185:
175:
165:
45:, defined using the mathematical possibility theory, used to represent the entire information associated to a measurement result. It has an internal possibility distribution and an external possibility distribution called membership functions. The internal distribution is the uncertainty contributions due to the systematic uncertainty and the bounds of the RFV are because of the random contributions. The external distribution gives the uncertainty bounds from all contributions.
1748:
54:
77:
An RFV can be seen in the figure. The external membership function is the distribution in blue and the internal membership function is the distribution in red. Both the membership functions are possibility distributions. Both the internal and external membership functions have a unitary value of
16:
In measurements, the measurement obtained can suffer from two types of uncertainties. The first is the random uncertainty which is due to the noise in the process and the measurement. The second contribution is due to the systematic uncertainty which may be present in the measuring instrument.
34:
introduced the concepts of fuzzy variables and fuzzy sets. Fuzzy variables are based on the theory of possibility and hence are possibility distributions. This makes them suitable to handle any type of uncertainty, i.e., both systematic and random contributions to the total uncertainty.
17:
Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed. But it can not be accurately known while using the instrument if there is a
633:
212:
is the internal distribution in the RFV which is the possibility distribution of the systematic contribution to the total uncertainty. This distribution can be built based on the information that is available about the measuring instrument and the process.
228:
But, in certain cases, it may be known that certain values have a higher or lower degrees of belief than certain other values. In this case, depending on the degrees of belief for the values, an appropriate possibility distribution could be constructed.
216:
The largest possible distribution is the uniform or rectangular possibility distribution. This means that every value in the specified interval is equally possible. This actually represents the state of total ignorance according to the
157:
The probability distribution can be modeled from the measurement data. Then, the probability distribution can be used to model an equivalent possibility distribution using the maximally specific probability-possibility transformation.
1711:
1509:
411:
27:
Statistical methods can be used to calculate the total uncertainty from both systematic and random contributions in a measurement. But, the computational complexity is very high and hence, are not desirable.
224:
This distribution is used for the systematic error when we have absolutely no idea about the systematic error except that it belongs to a particular interval of values. This is quite common in measurements.
543:
1617:
1563:
469:
677:
938:
906:
867:
835:
1414:
803:
2454:
Ferrero, Alessandro; Prioli, Marco; Salicone, Simona (2015). "Uncertainty propagation through non-linear measurement functions by means of joint Random-Fuzzy
Variables".
511:
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1295:
440:
2317:
1268:
1214:
1160:
1098:
1036:
2071:
2066:
1907:
Betta, Giovanni; Liguori, Consolatina; Pietrosanto, Antonio (2000-06-01). "Propagation of uncertainty in a discrete
Fourier transform algorithm".
2495:
Klement, Erich Peter; Mesiar, Radko; Pap, Endre (2004-04-01). "Triangular norms. Position paper I: basic analytical and algebraic properties".
1623:
1421:
2255:
1726:
A Random-Fuzzy variable is capable of giving a complete picture of the random and systematic contributions to the total uncertainty from the
139:
This is completely random in nature and is a normal probability distribution when several random contributions are combined according to the
1950:
Ferrero, A.; Lazzaroni, M.; Salicone, S. (2002). "A calibration procedure for a digital instrument for electric power quality measurement".
258:
2195:
Ferrero, A.; Salicone, S. (2003). "An innovative approach to the determination of uncertainty in measurements based on fuzzy variables".
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2340:
2293:
147:
132:
is the possibility distribution of the random contributions to the uncertainty. Any measurement instrument or process suffers from
2547:
184:
174:
2621:
2412:
1826:
164:
85:
which consists of just the internal membership function. Similarly, if there is no systematic error, then the RFV becomes a
2238:
Castillo, Oscar; Melin, Patricia; Kacprzyk, Janusz; Pedrycz, Witold (2007). "Type-2 Fuzzy Logic: Theory and
Applications".
