525:
951:
302:
597:
520:{\displaystyle {\begin{aligned}{\text{FVU}}&={{\text{VAR}}_{\text{err}} \over {\text{VAR}}_{\text{tot}}}={{\text{SS}}_{\text{err}}/N \over {\text{SS}}_{\text{tot}}/N}={{\text{SS}}_{\text{err}} \over {\text{SS}}_{\text{tot}}}\left(=1-{{\text{SS}}_{\text{reg}} \over {\text{SS}}_{\text{tot}}},{\text{ only true in some cases such as linear regression}}\right)\\&=1-R^{2}\end{aligned}}}
32:
946:{\displaystyle {\begin{aligned}{\text{SS}}_{\text{err}}&=\sum _{i=1}^{N}\;(y_{i}-{\widehat {y}}_{i})^{2}\\{\text{SS}}_{\text{tot}}&=\sum _{i=1}^{N}\;(y_{i}-{\bar {y}})^{2}\\{\text{SS}}_{\text{reg}}&=\sum _{i=1}^{N}\;({\widehat {y}}_{i}-{\bar {y}})^{2}{\text{ and}}\\{\bar {y}}&={\frac {1}{N}}\sum _{i=1}^{N}\;y_{i}.\end{aligned}}}
1020:
307:
602:
263:
1203:
1102:
290:
205:
178:
962:
42:
100:
72:
79:
86:
1322:
68:
57:
1239:
535:
1156:. But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where
1327:
1269:
1254:
210:
1249:
588:
93:
562:
1332:
1159:
1049:, the most naive regression function that we can think of is the constant function predicting the mean of
49:
1056:
296:
observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:
575:
146:
1264:
134:
17:
1030:
1259:
1298:
268:
183:
1045:
It is useful to consider the second definition to understand FVU. When trying to predict
163:
1316:
552:
are the variance of the residuals and the sample variance of the dependent variable.
1015:{\displaystyle {\text{FVU}}={\frac {\operatorname {MSE} (f)}{\operatorname {var} }}}
1244:
31:
956:
Alternatively, the fraction of variance unexplained can be defined as follows:
1149:
138:
122:
1302:
145:
which cannot be explained, i.e., which is not correctly predicted, by the
1289:
Achen, C. H. (1990). "'What Does "Explained
Variance" Explain?: Reply".
1133:
can be accounted for, and the FVU then has its maximum value of 1.
1104:. It follows that the MSE of this function equals the variance of
1136:
More generally, the FVU will be 1 if the explanatory variables
25:
561:(the sum of squared predictions errors, equivalently the
587:(the sum of squares of the regression, equivalently the
479: only true in some cases such as linear regression
53:
1162:
1059:
965:
600:
305:
271:
213:
186:
166:
1197:
1096:
1014:
945:
519:
284:
257:
199:
172:
8:
58:introducing citations to additional sources
258:{\displaystyle {\widehat {y}}_{i}=f(x_{i})}
160:Suppose we are given a regression function
1144:in the sense that the predicted values of
925:
818:
735:
645:
1189:
1176:
1165:
1164:
1161:
1083:
1082:
1070:
1058:
974:
966:
964:
930:
919:
908:
894:
876:
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866:
860:
845:
844:
835:
824:
823:
812:
801:
784:
779:
768:
753:
752:
743:
729:
718:
701:
696:
685:
675:
664:
663:
653:
639:
628:
611:
606:
601:
599:
507:
477:
466:
461:
454:
449:
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412:
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322:
310:
306:
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276:
270:
246:
227:
216:
215:
212:
191:
185:
165:
48:Relevant discussion may be found on the
16:For broader coverage of this topic, see
1281:
7:
1198:{\displaystyle {\hat {y}}_{i}=y_{i}}
1129:= 0. In this case, no variation in
1097:{\displaystyle f(x_{i})={\bar {y}}}
137:is the fraction of variance of the
69:"Fraction of variance unexplained"
14:
1033:of the regression function
127:fraction of variance unexplained
41:relies largely or entirely on a
30:
1170:
1088:
1076:
1063:
1006:
1000:
989:
983:
881:
857:
850:
819:
765:
758:
736:
682:
646:
252:
239:
1:
1240:Coefficient of determination
536:coefficient of determination
1349:
1270:Mean absolute scaled error
1255:Lack-of-fit sum of squares
15:
1250:Explained sum of squares
589:explained sum of squares
563:residual sum of squares
1199:
1140:tell us nothing about
1098:
1016:
947:
924:
817:
734:
644:
521:
286:
259:
201:
174:
133:) in the context of a
1323:Parametric statistics
1200:
1099:
1017:
948:
904:
797:
714:
624:
522:
292:is the vector of the
287:
285:{\displaystyle x_{i}}
260:
202:
200:{\displaystyle y_{i}}
175:
147:explanatory variables
141:(dependent variable)
1230:, and the FVU is 0.
