38:-based measures is not affordable. Like all screening, the EE method provides qualitative sensitivity analysis measures, i.e. measures which allow the identification of non-influential inputs or which allow to rank the input factors in order of importance, but do not quantify exactly the relative importance of the inputs.
1543:
636:
1861:
Iooss, Bertrand, and Paul Lemaître. 2015. “A Review on Global
Sensitivity Analysis Methods.” In Uncertainty Management in Simulation-Optimization of Complex Systems, edited by G. Dellino and C. Meloni, 101–22. Boston, MA: Springer, Boston, MA.
949:
1736:
1796:
An efficient technical scheme to construct the trajectories used in the EE method is presented in the original paper by Morris while an improvement strategy aimed at better exploring the input space is proposed by
Campolongo et al..
1549:
These two measures need to be read together (e.g. on a two-dimensional graph) in order to rank input factors in order of importance and identify those inputs which do not influence the output variability. Low values of both
1398:
1233:
In case input factors are not uniformly distributed, the best practice is to sample in the space of the quantiles and to obtain the inputs values using inverse cumulative distribution functions. Note that in this case
1098:, since a high number of levels to be explored needs to be balanced by a high number of trajectories, in order to obtain an exploratory sample. It is demonstrated that a convenient choice for the
716:
1405:
1049:
475:
460:
167:
797:
823:
1767:
1618:
1631:
1588:
1295:
1252:
1179:
1139:
756:
736:
364:
269:
217:
1888:
Wei, Pengfei, Zhenzhou Lu, and
Jingwen Song. 2015. “Variable Importance Analysis: A Comprehensive Review.” Reliability Engineering & System Safety 142: 399–432.
344:
1791:
1568:
1275:
194:
1228:
1159:
1119:
1096:
1072:
972:
818:
309:
289:
249:
84:
64:
1848:
Borgonovo, Emanuele, and Elmar
Plischke. 2016. “Sensitivity Analysis: A Review of Recent Advances.” European Journal of Operational Research 248 (3): 869–87.
1308:
1875:
Norton, J.P. 2015. “An
Introduction to Sensitivity Assessment of Simulation Models.” Environmental Modelling & Software 69 (C): 166–74.
231:
in the space of the inputs, where inputs are randomly moved One-At-a-Time (OAT). In this design, each model input is assumed to vary across
1925:
1901:
Campolongo, F., J. Cariboni, and A. Saltelli (2007). An effective screening design for sensitivity analysis of large models.
1620:, which on its own is sufficient to provide a reliable ranking of the input factors. The revised measure is the mean of the
34:
or for a model with a large number of inputs, where the costs of estimating other sensitivity analysis measures such as the
1930:
1538:{\displaystyle \sigma _{i}={\sqrt {{\frac {1}{(r-1)}}\sum _{j=1}^{r}\left(d_{i}\left(X^{(j)}\right)-\mu _{i}\right)^{2}}}}
645:
631:{\displaystyle d_{i}(X)={\frac {Y(X_{1},\ldots ,X_{i-1},X_{i}+\Delta ,X_{i+1},\ldots ,X_{k})-Y(\mathbf {X} )}{\Delta }}}
977:
1621:
369:
92:
1817:
944:{\displaystyle d_{i}\left(X^{(1)}\right),d_{i}\left(X^{(2)}\right),\ldots ,d_{i}\left(X^{(r)}\right)}
24:
761:
1770:
1298:
1075:
31:
1731:{\displaystyle \mu _{i}^{*}={\frac {1}{r}}\sum _{j=1}^{r}\left|d_{i}\left(X^{(j)}\right)\right|}
1593:
An improvement of this method was developed by
Campolongo et al. who proposed a revised measure
1745:
1596:
1573:
1280:
1237:
1164:
1124:
741:
721:
349:
254:
202:
220:
317:
172:
The original EE method of Morris provides two sensitivity measures for each input factor:
1831:
Morris, M. D. (1991). Factorial sampling plans for preliminary computational experiments.
