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out-of-sample data points is viewed more favorably, regardless of the models’ relative in-sample performances. The out-of-sample MSPE in this context is exact for the out-of-sample data points that it was computed over, but is merely an estimate of the model’s MSPE for the mostly unobserved
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held-back points. If the increase in the MSPE out of sample compared to in sample is relatively slight, that results in the model being viewed favorably. And if two models are to be compared, the one with the lower MSPE over the
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data points with the specific purpose of using them to compute the estimated model’s MSPE out of sample (i.e., not using data that were used in the model estimation process). Since the regression process is tailored to the
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1883:{\displaystyle n\cdot \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-\operatorname {tr} \left\right).}
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1425:{\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left\right)^{2}=\operatorname {E} \left-\sigma ^{2}\operatorname {tr} \left.}
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Second, as time goes on more data may become available to the data analyst, and then the MSPE can be computed over these new data.
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2053:{\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2p.}
1616:{\displaystyle n\cdot \operatorname {MSPE} (L)=\operatorname {E} \left-\sigma ^{2}\left(n-\operatorname {tr} \left\right).}
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1123:{\displaystyle n\cdot \operatorname {MSPE} (L)=g^{\text{T}}(I-L)^{\text{T}}(I-L)g+\sigma ^{2}\operatorname {tr} \left.}
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When the model has been estimated over all available data with none held back, the MSPE of the model over the entire
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is computed from the version of the model that includes all possible regressors. That concludes this proof.
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in-sample points, normally the in-sample MSPE will be smaller than the out-of-sample one computed over the
429:{\displaystyle \operatorname {MSPE} =\operatorname {E} \left=\sum _{i=1}^{n}\operatorname {PE} _{i}^{2}/n.}
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would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated.
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The mean squared prediction error can be computed exactly in two contexts. First, with a
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485:{\displaystyle \operatorname {MSPE} =\operatorname {ME} ^{2}+\operatorname {VAR} ,}
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Using in-sample data values, the first term on the right side is equivalent to
701:{\displaystyle \operatorname {VAR} =\operatorname {E} \left\right)^{2}\right].}
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advocated this method in the construction of his model selection statistic
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321:{\displaystyle \operatorname {PE_{i}} =g(x_{i})-{\widehat {g}}(x_{i}),}
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between the fitted values implied by the predictive function
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575:{\displaystyle \operatorname {ME} =\operatorname {E} \left}
984:{\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}
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of mostly unobserved data can be estimated as follows.
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The MSPE can be decomposed into two terms: the squared
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835:. Please help to ensure that disputed statements are
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2187:(3rd ed.). New York: McGraw-Hill. pp.
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2185:Econometric Models & Economic Forecasts
798:population from which the data were drawn.
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855:Learn how and when to remove this message
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2100:{\displaystyle {\widehat {\sigma }}^{2}}
1682:{\displaystyle {\widehat {\sigma }}^{2}}
831:Relevant discussion may be found on the
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805:Estimation of MSPE over the population
2181:"Forecasting with Time-Series Models"
103:and the values of the (unobservable)
41:mean squared error of the predictions
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2626:Statistical deviation and dispersion
237:then PE and MSPE are formulated as:
152:of an estimated model. Knowledge of
2067:the number of estimated parameters
2132:Errors and residuals in statistics
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148:and can be used in the process of
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2587:Pearson correlation coefficient
141:{\displaystyle {\widehat {g}},}
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1653:is known or well-estimated by
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230:{\displaystyle {\hat {y}}=Ly,}
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96:{\displaystyle {\widehat {g}}}
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2526:Deep Learning Related Metrics
747:Cross-validation (statistics)
33:mean squared prediction error
2621:Point estimation performance
2117:Akaike information criterion
727:is the square root of MSPE:
721:sum squared prediction error
2370:Sensitivity and specificity
1646:{\displaystyle \sigma ^{2}}
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779:), holding back the other
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771:of the data points (with
2398:Calinski-Harabasz index
2122:Bias-variance tradeoff
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2561:Intra-list Similarity
2137:Law of total variance
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745:Further information:
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2177:Rubinfeld, Daniel L.
