1735:
997:
1730:{\displaystyle {\begin{aligned}&f_{\theta _{1}}(x)=f_{\theta _{2}}(x)\\\Longleftrightarrow {}&{\frac {1}{{\sqrt {2\pi }}\sigma _{1}}}\exp \left(-{\frac {1}{2\sigma _{1}^{2}}}(x-\mu _{1})^{2}\right)={\frac {1}{{\sqrt {2\pi }}\sigma _{2}}}\exp \left(-{\frac {1}{2\sigma _{2}^{2}}}(x-\mu _{2})^{2}\right)\\\Longleftrightarrow {}&{\frac {1}{\sigma _{1}^{2}}}(x-\mu _{1})^{2}+\ln \sigma _{1}={\frac {1}{\sigma _{2}^{2}}}(x-\mu _{2})^{2}+\ln \sigma _{2}\\\Longleftrightarrow {}&x^{2}\left({\frac {1}{\sigma _{1}^{2}}}-{\frac {1}{\sigma _{2}^{2}}}\right)-2x\left({\frac {\mu _{1}}{\sigma _{1}^{2}}}-{\frac {\mu _{2}}{\sigma _{2}^{2}}}\right)+\left({\frac {\mu _{1}^{2}}{\sigma _{1}^{2}}}-{\frac {\mu _{2}^{2}}{\sigma _{2}^{2}}}+\ln \sigma _{1}-\ln \sigma _{2}\right)=0\end{aligned}}}
986:
384:
779:
61:
if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameters must generate different
658:
2102:
266:
981:{\displaystyle {\mathcal {P}}={\Big \{}\ f_{\theta }(x)={\tfrac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2\sigma ^{2}}}(x-\mu )^{2}}\ {\Big |}\ \theta =(\mu ,\sigma ):\mu \in \mathbb {R} ,\,\sigma \!>0\ {\Big \}}.}
1002:
1917:
169:
89:. In some cases, even though a model is non-identifiable, it is still possible to learn the true values of a certain subset of the model parameters. In this case we say that the model is
720:
515:
254:
538:
2003:
1838:
764:
217:
1967:
193:
66:
of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the
93:. In other cases it may be possible to learn the location of the true parameter up to a certain finite region of the parameter space, in which case the model is
2018:
379:{\displaystyle P_{\theta _{1}}=P_{\theta _{2}}\quad \Rightarrow \quad \theta _{1}=\theta _{2}\quad \ {\text{for all }}\theta _{1},\theta _{2}\in \Theta .}
119:
1850:
435:(pdfs), then two pdfs should be considered distinct only if they differ on a set of non-zero measure (for example two functions ƒ
2269:"Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood"
2468:
517:
is equivalent to being able to learn the model's true parameter if the model can be observed indefinitely long. Indeed, if {
2119:) are jointly normal independent random variables with zero expected value and unknown variances, and only the variables (
2416:
2375:
722:
be invertible, we will also be able to find the true value of the parameter which generated given distribution
2397:
31:
692:
487:
226:
2513:
17:
653:{\displaystyle {\frac {1}{T}}\sum _{t=1}^{T}\mathbf {1} _{\{X_{t}\in A\}}\ {\xrightarrow {\text{a.s.}}}\ \Pr,}
2425:
Reiersøl, Olav (1950), "Identifiability of a linear relation between variables which are subject to error",
2242:
682:). Thus, with an infinite number of observations we will be able to find the true probability distribution
432:
2267:
Raue, A.; Kreutz, C.; Maiwald, T.; Bachmann, J.; Schilling, M.; Klingmuller, U.; Timmer, J. (2009-08-01).
529:
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2232:
35:
86:
2027:
1984:
1819:
745:
198:
105:
63:
1933:
1841:
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104:
can be referred to in a wider scope when a model is tested with experimental data sets, using
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770:
178:
82:
54:
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8:
2140:
2006:
767:
2601:
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2444:
679:
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2412:
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2097:{\displaystyle {\begin{cases}y=\beta x^{*}+\varepsilon ,\\x=x^{*}+\eta ,\end{cases}}}
172:
50:
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is restricted to be greater than zero, we conclude that the model is identifiable: ƒ
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2522:
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478:
94:
689:
in the model, and since the identifiability condition above requires that the map
2147:
cannot be learned, we can guarantee that it must lie somewhere in the interval (
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1744:
only when all its coefficients are equal to zero, which is only possible when |
34:. For the concept of identifiability in the area of system identification, see
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2534:
2504:
2237:
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2427:
2294:
2009:
27:
Statistical property which a model must satisfy to allow precise inference
2526:
2485:
2127:) are observed. Then this model is not identifiable, only the product βσ²
2605:
2448:
42:
2553:(2013). "Nonparametric Identification in Structural Economic Models".