1773:
628:{\displaystyle F_{\alpha }=\{a\,\vert \,\mu _{\rm {F}}(a)\geq \alpha \}\qquad {\textit {where}}\qquad 0\leq \alpha \leq 1}
42:
24:
This systematic error can be approximately modeled based on our past data about the measuring instrument and the process.
2005:
1569:
1515:
1783:
218:
89:
with just the random contributions and therefore, is just the possibility distribution of the random contributions.
2117:
Mauris, G.; Berrah, L.; Foulloy, L.; Haurat, A. (2000). "Fuzzy handling of measurement errors in instrumentation".
161:
Some common probability distributions and the corresponding possibility distributions can be seen in the figures.
21:
and if there is, how much? Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature.
2152:
Urbanski, Michał K.; Wa̧sowski, Janusz (2003-07-01). "Fuzzy approach to the theory of measurement inexactness".
1798:
1752:
Construction of an external membership function and the RFV from internal and random possibility distributions.
1869:
63:
A Random-fuzzy
Variable (RFV) is defined as a type 2 fuzzy variable which satisfies the following conditions:
73:
Both the internal and external functions have a unitary value for possibility to the same interval of values.
445:
2364:
KLIR†, GEORGE J.; PARVIZ, BEHZAD (1992-08-01). "Probability-Possibility
Transformations: A Comparison".
140:
2064:(January 1973). "Outline of a New Approach to the Analysis of Complex Systems and Decision Processes".
644:
1870:"Structured approach to estimate the measurement uncertainty in digital signal elaboration algorithms"
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2204:
2161:
2126:
1959:
1916:
911:
879:
840:
808:
1308:
697:
1778:
244:
After modeling the random and internal possibility distribution, the external membership function,
78:
possibility only in the rectangular part of the RFV. So, all three conditions have been satisfied.
2673:
2639:
2575:
2477:
2456:
2015 IEEE International
Instrumentation and Measurement Technology Conference (I2MTC) Proceedings
2430:
2311:
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1932:
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2014:
1967:
1924:
1881:
1819:
An introduction to error analysis : the study of uncertainties in physical measurements
18:
1273:
418:
146:
But, there can also be random contributions from other probability distributions such as a
2587:
2095:
2036:
70:
Both the internal and the external functions are modeled as possibility distributions(pd).
2208:
2165:
2130:
1963:
1920:
2533:
2401:
2061:
1996:
1747:
1744:) and the RFV from a random PD and an internal PD can be seen in the following figure.
31:
2173:
2019:
2000:
1928:
1219:
1165:
1103:
1041:
979:
97:
A Random-fuzzy variable can be constructed using an
Internal possibility distribution(
81:
If there are only systematic errors in the measurement, then the RFV simply becomes a
53:
2662:
2579:
2481:
527:
RFV can also be built from the internal and random distributions by considering the
221:
which means it represents a scenario in which there is maximum lack of information.
2265:
133:
1737:
An example for the construction of the corresponding external membership function(
940:
are the lower and upper bounds respectively of the internal membership function(
869:
are the lower and upper bounds respectively of the external membership function(
2463:
2508:
2377:
2303:
2079:
1885:
2571:
2563:
2516:
2385:
2350:
2224:
2181:
2087:
2028:
1979:
1936:
1893:
683:. So, this gives the upper and lower bounds of the fuzzy variable F for each
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2216:
1971:
1836:
1763:
86:
82:
2422:
2595:
2103:
2044:
67:
Both the internal and the external functions of the RFV can be identified.
2616:. Gupta, Madan M. ( ed.). New York, N.Y.: Van Nostrand Reinhold Co.
