1277:
764:
1272:{\displaystyle {\frac {\left|{\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}-\mu \right|}{\sigma }}\leq \left\{{\begin{array}{cl}{\frac {{\sqrt{{\frac {4}{9(1-\alpha )}}-1}}{\text{ }}+{\text{ }}{\sqrt{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left({\frac {1}{6}},{\frac {5}{6}}\right)\!,\\{\frac {{\sqrt{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt{{\frac {4}{9\alpha }}-1}}}{2}}&{\text{for }}\alpha \in \left(0,{\frac {1}{6}}\right]\!.\end{array}}\right.}
51:
76:
477:. The Chebyshev inequality guarantees that in any probability distribution, "nearly all" the values are "close to" the mean value. The Vysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is continuous and unimodal. Further results were shown by Sellke and Sellke.
65:
2276:
Gauss C.F. Theoria
Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for
191:
is unimodal, as well as any other distribution in which the maximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Usually this definition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any single value is zero, while
433:
are given below which are only valid for unimodal distributions. Thus, it is important to assess whether or not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the article on
419:
685:
1529:
1465:
1669:
584:
1600:
756:
521:
280:
109:
If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates
1837:, a method for doing numerical optimization, is often demonstrated with such a function. It can be said that a unimodal function under this extension is a function with a single local
1752:
Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to be suitable for simple functions only. A general method based on
206:
Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of differences of the probabilities. A discrete distribution with a
1613:
is the skewness. Klaassen, Mokveld, and van Es showed that this only applies in certain settings, such as the set of unimodal distributions where the mode and mean coincide.
1397:
1339:
1365:
1306:
1904:
461:. Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends on unimodality.
1924:
2578:
196:
285:
697:
In 2020, Bernard, Kazzi, and
Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average
2381:
Klaassen, Chris A.J.; Mokveld, Philip J.; Van Es, Bert (2000). "Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions".
470:
630:
2476:
1674:
This bound is sharp, as it is reached by the equal-weights mixture of the uniform distribution on and the discrete distribution at {0}.
1833:
Depending on context, unimodal function may also refer to a function that has only one local minimum, rather than maximum. For example,
1476:
1412:
1857:
1622:
102:
which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of
2143:
532:
188:
156:
1553:
200:
82:
A bimodal distribution. Note that only the largest peak would correspond to a mode in the strict sense of the definition of mode
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700:
2121:
57:
2001:
and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional
2116:
1992:
491:
213:
118:
606:
429:
One reason for the importance of distribution unimodality is that it allows for several important results. Several
1933:
if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function.
447:
207:
141:
can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability.
430:
130:
126:
99:
1341:), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when
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435:
2111:
1930:
1683:
2242:
1877:
1849:
134:
458:
2463:. Lecture Notes in Computer Science. Vol. 3669. Berlin, Heidelberg: Springer. pp. 887–898.
2221:
D. F. Vysochanskij, Y. I. Petunin (1980). "Justification of the 3σ rule for unimodal distributions".
2199:
2019:
2014:
1998:
1827:
1760:
474:
138:
1953:
1370:
114:
110:
39:
38:. More generally, unimodality means there is only a single highest value, somehow defined, of some
1311:
2545:
2510:
2351:
Rohatgi, Vijay K.; Székely, Gábor J. (1989). "Sharp inequalities between skewness and kurtosis".
2259:
2181:
1344:
1285:
621:
1826:. An example of a weakly unimodal function which is not strongly unimodal is every other row in
1686:, the definitions above do not apply. The definition of "unimodal" was extended to functions of
2472:
2139:
2064:
2039:
103:
35:
2408:
1883:
1682:
As the term "modal" applies to data sets and probability distribution, and not in general to
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this definition allows for a non-zero probability, or an "atom of probability", at the mode.
2537:
2502:
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2390:
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1845:
1399:, which is the maximum distance between the median and the mean of a unimodal distribution.
172:
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2002:
160:
2435:
2456:
2240:
Sellke, T.M.; Sellke, S.H. (1997). "Chebyshev inequalities for unimodal distributions".