484:
Identifiability of the model in the sense of invertibility of the map
2576:
Rothenberg, Thomas J. (1971). "Identification in
Parametric Models".
1923:
257:
100:
Aside from strictly theoretical exploration of the model properties,
2589:
2509:"The Identification Zoo: Meanings of Identification in Econometrics"
2440:
612:
2385:
164:{\displaystyle {\mathcal {P}}=\{P_{\theta }:\theta \in \Theta \}}
2206:
normally distributed, retaining only the independence condition
1912:{\displaystyle y=\beta 'x+\varepsilon ,\quad \mathrm {E} =0}
528:
is the sequence of observations from the model, then by the
393:
should correspond to distinct probability distributions: if
2090:
2490:
Identification of
Parametric Models from Experimental Data
2266:
2198:
If we abandon the normality assumption and require that
481:
zero — and thus cannot be considered as distinct pdfs).
826:
2021:
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73:
A model that fails to be identifiable is said to be
431:. If the distributions are defined in terms of the
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1997:
1961:
1911:
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163:
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1740:This expression is equal to zero for almost all
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2484:
2403:
2307:
389:This definition means that distinct values of
2521:(4). American Economic Association: 835–903.
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2362:
2345:
2319:
599:
580:
158:
133:
2575:
2396:, Handbook of Econometrics, Vol. 1, Ch.4,
1930:is identifiable if and only if the matrix
715:{\displaystyle \theta \mapsto P_{\theta }}
510:{\displaystyle \theta \mapsto P_{\theta }}
249:{\displaystyle \theta \mapsto P_{\theta }}
30:For the related problem in economics, see
2330:
2328:
2284:
1899:
1889:
954:
947:
2424:
2334:
2187:is the coefficient in OLS regression of
2135:is the variance of the latent regressor
2569:10.1146/annurev-economics-082912-110231
2549:
2218:, then the model becomes identifiable.
14:
2615:
2503:
2325:
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2143:model: although the exact value of
24:
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1990:
1938:
1882:
1825:
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370:
204:
182:
155:
125:
25:
2634:
1969:is invertible. Thus, this is the
2398:North-Holland Publishing Company
2139:). This is also an example of a
575:
32:Parameter identification problem
2497:
1880:
1772:. Since in the scale parameter
335:
311:
307:
2514:Journal of Economic Literature
2339:
2313:
2301:
2260:
1998:{\displaystyle {\mathcal {P}}}
1956:
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1833:{\displaystyle {\mathcal {P}}}
1441:
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922:
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819:
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759:{\displaystyle {\mathcal {P}}}
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644:
625:
494:
473:differ only at a single point
308:
233:
212:{\displaystyle {\mathcal {P}}}
13:
1:
2455:van der Vaart, A. W. (1998),
2286:10.1093/bioinformatics/btp358
2248:
433:probability density functions
111:
2253:
2243:Simultaneous equations model
1976:
1962:{\displaystyle \mathrm {E} }
1811:
737:
7:
2366:; Berger, Roger L. (2002),
2221:
732:
530:strong law of large numbers
57:to be possible. A model is
10:
2639:
2556:Annual Review of Economics
2461:Cambridge University Press
2411:(2nd ed.), Springer,
2409:Theory of Point Estimation
2355:
2308:Lehmann & Casella 1998
2233:Structural identifiability
477: = 1 — a set of
87:observationally equivalent
36:Structural identifiability
29:
2346:Casella & Berger 2002
663:for every measurable set
68:identification conditions
64:probability distributions
53:must satisfy for precise
1971:identification condition
1922:(where ′ denotes matrix
106:identifiability analysis
2488:; Pronzato, L. (1997),
2310:, Ch. 1, Definition 5.2
1842:linear regression model
188:{\displaystyle \Theta }
2407:; Casella, G. (1998),
2168:is the coefficient in
2098:
1999:
1963:
1926:). Then the parameter
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982:
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572:
511:
380:
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213:
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165:
91:partially identifiable
49:is a property which a
2457:Asymptotic Statistics
2392:Hsiao, Cheng (1983),
2368:Statistical Inference
2228:System identification
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2000:
1964:
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771:location-scale family
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175:with parameter space
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2527:10.1257/jel.20181361
2019:
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2007:errors-in-variables
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2623:Estimation theory
2470:978-0-521-49603-2
2279:(15): 1923–1929.