2538:
2247:
1706:{\displaystyle X_{d}^{\alpha }=X_{UI}^{\alpha }-(X_{UR}^{\alpha }-x^{*})}
1504:{\displaystyle X_{a}^{\alpha }=X_{LI}^{\alpha }-(x^{*}-X_{LR}^{\alpha })}
641:-cut is the set of values for which the value of the membership function
2555:
2333:
Introduction to
Probability and Statistics for Engineers and Scientists
2138:
1730:-cuts for any confidence level as the confidence level is nothing but
1768:
521:
406:{\displaystyle r_{\textit {external}}(x)=\sup _{x^{\prime }}T_{min}}
2240:
2007 IEEE International
Conference on Granular Computing (GRC 2007)
1746:
183:
173:
163:
52:
253:, of the RFV can be constructed by using the following equation:
2614:
Introduction to fuzzy arithmetic : theory and applications
694:-cut of an RFV, however, has 4 specific bounds and is given by
604:
455:
378:
328:
268:
1821:(2nd ed.). Sausalito, Calif.: University Science Books.
392:
348:
296:
2536:(September 1975). "Fuzzy logic and approximate reasoning".
1100:
for the random and internal distributions respectively.
189:
Triangular distribution in probability and possibility.
136:
contributions due to intrinsic noise or other effects.
1868:
Pietrosanto, A.; Betta, G.; Liguori, C. (1999-01-01).
1874:
IEE Proceedings - Science, Measurement and
Technology
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477:
448:
421:
261:
2197:
IEEE Transactions on Instrumentation and Measurement
2119:
IEEE Transactions on Instrumentation and Measurement
1952:
IEEE Transactions on Instrumentation and Measurement
972:. This gives the lower and upper bounds for the two
179:
Uniform distribution in probability and possibility.
2286:
Measuring uncertainty within the theory of evidence
169:
Normal distribution in probability and possibility.
2400:
2067:IEEE Transactions on Systems, Man, and Cybernetics
1705:
1611:
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861:
829:
797:
671:
627:
505:
471:, which is the peak in the membership function of
463:
434:
405:
531:-cuts of the two possibility distributions(PDs).
1612:{\displaystyle X_{c}^{\alpha }=X_{UI}^{\alpha }}
1558:{\displaystyle X_{b}^{\alpha }=X_{LI}^{\alpha }}
287:
2407:. Princeton, N.J.: Princeton University Press.
2279:
2277:
2275:
2335:(4th ed.). Burlington: Elsevier Science.
538:-cut of a fuzzy variable F can be defined as
233:The construction of the external distribution(
1817:Taylor, John R. (John Robert), 1939- (1997).
1297:is the mode of the fuzzy variable. Then, the
8:
2644:: CS1 maint: multiple names: authors list (
2435:: CS1 maint: multiple names: authors list (
1849:: CS1 maint: multiple names: authors list (
1162:can be again divided into two sub-intervals
598:
567:
560:
2648:) CS1 maint: numeric names: authors list (
2607:
2605:
2439:) CS1 maint: numeric names: authors list (
2316:: CS1 maint: location missing publisher (
2072:IEEE Systems, Man, and Cybernetics Society
1853:) CS1 maint: numeric names: authors list (
1723:which gives us the final plot of the RFV.
2528:
2526:
2458:. Pisa, Italy: IEEE. pp. 1723–1728.
2018:
1694:
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267:
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260:
2366:International Journal of General Systems
2056:
2054:
1991:
1989:
1719:-cuts are calculated for every value of
947:) which is a fuzzy variable on its own.
876:) which is a fuzzy variable on its own.
104:) and a random possibility distribution(
1809:
1301:-cut for the RFV for the same value of
2637:
2428:
2309:
1842:
950:To build the RFV, let us consider the
679:of the fuzzy variable is greater than
2612:Kaufmann, A. (Arnold), 1911- (1991).
7:
2288:. Prioli, Marco. Cham, Switzerland.
464:{\displaystyle r_{\textit {random}}}
2284:Salicone, Simona (23 April 2018).
