1288:
775:
1283:{\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.}
62:
87:
488:. 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.
76:
2287:
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
202:
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
444:
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
430:
696:
1540:
1476:
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595:
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120:
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
1848:, 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
1763:
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
217:
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
1624:
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.
1408:
1350:
1376:
1317:
1915:
472:. 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.
1935:
2589:
207:
296:
708:
In 2020, Bernard, Kazzi, and
Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average
2392:
Klaassen, Chris A.J.; Mokveld, Philip J.; Van Es, Bert (2000). "Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions".
481:
641:
2487:
1685:
This bound is sharp, as it is reached by the equal-weights mixture of the uniform distribution on and the discrete distribution at {0}.
1844:
Depending on context, unimodal function may also refer to a function that has only one local minimum, rather than maximum. For example,
1487:
1423:
1868:
1633:
113:
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
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543:
199:
167:
1564:
211:
93:
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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711:
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68:
2012:
and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional
2127:
2003:
502:
224:
129:
617:
440:
One reason for the importance of distribution unimodality is that it allows for several important results. Several
1944:
if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function.
458:
218:
152:
can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability.
441:
141:
137:
110:
1352:), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when
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446:
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1941:
1694:
2253:
1888:
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145:
469:
2474:. Lecture Notes in Computer Science. Vol. 3669. Berlin, Heidelberg: Springer. pp. 887–898.
2232:
D. F. Vysochanskij, Y. I. Petunin (1980). "Justification of the 3σ rule for unimodal distributions".
2210:
2030:
2025:
2009:
1838:
1771:
485:
149:
1964:
1381:
125:
121:
50:
49:. More generally, unimodality means there is only a single highest value, somehow defined, of some
1322:
2556:
2521:
2362:
Rohatgi, Vijay K.; Székely, Gábor J. (1989). "Sharp inequalities between skewness and kurtosis".
2270:
2192:
1355:
1296:
632:
1837:. An example of a weakly unimodal function which is not strongly unimodal is every other row in
1697:, the definitions above do not apply. The definition of "unimodal" was extended to functions of
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2150:
2075:
2050:
114:
46:
2419:
1894:
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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.
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2513:
2475:
2401:
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1410:, which is the maximum distance between the median and the mean of a unimodal distribution.
183:
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2013:
171:
2446:
2467:
2251:
Sellke, T.M.; Sellke, S.H. (1997). "Chebyshev inequalities for unimodal distributions".
61:
1967:
1920:
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702:
2405:
2213:(1823). "Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior".
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2375:
2347:
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2196:
2172:
1855:
One important property of unimodal functions is that the extremum can be found using
30:"Unimodal" redirects here. For the company that promotes personal rapid transit, see
2560:
2420:"On the unimodality of METRIC Approximation subject to normally distributed demands"
166:
In continuous distributions, unimodality can be defined through the behavior of the
86:
2504:
John
Guckenheimer; Stewart Johnson (July 1990). "Distortion of S-Unimodal Maps".
2331:"Range Value-at-Risk bounds for unimodal distributions under partial information"
1698:
854:
425:{\displaystyle \dots ,p_{-2}-p_{-1},p_{-1}-p_{0},p_{0}-p_{1},p_{1}-p_{2},\dots }
38:
2315:
2053:
1764:
1627:
They derived a weaker inequality which applies to all unimodal distributions:
98:
2539:
Godfried T. Toussaint (June 1984). "Complexity, convexity, and unimodality".
2083:
2058:
1721:
1481:
It can also be shown that the mean and the mode lie within 3 of each other:
1767:
exists, but it does not succeed for every function despite its simplicity.
2300:"The Mean, Median, and Mode of Unimodal Distributions: A Characterization"
2078:
1849:
1775:
1555:
1551:
691:{\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {\frac {3}{5}}}}
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17:
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2525:
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2188:
1741:
163:
Other definitions of unimodality in distribution functions also exist.