2005:is the classical
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16:(Redirected from
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2551:Matzkin, Rosa L.
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2370:(2nd ed.),
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2141:set identifiable
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95:set identifiable
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2479:Further reading
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2441:10.2307/1907835
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2364:Casella, George
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47:identifiability
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2584:(3): 577–591.
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2563:(1): 457–486.
2547:
2507:(2019-12-01).
2505:Lewbel, Arthur
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2435:(4): 375–389,
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2405:Lehmann, E. L.
2401:
2394:Identification
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1034:
1031:
1028:
1025:
1018:
1014:
1009:
1005:
1003:
989:
988:
977:
972:
964:
961:
957:
953:
949:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
910:
898:
894:
890:
887:
884:
881:
873:
869:
865:
861:
856:
852:
844:
839:
836:
830:
824:
821:
818:
815:
810:
806:
797:
792:
787:
753:
739:
736:
734:
731:
726:
709:
705:
701:
698:
686:
675:
661:
660:
649:
646:
643:
640:
635:
631:
627:
624:
611:
601:
598:
595:
590:
586:
582:
577:
570:
565:
562:
559:
555:
549:
546:
524:} ⊆
520:
504:
500:
496:
493:
471: ≤ 1
467:0 ≤
466:
462:) =
455:
448:0 ≤
447:
443:) =
436:
427:
423:
415:
411:
404:
397:
387:
386:
375:
372:
369:
364:
360:
356:
351:
347:
332:
328:
324:
319:
315:
310:
302:
298:
293:
289:
282:
278:
273:
243:
239:
235:
232:
206:
195:. We say that
184:
160:
157:
154:
151:
148:
143:
139:
135:
132:
127:
113:
110:
81:: two or more
79:unidentifiable
26:
18:Unidentifiable
9:
6:
4:
3:
2:
2635:
2624:
2621:
2620:
2618:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2579:
2574:
2570:
2566:
2562:
2558:
2557:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2515:
2510:
2506:
2502:
2501:
2491:
2487:
2483:
2482:
2472:
2466:
2462:
2458:
2453:
2450:
2446:
2442:
2438:
2434:
2430:
2429:
2423:
2420:
2418:0-387-98502-6
2414:
2410:
2406:
2402:
2399:
2395:
2390:
2387:
2383:
2379:
2377:0-534-24312-6
2373:
2369:
2365:
2361:
2360:
2348:, p. 583
2347:
2342:
2336:
2335:Reiersøl 1950
2331:
2329:
2321:
2316:
2309:
2304:
2296:
2292:
2287:
2282:
2278:
2274:
2270:
2263:
2259:
2244:
2241:
2239:
2238:Observability
2236:
2234:
2231:
2229:
2226:
2225:
2219:
2217:
2214: ⊥
2213:
2210: ⊥
2209:
2205:
2201:
2196:
2194:
2190:
2183:
2179:
2175:
2171:
2164:
2157:
2150:
2146:
2142:
2138:
2126:
2122:
2118:
2114:
2110:
2084:
2081:
2078:
2073:
2069:
2065:
2062:
2055:
2052:
2049:
2044:
2040:
2036:
2033:
2030:
2024:
2015:
2014:
2013:
2011:
2008:
1974:
1972:
1952:
1949:
1945:
1929:
1925:
1906:
1903:
1896:
1893:
1890:
1877:
1874:
1871:
1868:
1864:
1861:
1857:
1854:
1847:
1846:
1845:
1843:
1809:
1804:
1801: =
1797:
1789:
1780:
1775:
1768:
1761:
1754:
1747:
1743:
1720:
1717:
1713:
1707:
1703:
1699:
1696:
1693:
1688:
1684:
1680:
1677:
1674:
1667:
1662:
1658:
1652:
1647:
1643:
1637:
1630:
1625:
1621:
1615:
1610:
1606:
1599:
1595:
1591:
1583:
1578:
1574:
1568:
1564:
1558:
1551:
1546:
1542:
1536:
1532:
1525:
1521:
1518:
1515:
1511:
1503:
1498:
1494:
1490:
1485:
1478:
1473:
1469:
1465:
1459:
1453:
1449:
1432:
1428:
1424:
1421:
1418:
1413:
1403:
1399:
1395:
1392:
1382:
1377:
1373:
1369:
1364:
1359:
1355:
1351:
1348:
1345:
1340:
1330:
1326:
1322:
1319:
1309:
1304:
1300:
1296:
1279:
1273:
1263:
1259:
1255:
1252:
1241:
1236:
1232:
1228:
1224:
1219:
1215:
1211:
1208:
1200:
1196:
1190:
1187:
1181:
1176:
1172:
1166:
1156:
1152:
1148:
1145:
1134:
1129:
1125:
1121:
1117:
1112:
1108:
1104:
1101:
1093:
1089:
1083:
1080:
1074:
1055:
1045:
1041:
1036:
1032:
1026:
1016:
1012:
1007:
994:
993:
992:
975:
962:
959:
955:
951:
943:
940:
937:
931:
928:
925:
919:
916:
896:
888:
885:
882:
871:
867:
863:
859:
854:
850:
842:
837:
834:
828:
822:
816:
808:
804:
790:
776:
775:
774:
772:
769:
730:
725:
707:
703:
696:
685:
681:
674:
670:
667: ⊆
666:
647:
641:
638:
633:
629:
609:
596:
593:
588:
584:
568:
563:
560:
557:
553:
547:
544:
535:
534:
533:
531:
527:
523:
502:
498:
491:
482:
480:
476:
470:
465:
461:
451:
446:
442:
434:
426:
422:
414:
410:
403:
396:
392:
373:
367:
362:
358:
354:
349:
345:
341:for all
330:
326:
322:
317:
313:
300:
296:
291:
287:
280:
276:
271:
263:
262:
261:
259:
241:
237:
230:
222:
174:
152:
149:
146:
141:
137:
130:
109:
107:
103:
98:
96:
92:
88:
84:
80:
76:
71:
69:
65:
60:
56:
52:
48:
44:
37:
33:
19:
2581:
2578:Econometrica
2577:
2560:
2554:
2518:
2512:
2498:Econometrics
2489:
2456:
2432:
2428:Econometrica
2426:
2408:
2393:
2367:
2341:
2322:, p. 62
2315:
2303:
2276:
2272:
2262:
2215:
2211:
2207:
2203:
2199:
2197:
2192:
2188:
2181:
2177:
2173:
2162:
2155:
2148:
2144:
2136:
2131:is (where σ²
2124:
2120:
2116:
2112:
2108:
2106:
2010:linear model
1980:
1970:
1927:
1921:
1815:
1802:
1795:
1787:
1778:
1773:
1766:
1759:
1752:
1745:
1741:
1739:
990:
741:
723:
683:
672:
668:
664:
662:
525:
518:
483:
474:
468:
463:
459:
449:
444:
440:
424:
420:
412:
408:
407:, then also
401:
394:
390:
388:
221:identifiable
220:
115:
101:
99:
90:
78:
74:
72:
67:
59:identifiable
58:
46:
40:
2492:, Springer
2486:Walter, É.
2386:2001025794
2249:References
258:one-to-one
112:Definition
43:statistics
2598:0012-9682
2543:125792293
2535:0022-0515
2254:Citations
2161:), where
2082:η
2074:∗
2053:ε
2045:∗
2037:β
1977:Example 3
1924:transpose
1894:∣
1891:ε
1875:ε
1862:β
1812:Example 2
1704:σ
1700:
1694:−
1685:σ
1681:
1659:σ
1644:μ
1638:−
1622:σ
1607:μ
1575:σ
1565:μ
1559:−
1543:σ
1533:μ
1516:−
1495:σ
1486:−
1470:σ
1442:⟺
1429:σ
1425:
1400:μ
1396:−
1374:σ
1356:σ
1352:
1327:μ
1323:−
1301:σ
1288:⟺
1260:μ
1256:−
1233:σ
1220:−
1212:
1197:σ
1191:π
1153:μ
1149:−
1126:σ
1113:−
1105:
1090:σ
1084:π
1066:⟺
1042:θ
1013:θ
956:σ
944:∈
941:μ
932:σ
926:μ
917:θ
889:μ
886:−
868:σ
855:−
843:σ
838:π
809:θ
738:Example 1
708:θ
700:↦
697:θ
639:∈
594:∈
554:∑
503:θ
495:↦
492:θ
371:Θ
368:∈
359:θ
346:θ
327:θ
314:θ
309:⇒
297:θ
277:θ
242:θ
234:↦
231:θ
183:Θ
156:Θ
153:∈
150:θ
142:θ
55:inference
2617:Category
2295:19505944
2222:See also
1981:Suppose
1953:′
1865:′
733:Examples
610:→
2606:1913267
2449:1907835
2356:Sources
2107:where (
766:be the
678:is the
479:measure
2604:
2596:
2541:
2533:
2467:
2447:
2415:
2384:
2374:
2293:
2180:, and
1758:| and
966:
914:
904:
801:
768:normal
671:(here
620:
605:
337:
2602:JSTOR
2539:S2CID
2445:JSTOR
2202:were
1751:| = |
991:Then
676:{...}
454:and ƒ
171:be a
51:model
2594:ISSN
2531:ISSN
2465:ISBN
2413:ISBN
2382:LCCN
2372:ISBN
2291:PMID
2154:, 1÷
1816:Let
960:>
742:Let
614:a.s.