654:
577:
14:
2403:A mathematical theory of evidence
672:{\displaystyle \mu _{\rm {F}}(a)}
1715:Using the above equations, the
933:{\displaystyle X_{c}^{\alpha }}
901:{\displaystyle X_{b}^{\alpha }}
862:{\displaystyle X_{d}^{\alpha }}
830:{\displaystyle X_{a}^{\alpha }}
609:
601:
2156:. Fundamental of Measurement.
1700:
1666:
1498:
1464:
1409:{\displaystyle RFV^{\alpha }=}
1403:
1331:
1257:
1223:
1203:
1169:
1149:
1107:
1087:
1045:
1025:
983:
798:{\displaystyle RFV^{\alpha }=}
792:
720:
666:
660:
589:
583:
400:
397:
384:
366:
334:
319:
280:
274:
1:
2399:Shafer, Glenn, 1946- (1976).
2174:10.1016/S0263-2241(03)00021-6
2020:10.1016/S0019-9958(65)90241-X
1929:10.1016/S0263-2241(99)00068-8
1774:Type-2 fuzzy sets and systems
2499:. Advances in Fuzzy Logic.
954:-cuts of the two PDs i.e.,
39:Random-fuzzy variable (RFV)
2690:
2464:10.1109/I2MTC.2015.7151540
506:{\displaystyle r_{random}}
195:The internal distribution(
2509:10.1016/j.fss.2003.06.007
2378:10.1080/03081079208945083
2331:Ross, Sheldon M. (2009).
2080:10.1109/TSMC.1973.5408575
2013:(3). San Diego: 338–353.
1799:Probability distribution
115:The random distribution(
2217:10.1109/TIM.2003.815993
2006:Information and Control
1972:10.1109/TIM.2002.803293
1886:10.1049/ip-smt:19990001
2497:Fuzzy Sets and Systems
1784:Dempster–Shafer theory
1754:
1707:
1613:
1559:
1505:
1410:
1291:
1264:
1210:
1156:
1094:
1032:
968:for the same value of
934:
902:
863:
831:
799:
673:
629:
507:
465:
436:
407:
191:
181:
171:
60:
1750:
1708:
1614:
1560:
1506:
1411:
1292:
1290:{\displaystyle x^{*}}
1265:
1211:
1157:
1095:
1033:
935:
903:
864:
832:
800:
674:
630:
508:
466:
437:
435:{\displaystyle x^{*}}
408:
187:
177:
167:
141:Central limit theorem
58:Random-Fuzzy Variable
56:
43:type 2 fuzzy variable
2248:10.1109/grc.2007.118
1624:
1570:
1516:
1422:
1309:
1274:
1220:
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1104:
1042:
980:
912:
880:
841:
809:
698:
645:
544:
475:
446:
419:
259:
148:uniform distribution
2209:2003ITIM...52.1174F
2166:2003Meas...34...67U
2131:2000ITIM...49...89M
1964:2002ITIM...51..716F
1921:2000Meas...27..231B
1779:Observational error
1686:
1662:
1641:
1608:
1587:
1554:
1533:
1497:
1460:
1439:
1416:can be defined by
1402:
1384:
1366:
1348:
1256:
1189:
1148:
1127:
1086:
1065:
1024:
1003:
976:-cuts. Let them be
929:
897:
858:
826:
791:
773:
755:
737:
637:So, essentially an
2556:10.1007/BF00485052
1794:Probability theory
1789:Possibility theory
1755:
1703:
1669:
1645:
1627:
1609:
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1007:
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898:
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827:
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795:
777:
759:
741:
723:
669:
625:
503:
461:
432:
403:
302:
219:theory of evidence
192:
182:
172:
152:gamma distribution
61:
2257:978-0-7695-3032-1
2139:10.1109/19.836316
606:
457:
380:
330:
286:
270:
2681:
2654:
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2328:
2322:
2321:
2315:
2307:
2281:
2270:
2269:
2235:
2229:
2228:
2203:(4): 1174–1181.