124:, which are unimodal. Other examples of unimodal distributions include
31:
2215:
Commentationes
Societatis Regiae Scientiarum Gottingensis Recentiores
601:
2517:
2266:
75:
1535:{\displaystyle {\frac {|\mu -\theta |}{\sigma }}\leq {\sqrt {3}}.}
1471:{\displaystyle {\frac {|\nu -\theta |}{\sigma }}\leq {\sqrt {3}}.}
85:
74:
60:
1887:) is "S-unimodal" (often referred to as "S-unimodal map") if its
1675:{\displaystyle \gamma ^{2}-\kappa \leq {\frac {186}{125}}=1.488}
1417:: they lie within 3 ≈ 1.732 standard deviations of each other:
590:{\displaystyle |\nu -\mu |\leq {\sqrt {\frac {3}{4}}}\omega ,}
1293:
It is worth noting that the maximum distance is minimized at
71:
of normal distributions, an example of unimodal distribution.
2468:"Optimizing a 2D Function Satisfying Unimodality Properties"
2329:
Bernard, Carole; Kazzi, Rodrigue; Vanduffel, Steven (2020).
623:
It can be shown for a unimodal distribution that the median
1277:
496:
Gauss also showed in 1823 that for a unimodal distribution
2541:
International
Journal of Computer and Information Sciences
1606:{\displaystyle \gamma ^{2}-\kappa \leq {\frac {6}{5}}=1.2}
1558:
of a unimodal distribution are related by the inequality:
1413:
A similar relation holds between the median and the mode
206:
Criteria for unimodality can also be defined through the
1319:(i.e., when the symmetric quantile average is equal to
762:{\displaystyle {\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}}
155:
Figure 2 and Figure 3 illustrate bimodal distributions.
2470:. In Brodal, Gerth Stølting; Leonardi, Stefano (eds.).
432:
has exactly one sign change (when zeroes don't count).
2145:
Vladimirovich
Gnedenko and Victor Yu Korolev (1996).
1923:
1897:
1833:) can be reached for a continuous range of values of
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2102:
A.Ya. Khinchin (1938). "On unimodal distributions".
1947:
A more general definition, applicable to a function
1809:
for which it is weakly monotonically increasing for
459:
Chebyshev's inequality § Unimodal distributions
198:
being the mode. Note that under this definition the
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2427:Method in appendix D, Example in theorem 2 page 5
2234:Theory of Probability and Mathematical Statistics
2147:Random summation: limit theorems and applications
1789:, from the fact that the monotonicity implied is
1774:functions with a negative quadratic coefficient,
1269:
1141:
997:
27:Property of having a unique mode or maximum value
2288:Industrial and Applied Mathematics, Philadelphia
527:{\displaystyle \sigma \leq \omega \leq 2\sigma }
286:{\displaystyle \{p_{n}:n=\dots ,-1,0,1,\dots \}}
2387:
2385:
2261:(1). American Statistical Association: 34–40.
2173:"On the unimodality of discrete distributions"
1704:A common definition is as follows: a function
8:
2466:Demaine, Erik D.; Langerman, Stefan (2005).
2304:Theory of Probability & Its Applications
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1817:and weakly monotonically decreasing for
2041:
1770:Examples of unimodal functions include
1760:) and there are no other local maxima.
1994:)) is convex. Usually one would want
1781:The above is sometimes related to as
1550:Rohatgi and Szekely claimed that the
293:, is called unimodal if the sequence
194:, then the distribution is unimodal,
7:
2394:Statistics & Probability Letters
2364:Statistics & Probability Letters
2335:Insurance: Mathematics and Economics
144:. Among discrete distributions, the
2590:Theory of probability distributions
2447:"Mathematical Programming Glossary"
1825:. In that case, the maximum value
210:of the distribution or through its
2006:with nonsingular Jacobian matrix.
1869:successive parabolic interpolation
25:
1732:and monotonically decreasing for
103:unimodal probability distribution
57:Unimodal probability distribution
2348:10.1016/j.insmatheco.2020.05.013
168:cumulative distribution function
117:which is usual in statistics.
2298:Basu, S.; Dasgupta, A. (1997).
2177:Periodica Mathematica Hungarica
482:Vysochanskiï–Petunin inequality
476:Vysochanskiï–Petunin inequality
2110:(2). University of Tomsk: 1–7.
1509:
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872:
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82:A simple bimodal distribution.
1:
2406:10.1016/S0167-7152(00)00090-0
2104:Trams. Res. Inst. Math. Mech.
1403:{\displaystyle {\sqrt {3/5}}}
2376:10.1016/0167-7152(89)90035-7
2171:Medgyessy, P. (March 1972).