50:
1956:
1909:
1853:
691:
2394:
2202:(1823). "Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior".
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2364:
2336:
2319:
2185:
2161:
1844:
One important property of unimodal functions is that the extremum can be found using
19:"Unimodal" redirects here. For the company that promotes personal rapid transit, see
2549:
2409:"On the unimodality of METRIC Approximation subject to normally distributed demands"
155:
In continuous distributions, unimodality can be defined through the behavior of the
75:
2493:
John
Guckenheimer; Stewart Johnson (July 1990). "Distortion of S-Unimodal Maps".
2320:"Range Value-at-Risk bounds for unimodal distributions under partial information"
1687:
843:
414:{\displaystyle \dots ,p_{-2}-p_{-1},p_{-1}-p_{0},p_{0}-p_{1},p_{1}-p_{2},\dots }
27:
2304:
2042:
1753:
1616:
They derived a weaker inequality which applies to all unimodal distributions:
87:
2528:
Godfried T. Toussaint (June 1984). "Complexity, convexity, and unimodality".
2072:
2047:
1710:
1470:
It can also be shown that the mean and the mode lie within 3 of each other:
1756:
exists, but it does not succeed for every function despite its simplicity.
2289:"The Mean, Median, and Mode of Unimodal Distributions: A Characterization"
2067:
1838:
1764:
1544:
1540:
680:{\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {\frac {3}{5}}}}
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2541:
2514:
2263:
2177:
1730:
152:
Other definitions of unimodality in distribution functions also exist.
113:, which are unimodal. Other examples of unimodal distributions include
20:
2204:
Commentationes
Societatis Regiae Scientiarum Gottingensis Recentiores
590:
2506:
2255:
64:
1524:{\displaystyle {\frac {|\mu -\theta |}{\sigma }}\leq {\sqrt {3}}.}
1460:{\displaystyle {\frac {|\nu -\theta |}{\sigma }}\leq {\sqrt {3}}.}
74:
63:
49:
1876:) is "S-unimodal" (often referred to as "S-unimodal map") if its
1664:{\displaystyle \gamma ^{2}-\kappa \leq {\frac {186}{125}}=1.488}
1406:: they lie within 3 ≈ 1.732 standard deviations of each other:
579:{\displaystyle |\nu -\mu |\leq {\sqrt {\frac {3}{4}}}\omega ,}
1282:
It is worth noting that the maximum distance is minimized at
60:
of normal distributions, an example of unimodal distribution.
2457:"Optimizing a 2D Function Satisfying Unimodality Properties"
2318:
Bernard, Carole; Kazzi, Rodrigue; Vanduffel, Steven (2020).
612:
It can be shown for a unimodal distribution that the median
1266:
485:
Gauss also showed in 1823 that for a unimodal distribution
2530:
International
Journal of Computer and Information Sciences
1595:{\displaystyle \gamma ^{2}-\kappa \leq {\frac {6}{5}}=1.2}
1547:
of a unimodal distribution are related by the inequality:
1402:
A similar relation holds between the median and the mode
195:
Criteria for unimodality can also be defined through the
1308:(i.e., when the symmetric quantile average is equal to
751:{\displaystyle {\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}}
144:
Figure 2 and Figure 3 illustrate bimodal distributions.
2459:. In Brodal, Gerth Stølting; Leonardi, Stefano (eds.).
421:
has exactly one sign change (when zeroes don't count).
2134:
Vladimirovich
Gnedenko and Victor Yu Korolev (1996).
1912:
1886:
1822:) can be reached for a continuous range of values of
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703:
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2091:
A.Ya. Khinchin (1938). "On unimodal distributions".
1936:
A more general definition, applicable to a function
1798:
for which it is weakly monotonically increasing for
448:
Chebyshev's inequality § Unimodal distributions
187:
being the mode. Note that under this definition the
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2416:Method in appendix D, Example in theorem 2 page 5
2223:Theory of Probability and Mathematical Statistics
2136:Random summation: limit theorems and applications
1778:, from the fact that the monotonicity implied is
1763:functions with a negative quadratic coefficient,
1258:
1130:
986:
16:Property of having a unique mode or maximum value
2277:Industrial and Applied Mathematics, Philadelphia
516:{\displaystyle \sigma \leq \omega \leq 2\sigma }
275:{\displaystyle \{p_{n}:n=\dots ,-1,0,1,\dots \}}
2376:
2374:
2250:(1). American Statistical Association: 34–40.
