4047:
544:
2572:
28:
323:
2341:
3990:
3775:
539:{\displaystyle d'={\sqrt {({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})'\mathbf {\Sigma } ^{-1}({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})}}=\lVert \mathbf {S} ^{-1}({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})\rVert =\lVert {\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b}\rVert /\sigma _{\boldsymbol {\mu }}}
2567:{\displaystyle {\boldsymbol {w}}={\begin{bmatrix}\sigma _{s}^{2}&-\sigma _{n}^{2}\end{bmatrix}},\;{\boldsymbol {k}}={\begin{bmatrix}1&1\end{bmatrix}},\;{\boldsymbol {\lambda }}={\frac {\mu _{s}-\mu _{n}}{\sigma _{s}^{2}-\sigma _{n}^{2}}}{\begin{bmatrix}\sigma _{s}^{2}&\sigma _{n}^{2}\end{bmatrix}}}
2191:
of a yes/no task between two univariate normal distributions with a single shifting criterion. It can also be computed from the ROC curve of any two distributions (in any number of variables) with a shifting likelihood-ratio, by locating the point on the ROC curve that is farthest from the diagonal.
3783:
We may sometimes want to scale the discriminability of two data distributions by moving them closer or farther apart. One such case is when we are modeling a detection or classification task, and the model performance exceeds that of the subject or observed data. In that case, we can move the model
88:
between two univariate histograms computed from their overlap area. Figure 2: Same computed from the overlap volume of two bivariate histograms. Figure 3: discriminability indices of two univariate normal distributions with unequal variances. The classification boundary is in black. Figure 4:
3778:
Scaling the discriminability of two distributions, by linearly interpolating the mean vector and sd matrix (square root of the covariance matrix) of one towards the other. Ellipses are the error ellipses of the two distributions. Black curve is a quadratic boundary that separates the two
3787:
There are several ways of doing this. One is to compute the mean vector and covariance matrix of the two distributions, then effect a linear transformation to interpolate the mean and sd matrix (square root of the covariance matrix) of one of the distributions towards the other.
852:
1817:
2296:
1330:
89:
discriminability indices of two bivariate normal distributions with unequal covariance matrices (ellipses are 1 sd error ellipses). Color-bar shows the relative contribution to the discriminability by each dimension. These are computed by numerical methods.
3791:
Another way that is by computing the decision variables of the data points (log likelihood ratio that a point belongs to one distribution vs another) under a multinormal model, then moving these decision variables closer together or farther apart.
612:
2154:
3058:
2981:
2718:
3583:
In general, the contribution to the total discriminability by each dimension or feature may be measured using the amount by which the discriminability drops when that dimension is removed. If the total Bayes discriminability is
263:
1949:
1056:
966:
2044:
1131:
2892:
3128:
3744:
2198:
2629:
1628:
1234:
3174:
3766:
when the covariance matrices are equal and diagonal, but in the other cases, this measure more accurately reflects the contribution of a dimension than its individual discriminability.
1356:
is the inverse cumulative distribution function of the standard normal. The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed (
1141:
When the two distributions have different standard deviations (or in general dimensions, different covariance matrices), there exist several contending indices, all of which reduce to
3784:
variable distributions closer together so that it matches the observed performance, while also predicting which specific data points should start overlapping and be misclassified.
695:
2332:
634:
1462:
289:
1580:
1507:
1853:
1620:
2639:
A common approximate (i.e. sub-optimal) discriminability index that has a closed-form is to take the average of the variances, i.e. the rms of the two standard deviations:
1420:
121:
The discriminability index is the separation between the means of two distributions (typically the signal and the noise distributions), in units of the standard deviation.
3660:
3543:
311:
3573:
3507:
3477:
3447:
3417:
3387:
3357:
3327:
3297:
3267:
3237:
3207:
2769:
2185:
1537:
1390:
86:
1174:
This is the maximum (Bayes-optimal) discriminability index for two distributions, based on the amount of their overlap, i.e. the optimal (Bayes) error of classification
553:
4031:
3607:
1164:
1083:
684:
659:
192:
2745:
1226:
1199:
876:
56:
3764:
3680:
3627:
2986:
2789:
1354:
167:
147:
2049:
2795:
curve (AUC) of a single-criterion observer. This index is extended to general dimensions as the
Mahalanobis distance using the pooled covariance, i.e. with
2905:
2642:
1862:
4088:
200:
971:
881:
4024:
2335:
1954:
1856:
1357:
1088:
3957:
3901:
3801:
2798:
2792:
2188:
4122:
2291:{\displaystyle a_{b}=p\left({\tilde {\chi }}_{{\boldsymbol {w}},{\boldsymbol {k}},{\boldsymbol {\lambda }},0,0}^{2}>0\right)}
4017:
4081:
1812:{\displaystyle p(A|a)=p({\chi '}_{1,v_{a}\lambda }^{2}>v_{b}c),\;\;p(B|b)=p({\chi '}_{1,v_{b}\lambda }^{2}<v_{a}c)}
1325:{\displaystyle d'_{b}=-2Z\left({\text{Bayes error rate }}e_{b}\right)=2Z\left({\text{best accuracy rate }}a_{b}\right)}
4117:
3947:
3774:
1393:
3685:
4112:
3071:
1392:
is a positive-definite statistical distance measure that is free of assumptions about the distributions, like the
4127:
4107:
3869:
Das, Abhranil; Wilson S Geisler (2020). "Methods to integrate multinormals and compute classification measures".