1345:{\displaystyle q_{0.5}=\nu }
468:A first important result is
69:Probability density function
2128:Encyclopedia of Mathematics
2004:continuously differentiable
1371:{\displaystyle \alpha =0.5}
1312:{\displaystyle \alpha =0.5}
635:of each other. In symbols,
212:Laplace–Stieltjes transform
2606:
1963:is unimodal if there is a
631:lie within (3/5) ≈ 0.7746
618:root mean square deviation
463:
456:
45:means possessing a unique
29:
2316:10.1137/S0040585X97975447
219:probability mass function
1805:if there exists a value
1803:weakly unimodal function
1378:, the bound is equal to
142:exponential distribution
138:chi-squared distribution
111:probability distribution
2123:"Unimodal distribution"
2121:Ushakov, N.G. (2001) ,
1955:) of a vector variable
1937:is the critical point.
1910:{\displaystyle x\neq c}
1846:local unimodal sampling
447:multimodal distribution
208:characteristic function
2585:Mathematical relations
2580:Functions and mappings
1942:computational geometry
1931:
1911:
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94:
83:
72:
2506:Annals of Mathematics
2472:Algorithms – ESA 2005
2254:American Statistician
2010:Quasiconvex functions
1932:
1912:
1889:Schwarzian derivative
1861:golden section search
1778:functions, and more.
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1608:
1546:Skewness and kurtosis
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492:Mode, median and mean
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170:(cdf). If the cdf is
146:binomial distribution
107:unimodal distribution
89:
78:
64:
2026:Bimodal distribution
1921:
1895:
1891:is negative for all
1772:quadratic polynomial
1740:. In that case, the
1634:
1620:is the kurtosis and
1565:
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712:
642:
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486:Chebyshev inequality
297:
225:
200:uniform distribution
150:Poisson distribution
122:normal distributions
2480:10.1007/11561071_78
1791:strong monotonicity
701:where | . | is the
633:standard deviations
126:Cauchy distribution
51:mathematical object
2553:10.1007/bf00979872
2189:10.1007/bf02018665
2076:Weisstein, Eric W.
2051:Weisstein, Eric W.
1927:
1907:
1785:strong unimodality
1716:if for some value
1672:
1603:
1532:
1468:
1400:
1368:
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688:
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470:Gauss's inequality
464:Gauss's inequality
422:
283:
95:
84:
73:
2508:. Second Series.
2489:978-3-540-31951-1
2031:Read's conjecture
1930:{\displaystyle c}
1857:search algorithms
1839:Pascal's triangle
1714:unimodal function
1689:Unimodal function
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159:Other definitions
16:(Redirected from
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2183:(1–4): 245–257.
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11:
5:
2603:
2601:
2593:
2592:
2587:
2582:
2572:
2571:
2567:
2566:
2547:(3): 197–217.
2531:
2495:
2488:
2458:
2438:
2411:
2400:(2): 131–135.
2381:
2370:(4): 297–299.
2354:
2321:
2310:(2): 210–223.
2290:
2280:
2243:
2224:
2202:
2163:
2155:
2137:
2113:
2106:(in Russian).
2091:
2066:
2040:
2038:
2035:
2034:
2033:
2028:
2021:
2018:
1968:differentiable
1926:
1906:
1903:
1900:
1876:
1873:
1865:ternary search
1690:
1687:
1683:
1682:
1671:
1668:
1663:
1660:
1655:
1652:
1649:
1644:
1640:
1614:
1613:
1602:
1599:
1594:
1591:
1586:
1583:
1580:
1575:
1571:
1547:
1544:
1543:
1542:
1531:
1526:
1521:
1516:
1511:
1507:
1504:
1501:
1497:
1479:
1478:
1467:
1462:
1457:
1452:
1447:
1443:
1440:
1437:
1433:
1397:
1393:
1389:
1367:
1364:
1361:
1341:
1338:
1333:
1329:
1308:
1305:
1302:
1291:
1290:
1278:
1272:
1267:
1261:
1258:
1253:
1250:
1246:
1242:
1239:
1231:
1227:
1218:
1215:
1209:
1206:
1202:
1189:
1175:
1172:
1169:
1166:
1161:
1158:
1148:
1147:
1144:
1139:
1133:
1130:
1125:
1120:
1117:
1111:
1107:
1104:
1096:
1092:
1082:
1079:
1076:
1072:
1068:
1063:
1060:
1057:
1045:
1031:
1028:
1025:
1022:
1017:
1014:
1004:
1003:
1000:
995:
991:
988:
983:
980:
974:
970:
967:
959:
955:
945:
942:
939:
935:
931:
926:
923:
920:
908:
895:
892:
886:
883:
880:
877:
874:
871:
867:
856:
855:
852:
848:
843:
839:
835:
832:
827:
821:
818:
815:
812:
809:
805:
801:
796:
792:
784:
769:and the mean,
756:
750:
747:
744:
741:
738:
734:
730:
725:
721:
703:absolute value
699:
698:
684:
681:
675:
670:
665:
661:
658:
655:
651:
627:and the mean
608:, the mean is
598:
597:
586:
583:
577:
574:
568:
564:
560:
557:
554:
550:
535:
534:
523:
520:
517:
514:
511:
508:
493:
490:
477:
474:
465:
462:
454:
451:
437:
434:
421:
418:
413:
409:
405:
400:
396:
392:
387:
383:
379:
374:
370:
366:
361:
357:
353:
348:
345:
341:
337:
332:
329:
325:
321:
316:
313:
309:
305:
302:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
238:
234:
230:
160:
157:
58:
55:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2602:
2591:
2588:
2586:
2583:
2581:
2578:
2577:
2575:
2562:
2558:
2554:
2550:
2546:
2542:
2535:
2532:
2527:
2523:
2519:
2515:
2512:(1): 71–130.
2511:
2507:
2499:
2496:
2491:
2485:
2481:
2477:
2473:
2469:
2462:
2459:
2448:
2442:
2439:
2428:
2421:
2415:
2412:
2407:
2403:
2399:
2395:
2388:
2386:
2382:
2377:
2373:
2369:
2365:
2358:
2355:
2349:
2344:
2340:
2336:
2332:
2325:
2322:
2317:
2313:
2309:
2305:
2301:
2294:
2291:
2284:
2281:
2276:
2272:
2268:
2264:
2260:
2256:
2255:
2247:
2244:
2239:
2235:
2228:
2225:
2220:
2216:
2212:
2206:
2203:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2167:
2164:
2158:
2156:0-8493-2875-6
2152:
2149:. CRC-Press.
2148:
2141:
2138:
2134:
2130:
2129:
2124:
2117:
2114:
2109:
2105:
2098:
2096:
2092:
2086:
2085:
2080:
2077:
2070:
2067:
2061:
2060:
2055:
2052:
2045:
2042:
2036:
2032:
2029:
2027:
2024:
2023:
2019:
2017:
2015:
2011:
2007:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1966:
1962:
1958:
1954:
1950:
1945:
1943:
1938:
1924:
1904:
1901:
1898:
1890:
1886:
1882:
1874:
1872:
1870:
1866:
1862:
1858:
1853:
1851:
1847:
1842:
1840:
1836:
1832:
1828:
1824:
1821: ≥
1820:
1816:
1813: ≤
1812:
1808:
1804:
1800:
1796:
1793:. A function
1792:
1788:
1779:
1777:
1773:
1768:
1766:
1761:
1759:
1755:
1751:
1747:
1743:
1739:
1736: ≥
1735:
1731:
1728: ≤
1727:
1723:
1722:monotonically
1719:
1715:
1711:
1707:
1702:
1700:
1696:
1688:
1686:
1669:
1666:
1661:
1658:
1653:
1650:
1647:
1642:
1638:
1630:
1629:
1628:
1625:
1623:
1619:
1600:
1597:
1592:
1589:
1584:
1581:
1578:
1573:
1569:
1561:
1560:
1559:
1557:
1553:
1545:
1529:
1524:
1519:
1514:
1505:
1502:
1499:
1484:
1483:
1482:
1465:
1460:
1455:
1450:
1441:
1438:
1435:
1420:
1419:
1418:
1416:
1411:
1395:
1391:
1387:
1365:
1362:
1359:
1339:
1336:
1331:
1327:
1306:
1303:
1300:
1270:
1265:
1259:
1256:
1251:
1248:
1244:
1240:
1237:
1225:
1216:
1213:
1207:
1204:
1200:
1187:
1173:
1170:
1167:
1164:
1159:
1156:
1142:
1137:
1131:
1128:
1123:
1118:
1115:
1109:
1105:
1102:
1090:
1080:
1077:
1074:
1070:
1066:
1061:
1058:
1055:
1043:
1029:
1026:
1023:
1020:
1015:
1012:
998:
993:
989:
986:
981:
978:
972:
968:
965:
953:
943:
940:
937:
933:
929:
924:
921:
918:
906:
893:
890:
881:
878:
875:
869:
865:
850:
846:
841:
837:
833:
830:
825:
816:
813:
810:
803:
799:
794:
790:
782:
772:
771:
770:
754:
745:
742:
739:
732:
728:
723:
719:
706:
704:
682:
679:
673:
668:
659:
656:
653:
638:
637:
636:
634:
630:
626:
621:
619:
615:
611:
607:
603:
584:
581:
575:
572:
566:
558:
555:
552:
540:
539:
538:
521:
518:
515:
512:
509:
506:
499:
498:
497:
491:
489:
487:
483:
475:
473:
471:
460:
452:
450:
448:
443:
435:
433:
419:
416:
411:
407:
403:
398:
394:
390:
385:
381:
377:
372:
368:
364:
359:
355:
351:
346:
343:
339:
335:
330:
327:
323:
319:
314:
311:
307:
303:
300:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
236:
232:
220:
215:
213:
209:
204:
201:
197:
193:
189:
185:
181:
177:
173:
169:
164:
158:
156:
153:
151:
147:
143:
139:
135:
134:-distribution
133:
127:
123:
118:
116:
112:
108:
104:
100:
92:
88:
81:
77:
70:
67:
63:
56:
54:
52:
48:
44:
40:
33:
19:
2544:
2540:
2534:
2509:
2505:
2498:
2471:
2461:
2450:. Retrieved
2441:
2430:. Retrieved
2426:
2414:
2397:
2393:
2367:
2363:
2357:
2338:
2334:
2324:
2307:
2303:
2293:
2283:
2258:
2252:
2246:
2237:
2233:
2227:
2218:
2214:
2211:Gauss, C. F.
2205:
2180:
2176:
2166:
2146:
2140:
2126:
2116:
2107:
2103:
2082:
2069:
2057:
2044:
2008:
1999:
1995:
1991:
1987:
1983:
1982:) such that
1979:
1975:
1971:
1960:
1956:
1952:
1948:
1946:
1939:
1884:
1880:
1878:
1854:
1843:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1782:
1780:
1769:
1762:
1757:
1753:
1749:
1745:
1737:
1733:
1729:
1725:
1717:
1713:
1709:
1705:
1703:
1699:real numbers
1692:
1684:
1626:
1621:
1617:
1615:
1549:
1480:
1414:
1412:
1292:
707:
700:
628:
624:
622:
613:
609:
605:
599:
536:
495:
479:
467:
453:Inequalities
442:inequalities
439:
216:
205:
195:
191:
187:
179:
175:
165:
162:
154:
131:
119:
106:
102:
96:
90:
79:
65:
42:
36:
1879:A function
1765:derivatives
1701:as well.
43:unimodality
39:mathematics
2574:Categories
2452:2020-03-29
2432:2013-08-28
2161:p. 31
2054:"Unimodal"
2037:References
1965:one-to-one
600:where the
457:See also:
130:Student's
99:statistics
2502:See e.g.
2197:119817256
2133:EMS Press
2084:MathWorld
2059:MathWorld
1902:≠
1744:value of
1695:functions
1654:≤
1651:κ
1648:−
1639:γ
1585:≤
1582:κ
1579:−
1570:γ
1520:≤
1515:σ
1506:θ
1503:−
1500:μ
1456:≤
1451:σ
1442:θ
1439:−
1436:ν
1360:α
1340:ν
1301:α
1241:∈
1238:α
1234:for
1214:−
1208:α
1174:α
1168:−
1160:α
1106:∈
1103:α
1099:for
1081:α
1062:α
1059:−
1030:α
1024:−
1016:α
969:∈
966:α
962:for
944:α
925:α
922:−
891:−
882:α
879:−
847:≤
842:σ
834:μ
831:−
817:α
814:−
795:α
746:α
743:−
724:α
674:≤
669:σ
660:μ
657:−
654:ν
582:ω
567:≤
559:μ
556:−
553:ν
522:σ
516:≤
513:ω
510:≤
507:σ
420:…
404:−
378:−
352:−
344:−
328:−
320:−
312:−
301:…
278:…
257:−
251:…
91:Figure 3.