1160:
1057:
963:
598:
576:total sum of squares
303:
269:
211:
184:
164:
54:improve this article
1303:10.1093/pan/2.1.173
1265:Regression analysis
18:Explained variation
1328:Statistical ratios
1291:Political Analysis
1195:
1094:
1031:mean squared error
1012:
943:
941:
517:
515:
282:
255:
197:
180:yielding for each
170:
1260:Linear regression
1173:
1091:
1010:
969:
902:
884:
869:
853:
832:
787:
782:
761:
704:
699:
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410:
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387:
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365:
360:
348:
345:
340:
333:
328:
313:
224:
173:{\displaystyle f}
156:Formal definition
119:
118:
104:
23:Statistical noise
1340:
1307:
1306:
1286:
1209:, the MSE is 0,
1204:
1202:
1201:
1196:
1194:
1193:
1181:
1180:
1175:
1174:
1166:
1103:
1101:
1100:
1095:
1093:
1092:
1084:
1075:
1074:
1021:
1019:
1018:
1013:
1011:
1009:
992:
975:
970:
967:
952:
950:
949:
944:
942:
935:
934:
923:
918:
903:
895:
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867:
865:
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834:
833:
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816:
811:
789:
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762:
754:
748:
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733:
728:
706:
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526:
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114:
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105:
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62:
34:
26:
1348:
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1313:
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1287:
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1215:
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1163:
1158:
1157:
1128:
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1114:
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993:
976:
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822:
790:
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774:
764:
739:
707:
695:
692:
691:
681:
662:
649:
617:
605:
596:
595:
591:) are given by
586:
573:
560:
551:
544:
514:
513:
503:
488:
487:
460:
448:
436:
432:
418:
406:
378:
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356:
355:
336:
324:
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301:
300:
272:
267:
266:
242:
214:
209:
208:
187:
182:
181:
162:
161:
158:
135:regression task
115:
109:
106:
63:
61:
47:
35:
24:
21:
12:
11:
5:
1346:
1344:
1336:
1335:
1330:
1325:
1315:
1314:
1309:
1308:
1297:(1): 173–184.
1280:
1279:
1277:
1274:
1273:
1272:
1267:
1262:
1257:
1252:
1247:
1242:
1235:
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1227:
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1213:
1192:
1188:
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1179:
1172:
1169:
1126:
1119:
1112:
1090:
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1073:
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1062:
1042:
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1005:
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996:
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988:
985:
982:
979:
973:
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953:
938:
933:
929:
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917:
914:
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901:
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890:
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863:
859:
852:
849:
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838:
831:
828:
821:
815:
810:
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771:
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760:
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751:
746:
742:
738:
732:
727:
724:
721:
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713:
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708:
694:
693:
688:
684:
678:
671:
668:
661:
656:
652:
648:
642:
637:
634:
631:
627:
623:
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604:
603:
582:
569:
556:
549:
542:
528:
527:
510:
506:
502:
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491:
489:
485:
476:
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403:
397:
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375:
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352:
321:
318:
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309:
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279:
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254:
249:
245:
241:
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194:
190:
169:
157:
154:
117:
116:
52:. Please help
38:
36:
29:
22:
13:
10:
9:
6:
4:
3:
2:
1345:
1334:
1333:Least squares
1331:
1329:
1326:
1324:
1321:
1320:
1318:
1304:
1300:
1296:
1292:
1285:
1282:
1275:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1237:
1233:
1231:
1226:
1219:
1212:
1208:
1190:
1186:
1182:
1177:
1167:
1155:
1151:
1147:
1143:
1139:
1134:
1132:
1125:
1118:
1111:
1107:
1085:
1079:
1071:
1067:
1060:
1052:
1048:
1040:
1038:
1036:
1032:
1028:
1003:
997:
994:
986:
980:
977:
971:
959:
958:
957:
936:
931:
927:
920:
915:
912:
909:
905:
899:
896:
891:
889:
878:
861:
847:
841:
836:
829:
826:
813:
808:
805:
802:
798:
794:
792:
769:
755:
749:
744:
740:
730:
725:
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719:
715:
711:
709:
686:
676:
669:
666:
659:
654:
650:
640:
635:
632:
629:
625:
621:
619:
594:
593:
592:
590:
585:
581:
577:
572:
568:
564:
559:
555:
548:
541:
537:
533:
508:
504:
500:
497:
494:
492:
483:
474:
443:
440:
437:
433:
401:
395:
391:
373:
369:
350:
319:
317:
299:
298:
297:
295:
277:
273:
247:
243:
236:
233:
228:
221:
218:
192:
188:
167:
155:
153:
151:
148:
144:
140:
136:
132:
128:
124:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
51:
45:
44:
43:single source
39:This article
37:
33:
28:
27:
19:
1294:
1290:
1284:
1224:
1217:
1210:
1206:
1153:
1145:
1141:
1137:
1135:
1130:
1123:
1116:
1109:
1105:
1050:
1046:
1044:
1034:
1026:
1024:
955:
583:
579:
570:
566:
557:
553:
546:
539:
531:
529:
293:
207:an estimate
159:
149:
142:
130:
126:
120:
107:
97:
90:
83:
76:
64:
40:
1245:Correlation
1108:; that is,
1041:Explanation
1317:Categories
1276:References
1025:where MSE(
139:regressand
123:statistics
80:newspapers
1171:^
1089:¯
1029:) is the
998:
981:
906:∑
882:¯
868: and
851:¯
842:−
830:^
799:∑
759:¯
750:−
716:∑
670:^
660:−
626:∑
501:−
444:−
222:^
110:June 2020
50:talk page
1234:See also
1205:for all
1053:, i.e.,
1148:do not
578:), and
534:is the
94:scholar
1150:covary
1122:, and
1035:ƒ
530:where
265:where
125:, the
96:
89:
82:
75:
67:
1216:= 0,
1152:with
574:(the
101:JSTOR
87:books
545:and
538:and
73:news
1299:doi
1228:tot
1221:reg
1214:err
1127:reg
1120:tot
1113:err
995:var
978:MSE
968:FVU
786:reg
703:tot
613:err
584:reg
571:tot
565:),
558:err
550:tot
547:VAR
543:err
540:VAR
468:tot
456:reg
426:tot
414:err
386:tot
364:err
344:tot
339:VAR
332:err
327:VAR
312:FVU
131:FVU
121:In
56:by
1319::
1293:.
1225:SS
1223:=
1218:SS
1211:SS
1124:SS
1117:SS
1115:=
1110:SS
1037:.
781:SS
698:SS
608:SS
580:SS
567:SS
554:SS
463:SS
451:SS
421:SS
409:SS
381:SS
359:SS
152:.
1305:.
1301::
1295:2
1207:i
1191:i
1187:y
1183:=
1178:i
1168:y
1154:Y
1146:Y
1142:Y
1138:X
1131:Y
1106:Y
1086:y
1080:=
1077:)
1072:i
1068:x
1064:(
1061:f
1051:Y
1047:Y
1027:f
1007:]
1004:Y
1001:[
990:)
987:f
984:(
972:=
937:.
932:i
928:y
921:N
916:1
913:=
910:i
900:N
897:1
892:=
879:y
862:2
858:)
848:y
837:i
827:y
820:(
814:N
809:1
806:=
803:i
795:=
770:2
766:)
756:y
745:i
741:y
737:(
731:N
726:1
723:=
720:i
712:=
687:2
683:)
677:i
667:y
655:i
651:y
647:(
641:N
636:1
633:=
630:i
622:=
532:R
509:2
505:R
498:1
495:=
484:)
475:,
441:1
438:=
434:(
402:=
396:N
392:/
374:N
370:/
351:=
320:=
294:i
278:i
274:x
253:)
248:i
244:x
240:(
237:f
234:=
229:i
219:y
193:i
189:y
168:f
150:X
143:Y
129:(
112:)
108:(
98:·
91:·
84:·
77:·
60:.
46:.
20:.
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