1776:
1553:
1260:
952:
227:
These two measures are obtained through a design based on the construction of a series of
179:
1184:
1769:
solves the problem of the effects of opposite signs which occurs when the model is non-
1144:
1104:
1081:
1057:
957:
803:
294:
274:
234:
69:
49:
1919:
1813:
251:
selected levels in the space of the input factors. The region of experimentation
1876:
46:
To exemplify the EE method, let us assume to consider a mathematical model with
1393:{\displaystyle \mu _{i}={\frac {1}{r}}\sum _{j=1}^{r}d_{i}\left(X^{(j)}\right)}
1863:
228:
30:
EE is applied to identify non-influential inputs for a computationally costly
1889:
1849:
1099:
196:, assessing the overall importance of an input factor on the model output;
35:
1773:
and which can cancel each other out, thus resulting in a low value for
1624:
of the absolute values of the elementary effects of the input factors:
1230:, as this ensures equal probability of sampling in the input space.
1254:
equals the step taken by the inputs in the space of the quantiles.
1301:
of the distribution of the elementary effects of each input:
1074:~ 4-10, depending on the number of input factors, on the
1078:
of the model and on the choice of the number of levels
1779:
1748:
1634:
1599:
1576:
1556:
1408:
1311:
1283:
1263:
1240:
1187:
1167:
1147:
1127:
1107:
1084:
1060:
980:
960:
826:
806:
764:
744:
724:
648:
478:
372:
352:
346:
points since input factors move one by one of a step
320:
297:
277:
257:
237:
205:
182:
95:
86:
be the output of interest (a scalar for simplicity):
72:
52:
711:{\displaystyle \mathbf {X} =(X_{1},X_{2},...X_{k})}
1785:
1761:
1730:
1612:
1582:
1562:
1537:
1392:
1289:
1269:
1246:
1222:
1173:
1153:
1133:
1113:
1090:
1066:
1043:
966:
943:
812:
791:
750:
730:
710:
630:
454:
358:
338:
303:
283:
263:
243:
211:
188:
161:
78:
58:
820:elementary effects are estimated for each input
1044:{\displaystyle X^{(1)},X^{(2)},\ldots ,X^{(r)}}
1814:https://www.stat.iastate.edu/people/max-morris
1877:https://doi.org/10.1016/j.envsoft.2015.03.020
23:is one of the most used screening methods in
8:
738:such that the transformed point is still in
449:
373:
1864:https://doi.org/10.1007/978-1-4899-7547-8_5
455:{\displaystyle \{0,1/(p-1),2/(p-1),...,1\}}
1890:https://doi.org/10.1016/j.ress.2015.05.018
1850:https://doi.org/10.1016/J.EJOR.2015.06.032
162:{\displaystyle Y=f(X_{1},X_{2},...X_{k}).}
1778:
1753:
1747:
1707:
1693:
1678:
1667:
1653:
1644:
1639:
1633:
1604:
1598:
1575:
1555:
1527:
1516:
1493:
1479:
1463:
1452:
1424:
1422:
1413:
1407:
1374:
1360:
1350:
1339:
1325:
1316:
1310:
1282:
1262:
1239:
1191:
1186:
1166:
1146:
1126:
1106:
1083:
1059:
1029:
1004:
985:
979:
959:
925:
911:
882:
868:
845:
831:
825:
805:
763:
743:
723:
699:
677:
664:
649:
647:
614:
596:
571:
552:
533:
514:
501:
483:
477:
411:
385:
371:
351:
319:
296:
276:
256:
236:
204:
181:
147:
125:
112:
94:
71:
51:
1827:
1825:
1806:
469:for each input factor is defined as:
7:
1903:Environmental Modelling and Software
1590:correspond to a non-influent input.
465:Along each trajectory the so-called
19:Published in 1991 by Max Morris the
462:while all the others remain fixed.
1241:
1168:
1128:
745:
725:
623:
561:
353:
258:
14:
1297:are defined as the mean and the
650:
615:
314:Each trajectory is composed of
1816:Home Page of Max D. Morris at
1714:
1708:
1500:
1494:
1442:
1430:
1381:
1375:
1217:
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1202:
1196:
1036:
1030:
1011:
1005:
992:
986:
932:
926:
889:
883:
852:
846:
792:{\displaystyle i=1,\ldots ,k.}
705:
657:
619:
611:
602:
507:
495:
489:
428:
416:
402:
390:
333:
321:
153:
105:
21:elementary effects (EE) method
1:
1947:
718:is any selected value in
223:effects and interactions.