80:Figure 2.
66:Figure 1.
2561:11577312
2341:: 9–24.
2240:: 25–36.
2020:See also
2002:) to be
1970:mapping
1959:is that
1917:, where
1859:such as
1850:extremum
1776:tent map
1720:, it is
1556:kurtosis
1552:skewness
18:Unimodal
2526:1971501
2275:2684690
1801:) is a
1742:maximum
1712:) is a
616:is the
184:concave
32:SkyTran
2559:
2524:
2486:
2273:
2195:
2153:
2079:"Mode"
1616:where
1192:
1184:
1048:
1040:
911:
903:
602:median
172:convex
2557:S2CID
2522:JSTOR
2423:(PDF)
2271:JSTOR
2193:S2CID
1752:) is
1670:1.488
612:and
109:is a
2484:ISBN
2151:ISBN
1554:and
537:and
186:for
182:and
174:for
148:and
140:and
115:mode
101:, a
47:mode
2549:doi
2514:doi
2510:132
2476:doi
2402:doi
2372:doi
2343:doi
2312:doi
2263:doi
2185:doi
1940:In
1867:or
1662:125
1659:186
1601:1.2
1366:0.5
1332:0.5
1307:0.5
604:is
105:or
97:In
37:In
2576::
2555:.
2545:13
2543:.
2520:.
2482:.
2425:.
2398:50
2396:.
2384:^
2366:.
2339:94
2337:.
2333:.
2308:41
2306:.
2302:.
2269:.
2259:51
2257:.
2238:21
2236:.
2217:.
2191:.
2179:.
2175:.
2131:,
2125:,
2094:^
2081:.
2056:.
2016:.
1974:=
1871:.
1863:,
1852:.
1841:.
705:.
449:.
221:,
214:.
136:,
128:,
53:.
41:,
2563:.
2551::
2528:.
2516::
2492:.
2478::
2455:.
2435:.
2408:.
2404::
2378:.
2374::
2368:8
2351:.
2345::
2318:.
2314::
2277:.
2265::
2221:.
2219:5
2199:.
2187::
2181:2
2159:.
2108:2
2087:.
2062:.
2000:Z
1998:(
1996:G
1992:Z
1990:(
1988:G
1986:(
1984:f
1980:Z
1978:(
1976:G
1972:X
1961:f
1957:X
1953:X
1951:(
1949:f
1925:c
1905:c
1899:x
1885:x
1883:(
1881:f
1835:x
1831:m
1829:(
1827:f
1823:m
1819:x
1815:m
1811:x
1807:m
1799:x
1797:(
1795:f
1758:m
1756:(
1754:f
1750:x
1748:(
1746:f
1738:m
1734:x
1730:m
1726:x
1718:m
1710:x
1708:(
1706:f
1667:=
1643:2
1622:γ
1618:κ
1598:=
1593:5
1590:6
1574:2
1530:.
1525:3
1510:|
1496:|
1466:.
1461:3
1446:|
1432:|
1415:θ
1396:5
1392:/
1388:3
1363:=
1337:=
1328:q
1304:=
1271:.
1266:]
1260:6
1257:1
1252:,
1249:0
1245:(
1226:2
1217:1
1205:9
1201:4
1188:+
1171:3
1165:4
1157:3
1143:,
1138:)
1132:6
1129:5
1124:,
1119:6
1116:1
1110:(
1091:2
1078:+
1075:3
1071:/
1067:1
1056:1
1044:+
1027:3
1021:4
1013:3
999:,
994:)
990:1
987:,
982:6
979:5
973:[
954:2
941:+
938:3
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930:1
919:1
907:+
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876:1
873:(
870:9
866:4
851:{
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808:(
804:q
800:+
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755:2
749:)
740:1
737:(
733:q
729:+
720:q
683:5
680:3
664:|
650:|
629:μ
625:ν
614:ω
610:μ
606:ν
585:,
576:4
573:3
563:|
549:|
519:2
417:,
412:2
408:p
399:1
395:p
391:,
386:1
382:p
373:0
369:p
365:,
360:0
356:p
347:1
340:p
336:,
331:1
324:p
315:2
308:p
304:,
281:}
275:,
272:1
269:,
266:0
263:,
260:1
254:,
248:=
245:n
242::
237:n
233:p
229:{
196:m
192:m
188:x
180:m
176:x
132:t
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