2577:
4074:
3329:
by a maximum of approximately 30%. At the limit of high discriminability for univariate normal distributions,
847:{\displaystyle {d'}^{2}={\frac {1}{1-\rho ^{2}}}\left({d'}_{x}^{2}+{d'}_{y}^{2}-2\rho {d'}_{x}{d'}_{y}\right)}
110:
2301:
617:
1425:
272:
269:
In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix
3973:
3133:
1545:
4046:
314:
1585:
3389:. These results often hold true in higher dimensions, but not always. Simpson and Fitter promoted
1398:
607:{\displaystyle \sigma _{\boldsymbol {\mu }}=1/\lVert \mathbf {S} ^{-1}{\boldsymbol {\mu }}\rVert }
294:
3870:
3807:
2750:
1467:
1826:
3997:
3953:
3897:
4058:
4001:
3632:
3515:
3419:
as the best index, particularly for two-interval tasks, but Das and
Geisler have shown that
3929:
3548:
3482:
3452:
3422:
3392:
3362:
3332:
3302:
3272:
3242:
3212:
3182:
2160:
1512:
1365:
61:
2723:
1204:
1177:
861:
34:
3053:{\displaystyle \mathbf {S} _{\text{avg}}=\left(\mathbf {S} _{a}+\mathbf {S} _{b}\right)/2}
3587:
1542:
In particular, for a yes/no task between two univariate normal distributions with means
1360:), and may also be used as an approximation when the distributions are close to normal.
1144:
1063:
664:
639:
172:
3749:
3665:
3612:
2774:
2149:{\displaystyle d'_{b}=2Z\left({\frac {p\left(A|a\right)+p\left(B|b\right)}{2}}\right).}
1339:
152:
132:
27:
4101:
2976:{\displaystyle d'_{e}=\left\vert \mu _{a}-\mu _{b}\right\vert /\sigma _{\text{avg}}}
2713:{\displaystyle d'_{a}=\left\vert \mu _{a}-\mu _{b}\right\vert /\sigma _{\text{rms}}}
3920:
Simpson, A. J.; Fitter, M. J. (1973). "What is the best index of detectability?".
1944:{\displaystyle \lambda =\left({\frac {\mu _{a}-\mu _{b}}{v_{a}-v_{b}}}\right)^{2}}
689:
For two bivariate distributions with equal variance-covariance, this is given by:
3891:
3812:
4054:
258:{\displaystyle d'={\frac {\left\vert \mu _{a}-\mu _{b}\right\vert }{\sigma }}}
2195:
For a two-interval task between these distributions, the optimal accuracy is
1058:, i.e. including the signs of the mean differences instead of the absolute.
106:
1051:{\displaystyle d'_{y}={\frac {{\mu _{b}}_{y}-{\mu _{a}}_{y}}{\sigma _{y}}}}
961:{\displaystyle d'_{x}={\frac {{\mu _{b}}_{x}-{\mu _{a}}_{x}}{\sigma _{x}}}}
113:. A higher index indicates that the signal can be more readily detected.
3989:
1539:
does not satisfy the triangle inequality, so it is not a full metric.
17:
3933:
2039:{\displaystyle c=\lambda +{\frac {\ln v_{a}-\ln v_{b}}{v_{a}-v_{b}}}}
3875:
3746:. This is the same as the individual discriminability of dimension
3773:
3575:
at small discriminability, but greater at large discriminability.
