354:
352:
351:
346:
344:
343:
338:
323:
322:
317:
308:
307:
302:
283:
281:
280:
275:
267:
266:
248:
247:
235:
234:
215:
213:
212:
207:
205:
204:
193:
180:
178:
177:
172:
170:
169:
164:
151:
149:
148:
143:
141:
140:
131:
130:
112:
111:
99:
98:
83:
4001:
4000:
3996:
3995:
3994:
3992:
3991:
3990:
3966:
3965:
3964:
3963:
3932:
3928:
3881:
3877:
3868:
3866:
3858:
3857:
3848:
3817:
3813:
3774:
3767:
3744:10.2307/1268982
3728:
3724:
3719:
3706:hazard function
3683:
3680:
3679:
3642:
3639:
3638:
3618:
3566:
3565:
3541:
3530:
3529:
3528:
3520:
3509:
3508:
3507:
3505:
3479:
3478:
3454:
3443:
3442:
3441:
3439:
3436:
3435:
3407:
3404:
3403:
3387:
3384:
3383:
3348:
3345:
3344:
3319:
3316:
3315:
3238:
3235:
3234:
3182:
3179:
3178:
3161:
3160:
3137:
3134:
3133:
3131:random function
3101:
3097:
3073:
3069:
3051:
3047:
3045:
3042:
3041:
3025:
3022:
3021:
3018:
2963:
2952:
2950:
2949:
2948:
2940:
2929:
2928:
2927:
2925:
2909:
2907:
2892:
2881:
2879:
2878:
2877:
2875:
2872:
2871:
2852:
2850:
2847:
2846:
2830:
2827:
2826:
2806:
2795:
2793:
2792:
2791:
2782:
2771:
2770:
2769:
2748:
2737:
2735:
2734:
2733:
2724:
2713:
2712:
2711:
2696:
2685:
2683:
2682:
2681:
2672:
2661:
2660:
2659:
2654:
2651:
2650:
2634:
2632:
2629:
2628:
2607:
2605:
2603:
2600:
2599:
2583:
2581:
2578:
2577:
2561:
2559:
2556:
2555:
2539:
2537:
2534:
2533:
2517:
2514:
2513:
2497:
2494:
2493:
2476:
2471:
2470:
2455:
2450:
2449:
2440:
2435:
2434:
2432:
2429:
2428:
2425:
2414:
2387:
2373:
2370:
2369:
2353:
2350:
2349:
2299:
2298:
2271:
2267:
2259:
2255:
2253:
2211:
2207:
2205:
2202:
2201:
2182:
2179:
2178:
2153:
2150:
2149:
2133:
2130:
2129:
2100: for
2098:
2083:
2079:
2073:
2069:
2058:
2055:
2054:
2034:
2030:
2018:
2014:
1987:
1983:
1972:
1969:
1968:
1949:
1946:
1945:
1910:
1906:
1866:
1865:
1861:
1852:
1848:
1846:
1843:
1842:
1807:
1803:
1797:
1793:
1787:
1776:
1740:
1737:
1736:
1669:
1668:
1664:
1636:
1635:
1626:
1622:
1610:
1609:
1600:
1596:
1588:
1585:
1584:
1548:
1544:
1535:
1531:
1526:
1523:
1522:
1493:
1489:
1480:
1476:
1474:
1471:
1470:
1438:
1434:
1419:
1415:
1409:
1405:
1399:
1388:
1319:
1316:
1315:
1260:
1257:
1256:
1239:
1238:
1236:
1233:
1232:
1210:
1207:
1206:
1189:
1185:
1176:
1175:
1152:
1149:
1148:
1146:random function
1139:
1109:
1101:
1098:
1097:
1081:
1078:
1077:
1027:
1022:
1021:
1013:
1009:
1007:
996:
981:
976:
975:
973:
970:
969:
950:
947:
946:
930:
928:
925:
924:
908:
905:
904:
870:
852:
848:
842:
838:
827:
824:
823:
803:
799:
787:
783:
756:
752:
741:
738:
737:
717:
712:
711:
709:
706:
705:
689:
686:
685:
666:
658:
649:
644:
639:
629:
625:
623:
620:
619:
596:
591:
590:
584:
580:
574:
563:
551:
543:
541:
538:
537:
514:
512:
509:
508:
492:
490:
487:
486:
470:
468:
465:
464:
444:
442:
439:
438:
415:
410:
409:
404:
399:
391:
389:
386:
385:
366:
364:
361:
360:
339:
334:
333:
318:
313:
312:
303:
298:
297:
295:
292:
291:
262:
258:
243:
239:
230:
226:
224:
221:
220:
194:
189:
188:
186:
183:
182:
165:
160:
159:
157:
154:
153:
136:
132:
126:
122:
107:
103:
94:
90:
79:
77:
74:
73:
70:
51:
16:In statistics,
12:
11:
5:
3999:
3989:
3988:
3983:
3978:
3962:
3961:
3942:(2): 140–145.