2162:"On the unimodality of discrete distributions"
1693:A common definition is as follows: a function
8:
2455:Demaine, Erik D.; Langerman, Stefan (2005).
2293:Theory of Probability & Its Applications
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1806:and weakly monotonically decreasing for
2030:
1759:Examples of unimodal functions include
1749:) and there are no other local maxima.
1983:)) is convex. Usually one would want
1770:The above is sometimes related to as
1539:Rohatgi and Szekely claimed that the
282:, is called unimodal if the sequence
183:, then the distribution is unimodal,
7:
2383:Statistics & Probability Letters
2353:Statistics & Probability Letters
2324:Insurance: Mathematics and Economics
133:. Among discrete distributions, the
2579:Theory of probability distributions
2436:"Mathematical Programming Glossary"
1814:. In that case, the maximum value
199:of the distribution or through its
1995:with nonsingular Jacobian matrix.
1858:successive parabolic interpolation
14:
1721:and monotonically decreasing for
92:unimodal probability distribution
46:Unimodal probability distribution
2337:10.1016/j.insmatheco.2020.05.013
157:cumulative distribution function
106:which is usual in statistics.
2287:Basu, S.; Dasgupta, A. (1997).
2166:Periodica Mathematica Hungarica
471:Vysochanskiï–Petunin inequality
465:Vysochanskiï–Petunin inequality
2099:(2). University of Tomsk: 1–7.
1498:
1484:
1434:
1420:
873:
861:
808:
796:
737:
725:
652:
638:
551:
537:
71:A simple bimodal distribution.
1:
2395:10.1016/S0167-7152(00)00090-0
2093:Trams. Res. Inst. Math. Mech.
1392:{\displaystyle {\sqrt {3/5}}}
2365:10.1016/0167-7152(89)90035-7
2160:Medgyessy, P. (March 1972).
1334:{\displaystyle q_{0.5}=\nu }
457:A first important result is
58:Probability density function
2117:Encyclopedia of Mathematics
1993:continuously differentiable
1360:{\displaystyle \alpha =0.5}
1301:{\displaystyle \alpha =0.5}
624:of each other. In symbols,
201:Laplace–Stieltjes transform
2595:
1952:is unimodal if there is a
620:lie within (3/5) ≈ 0.7746
607:root mean square deviation
452:
445:
34:means possessing a unique
18:
2305:10.1137/S0040585X97975447
208:probability mass function
1794:if there exists a value
1792:weakly unimodal function
1367:, the bound is equal to
131:exponential distribution
127:chi-squared distribution
100:probability distribution
2112:"Unimodal distribution"
2110:Ushakov, N.G. (2001) ,
1944:) of a vector variable
1926:is the critical point.
1899:{\displaystyle x\neq c}
1835:local unimodal sampling
436:multimodal distribution
197:characteristic function
2574:Mathematical relations
2569:Functions and mappings
1931:computational geometry
1920:
1900:
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681:
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473:, a refinement of the
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276:
83:
72:
61:
2495:Annals of Mathematics
2461:Algorithms – ESA 2005
2243:American Statistician
1999:Quasiconvex functions
1921:
1901:
1878:Schwarzian derivative
1850:golden section search
1767:functions, and more.