2192:
2186:
2185:
2149:
2143:
2142:
2114:
2108:
2107:
2058:
2049:
2048:
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1993:
1984:
1983:
1947:
1941:
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1293:
1288:
1286:
1285:
1269:
1267:
1266:
1263:{\displaystyle }
1261:
1255:
1250:
1235:
1234:
1215:
1213:
1212:
1209:{\displaystyle }
1207:
1202:
1201:
1188:
1183:
1161:
1159:
1158:
1155:{\displaystyle }
1153:
1147:
1142:
1126:
1121:
1099:
1097:
1096:
1093:{\displaystyle }
1091:
1085:
1080:
1064:
1059:
1037:
1035:
1034:
1031:{\displaystyle }
1029:
1023:
1018:
1002:
997:
939:
937:
936:
931:
928:
923:
907:
905:
904:
899:
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891:
868:
866:
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836:
834:
833:
828:
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802:
801:
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772:
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754:
749:
736:
731:
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631:
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607:
582:
581:
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501:
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352:
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333:
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318:
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301:
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273:
272:
271:
19:systematic error
2689:
2688:
2684:
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2680:
2679:
2678:
2659:
2658:
2657:
2636:
2624:
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2358:
2343:
2330:
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2325:
2308:
2296:
2283:
2282:
2273:
2258:
2242:. p. 145.
2237:
2236:
2232:
2194:
2193:
2189:
2151:
2150:
2146:
2116:
2115:
2111:
2060:
2059:
2052:
1995:
1994:
1987:
1949:
1948:
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1906:
1905:
1901:
1867:
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1318:
1307:
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978:
977:
966:
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874:
839:
838:
807:
806:
707:
696:
695:
648:
643:
642:
635:
571:
547:
542:
541:
522:triangular norm
520:is the minimum
518:
478:
473:
472:
449:
444:
443:
442:is the mode of
422:
417:
416:
413:
387:
372:
356:
343:
322:
303:
291:
262:
257:
256:
250:
242:
238:
210:
204:
200:
190:
180:
170:
130:
124:
120:
109:
102:
95:
59:
51:
12:
11:
5:
2687:
2685:
2677:
2676:
2671:
2661:
2660:
2656:
2655:
2622:
2601:
2522:
2487:
2472:
2446:
2413:
2391:
2372:(3): 291–310.
2356:
2341:
2323:
2294:
2271:
2256:
2230:
2187:
2144:
2109:
2050:
1985:
1958:(4): 716–722.
1942:
1915:(4): 231–239.
1899:
1860:
1827:
1808:
1806:
1803:
1802:
1801:
1796:
1791:
1786:
1781:
1776:
1771:
1766:
1759:
1756:
1751:
1740:
1702:
1697:
1693:
1689:
1684:
1679:
1676:
1672:
1668:
1665:
1660:
1655:
1652:
1648:
1644:
1639:
1634:
1630:
1620:
1606:
1601:
1598:
1594:
1590:
1585:
1580:
1576:
1566:
1552:
1547:
1544:
1540:
1536:
1531:
1526:
1522:
1512:
1500:
1495:
1490:
1487:
1483:
1479:
1474:
1470:
1466:
1463:
1458:
1453:
1450:
1446:
1442:
1437:
1432:
1428:
1418:
1405:
1400:
1395:
1391:
1387:
1382:
1377:
1373:
1369:
1364:
1359:
1355:
1351:
1346:
1341:
1337:
1333:
1330:
1325:
1321:
1317:
1314:
1284:
1280:
1259:
1254:
1249:
1246:
1242:
1238:
1233:
1229:
1225:
1205:
1200:
1196:
1192:
1187:
1182:
1179:
1175:
1171:
1151:
1146:
1141:
1138:
1134:
1130:
1125:
1120:
1117:
1113:
1109:
1089:
1084:
1079:
1076:
1072:
1068:
1063:
1058:
1055:
1051:
1047:
1027:
1022:
1017:
1014:
1010:
1006:
1001:
996:
993:
989:
985:
964:
957:
943:
927:
922:
918:
895:
890:
886:
872:
856:
851:
847:
824:
819:
815:
794:
789:
784:
780:
776:
771:
766:
762:
758:
753:
748:
744:
740:
735:
730:
726:
722:
719:
714:
710:
706:
703:
668:
665:
662:
656:
651:
624:
621:
618:
615:
612:
600:
597:
594:
591:
588:
585:
579:
574:
569:
565:
562:
559:
554:
550:
540:
516:
500:
497:
494:
491:
488:
485:
481:
452:
429:
425:
402:
399:
394:
390:
386:
375:
371:
368:
363:
359:
355:
350:
346:
342:
339:
336:
325:
321:
316:
313:
310:
306:
298:
294:
289:
285:
282:
279:
276:
265:
255:
248:
241:
236:
231:
208:
203:
198:
193:
188:
178:
168:
128:
123:
118:
113:
107:
100:
94:
91:
87:fuzzy variable
83:fuzzy variable
75:
74:
71:
68:
57:
50:
47:
13:
10:
9:
6:
4:
3:
2:
2686:
2675:
2672:
2670:
2667:
2666:
2664:
2651:
2647:
2641:
2633:
2629:
2625:
2619:
2615:
2608:
2606:
2602:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2540:
2535:
2529:
2527:
2523:
2518:
2514:
2510:
2506:
2502:
2498:
2491:
2488:
2483:
2479:
2475:
2473:9781479961146
2469:
2465:
2461:
2457:
2450:
2447:
2442:
2438:
2432:
2424:
2420:
2416:
2410:
2405:
2404:
2395:
2392:
2387:
2383:
2379:
2375:
2371:
2367:
2360:
2357:
2352:
2348:
2344:
2342:9780080919379
2338:
2334:
2327:
2324:
2319:
2313:
2305:
2301:
2297:
2295:9783319741390
2291:
2287:
2280:
2278:
2276:
2272:
2267:
2263:
2259:
2253:
2249:
2245:
2241:
2234:
2231:
2226:
2222:
2218:
2214:
2210:
2206:
2202:
2198:
2191:
2188:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2148:
2145:
2140:
2136:
2132:
2128:
2124:
2120:
2113:
2110:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2070:. SMC-3 (1).
2069:
2068:
2063:
2057:
2055:
2051:
2046:
2042:
2038:
2034:
2030:
2026:
2021:
2016:
2012:
2008:
2007:
2002:
1999:(June 1965).
1998:
1992:
1990:
1986:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1946:
1943:
1938:
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1903:
1900:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1864:
1861:
1856:
1852:
1846:
1838:
1834:
1830:
1824:
1820:
1813:
1810:
1804:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1761:
1757:
1749:
1745:
1743:
1735:
1733:
1729:
1724:
1722:
1718:
1695:
1691:
1687:
1682:
1677:
1674:
1670:
1663:
1658:
1653:
1650:
1646:
1642:
1637:
1632:
1628:
1604:
1599:
1596:
1592:
1588:
1583:
1578:
1574:
1550:
1545:
1542:
1538:
1534:
1529:
1524:
1520:
1493:
1488:
1485:
1481:
1477:
1472:
1468:
1461:
1456:
1451:
1448:
1444:
1440:
1435:
1430:
1426:
1417:
1398:
1393:
1389:
1385:
1380:
1375:
1371:
1367:
1362:
1357:
1353:
1349:
1344:
1339:
1335:
1328:
1323:
1319:
1315:
1312:
1304:
1300:
1282:
1278:
1252:
1247:
1244:
1240:
1236:
1231:
1227:
1198:
1194:
1190:
1185:
1180:
1177:
1173:
1144:
1139:
1136:
1132:
1128:
1123:
1118:
1115:
1111:
1082:
1077:
1074:
1070:
1066:
1061:
1056:
1053:
1049:
1020:
1015:
1012:
1008:
1004:
999:
994:
991:
987:
975:
971:
967:
960:
953:
948:
946:
925:
920:
916:
893:
888:
884:
875:
854:
849:
845:
822:
817:
813:
787:
782:
778:
774:
769:
764:
760:
756:
751:
746:
742:
738:
733:
728:
724:
717:
712:
708:
704:
701:
693:
688:
686:
682:
663:
649:
640:
622:
619:
616:
613:
610:
595:
592:
586:
572:
563:
557:
552:
548:
539:
537:
532:
530:
525:
523:
519:
498:
495:
492:
489:
486:
483:
479:
450:
427:
423:
388:
373:
369:
361:
357:
353:
344:
340:
337:
323:
314:
311:
308:
304:
292:
283:
277:
263:
254:
252:
251:
240:) and the RFV
239:
232:
230:
226:
222:
220:
214:
211:
201:
194:
186:
176:
166:
162:
159:
155:
153:
149:
144:
142:
137:
135:
131:
121:
114:
112:
110:
103:
92:
90:
88:
84:
79:
72:
69:
66:
65:
64:
55:
48:
46:
44:
40:
36:
33:
29:
25:
22:
20:
2613:
2543:
2537:
2500:
2496:
2490:
2455:
2449:
2402:
2394:
2369:
2365:
2359:
2332:
2326:
2285:
2239:
2233:
2200:
2196:
2190:
2160:(1): 67–74.
2157:
2153:
2147:
2125:(1): 89–93.
2122:
2118:
2112:
2065:
2010:
2004:
2001:"Fuzzy sets"
1955:
1951:
1945:
1912:
1908:
1902:
1880:(1): 21–26.
1877:
1873:
1863:
1818:
1812:
1738:
1736:
1731:
1727:
1725:
1720:
1716:
1714:
1302:
1298:
973:
969:
962:
955:
951:
949:
941:
870:
691:
689:
684:
680:
638:
636:
535:
533:
528:
526:
514:
414:
246:
245:
243:
234:
227:
223:
215:
206:
205:
196:
160:
156:
145:
138:
134:random error
126:
125:
116:
105:
98:
96:
93:Construction
80:
76:
62:
38:
37:
30:
26:
23:
15:
2669:Fuzzy logic
2550:: 407–428.
2534:Zadeh, L.A.
2503:(1): 5–26.
2154:Measurement
2062:Zadeh, L.A.
1997:Zadeh, L.A.
1909:Measurement
154:and so on.
2663:Categories
2623:0442008996
2588:0319.02016
2414:0691081751
2304:1032810109
2096:0273.93002
2037:0139.24606
1828:0935702423
1805:References
49:Definition
2674:Metrology
2640:cite book
2596:Q57275767
2572:714993477
2564:0039-7857
2517:0165-0114
2431:cite book
2386:0308-1079
2351:761646775
2312:cite book
2225:0018-9456
2182:0263-2241
2104:Q56083455
2088:1083-4419
2074:: 28–44.
2045:Q25938993
2029:0019-9958
1980:0018-9456
1937:0263-2241
1894:1350-2344
1845:cite book
1764:Fuzzy set
1696:∗
1688:−
1683:α
1664:−
1659:α
1638:α
1605:α
1584:α
1551:α
1530:α
1494:α
1478:−
1473:∗
1462:−
1457:α
1436:α
1399:α
1381:α
1363:α
1345:α
1324:α
1283:∗
1253:α
1232:∗
1199:∗
1186:α
1145:α
1124:α
1083:α
1062:α
1021:α
1000:α
926:α
894:α
855:α
823:α
788:α
770:α
752:α
734:α
713:α
650:μ
620:≤
617:α
614:≤
596:α
593:≥
573:μ
553:α
428:∗
393:′
362:∗
349:′
341:−
297:′
32:L.A.Zadeh
2632:24309785
2592:Wikidata
2580:46975216
2548:Springer
2539:Synthese
2482:22811201
2100:Wikidata
2041:Wikidata
1837:34150960
1758:See also
1741:external
965:internal
944:internal
873:external
379:internal
269:external
249:external
237:external
209:internal
199:internal
101:internal
2546:(3–4).