116:Let
85:are
2586:doi
2565:doi
2523:doi
2437:doi
2281:doi
2204:not
2191:on
2176:on
2170:OLS
1209:exp
1102:exp
256:is
219:is
77:or
41:In
2619::
2600:.
2592:.
2582:39
2580:.
2559:.
2537:.
2529:.
2519:57
2517:.
2511:.
2463:,
2459:,
2443:,
2433:18
2431:,
2380:,
2327:^
2289:.
2277:25
2275:.
2271:.
2216:x*
2200:x*
2195:.
2185:xy
2166:yx
2159:xy
2152:yx
2137:x*
2117:x*
2012::
1844::
1808:.
1794:⇔
1765:=
1697:ln
1678:ln
1422:ln
1349:ln
773::
729:.
623:Pr
532:,
260::
108:.
97:.
70:.
45:,
2608:.
2588::
2571:.
2567::
2561:5
2545:.
2525::
2439::
2297:.
2283::
2212:η
2208:ε
2193:y
2189:x
2182:β
2178:x
2174:y
2163:β
2156:β
2149:β
2145:β
2133:∗
2129:∗
2125:y
2123:,
2121:x
2115:,
2113:η
2111:,
2109:ε
2085:,
2079:+
2070:x
2066:=
2063:x
2056:,
2050:+
2041:x
2034:=
2031:y
2025:{
1991:P
1957:]
1950:x
1946:x
1943:[
1939:E
1928:β
1907:0
1904:=
1901:]
1897:x
1887:[
1883:E
1878:,
1872:+
1869:x
1858:=
1855:y
1826:P
1806:2
1803:θ
1799:1
1796:θ
1791:2
1788:θ
1782:1
1779:θ
1774:σ
1770:2
1767:μ
1763:1
1760:μ
1756:2
1753:σ
1749:1
1746:σ
1742:x
1721:0
1718:=
1714:)
1708:2
1689:1
1675:+
1668:2
1663:2
1653:2
1648:2
1631:2
1626:1
1616:2
1611:1
1600:(
1596:+
1592:)
1584:2
1579:2
1569:2
1552:2
1547:1
1537:1
1526:(
1522:x
1519:2
1512:)
1504:2
1499:2
1491:1
1479:2
1474:1
1466:1
1460:(
1454:2
1450:x
1433:2
1419:+
1414:2
1410:)
1404:2
1393:x
1390:(
1383:2
1378:2
1370:1
1365:=
1360:1
1346:+
1341:2
1337:)
1331:1
1320:x
1317:(
1310:2
1305:1
1297:1
1280:)
1274:2
1270:)
1264:2
1253:x
1250:(
1242:2
1237:2
1229:2
1225:1
1216:(
1201:2
1188:2
1182:1
1177:=
1173:)
1167:2
1163:)
1157:1
1146:x
1143:(
1135:2
1130:1
1122:2
1118:1
1109:(
1094:1
1081:2
1075:1
1059:)
1056:x
1053:(
1046:2
1037:f
1033:=
1030:)
1027:x
1024:(
1017:1
1008:f
976:.
971:}
963:0
952:,
948:R
938::
935:)
929:,
923:(
920:=
909:|
897:2
893:)
883:x
880:(
872:2
864:2
860:1
851:e
835:2
829:1
823:=
820:)
817:x
814:(
805:f
796:{
791:=
786:P
752:P
727:0
724:P
704:P
687:0
684:P
673:1
669:S
665:A
648:,
645:]
642:A
634:t
630:X
626:[
600:}
597:A
589:t
585:X
581:{
576:1
569:T
564:1
561:=
558:t
548:T
545:1
526:S
521:t
519:X
499:P
475:x
469:x
464:1
460:x
458:(
456:2
450:x
445:1
441:x
439:(
437:1
428:2
425:θ
421:P
419:≠
416:1
413:θ
409:P
405:2
402:θ
400:≠
398:1
395:θ
391:θ
374:.
363:2
355:,
350:1
331:2
323:=
318:1
301:2
292:P
288:=
281:1
272:P
238:P
205:P
159:}
147::
138:P
134:{
131:=
126:P
38:.
20:)
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