291:, (whose symmetric square-root, the standard deviation matrix, is
1126:{\displaystyle Z({\text{hit rate}})-Z({\text{false alarm rate}})}
3068:
It has been shown that for two univariate normal distributions,
2887:{\displaystyle \mathbf {S} _{\text{rms}}=\left^{\frac {1}{2}}}
1201:
by an ideal observer, or its complement, the optimal accuracy
3545:, which uses the geometric mean of the sd's, is less than
31:
Figure 1: Bayes-optimal classification error probability
4062:
4005:
2521:
2421:
2358:
3752:
3688:
3668:
3635:
3615:
3590:
3551:
3518:
3485:
3455:
3425:
3395:
3365:
3335:
3305:
3275:
3245:
3215:
3185:
3136:
3074:
2989:
2908:
2801:
2777:
2753:
2726:
2645:
2580:
2344:
2304:
2201:
2163:
2052:
1957:
1865:
1829:
1631:
1588:
1548:
1515:
1470:
1428:
1401:
1368:
1342:
1237:
1207:
1180:
1147:
1091:
1066:
974:
884:
864:
698:
667:
642:
620:
556:
326:
297:
275:
203:
175:
155:
135:
64:
37:
1622:, the Bayes-optimal classification accuracies are:
169:with the same standard deviation, it is denoted by
3758:
3738:
3674:
3654:
3621:
3601:
3579:Contribution to discriminability by each dimension
3567:
3537:
3501:
3471:
3449:is the optimal discriminability in all cases, and
3441:
3411:
3381:
3351:
3321:
3291:
3261:
3231:
3201:
3168:
3122:
3052:
2975:
2886:
2783:
2763:
2739:
2712:
2623:
2566:
2326:
2290:
2179:
2148:
2038:
1943:
1847:
1811:
1614:
1574:
1531:
1501:
1456:
1414:
1384:
1348:
1324:
1220:
1193:
1158:
1125:
1077:
1050:
960:
870:
846:
678:
653:
628:
606:
538:
305:
283:
257:
186:
161:
141:
80:
50:
3770:Scaling the discriminability of two distributions
3479:is often a better closed-form approximation than
1509:is symmetric for the two distributions. However,
614:is the 1d slice of the sd along the unit vector
3739:{\displaystyle {\sqrt {d'^{2}-{d'_{-i}}^{2}}}}
3662:, we can define the contribution of dimension
3609:and the Bayes discriminability with dimension
4082:
4025:
3130:, and for multivariate normal distributions,
8:
3974:Interactive signal detection theory tutorial
3123:{\displaystyle d'_{a}\leq d'_{e}\leq d'_{b}}
601:
578:
518:
488:
482:
431:
3239:underestimate the maximum discriminability
2624:{\displaystyle d'_{b}=2Z\left(a_{b}\right)}
4089:
4075:
4032:
4018:
2443:
2407:
1722:
1721:
3874:
3751:
3728:
3715:
3710:
3699:
3689:
3687:
3667:
3640:
3634:
3614:
3589:
3556:
3550:
3523:
3517:
3490:
3484:
3460:
3454:
3430:
3424:
3400:
3394:
3370:
3364:
3340:
3334:
3310:
3304:
3280:
3274:
3250:
3244:
3220:
3214:
3190:
3184:
3157:
3141:
3135:
3111:
3095:
3079:
3073:
3042:
3031:
3026:
3016:
3011:
2996:
2991:
2988:
2967:
2958:
2947:
2934:
2913:
2907:
2873:
2860:
2849:
2844:
2834:
2829:
2808:
2803:
2800:
2776:
2754:
2752:
2731:
2725:
2704:
2695:
2684:
2671:
2650:
2644:
2611:
2585:
2579:
2550:
2545:
2533:
2528:
2516:
2507:
2502:
2489:
2484:
2472:
2459:
2452:
2444:
2416:
2408:
2390:
2385:
2370:
2365:
2353:
2345:
2343:
2318:
2307:
2306:
2303:
2271:
2253:
2245:
2237:
2236:
2225:
2224:
2206:
2200:
2168:
2162:
2120:
2093:
2079:
2057:
2051:
2027:
2014:
2002:
1983:
1970:
1956:
1935:
1922:
1909:
1897:
1884:
1877:
1864:
1838:
1828:
1797:
1784:
1774:
1763:
1753:
1732:
1706:
1693:
1683:
1672:
1662:
1641:
1630:
1606:
1593:
1587:
1566:
1553:
1547:
1520:
1514:
1475:
1469:
1433:
1427:
1406:
1400:
1373:
1367:
1341:
1311:
1302:
1277:
1268:
1242:
1236:
1212:
1206:
1185:
1179:
1146:
1115:
1098:
1090:
1065:
1040:
1029:
1022:
1017:
1007:
1000:
995:
991:
979:
973:
950:
939:
932:
927:
917:
910:
905:
901:
889:
883:
878:is the correlation coefficient, and here
863:
833:
823:
816:
806:
790:
785:
775:
765:
760:
750:
735:
719:
710:
700:
697:
666:
641:
621:
619:
596:
587:
582:
573:
561:
555:
530:
521:
512:
507:
497:
492:
473:
468:
458:
453:
440:
435:
417:
412:
402:
397:
384:
379:
364:
359:
349:
344:
338:
325:
298:
296:
276:
274:
239:
226:
215:
202:
174:
154:
134:
69:
63:
42:
36:
26:
3915:
3913:
3824:
3696:
2983:, extended to general dimensions using
2445:
2409:
2346:
2254:
2246:
2238:
1835:
622:
597:
562:
531:
508:
493:
469:
454:
413:
398:
360:
345:
3864:
3862:
3860:
3858:
3856:
3854:
3852:
3850:
3848:
686:along the 1d slice through the means.
3846:
3844:
3842:
3840:
3838:
3836:
3834:
3832:
3830:
3828:
7:
4043:
4041:
3986:
3984:
3890:MacMillan, N.; Creelman, C. (2005).