3926:
3891:(4): 429–450.
3875:
3846:
3827:(5): 391–406.
3811:
3790:(1): 257–295.
3765:
3721:
3720:
3718:
3715:
3690:
3687:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3617:
3614:
3613:
3612:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3569:
3564:
3561:
3558:
3555:
3552:
3549:
3544:
3537:
3534:
3523:
3516:
3513:
3504:
3501:
3498:
3495:
3492:
3486:
3483:
3477:
3474:
3471:
3468:
3463:
3460:
3457:
3450:
3447:
3420:
3417:
3414:
3411:
3391:
3367:
3364:
3361:
3358:
3355:
3352:
3332:
3329:
3326:
3323:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3164:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3115:
3112:
3109:
3104:
3100:
3096:
3093:
3090:
3087:
3084:
3081:
3076:
3072:
3068:
3065:
3062:
3059:
3054:
3050:
3029:
3017:
3011:
3010:
3009:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2966:
2959:
2955:
2943:
2936:
2933:
2924:
2921:
2916:
2912:
2906:
2901:
2898:
2895:
2888:
2884:
2855:
2834:
2814:
2809:
2802:
2798:
2790:
2785:
2778:
2775:
2768:
2765:
2762:
2759:
2756:
2751:
2744:
2740:
2732:
2727:
2720:
2717:
2710:
2707:
2704:
2699:
2692:
2688:
2680:
2675:
2668:
2665:
2658:
2637:
2614:
2610:
2586:
2564:
2542:
2521:
2501:
2479:
2474:
2469:
2466:
2463:
2458:
2453:
2448:
2443:
2438:
2424:
2418:
2413:
2410:
2397:
2383:
2380:
2377:
2368:, usually for
2357:
2346:
2345:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2307:
2302:
2297:
2294:
2288:
2285:
2282:
2279:
2274:
2270:
2262:
2258:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2220:
2217:
2214:
2210:
2186:
2166:
2163:
2160:
2157:
2137:
2126:
2125:
2114:
2111:
2108:
2105:
2097:
2094:
2091:
2086:
2082:
2076:
2072:
2068:
2065:
2062:
2042:
2037:
2033:
2029:
2026:
2021:
2017:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1990:
1986:
1982:
1979:
1976:
1953:
1942:
1941:
1930:
1927:
1924:
1921:
1918:
1913:
1909:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1869:
1864:
1860:
1855:
1851:
1836:
1835:
1824:
1821:
1818:
1815:
1810:
1806:
1800:
1796:
1790:
1785:
1782:
1779:
1775:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1726:
1725:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1672:
1667:
1663:
1660:
1657:
1654:
1651:
1647:
1644:
1639:
1634:
1629:
1625:
1621:
1618:
1613:
1608:
1603:
1599:
1595:
1592:
1562:
1559:
1556:
1551:
1547:
1543:
1538:
1534:
1530:
1510:
1507:
1504:
1501:
1496:
1492:
1488:
1483:
1479:
1467:
1466:
1455:
1452:
1449:
1446:
1441:
1437:
1433:
1430:
1427:
1422:
1418:
1412:
1408:
1402:
1397:
1394:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1242:
1220:
1217:
1214:
1192:
1188:
1184:
1179:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1138:
1132:
1119:
1105:
1085:
1074:
1073:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1030:
1025:
1016:
1012:
1006:
1003:
999:
995:
990:
987:
984:
979:
954:
933:
912:
901:
900:
889:
886:
883:
880:
866:
863:
860:
855:
851:
845:
841:
837:
834:
831:
811:
806:
802:
798:
795:
790:
786:
782:
779:
776:
773:
770:
767:
764:
759:
755:
751:
748:
745:
720:
715:
693:
673:
669:
665:
661:
657:
652:
647:
642:
637:
632:
628:
616:
615:
604:
599:
594:
587:
583:
577:
572:
569:
566:
562:
558:
554:
550:
546:
517:
495:
473:
447:
435:
434:
423:
418:
413:
407:
402:
398:
394:
369:
342:
337:
332:
329:
326:
321:
316:
311:
306:
301:
273:
270:
265:
261:
257:
254:
251:
246:
242:
238:
233:
229:
203:
200:
197:
192:
168:
163:
139:
135:
129:
125:
121:
118:
115:
110:
106:
102:
97:
93:
89:
86:
82:
69:
63:
50:
47:
31:eigenfunctions
9:
6:
4:
3:
2:
3998:
3987:
3984:
3982:
3979:
3977:
3974:
3973:
3971:
3957:
3953:
3949:
3945:
3941:
3937:
3930:
3922:
3918:
3914:
3910:
3906:
3902:
3898:
3894:
3890:
3886:
3879:
3865:
3861:
3855:
3853:
3851:
3842:
3838:
3834:
3830:
3826:
3822:
3815:
3807:
3803:
3798:
3793:
3789:
3785:
3784:
3779:
3772:
3770:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3732:Technometrics
3726:
3722:
3714:
3710:
3707:
3702:
3688:
3685:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3630:
3622:
3599:
3593:
3590:
3587:
3584:
3578:
3575:
3572:
3562:
3559:
3556:
3550:
3542:
3532:
3521:
3511:
3502:
3499:
3493:
3481:
3475:
3469:
3461:
3458:
3455:
3445:
3434:
3433:
3432:
3415:
3409:
3389:
3381:
3380:interpolation
3362:
3359:
3356:
3350:
3327:
3321:
3313:
3291:
3285:
3282:
3276:
3270:
3264:
3261:
3258:
3252:
3249:
3246:
3240:
3214:
3208:
3202:
3196:
3190:
3184:
3157:
3154:
3151:
3145:
3139:
3132:
3129:
3110:
3102:
3098:
3094:
3091:
3088:
3082:
3074:
3070:
3066:
3060:
3052:
3048:
3040:realizations
3027:
3016:
2996:
2990:
2987:
2984:
2981:
2975:
2972:
2969:
2964:
2941:
2931:
2922:
2919:
2904:
2899:
2896:
2893:
2870:
2869:
2868:
2832:
2807:
2788:
2783:
2773:
2763:
2760:
2757:
2749:
2730:
2725:
2715:
2705:
2697:
2678:
2673:
2663:
2519:
2499:
2477:
2467:
2464:
2461:
2456:
2446:
2441:
2423:
2417:
2409:
2395:
2381:
2378:
2375:
2355:
2329:
2326:
2323:
2320:
2314:
2311:
2305:
2295:
2292:
2286:
2280:
2272:
2268:
2260:
2256:
2250:
2247:
2241:
2235:
2232:
2226:
2218:
2215:
2212:
2208:
2200:
2199:
2198:
2184:
2161:
2155:
2135:
2112:
2109:
2106:
2103:
2095:
2092:
2084:
2080:
2074:
2070:
2063:
2040:
2035:
2031:
2027:
2019:
2015:
2008:
2005:
2002:
1999:
1996:
1988:
1984:
1977:
1967:
1966:
1965:
1951:
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289:eigenvectors
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3970:Categories
3869:2020-03-12
3738:(4): 329.
3717:References
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2412:Estimation
3956:0003-1305
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