1666:
1597:
1535:Skewness and kurtosis
1526:
1462:
1394:
1362:
1336:
1303:
1274:
753:
682:
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481:Mode, median and mean
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159:(cdf). If the cdf is
135:binomial distribution
96:unimodal distribution
78:
67:
53:
2015:Bimodal distribution
1910:
1884:
1880:is negative for all
1761:quadratic polynomial
1729:. In that case, the
1623:
1609:is the kurtosis and
1554:
1477:
1413:
1371:
1345:
1312:
1286:
765:
701:
631:
533:
492:
475:Chebyshev inequality
286:
214:
189:uniform distribution
139:Poisson distribution
111:normal distributions
2469:10.1007/11561071_78
1780:strong monotonicity
690:where | . | is the
622:standard deviations
115:Cauchy distribution
40:mathematical object
2542:10.1007/bf00979872
2178:10.1007/bf02018665
2065:Weisstein, Eric W.
2040:Weisstein, Eric W.
1916:
1896:
1774:strong unimodality
1705:if for some value
1661:
1592:
1521:
1457:
1389:
1357:
1331:
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1264:
748:
677:
576:
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459:Gauss's inequality
453:Gauss's inequality
411:
272:
84:
73:
62:
2497:. Second Series.
2478:978-3-540-31951-1
2020:Read's conjecture
1919:{\displaystyle c}
1846:search algorithms
1828:Pascal's triangle
1703:unimodal function
1678:Unimodal function
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148:Other definitions
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2172:(1–4): 245–257.
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2003:Euclidean spaces
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1864:Other extensions
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469:A second is the
425:Uses and results
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167: <
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2011:
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1713:increasing for
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791:
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609:from the mode.
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2581:
2576:
2571:
2561:
2560:
2556:
2555:
2536:(3): 197–217.
2520:
2484:
2477:
2447:
2427:
2400:
2389:(2): 131–135.
2370:
2359:(4): 297–299.
2343:
2310:
2299:(2): 210–223.
2279:
2269:
2232:
2213:
2191:
2152:
2144:
2126:
2102:
2095:(in Russian).
2080:
2055:
2029:
2027:
2024:
2023:
2022:
2017:
2010:
2007:
1957:differentiable
1915:
1895:
1892:
1889:
1865:
1862:
1854:ternary search
1679:
1676:
1672:
1671:
1660:
1657:
1652:
1649:
1644:
1641:
1638:
1633:
1629:
1603:
1602:
1591:
1588:
1583:
1580:
1575:
1572:
1569:
1564:
1560:
1536:
1533:
1532:
1531:
1520:
1515:
1510:
1505:
1500:
1496:
1493:
1490:
1486:
1468:
1467:
1456:
1451:
1446:
1441:
1436:
1432:
1429:
1426:
1422:
1386:
1382:
1378:
1356:
1353:
1350:
1330:
1327:
1322:
1318:
1297:
1294:
1291:
1280:
1279:
1267:
1261:
1256:
1250:
1247:
1242:
1239:
1235:
1231:
1228:
1220:
1216:
1207:
1204:
1198:
1195:
1191:
1178:
1164:
1161:
1158:
1155:
1150:
1147:
1137:
1136:
1133:
1128:
1122:
1119:
1114:
1109:
1106:
1100:
1096:
1093:
1085:
1081:
1071:
1068:
1065:
1061:
1057:
1052:
1049:
1046:
1034:
1020:
1017:
1014:
1011:
1006:
1003:
993:
992:
989:
984:
980:
977:
972:
969:
963:
959:
956:
948:
944:
934:
931:
928:
924:
920:
915:
912:
909:
897:
884:
881:
875:
872:
869:
866:
863:
860:
856:
845:
844:
841:
837:
832:
828:
824:
821:
816:
810:
807:
804:
801:
798:
794:
790:
785:
781:
773:
758:and the mean,
745:
739:
736:
733:
730:
727:
723:
719:
714:
710:
692:absolute value
688:
687:
673:
670:
664:
659:
654:
650:
647:
644:
640:
616:and the mean
597:, the mean is
587:
586:
575:
572:
566:
563:
557:
553:
549:
546:
543:
539:
524:
523:
512:
509:
506:
503:
500:
497:
482:
479:
466:
463:
454:
451:
443:
440:
426:
423:
410:
407:
402:
398:
394:
389:
385:
381:
376:
372:
368:
363:
359:
355:
350:
346:
342:
337:
334:
330:
326:
321:
318:
314:
310:
305:
302:
298:
294:
291:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
227:
223:
219:
149:
146:
47:
44:
15:
13:
10:
9:
6:
4:
3:
2:
2591:
2580:
2577:
2575:
2572:
2570:
2567:
2566:
2564:
2551:
2547:
2543:
2539:
2535:
2531:
2524:
2521:
2516:
2512:
2508:
2504:
2501:(1): 71–130.