3269:of univariate normal distributions.
2336:generalized chi-squared distribution
2327:{\displaystyle {\tilde {\chi }}^{2}}
1857:non-central chi-squared distribution
629:{\displaystyle {\boldsymbol {\mu }}}
3952:. OUP USA. ch. 2, p. 20.
4061:. You can help Knowledge (XXG) by
4004:. You can help Knowledge (XXG) by
3949:Elementary Signal Detection Theory
1457:{\displaystyle D_{\text{KL}}(a,b)}
284:{\displaystyle \mathbf {\Sigma } }
25:
3802:Receiver operating characteristic
3169:{\displaystyle d'_{a}\leq d'_{e}}
2898:Average sd discriminability index
2793:receiver operating characteristic
1575:{\displaystyle \mu _{a},\mu _{b}}
129:For two univariate distributions
58:and Bayes discriminability index
4045:
3988:
3893:Detection Theory: A User's Guide
3027:
3012:
2992:
2845:
2830:
2804:
1166:for equal variance/covariance.
583:
436:
380:
299:
277:
3896:. Lawrence Erlbaum Associates.
3509:, even for two-interval tasks.
317:between the two distributions:
2312:
2230:
2187:can also be computed from the
2121:
2094:
1806:
1749:
1740:
1733:
1726:
1715:
1658:
1649:
1642:
1635:
1615:{\displaystyle v_{a}>v_{b}}
1496:
1484:
1451:
1439:
1120:
1112:
1103:
1095:
479:
449:
423:
393:
371:
340:
1:
2791:-score of the area under the
2635:RMS sd discriminability index
2574:. The Bayes discriminability
2046:. The Bayes discriminability
1415:{\displaystyle D_{\text{KL}}}
1137:Unequal variances/covariances
1170:Bayes discriminability index
636:through the means, i.e. the
306:{\displaystyle \mathbf {S} }
3946:Wickens, Thomas D. (2001).
2764:{\displaystyle {\sqrt {2}}}
1502:{\displaystyle d'_{b}(a,b)}
1394:Kullback-Leibler divergence
313:), this generalizes to the
125:Equal variances/covariances
4144:
4040:
3983:
1848:{\displaystyle \chi '^{2}}
3976:including calculation of
3064:Comparison of the indices
3060:as the common sd matrix.
2894:as the common sd matrix.
1304:best accuracy rate
4123:Signal processing stubs
3655:{\displaystyle d'_{-i}}
3538:{\displaystyle d'_{gm}}
1464:is asymmetric, whereas
111:signal detection theory
4057:-related article is a
4000:-related article is a
3922:Psychological Bulletin
3780:
3760:
3740:
3676:
3656:
3623:
3603:
3569:
3568:{\displaystyle d'_{b}}
3539:
3512:The approximate index
3503:
3502:{\displaystyle d'_{a}}
3473:
3472:{\displaystyle d'_{e}}
3443:
3442:{\displaystyle d'_{b}}
3413:
3412:{\displaystyle d'_{a}}
3383:
3382:{\displaystyle d'_{b}}
3353:
3352:{\displaystyle d'_{e}}
3323:
3322:{\displaystyle d'_{b}}
3293:
3292:{\displaystyle d'_{a}}
3263:
3262:{\displaystyle d'_{b}}
3233:
3232:{\displaystyle d'_{e}}
3203:
3202:{\displaystyle d'_{a}}
3170:
3124:
3054:
2977:
2888:
2785:
2765:
2741:
2714:
2625:
2568:
2328:
2292:
2181:
2180:{\displaystyle d'_{b}}
2150:
2040:
1945:
1849:
1813:
1616:
1576:
1533:
1532:{\displaystyle d'_{b}}
1503:
1458:
1416:
1386:
1385:{\displaystyle d'_{b}}
1350:
1326:
1270:Bayes error rate
1222:
1195:
1160:
1127:
1079:
1052:
962:
872:
848:
680:
655:
630:
608:
540:
307:
285:
259:
188:
163:
143:
99:discriminability index
90:
82:
81:{\displaystyle d'_{b}}
52:
3777:
3761:
3741:
3677:
3657:
3624:
3604:
3570:
3540:
3504:
3474:
3444:
3414:
3384:
3354:
3324:
3294:
3264:
3234:
3204:
3171:
3125:
3055:
2978:
2889:
2786:
2766:
2742:
2740:{\displaystyle d_{a}}