2423:1859710
2266:1942035
2205:Bibcode
2162:Bibcode
2127:Bibcode
1960:Bibcode
1917:Bibcode
2630:
2620:
2594:
2586:
2578:
2570:
2562:
2515:
2480:
2470:
2421:
2411:
2384:
2349:
2339:
2302:
2292:
2264:
2254:
2223:
2180:
2102:
2094:
2086:
2043:
2035:
2027:
1978:
1935:
1892:
1835:
1825:
1769:T-norm
1270:where
958:random
687:-cut.
456:random
415:where
329:random
129:random
119:random
108:random
2576:S2CID
2478:S2CID
2262:S2CID
605:where
41:is a
2650:link
2646:link
2628:OCLC
2618:ISBN
2568:OCLC
2560:ISSN
2513:ISSN
2468:ISBN
2441:link
2437:link
2419:OCLC
2409:ISBN
2382:ISSN
2347:OCLC
2337:ISBN
2318:link
2300:OCLC
2290:ISBN
2252:ISBN
2221:ISSN
2178:ISSN
2084:ISSN
2025:ISSN
1976:ISSN
1933:ISSN
1890:ISSN
1855:link
1851:link
1833:OCLC
1823:ISBN
1216:and
1038:and
961:and
908:and
837:and
690:The
513:and
2584:Zbl
2552:doi
2505:doi
2501:143
2460:doi
2374:doi
2244:doi
2213:doi
2170:doi
2135:doi
2092:Zbl
2076:doi
2033:Zbl
2015:doi
1968:doi
1925:doi
1882:doi
1878:146
1732:1-α
534:An
517:min
288:sup
111:).
2665::
2642:}}
2638:{{
2626:.
2604:^
2590:.
2582:.
2574:.
2566:.
2558:.
2544:30
2542:.
2525:^
2511:.
2476:.
2466:.
2433:}}
2429:{{
2417:.
2380:.
2370:21
2368:.
2345:.
2314:}}
2310:{{
2298:.
2274:^
2260:.
2250:.
2219:.
2211:.
2201:52
2199:.
2176:.
2168:.
2158:34
2133:.
2123:49
2121:.
2098:.
2090:.
2082:.
2053:^
2039:.
2031:.
2023:.
2009:.
2003:.
1988:^
1974:.
1966:.
1956:51
1954:.
1931:.
1923:.
1913:27
1911:.
1888:.
1876:.
1872:.
1847:}}
1843:{{
1831:.
1734:.
1305:,
805:.
524:.
150:,
143:.
2652:)
2634:.
2598:.
2554::
2519:.
2507::
2484:.
2462::
2443:)
2425:.
2388:.
2376::
2353:.
2320:)
2306:.
2268:.
2246::
2227:.
2215::
2207::
2184:.
2172::
2164::
2141:.
2137::
2129::
2106:.
2078::
2047:.
2017::
2011:8
1982:.
1970::
1962::
1939:.
1927::
1919::
1896:.
1884::
1857:)
1839:.
1739:r
1728:α
1721:α
1717:α
1701:)
1692:x
1678:R
1675:U
1671:X
1667:(
1654:I
1651:U
1647:X
1643:=
1633:d
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1589:=
1579:c
1575:X
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1535:=
1525:b
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1499:)
1489:R
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1482:X
1469:x
1465:(
1452:I
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1441:=
1431:a
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1404:]
1394:d
1390:X
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1368:,
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1350:,
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1332:[
1329:=
1320:V
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1195:x
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1170:[
1150:]
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1108:[
1088:]
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793:]
783:d
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721:[
718:=
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