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824:factual accuracy is
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168:(i.e., hat matrix)
2582:Euclidean distance
2548:Recommender system
2428:Similarity measure
2242:evaluation metrics
2173:Pindyck, Robert S.
2127:Mean squared error
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2577:Cosine similarity
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185:{\displaystyle y}
166:projection matrix
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113:explanatory power
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72:square difference
64:prediction errors
55:procedure is the
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16:(Redirected from
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2600:Confusion matrix
2375:Logarithmic Loss
2240:Machine learning
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991:, one may write
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837:reliably sourced
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18:Prediction error
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2533:Inception score
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2476:Computer Vision
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2153:Model selection
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822:This article's
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2631:Loss functions
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2317:Classification
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502:
492:
481:
478:
475:
470:
466:
462:
459:
437:
436:
425:
422:
418:
414:
409:
404:
400:
394:
389:
386:
383:
379:
375:
371:
366:
361:
357:
353:
349:
346:
343:
340:
329:
328:
317:
314:
309:
305:
301:
295:
292:
286:
283:
278:
274:
270:
267:
264:
258:
254:
250:
226:
223:
220:
217:
211:
208:
181:
161:
158:
137:
131:
128:
89:
86:
57:expected value
24:
14:
13:
10:
9:
6:
4:
3:
2:
2643:
2632:
2629:
2627:
2624:
2622:
2619:
2618:
2616:
2601:
2598:
2597:
2594:
2588:
2585:
2583:
2580:
2578:
2575:
2574:
2572:
2568:
2562:
2559:
2557:
2554:
2553:
2551:
2549:
2545:
2539:
2536:
2534:
2531:
2530:
2528:
2524:
2518:
2515:
2513:
2510:
2509:
2507:
2505:
2501:
2495:
2492:
2490:
2487:
2485:
2482:
2481:
2479:
2477:
2473:
2467:
2464:
2462:
2459:
2457:
2454:
2453:
2451:
2449:
2445:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2418:Jaccard index
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2390:
2388:
2386:
2382:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2322:
2320:
2318:
2314:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2254:
2252:
2250:
2246:
2241:
2234:
2229:
2227:
2222:
2220:
2215:
2214:
2211:
2200:
2198:0-07-050098-3
2194:
2190:
2186:
2182:
2178:
2174:
2168:
2165:
2158:
2154:
2151:
2149:
2148:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2114:
2110:
2108:
2092:
2085:
2082:
2070:
2066:
2047:
2044:
2041:
2038:
2035:
2032:
2025:
2018:
2015:
2005:
2000:
1991:
1987:
1977:
1974:
1968:
1963:
1959:
1954:
1947:
1942:
1939:
1936:
1932:
1925:
1920:
1916:
1908:
1907:
1906:
1904:
1903:
1895:
1894:Colin Mallows
1877:
1873:
1868:
1865:
1862:
1858:
1855:
1852:
1849:
1845:
1839:
1832:
1829:
1822:
1817:
1812:
1803:
1799:
1789:
1786:
1780:
1775:
1771:
1766:
1759:
1754:
1751:
1748:
1744:
1740:
1734:
1728:
1722:
1702:
1699:
1692:
1691:
1690:
1674:
1667:
1664:
1638:
1634:
1610:
1606:
1601:
1598:
1595:
1591:
1588:
1585:
1582:
1578:
1572:
1568:
1564:
1560:
1554:
1549:
1540:
1536:
1526:
1523:
1517:
1512:
1508:
1503:
1496:
1491:
1488:
1485:
1481:
1476:
1472:
1466:
1460:
1454:
1451:
1448:
1445:
1438:
1437:
1436:
1419:
1415:
1410:
1406:
1403:
1400:
1396:
1390:
1385:
1381:
1378:
1375:
1371:
1365:
1361:
1358:
1353:
1349:
1345:
1341:
1335:
1330:
1321:
1317:
1307:
1304:
1298:
1293:
1289:
1284:
1277:
1272:
1269:
1266:
1262:
1257:
1253:
1247:
1242:
1237:
1232:
1223:
1219:
1209:
1206:
1200:
1192:
1188:
1181:
1177:
1173:
1166:
1159:
1154:
1151:
1148:
1144:
1136:
1135:
1134:
1117:
1113:
1109:
1100:
1095:
1091:
1088:
1083:
1079:
1075:
1072:
1066:
1063:
1060:
1044:
1041:
1038:
1026:
1022:
1016:
1010:
1007:
1004:
1001:
994:
993:
992:
975:
972:
969:
956:
951:
947:
924:
920:
916:
913:
905:
901:
894:
891:
886:
882:
872:
870:
859:
856:
848:
838:
834:
828:
827:
820:
811:
810:
804:
802:
799:
796:
791:
787:
782:
778:
774:
770:
766:
762:
758:
754:
748:
740:
738:
726:
722:
716:
711:The quantity
695:
691:
686:
681:
676:
667:
663:
655:
650:
646:
640:
632:
628:
618:
615:
608:
603:
599:
593:
590:
583:
568:
559:
555:
548:
545:
537:
533:
523:
520:
513:
509:
503:
500:
493:
479:
476:
473:
468:
464:
460:
457:
450:
449:
448:
446:
442:
423:
420:
416:
412:
407:
402:
398:
392:
387:
384:
381:
377:
373:
369:
364:
359:
355:
351:
347:
341:
338:
331:
330:
315:
307:
303:
293:
290:
284:
276:
272:
265:
262:
240:
239:
238:
224:
221:
218:
215:
206:
195:
179:
171:
167:
159:
157:
155:
151:
135:
129:
126:
115:
114:
109:
106:
87:
84:
73:
69:
65:
62:
58:
54:
50:
49:curve fitting
46:
42:
38:
34:
30:
19:
2281:
2184:
2167:
2143:
2068:
2064:
2062:
1898:
1892:
1625:
1434:
1132:
873:
866:
851:
842:
823:
800:
794:
789:
785:
780:
776:
772:
768:
763:may run the
761:data analyst
756:
750:
724:
720:
714:
710:
438:
169:
163:
153:
112:
111:
107:
67:
63:
40:
36:
32:
26:
753:data sample
160:Formulation
2615:Categories
2570:Similarity
2512:Perplexity
2423:Rand index
2408:Dunn index
2393:Silhouette
2385:Clustering
2249:Regression
2159:References
2142:Mallows's
869:population
767:over only
765:regression
755:of length
719:is called
105:true value
53:regression
29:statistics
2340:Precision
2292:RMSE/RMSD
2086:^
2083:σ
2033:−
2019:^
2016:σ
1978:^
1969:−
1933:∑
1859:
1853:−
1833:^
1830:σ
1823:−
1790:^
1781:−
1745:∑
1729:
1723:^
1703:⋅
1668:^
1665:σ
1635:σ
1592:
1586:−
1569:σ
1565:−
1527:^
1518:−
1482:∑
1473:
1455:
1449:⋅
1404:−
1379:−
1362:
1350:σ
1346:−
1308:^
1299:−
1263:∑
1254:
1210:^
1201:−
1174:
1145:∑
1092:
1080:σ
1064:−
1042:−
1011:
1005:⋅
957:∼
948:ε
921:ε
917:σ
833:talk page
647:
641:−
619:^
600:
546:−
524:^
510:
413:
378:∑
348:
294:^
285:−
210:^
130:^
88:^
45:smoothing
2556:Coverage
2335:Accuracy
2179:(1991).
2111:See also
845:May 2018
826:disputed
445:variance
2448:Ranking
2438:SimHash
2325:F-score
2189:516–535
731:√
196:vector
70:), the
61:squared
59:of the
43:, of a
2345:Recall
2195:
2063:where
1435:Thus,
939:where
759:, the
729:RMSPE=
723:. The
2350:Kappa
2267:sMAPE
795:n – q
790:n – q
781:n – q
775:<
713:SSPE=
51:, or
2517:BLEU
2489:SSIM
2484:PSNR
2461:NDCG
2282:MSPE
2277:MASE
2272:MAPE
2193:ISBN
2071:and
1452:MSPE
1008:MSPE
733:MSPE
717:MSPE
458:MSPE
441:bias
339:MSPE
37:MSPE
31:the
2538:FID
2504:NLP
2494:IoU
2456:MRR
2433:SMC
2365:ROC
2360:AUC
2355:MCC
2307:MAD
2302:MDA
2287:RMS
2262:MAE
2257:MSE
1626:If
591:VAR
477:VAR
192:to
116:of
27:In
2617::
2466:AP
2330:P4
2191:.