2715:
2626:
2569:
2329:
2293:
2182:
2151:
2041:
1946:
1850:
1814:
1617:
1577:
1534:
1504:
1459:
1417:
1387:
1351:
1327:
1223:
1221:{\displaystyle a_{b}}
1196:
1194:{\displaystyle e_{b}}
1161:
1128:
1085:is also estimated as
1080:
1053:
963:
873:
871:{\displaystyle \rho }
849:
681:
656:
631:
609:
541:
308:
286:
260:
189:
164:
144:
83:
53:
51:{\displaystyle e_{b}}
30:
3750:
3686:
3666:
3633:
3613:
3588:
3549:
3516:
3483:
3453:
3423:
3393:
3363:
3333:
3303:
3273:
3243:
3213:
3183:
3134:
3072:
2987:
2906:
2799:
2775:
2751:
2724:
2643:
2578:
2342:
2302:
2199:
2161:
2050:
1955:
1863:
1827:
1629:
1586:
1546:
1513:
1468:
1426:
1399:
1366:
1340:
1235:
1205:
1178:
1145:
1089:
1064:
972:
882:
862:
696:
665:
640:
618:
554:
324:
315:Mahalanobis distance
295:
273:
201:
173:
153:
133:
62:
35:
3726:
3651:
3564:
3534:
3498:
3468:
3438:
3408:
3378:
3348:
3318:
3288:
3258:
3228:
3198:
3165:
3149:
3119:
3103:
3087:
2921:
2658:
2593:
2555:
2538:
2512:
2494:
2395:
2375:
2276:
2176:
2065:
1789:
1698:
1528:
1483:
1381:
1250:
987:
897:
795:
770:
105:is a dimensionless
103:detectability index
77:
4118:Summary statistics
3808:Summary statistics
3781:
3756:
3736:
3711:
3672:
3652:
3636:
3619:
3602:{\displaystyle d'}
3599:
3565:
3552:
3535:
3519:
3499:
3486:
3469:
3456:
3439:
3426:
3409:
3396:
3379:
3366:
3349:
3336:
3319:
3306:
3299:can underestimate
3289:
3276:
3259:
3246:
3229:
3216:
3199:
3186:
3166:
3153:
3137:
3120:
3107:
3091:
3075:
3050:
2973:
2909:
2884:
2781:
2761:
2737:
2710:
2646:
2621:
2581:
2564:
2558:
2541:
2524:
2498:
2480:
2434:
2398:
2381:
2361:
2324:
2288:
2223:
2177:
2164:
2146:
2053:
2036:
1941:
1845:
1809:
1752:
1661:
1612:
1572:
1529:
1516:
1499:
1471:
1454:
1412:
1382:
1369:
1346:
1322:
1238:
1218:
1191:
1159:{\displaystyle d'}
1156:
1123:
1078:{\displaystyle d'}
1075:
1048:
975:
958:
885:
868:
844:
774:
749:
679:{\displaystyle d'}
676:
654:{\displaystyle d'}
651:
626:
604:
536:
303:
281:
255:
187:{\displaystyle d'}
184:
159:
139:
91:
78:
65:
48:
4113:Signal processing
4070:
4069:
4013:
4012:
3998:signal processing
3759:{\displaystyle i}
3734:
3675:{\displaystyle i}
3622:{\displaystyle i}
2999:
2970:
2902:Another index is
2881:
2811:
2784:{\displaystyle z}
2759:
2720:(also denoted by
2707:
2514:
2315:
2233:
2137:
2034:
1929:
1436:
1409:
1349:{\displaystyle Z}
1305:
1271:
1118:
1101:
1046:
956:
742:
426:
253:
162:{\displaystyle b}
142:{\displaystyle a}
95:sensitivity index
16:(Redirected from
4135:
4128:Statistics stubs
4108:Detection theory
4091:
4084:
4077:
4049:
4042:
4034:
4027:
4020:
3992:
3985:
3963:
3938:
3937:
3934:10.1037/h0035203
3917:
3908:
3907:
3887:
3881:
3880:
3878:
3866:
3765:
3763:
3762:
3757:
3745:
3743:
3742:
3737:
3735:
3733:
3732:
3727:
3722:
3705:
3704:
3703:
3690:
3681:
3679:
3678:
3673:
3661:
3659:
3658:
3653:
3647:
3628:
3626:
3625:
3620:
3608:
3606:
3605:
3600:
3598:
3574:
3572:
3571:
3566:
3560:
3544:
3542:
3541:
3536:
3530:
3508:
3506:
3505:
3500:
3494:
3478:
3476:
3475:
3470:
3464:
3448:
3446:
3445:
3440:
3434:
3418:
3416:
3415:
3410:
3404:
3388:
3386:
3385:
3380:
3374:
3358:
3356:
3355:
3350:
3344:
3328:
3326:
3325:
3320:
3314:
3298:
3296:
3295:
3290:
3284:
3268:
3266:
3265:
3260:
3254:
3238:
3236:
3235:
3230:
3224:
3208:
3206:
3205:
3200:
3194:
3175:
3173:
3172:
3167:
3161:
3145:
3129:
3127:
3126:
3121:
3115:
3099:
3083:
3059:
3057:
3056:
3051:
3046:
3041:
3037:
3036:
3035:
3030:
3021:
3020:
3015:
3001:
3000:
2997:
2995:
2982:
2980:
2979:
2974:
2972:
2971:
2968:
2962:
2957:
2953:
2952:
2951:
2939:
2938:
2917:
2893:
2891:
2890:
2885:
2883:
2882:
2874:
2872:
2868:
2864:
2859:
2855:
2854:
2853:
2848:
2839:
2838:
2833:
2813:
2812:
2809:
2807:
2790:
2788:
2787:
2782:
2770:
2768:
2767:
2762:
2760:
2755:
2746:
2744:
2743:
2738:
2736:
2735:
2719:
2717:
2716:
2711:
2709:
2708:
2705:
2699:
2694:
2690:
2689:
2688:
2676:
2675:
2654:
2630:
2628:
2627:
2622:
2620:
2616:
2615:
2589:
2573:
2571:
2570:
2565:
2563:
2562:
2554:
2549:
2537:
2532:
2515:
2513:
2511:
2506:
2493:
2488:
2478:
2477:
2476:
2464:
2463:
2453:
2448:
2439:
2438:
2412:
2403:
2402:
2394:
2389:
2374:
2369:
2349:
2333:
2331:
2330:
2325:
2323:
2322:
2317:
2316:
2308:
2297:
2295:
2294:
2289:
2287:
2283:
2275:
2270:
2257:
2249:
2241:
2235:
2234:
2226:
2211:
2210:
2186:
2184:
2183:
2178:
2172:
2155:
2153:
2152:
2147:
2142:
2138:
2133:
2132:
2128:
2124:
2105:
2101:
2097:
2080:
2061:
2045:
2043:
2042:
2037:
2035:
2033:
2032:
2031:
2019:
2018:
2008:
2007:
2006:
1988:
1987:
1971:
1950:
1948:
1947:
1942:
1940:
1939:
1934:
1930:
1928:
1927:
1926:
1914:
1913:
1903:
1902:
1901:
1889:
1888:
1878:
1854:
1852:
1851:
1846:
1844:
1843:
1842:
1818:
1816:
1815:
1810:
1802:
1801:
1788:
1783:
1779:
1778:
1762:
1761:
1736:
1711:
1710:
1697:
1692:
1688:
1687:
1671:
1670:
1645:
1621:
1619:
1618:
1613:
1611:
1610:
1598:
1597:
1581:
1579:
1578:
1573:
1571:
1570:
1558:
1557:
1538:
1536:
1535:
1530:
1524:
1508:
1506:
1505:
1500:
1479:
1463:
1461:
1460:
1455:
1438:
1437:
1434:
1421:
1419:
1418:
1413:
1411:
1410:
1407:
1391:
1389:
1388:
1383:
1377:
1355:
1353:
1352:
1347:
1331:
1329:
1328:
1323:
1321:
1317:
1316:
1315:
1306:
1303:
1287:
1283:
1282:
1281:
1272:
1269:
1246:
1227:
1225:
1224:
1219:
1217:
1216:
1200:
1198:
1197:
1192:
1190:
1189:
1165:
1163:
1162:
1157:
1155:
1132:
1130:
1129:
1124:
1119:
1117:false alarm rate
1116:
1102:
1099:
1084:
1082:
1081:
1076:
1074:
1057:
1055:
1054:
1049:
1047:
1045:
1044:
1035:
1034:
1033:
1028:
1027:
1026:
1012:
1011:
1006:
1005:
1004:
992:
983:
967:
965:
964:
959:
957:
955:
954:
945:
944:
943:
938:
937:
936:
922:
921:
916:
915:
914:
902:
893:
877:
875:
874:
869:
853:
851:
850:
845:
843:
839:
838:
837:
832:
831:
821:
820:
815:
814:
794:
789:
784:
783:
769:
764:
759:
758:
743:
741:
740:
739:
720:
715:
714:
709:
708:
685:
683:
682:
677:
675:
660:
658:
657:
652:
650:
635:
633:
632:
627:
625:
613:
611:
610:
605:
600:
595:
594:
586:
577:
566:
565:
545:
543:
542:
537:
535:
534:
525:
517:
516:
511:
502:
501:
496:
478:
477:
472:
463:
462:
457:
448:
447:
439:
427:
422:
421:
416:
407:
406:
401:
392:
391:
383:
377:
369:
368:
363:
354:
353:
348:
339:
334:
312:
310:
309:
304:
302:
290:
288:
287:
282:
280:
264:
262:
261:
256:
254:
249:
245:
244:
243:
231:
230:
216:
211:
193:
191:
190:
185:
183:
168:
166:
165:
160:
148:
146:
145:
140:
87:
85:
84:
79:
73:
57:
55:
54:
49:
47:
46:
21:
4143:
4142:
4138:
4137:
4136:
4134:
4133:
4132:
4098:
4097:
4096:
4095:
4039:
4038:
3970:
3960:
3945:
3942:
3941:
3919:
3918:
3911:
3904:
3889:
3888:
3884:
3868:
3867:
3826:
3821:
3798:
3779:distributions.