2500:
2496:
2488:
2485:
2480:
2474:
2470:
2466:
2462:
2458:
2451:
2448:
2437:
2431:
2428:
2417:
2410:
2404:
2401:
2396:
2392:
2388:
2384:
2377:
2375:
2371:
2366:
2362:
2358:
2354:
2347:
2344:
2338:
2333:
2329:
2325:
2321:
2314:
2311:
2306:
2302:
2298:
2294:
2290:
2283:
2280:
2273:
2270:
2265:
2261:
2257:
2253:
2249:
2245:
2244:
2236:
2233:
2228:
2224:
2217:
2214:
2209:
2205:
2201:
2195:
2192:
2187:
2183:
2179:
2175:
2171:
2167:
2163:
2156:
2153:
2147:
2145:0-8493-2875-6
2141:
2138:. CRC-Press.
2137:
2130:
2127:
2123:
2119:
2118:
2113:
2106:
2103:
2098:
2094:
2087:
2085:
2081:
2075:
2074:
2069:
2066:
2059:
2056:
2050:
2049:
2044:
2041:
2034:
2031:
2025:
2021:
2018:
2016:
2013:
2012:
2008:
2006:
2004:
2000:
1996:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1955:
1951:
1947:
1943:
1939:
1934:
1932:
1927:
1913:
1893:
1890:
1887:
1879:
1875:
1871:
1863:
1861:
1859:
1855:
1851:
1847:
1842:
1840:
1836:
1831:
1829:
1825:
1821:
1817:
1813:
1810: ≥
1809:
1805:
1802: ≤
1801:
1797:
1793:
1789:
1785:
1782:. A function
1781:
1777:
1768:
1766:
1762:
1757:
1755:
1750:
1748:
1744:
1740:
1736:
1732:
1728:
1725: ≥
1724:
1720:
1717: ≤
1716:
1712:
1711:monotonically
1708:
1704:
1700:
1696:
1691:
1689:
1685:
1677:
1675:
1658:
1655:
1650:
1647:
1642:
1639:
1636:
1631:
1627:
1619:
1618:
1617:
1614:
1612:
1608:
1589:
1586:
1581:
1578:
1573:
1570:
1567:
1562:
1558:
1550:
1549:
1548:
1546:
1542:
1534:
1518:
1513:
1508:
1503:
1494:
1491:
1488:
1473:
1472:
1471:
1454:
1449:
1444:
1439:
1430:
1427:
1424:
1409:
1408:
1407:
1405:
1400:
1384:
1380:
1376:
1354:
1351:
1348:
1328:
1325:
1320:
1316:
1295:
1292:
1289:
1259:
1254:
1248:
1245:
1240:
1237:
1233:
1229:
1226:
1214:
1205:
1202:
1196:
1193:
1189:
1176:
1162:
1159:
1156:
1153:
1148:
1145:
1131:
1126:
1120:
1117:
1112:
1107:
1104:
1098:
1094:
1091:
1079:
1069:
1066:
1063:
1059:
1055:
1050:
1047:
1044:
1032:
1018:
1015:
1012:
1009:
1004:
1001:
987:
982:
978:
975:
970:
967:
961:
957:
954:
942:
932:
929:
926:
922:
918:
913:
910:
907:
895:
882:
879:
870:
867:
864:
858:
854:
839:
835:
830:
826:
822:
819:
814:
805:
802:
799:
792:
788:
783:
779:
771:
761:
760:
759:
743:
734:
731:
728:
721:
717:
712:
708:
695:
693:
671:
668:
662:
657:
648:
645:
642:
627:
626:
625:
623:
619:
615:
610:
608:
604:
600:
596:
592:
573:
570:
564:
561:
555:
547:
544:
541:
529:
528:
527:
510:
507:
504:
501:
498:
495:
488:
487:
486:
480:
478:
476:
472:
464:
462:
460:
449:
441:
439:
437:
432:
424:
422:
408:
405:
400:
396:
392:
387:
383:
379:
374:
370:
366:
361:
357:
353:
348:
344:
340:
335:
332:
328:
324:
319:
316:
312:
308:
303:
300:
296:
292:
289:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
225:
221:
209:
204:
202:
198:
193:
190:
186:
182:
178:
174:
170:
166:
162:
158:
153:
147:
145:
142:
140:
136:
132:
128:
124:
123:-distribution
122:
116:
112:
107:
105:
101:
97:
93:
89:
81:
77:
70:
66:
59:
56:
52:
45:
43:
41:
37:
33:
29:
22:
2533:
2529:
2523:
2498:
2494:
2487:
2460:
2450:
2439:. Retrieved
2430:
2419:. Retrieved
2415:
2403:
2386:
2382:
2356:
2352:
2346:
2327:
2323:
2313:
2296:
2292:
2282:
2272:
2247:
2241:
2235:
2226:
2222:
2216:
2207:
2203:
2200:Gauss, C. F.
2194:
2169:
2165:
2155:
2135:
2129:
2115:
2105:
2096:
2092:
2071:
2058:
2046:
2033:
1997:
1988:
1984:
1980:
1976:
1972:
1971:) such that
1968:
1964:
1960:
1949:
1945:
1941:
1937:
1935:
1928:
1873:
1869:
1867:
1843:
1832:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1771:
1769:
1758:
1751:
1746:
1742:
1738:
1734:
1726:
1722:
1718:
1714:
1706:
1702:
1698:
1694:
1692:
1688:real numbers
1681:
1673:
1615:
1610:
1606:
1604:
1538:
1469:
1403:
1401:
1281:
696:
689:
617:
613:
611:
602:
598:
594:
588:
525:
484:
468:
456:
442:Inequalities
431:inequalities
428:
205:
194:
184:
180:
176:
168:
164:
154:
151:
143:
120:
108:
95:
91:
85:
79:
68:
54:
31:
25:
1868:A function
1754:derivatives
1690:as well.
32:unimodality
28:mathematics
2563:Categories
2441:2020-03-29
2421:2013-08-28
2150:p. 31
2043:"Unimodal"
2026:References
1954:one-to-one
589:where the
446:See also:
119:Student's
88:statistics
2491:See e.g.
2186:119817256
2122:EMS Press
2073:MathWorld
2048:MathWorld
1891:≠
1733:value of
1684:functions
1643:≤
1640:κ
1637:−
1628:γ
1574:≤
1571:κ
1568:−
1559:γ
1509:≤
1504:σ
1495:θ
1492:−
1489:μ
1445:≤
1440:σ
1431:θ
1428:−
1425:ν
1349:α
1329:ν
1290:α
1230:∈
1227:α
1223:for
1203:−
1197:α
1163:α
1157:−
1149:α
1095:∈
1092:α
1088:for
1070:α
1051:α
1048:−
1019:α
1013:−
1005:α
958:∈
955:α
951:for
933:α
914:α
911:−
880:−
871:α
868:−
836:≤
831:σ
823:μ
820:−
806:α
803:−
784:α
735:α
732:−
713:α
663:≤
658:σ
649:μ
646:−
643:ν
571:ω
556:≤
548:μ
545:−
542:ν
511:σ
505:≤
502:ω
499:≤
496:σ
409:…
393:−
367:−
341:−
333:−
317:−
309:−
301:−
290:…
267:…
246:−
240:…
80:Figure 3.
69:Figure 2.
55:Figure 1.
2550:11577312
2330:: 9–24.
2229:: 25–36.