2183:.
2175:;
1856:tr
1589:tr
1359:tr
1089:tr
737:.
501:ME
465:ME
399:PE
356:PE
68:PE
47:,
2297:R
2232:e
2225:t
2218:v
2201:.
2146:p
2144:C
2093:2
2069:p
2065:p
2048:.
2045:p
2042:2
2039:+
2036:n
2026:2
2006:2
2001:)
1997:)
1992:i
1988:x
1984:(
1975:g
1964:i
1960:y
1955:(
1948:n
1943:1
1940:=
1937:i
1926:=
1921:p
1917:C
1901:p
1899:C
1878:.
1874:)
1869:]
1866:L
1863:[
1850:n
1846:(
1840:2
1818:2
1813:)
1809:)
1804:i
1800:x
1796:(
1787:g
1776:i
1772:y
1767:(
1760:n
1755:1
1752:=
1749:i
1741:=
1738:)
1735:L
1732:(
1719:E
1716:P
1713:S
1710:M
1700:n
1675:2
1639:2
1611:.
1607:)
1602:]
1599:L
1596:[
1583:n
1579:(
1573:2
1561:]
1555:2
1550:)
1546:)
1541:i
1537:x
1533:(
1524:g
1513:i
1509:y
1504:(
1497:n
1492:1
1489:=
1486:i
1477:[
1470:E
1467:=
1464:)
1461:L
1458:(
1446:n
1420:.
1416:]
1411:)
1407:L
1401:I
1397:(
1391:T
1386:)
1382:L
1376:I
1372:(
1366:[
1354:2
1342:]
1336:2
1331:)
1327:)
1322:i
1318:x
1314:(
1305:g
1294:i
1290:y
1285:(
1278:n
1273:1
1270:=
1267:i
1258:[
1251:E
1248:=
1243:2
1238:)
1233:]
1229:)
1224:i
1220:x
1216:(
1207:g
1198:)
1193:i
1189:x
1185:(
1182:g
1178:[
1171:E
1167:(
1160:n
1155:1
1152:=
1149:i
1118:.
1114:]
1110:L
1105:T
1101:L
1096:[
1084:2
1076:+
1073:g
1070:)
1067:L
1061:I
1058:(
1053:T
1049:)
1045:L
1039:I
1036:(
1031:T
1027:g
1023:=
1020:)
1017:L
1014:(
1002:n
979:)
976:1
973:,
970:0
967:(
962:N
952:i
925:i
914:+
911:)
906:i
902:x
898:(
895:g
892:=
887:i
883:y
858:)
852:(
847:)
843:(
839:.
829:.
786:q
777:n
773:q
769:q
757:n
715:n
696:.
692:]
687:2
682:)
677:]
673:)
668:i
664:x
660:(
656:g
651:[
644:E
638:)
633:i
629:x
625:(
616:g
609:(
604:[
597:E
594:=
569:]
565:)
560:i
556:x
552:(
549:g
543:)
538:i
534:x
530:(
521:g
514:[
507:E
504:=
480:,
474:+
469:2
461:=
424:.
421:n
417:/
408:2
403:i
393:n
388:1
385:=
382:i
374:=
370:]
365:2
360:i
352:[
345:E
342:=
316:,
313:)
308:i
304:x
300:(
291:g
282:)
277:i
273:x
269:(
266:g
263:=
257:i
253:E
249:P
225:,
222:y
219:L
216:=
207:y
180:y
170:L
154:g
136:,
127:g
108:g
85:g
66:(
35:(
20:)
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