3772:
3748:
3747:
3709:
3695:
3691:
3684:
3683:
3664:
3663:
3631:
3630:
3611:
3610:
3591:
3586:
3585:
3581:
3547:
3546:
3514:
3513:
3481:
3480:
3451:
3450:
3421:
3420:
3391:
3390:
3361:
3360:
3331:
3330:
3301:
3300:
3271:
3270:
3241:
3240:
3211:
3210:
3181:
3180:
3132:
3131:
3070:
3069:
3066:
3025:
3010:
3009:
3005:
2990:
2985:
2984:
2963:
2943:
2930:
2929:
2925:
2904:
2903:
2900:
2843:
2828:
2827:
2823:
2822:
2818:
2817:
2802:
2797:
2796:
2773:
2772:
2749:
2748:
2727:
2722:
2721:
2700:
2680:
2667:
2666:
2662:
2641:
2640:
2637:
2607:
2603:
2576:
2575:
2557:
2556:
2539:
2517:
2479:
2468:
2455:
2454:
2433:
2432:
2427:
2417:
2397:
2396:
2376:
2354:
2340:
2339:
2305:
2300:
2299:
2222:
2218:
2202:
2197:
2196:
2159:
2158:
2116:
2112:
2089:
2085:
2081:
2075:
2048:
2047:
2023:
2010:
2009:
1998:
1979:
1972:
1953:
1952:
1918:
1905:
1904:
1893:
1880:
1879:
1873:
1872:
1861:
1860:
1834:
1830:
1825:
1824:
1793:
1770:
1754:
1702:
1679:
1663:
1627:
1626:
1602:
1589:
1584:
1583:
1562:
1549:
1544:
1543:
1511:
1510:
1466:
1465:
1429:
1424:
1423:
1402:
1397:
1396:
1364:
1363:
1338:
1337:
1307:
1301:
1297:
1273:
1267:
1263:
1233:
1232:
1208:
1203:
1202:
1181:
1176:
1175:
1172:
1148:
1143:
1142:
1139:
1087:
1086:
1067:
1062:
1061:
1036:
1018:
1016:
996:
994:
993:
970:
969:
946:
928:
926:
906:
904:
903:
880:
879:
860:
859:
824:
822:
807:
805:
776:
751:
748:
744:
731:
724:
701:
699:
694:
693:
668:
663:
662:
643:
638:
637:
616:
615:
581:
557:
552:
551:
526:
506:
491:
467:
452:
434:
411:
396:
378:
370:
358:
343:
327:
322:
321:
293:
292:
271:
270:
235:
222:
221:
217:
204:
199:
198:
194:('dee-prime'):
176:
171:
170:
151:
150:
131:
130:
127:
119:
60:
59:
38:
33:
32:
23:
22:
15:
12:
11:
5:
4141:
4139:
4131:
4130:
4125:
4120:
4115:
4110:
4100:
4099:
4094:
4093:
4086:
4079:
4071:
4068:
4067:
4050:
4037:
4036:
4029:
4022:
4014:
4011:
4010:
3993:
3982:
3981:
3969:
3968:External links
3966:
3965:
3964:
3958:
3940:
3939:
3928:(6): 481–488.
3909:
3902:
3882:
3823:
3822:
3820:
3817:
3816:
3815:
3810:
3805:
3797:
3794:
3771:
3768:
3755:
3731:
3725:
3721:
3718:
3714:
3708:
3702:
3698:
3694:
3671:
3650:
3646:
3643:
3639:
3618:
3597:
3594:
3580:
3577:
3563:
3559:
3555:
3533:
3529:
3526:
3522:
3497:
3493:
3489:
3467:
3463:
3459:
3437:
3433:
3429:
3407:
3403:
3399:
3377:
3373:
3369:
3347:
3343:
3339:
3317:
3313:
3309:
3287:
3283:
3279:
3257:
3253:
3249:
3227:
3223:
3219:
3197:
3193:
3189:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3065:
3062:
3049:
3045:
3040:
3034:
3029:
3024:
3019:
3014:
3008:
3004:
2994:
2966:
2961:
2956:
2950:
2946:
2942:
2937:
2933:
2928:
2924:
2920:
2916:
2912:
2899:
2896:
2880:
2877:
2871:
2867:
2863:
2858:
2852:
2847:
2842:
2837:
2832:
2826:
2821:
2816:
2806:
2780:
2758:
2734:
2730:
2703:
2698:
2693:
2687:
2683:
2679:
2674:
2670:
2665:
2661:
2657:
2653:
2649:
2636:
2633:
2619:
2614:
2610:
2606:
2602:
2599:
2596:
2592:
2588:
2584:
2561:
2553:
2548:
2544:
2540:
2536:
2531:
2527:
2523:
2522:
2520:
2510:
2505:
2501:
2497:
2492:
2487:
2483:
2475:
2471:
2467:
2462:
2458:
2451:
2447:
2442:
2437:
2431:
2428:
2426:
2423:
2422:
2420:
2415:
2411:
2406:
2401:
2393:
2388:
2384:
2380:
2377:
2373:
2368:
2364:
2360:
2359:
2357:
2352:
2348:
2321:
2314:
2311:
2286:
2282:
2279:
2274:
2269:
2266:
2263:
2260:
2256:
2252:
2248:
2244:
2240:
2232:
2229:
2221:
2217:
2214:
2209:
2205:
2175:
2171:
2167:
2145:
2141:
2136:
2131:
2127:
2123:
2119:
2115:
2111:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2078:
2074:
2071:
2068:
2064:
2060:
2056:
2030:
2026:
2022:
2017:
2013:
2005:
2001:
1997:
1994:
1991:
1986:
1982:
1978:
1975:
1969:
1966:
1963:
1960:
1938:
1933:
1925:
1921:
1917:
1912:
1908:
1900:
1896:
1892:
1887:
1883:
1876:
1871:
1868:
1841:
1837:
1833:
1821:
1820:
1808:
1805:
1800:
1796:
1792:
1787:
1782:
1777:
1773:
1769:
1766:
1760:
1757:
1751:
1748:
1745:
1742:
1739:
1735:
1731:
1728:
1725:
1720:
1717:
1714:
1709:
1705:
1701:
1696:
1691:
1686:
1682:
1678:
1675:
1669:
1666:
1660:
1657:
1654:
1651:
1648:
1644:
1640:
1637:
1634:
1609:
1605:
1601:
1596:
1592:
1582:and variances