2009:See also
1991:) to be
1959:mapping
1948:is that
1906:, where
1848:such as
1839:extremum
1765:tent map
1709:, it is
1545:kurtosis
1541:skewness
2515:1971501
2264:2684690
1790:) is a
1731:maximum
1701:) is a
605:is the
173:concave
21:SkyTran
2548:
2513:
2475:
2262:
2184:
2142:
2068:"Mode"
1605:where
1181:
1173:
1037:
1029:
900:
892:
591:median
161:convex
2546:S2CID
2511:JSTOR
2412:(PDF)
2260:JSTOR
2182:S2CID
1741:) is
1659:1.488
601:and
98:is a
2473:ISBN
2140:ISBN
1543:and
526:and
175:for
171:and
163:for
137:and
129:and
104:mode
90:, a
36:mode
2538:doi
2503:doi
2499:132
2465:doi
2391:doi
2361:doi
2332:doi
2301:doi
2252:doi
2174:doi
1929:In
1856:or
1651:125
1648:186
1590:1.2
1355:0.5
1321:0.5
1296:0.5
593:is
94:or
86:In
26:In
2565::
2544:.
2534:13
2532:.
2509:.
2471:.
2414:.
2387:50
2385:.
2373:^
2355:.
2328:94
2326:.
2322:.
2297:41
2295:.
2291:.
2258:.
2248:51
2246:.
2227:21
2225:.
2206:.
2180:.
2168:.
2164:.
2120:,
2114:,
2083:^
2070:.
2045:.
2005:.
1963:=
1860:.
1852:,
1841:.
1830:.
694:.
438:.
210:,
203:.
125:,
117:,
42:.
30:,
2552:.
2540::
2517:.
2505::
2481:.
2467::
2444:.
2424:.
2397:.
2393::
2367:.
2363::
2357:8
2340:.
2334::
2307:.
2303::
2266:.
2254::
2210:.
2208:5
2188:.
2176::
2170:2
2148:.
2097:2
2076:.
2051:.
1989:Z
1987:(
1985:G
1981:Z
1979:(
1977:G
1975:(
1973:f
1969:Z
1967:(
1965:G
1961:X
1950:f
1946:X
1942:X
1940:(
1938:f
1914:c
1894:c
1888:x
1874:x
1872:(
1870:f
1824:x
1820:m
1818:(
1816:f
1812:m
1808:x
1804:m
1800:x
1796:m
1788:x
1786:(
1784:f
1747:m
1745:(
1743:f
1739:x
1737:(
1735:f
1727:m
1723:x
1719:m
1715:x
1707:m
1699:x
1697:(
1695:f
1656:=
1632:2
1611:γ
1607:κ
1587:=
1582:5
1579:6
1563:2
1519:.
1514:3
1499:|
1485:|
1455:.
1450:3
1435:|
1421:|
1404:θ
1385:5
1381:/
1377:3
1352:=
1326:=
1317:q
1293:=
1260:.
1255:]
1249:6
1246:1
1241:,
1238:0
1234:(
1215:2
1206:1
1194:9
1190:4
1177:+
1160:3
1154:4
1146:3
1132:,
1127:)
1121:6
1118:5
1113:,
1108:6
1105:1
1099:(
1080:2
1067:+
1064:3
1060:/
1056:1
1045:1
1033:+
1016:3
1010:4
1002:3
988:,
983:)
979:1
976:,
971:6
968:5
962:[
943:2
930:+
927:3
923:/
919:1
908:1
896:+
883:1
874:)
865:1
862:(
859:9
855:4
840:{
827:|
815:2
809:)
800:1
797:(
793:q
789:+
780:q
772:|
744:2
738:)
729:1
726:(
722:q
718:+
709:q
672:5
669:3
653:|
639:|
618:μ
614:ν
603:ω
599:μ
595:ν
574:,
565:4
562:3
552:|
538:|
508:2
406:,
401:2
397:p
388:1
384:p
380:,
375:1
371:p
362:0
358:p
354:,
349:0
345:p
336:1
329:p
325:,
320:1
313:p
304:2
297:p
293:,
270:}
264:,
261:1
258:,
255:0
252:,
249:1
243:,
237:=
234:n
231::
226:n
222:p
218:{
185:m
181:m
177:x
169:m
165:x
121:t
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