1762:{\displaystyle \mu ^{*}}
1613:{\displaystyle \mu ^{*}}
1583:{\displaystyle \sigma }
1290:{\displaystyle \sigma }
1247:{\displaystyle \Delta }
1174:{\displaystyle \Delta }
1134:{\displaystyle \Delta }
751:{\displaystyle \Omega }
731:{\displaystyle \Omega }
359:{\displaystyle \Delta }
264:{\displaystyle \Omega }
212:{\displaystyle \sigma }
1787:
1763:
1732:
1683:
1614:
1584:
1564:
1539:
1468:
1394:
1355:
1291:
1271:
1248:
1224:
1175:
1155:
1135:
1115:
1092:
1068:
1045:
968:
945:
814:
793:
752:
732:
712:
632:
456:
360:
340:
305:
285:
265:
245:
213:
190:
163:
80:
60:
1926:Mathematical modeling
1818:Iowa State University
1788:
1764:
1733:
1663:
1615:
1585:
1565:
1540:
1448:
1395:
1335:
1292:
1272:
1249:
1225:
1176:
1156:
1136:
1116:
1093:
1069:
1046:
969:
946:
815:
794:
753:
733:
713:
633:
457:
361:
341:
339:{\displaystyle (k+1)}
306:
286:
266:
246:
214:
191:
164:
81:
61:
1931:Sensitivity analysis
1786:{\displaystyle \mu }
1777:
1746:
1632:
1597:
1574:
1563:{\displaystyle \mu }
1554:
1406:
1309:
1281:
1270:{\displaystyle \mu }
1261:
1238:
1185:
1165:
1145:
1125:
1105:
1082:
1058:
978:
958:
824:
804:
762:
742:
722:
646:
476:
370:
350:
318:
295:
275:
255:
235:
203:
189:{\displaystyle \mu }
180:
93:
70:
50:
25:sensitivity analysis
1649:
66:input factors. Let
1909:, 1509–1518.
1783:
1759:
1728:
1635:
1610:
1580:
1560:
1535:
1390:
1299:standard deviation
1287:
1267:
1244:
1223:{\displaystyle p/}
1220:
1171:
1151:
1131:
1111:
1088:
1076:computational cost
1064:
1041:
964:
941:
810:
789:
748:
728:
708:
628:
452:
356:
336:
301:
281:
261:
241:
209:
186:
159:
76:
56:
32:mathematical model
1661:
1533:
1446:
1333:
1257:The two measures
1154:{\displaystyle p}
1114:{\displaystyle p}
1091:{\displaystyle p}
1067:{\displaystyle r}
967:{\displaystyle r}
953:randomly sampling
813:{\displaystyle r}
626:
467:elementary effect
304:{\displaystyle p}
284:{\displaystyle k}
244:{\displaystyle p}
79:{\displaystyle Y}
59:{\displaystyle k}
1938:
1910:
1899:
1893:
1886:
1880:
1873:
1867:
1859:
1853:
1846:
1840:
1829:
1820:
1811:
1792:
1790:
1789:
1784:
1768:
1766:
1765:
1760:
1758:
1757:
1737:
1735:
1734:
1729:
1727:
1723:
1722:
1718:
1717:
1698:
1697:
1682:
1677:
1662:
1654:
1648:
1643:
1619:
1617:
1616:
1611:
1609:
1608:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1544:
1542:
1541:
1536:
1534:
1532:
1531:
1526:
1522:
1521:
1520:
1508:
1504:
1503:
1484:
1483:
1467:
1462:
1447:
1445:
1425:
1423:
1418:
1417:
1399:
1397:
1396:
1391:
1389:
1385:
1384:
1365:
1364:
1354:
1349:
1334:
1326:
1321:
1320:
1296:
1294:
1293:
1288:
1276:
1274:
1273:
1268:
1253:
1251:
1250:
1245:
1229:
1227:
1226:
1221:
1195:
1180:
1178:
1177:
1172:
1160:
1158:
1157:
1152:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1097:
1095:
1094:
1089:
1073:
1071:
1070:
1065:
1050:
1048:
1047:
1042:
1040:
1039:
1015:
1014:
996:
995:
973:
971:
970:
965:
950:
948:
947:
942:
940:
936:
935:
916:
915:
897:
893:
892:
873:
872:
860:
856:
855:
836:
835:
819:
817:
816:
811:
798:
796:
795:
790:
757:
755:
754:
749:
737:
735:
734:
729:
717:
715:
714:
709:
704:
703:
682:
681:
669:
668:
653:
637:
635:
634:
629:
627:
622:
618:
601:
600:
582:
581:
557:
556:
544:
543:
519:
518:
502:
488:
487:
461:
459:
458:
453:
415:
389:
365:
363:
362:
357:
345:
343:
342:
337:
310:
308:
307:
302:
290:
288:
287:
282:
270:
268:
267:
262:
250:
248:
247:
242:
218:
216:
215:
210:
195:
193:
192:
187:
168:
166:
165:
160:
152:
151:
130:
129:
117:
116:
85:
83:
82:
77:
65:
63:
62:
57:
16:Screening method
1946:
1945:
1941:
1940:
1939:
1937:
1936:
1935:
1916:
1915:
1914:
1913:
1900:
1896:
1887:
1883:
1874:
1870:
1860:
1856:
1847:
1843:
1830:
1823:
1812:
1808:
1803:
1775:
1774:
1749:
1744:
1743:
1703:
1699:
1689:
1688:
1684:
1630:
1629:
1625:
1600:
1595:
1594:
1572:
1571:
1552:
1551:
1512:
1489:
1485:
1475:
1474:
1470:
1469:
1429:
1409:
1404:
1403:
1370:
1366:
1356:
1312:
1307:
1306:
1302:
1279:
1278:
1259:
1258:
1236:
1235:
1183:
1182:
1163:
1162:
1143:
1142:
1123:
1122:
1103:
1102:
1080:
1079:
1056:
1055:
1025:
1000:
981:
976:
975:
956:
955:
921:
917:
907:
878:
874:
864:
841:
837:
827:
822:
821:
802:
801:
760:
759:
758:for each index
740:
739:
720:
719:
695:
673:
660:
644:
643:
592:
567:
548:
529:
510:
503:
479:
474:
473:
368:
367:
348:
347:
316:
315:
293:
292:
273:
272:
253:
252:
233:
232:
201:
200:
178:
177:
143:
121:
108:
91:
90:
68:
67:
48:
47:
44:
17:
12:
11:
5:
1944:
1942:
1934:
1933:
1928:
1918:
1917:
1912:
1911:
1894:
1881:
1868:
1854:
1841:
1821:
1805:
1804:
1802:
1799:
1782:
1756:
1752:
1740:
1739:
1726:
1721:
1716:
1713:
1710:
1706:
1702:
1696:
1692:
1687:
1681:
1676:
1673:
1670:
1666:
1660:
1657:
1652:
1647:
1642:
1638:
1607:
1603:
1579:
1559:
1547:
1546:
1530:
1525:
1519:
1515:
1511:
1507:
1502:
1499:
1496:
1492:
1488:
1482:
1478:
1473:
1466:
1461:
1458:
1455:
1451:
1444:
1441:
1438:
1435:
1432:
1428:
1421:
1416:
1412:
1401:
1388:
1383:
1380:
1377:
1373:
1369:
1363:
1359:
1353:
1348:
1345:
1342:
1338:
1332:
1329:
1324:
1319:
1315:
1286:
1266:
1243:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1194:
1190:
1170:
1150:
1130:
1110:
1087:
1063:
1038:
1035:
1032:
1028:
1024:
1021:
1018:
1013:
1010:
1007:
1003:
999:
994:
991:
988:
984:
963:
939:
934:
931:
928:
924:
920:
914:
910:
906:
903:
900:
896:
891:
888:
885:
881:
877:
871:
867:
863:
859:
854:
851:
848:
844:
840:
834:
830:
809:
788:
785:
782:
779:
776:
773:
770:
767:
747:
727:
707:
702:
698:
694:
691:
688:
685:
680:
676:
672:
667:
663:
659:
656:
652:
640:
639:
625:
621:
617:
613:
610:
607:
604:
599:
595:
591:
588:
585:
580:
577:
574:
570:
566:
563:
560:
555:
551:
547:
542:
539:
536:
532:
528:
525:
522:
517:
513:
509:
506:
500:
497:
494:
491:
486:
482:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
414:
410:
407:
404:
401:
398:
395:
392:
388:
384:
381:
378:
375:
355:
335:
332:
329:
326:
323:
300:
280:
260:
240:
225:
224:
208:
197:
185:
170:
169:
158:
155:
150:
146:
142:
139:
136:
133:
128:
124:
120:
115:
111:
107:
104:
101:
98:
75:
55:
43:
40:
15:
13:
10:
9:
6:
4:
3:
2:
1943:
1932:
1929:
1927:
1924:
1923:
1921:
1908:
1904:
1898:
1895:
1891:
1885:
1882:
1878:
1872:
1869:
1865:
1858:
1855:
1851:
1845:
1842:
1838:
1834:
1833:Technometrics
1828:
1826:
1822:
1819:
1815:
1810:
1807:
1800:
1798:
1794:
1780:
1772:
1754:
1750:
1724:
1719:
1711:
1704:
1700:
1694:
1690:
1685:
1679:
1674:
1671:
1668:
1664:
1658:
1655:
1650:
1645:
1640:
1636:
1628:
1627:
1626:
1623:
1605:
1601:
1591:
1577:
1557:
1528:
1523:
1517:
1513:
1509:
1505:
1497:
1490:
1486:
1480:
1476:
1471:
1464:
1459:
1456:
1453:
1449:
1439:
1436:
1433:
1426:
1419:
1414:
1410:
1402:
1386:
1378:
1371:
1367:
1361:
1357:
1351:
1346:
1343:
1340:
1336:
1330:
1327:
1322:
1317:
1313:
1305:
1304:
1303:
1300:
1284:
1264:
1255:
1231:
1211:
1208:
1205:
1199:
1192:
1188:
1148:
1108:
1101:
1085:
1077:
1061:
1052:
1033:
1026:
1022:
1019:
1016:
1008:
1001:
997:
989:
982:
961:
954:
937:
929:
922:
918:
912:
908:
904:
901:
898:
894:
886:
879:
875:
869:
865:
861:
857:
849:
842:
838:
832:
828:
807:
799:
786:
783:
780:
777:
774:
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768:
765:
700:
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686:
683:
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674:
670:
665:
661:
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597:
593:
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586:
583:
578:
575:
572:
568:
564:
558:
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545:
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530:
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511:
504:
498:
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484:
480:
472:
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463:
446:
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437:
434:
431:
425:
422:
419:
412:
408:
405:
399:
396:
393:
386:
382:
379:
376:
330:
327:
324:
312:
311:-level grid.
298:
291:-dimensional
278:
238:
230:
222:
219:, describing
206:
198:
183:
175:
174:
173:
156:
148:
144:
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134:
131:
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89:
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41:
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28:
26:
22:
1906:
1902:
1897:
1884:
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1836:
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1795:
1741:
1622:distribution
1592:
1548:
1256:
1232:
1053:
800:
641:
466:
464:
313:
229:trajectories
226:
199:the measure
176:the measure
171:
45:
29:
20:
18:
1742:The use of
42:Methodology
1920:Categories
1839:, 161–174.
1801:References
1100:parameters
271:is thus a
221:non-linear
1781:μ
1771:monotonic
1755:∗
1751:μ
1665:∑
1646:∗
1637:μ
1606:∗
1602:μ
1578:σ
1558:μ
1514:μ
1510:−
1450:∑
1437:−
1411:σ
1337:∑
1314:μ
1285:σ
1265:μ
1242:Δ
1209:−
1181:equal to
1169:Δ
1161:even and
1129:Δ
1020:…
902:…
778:…
746:Ω
726:Ω
624:Δ
606:−
587:…
562:Δ
538:−
524:…
423:−
397:−
354:Δ
259:Ω
207:σ
184:μ
1054:Usually
36:variance
974:points
642:where
1570:and
1277:and
1121:and
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951:by
366:in
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