1569:
1565:
1561:
1556:
1552:
1527:
1523:
1519:
1498:
1495:
1492:
1489:
1486:
1482:
1478:
1474:
1453:
1450:
1447:
1444:
1441:
1432:
1405:
1380:
1376:
1372:
1345:
1334:
1333:
1320:
1314:
1310:
1300:
1296:
1293:
1290:
1286:
1280:
1276:
1266:
1262:
1259:
1256:
1253:
1249:
1245:
1241:
1215:
1211:
1188:
1184:
1171:
1168:
1154:
1151:
1138:
1135:
1122:
1114:
1111:
1108:
1105:
1097:
1094:
1073:
1070:
1043:
1039:
1032:
1025:
1021:
1015:
1010:
1003:
999:
990:
986:
982:
978:
953:
949:
942:
935:
931:
925:
920:
913:
909:
900:
896:
892:
888:
867:
856:
855:
842:
836:
830:
827:
819:
813:
810:
804:
801:
798:
793:
788:
782:
779:
773:
768:
763:
757:
754:
747:
738:
734:
730:
727:
723:
718:
713:
707:
704:
674:
671:
649:
646:
624:
603:
599:
593:
590:
585:
580:
576:
572:
569:
564:
560:
548:
547:
533:
529:
524:
520:
515:
510:
505:
500:
495:
490:
487:
484:
481:
476:
471:
466:
461:
456:
451:
446:
443:
438:
433:
430:
425:
420:
415:
410:
405:
400:
395:
390:
387:
382:
376:
373:
367:
362:
357:
352:
347:
342:
337:
333:
330:
301:
279:
267:
266:
252:
248:
242:
238:
234:
229:
225:
220:
214:
210:
207:
182:
179:
158:
138:
126:
123:
118:
115:
76:
72:
68:
45:
41:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4140:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4105:
4103:
4092:
4087:
4085:
4080:
4078:
4073:
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3959:0-19-509250-3
3955:
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2020:
2015:
2011:
2003:
1999:
1995:
1992:
1989:
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1980:
1976:
1973:
1967:
1964:
1961:
1958:
1936:
1931:
1923:
1919:
1915:
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3813:Effect size
3629:removed is
1358:Matlab code
661:equals the
4102:Categories
4055:statistics
3876:2012.14331
3819:References
2771:times the
117:Definition
3717:−
3707:−
3642:−
3151:≤
3105:≤
3089:≤
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2831:Σ
2747:). It is
2702:σ
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2526:σ
2500:σ
2496:−
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2313:~
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2189:ROC curve
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1996:
1990:−
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1965:λ
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924:−
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866:ρ
803:ρ
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381:Σ
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107:statistic
3796:See also
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75:′
3176:still.
18:D prime
3956:
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3179:Thus,
1951:, and
1823:where
1336:where
858:where
550:where
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3996:This
3871:arXiv
3804:(ROC)
4059:stub
4002:stub
3954:ISBN
3898:ISBN
3209:and
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93:The
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1974:ln
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2055:d
2029:b
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2016:a
2012:v
2004:b
2000:v
1985:a
1981:v
1968:+
1962:=
1959:c
1937:2
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1920:v
1911:a
1907:v
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1875:(
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1024:a
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486:=
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475:b
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450:(
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389:1
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341:(
336:=
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300:S
265:.
247:|
241:b
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219:|
213:=
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157:b
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71:b
67:d
44:b
40:e
20:)
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