2037:
6418:
1666:
4190:
6428:
1200:
564:
93:
1444:
1923:
698:
distribution, which, analogously to the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations. In fact, the von Mises distribution is often known as the "circular normal" distribution because of its ease of use and its close
693:
is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable; however, for statistical
3919:
956:
124:
of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week,
2023:
The projected normal distribution is a circular distribution representing the direction of a random variable with multivariate normal distribution, obtained by radial projection of the variable over the unit (n-1)-sphere. Due to this, and unlike other commonly used circular distributions, it is not
3965:
3118:
with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular
3718:
379:
694:
purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the
2758:
The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.
1736:
3116:
806:
3355:
1341:
1661:{\displaystyle WC(\theta ;\theta _{0},\gamma )=\sum _{n=-\infty }^{\infty }{\frac {\gamma }{\pi (\gamma ^{2}+(\theta +2\pi n-\theta _{0})^{2})}}={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\theta _{0})}}}
2239:
4185:{\displaystyle P({\overline {C}},{\overline {S}})\,d{\overline {C}}\,d{\overline {S}}=P({\overline {R}},{\overline {\theta }})\,d{\overline {R}}\,d{\overline {\theta }}=\int _{\Gamma }\cdots \int _{\Gamma }\prod _{n=1}^{N}\left}
3795:
3119:
distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution. Note that for small
2105: = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For
3569:
352:
2475:
3784:
1195:{\displaystyle WN(\theta ;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\sum _{k=-\infty }^{\infty }\exp \left={\frac {1}{2\pi }}\vartheta \left({\frac {\theta -\mu }{2\pi }},{\frac {i\sigma ^{2}}{2\pi }}\right)}
650:
4349:
The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.
2854:
3496:
260:
3558:
559:{\displaystyle p_{w}({\boldsymbol {\theta }})=\sum _{k_{1}=-\infty }^{\infty }\cdots \sum _{k_{F}=-\infty }^{\infty }{p({\boldsymbol {\theta }}+2\pi k_{1}\mathbf {e} _{1}+\dots +2\pi k_{F}\mathbf {e} _{F})}}
3002:
2618:
3273:
3203:
3006:
905:
2726:
2898:
705:
1967:
1235:
1380:
1422:
3418:
4344:
3949:
3277:
1244:
5076:
4317:
4290:
4263:
2354:
2674:
2514:
1713:
2383:
2314:
2560:
2036:
933:
4358:
4213:
2262:
1686:
265:
177:
4236:
2285:
833:
3146:
2007:
1918:{\displaystyle f_{WL}(\theta ;\mu ,c)=\sum _{n=-\infty }^{\infty }{\sqrt {\frac {c}{2\pi }}}\,{\frac {e^{-c/2(\theta +2\pi n-\mu )}}{(\theta +2\pi n-\mu )^{3/2}}}}
1987:
670:
374:
4405:
5205:
4818:. In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Quantitative Biology, Shape Analysis, and Wavelets, pp. 57â60. Leeds, Leeds University Press
3914:{\displaystyle {\overline {R}}={\sqrt {{\overline {C}}^{2}+{\overline {S}}^{2}}}{\text{ and }}{\overline {\theta }}=\arctan({\overline {S}}/{\overline {C}}).}
2150:
2799:
6467:
5688:
3713:{\displaystyle {\overline {C}}={\frac {1}{N}}\sum _{n=1}^{N}\cos(\theta _{n}){\text{ and }}{\overline {S}}={\frac {1}{N}}\sum _{n=1}^{N}\sin(\theta _{n})}
186:
5596:
2075:
6383:
2913:
2121: = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von MisesâFisher distribution.
6249:
5461:
5220:
5069:
2394:
4476:"Hamelryck, T., Kent, J., Krogh, A. (2006) Sampling realistic protein conformations using local structural bias. PLoS Comput. Biol., 2(9): e131"
6144:
5908:
3729:
2385:
is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.
4583:
Mardia, KM. Taylor; CC; Subramaniam, GK. (2007). "Protein
Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data".
569:
108:
of such points for a large collection of protein structures. The statistical treatment of such data is in the realm of directional statistics.
5582:
5045:
5903:
5847:
5507:
5145:
5653:
6189:
5923:
5776:
5451:
5195:
3430:
6431:
5648:
3218:
6421:
6093:
6069:
5062:
3507:
6290:
5918:
2060:
6167:
6128:
6100:
6074:
5992:
5341:
5089:
5019:
4991:
4970:
4944:
4918:
860:
6278:
6244:
6110:
6105:
5950:
5758:
5456:
5210:
4442:
2859:
2068:
6457:
6452:
6028:
5941:
5913:
5822:
5771:
5745:
5643:
5426:
5391:
4533:
2568:
6042:
5959:
5796:
5543:
5421:
5396:
5260:
5255:
5250:
4954:
4928:
5720:
5230:
5225:
6358:
6224:
5932:
5781:
5713:
5698:
5591:
5565:
5497:
5336:
5167:
5152:
4828:
Boomsma, Wouter; Mardia, Kanti V.; Taylor, Charles C.; Ferkinghoff-Borg, Jesper; Krogh, Anders; Hamelryck, Thomas (2008).
4408:
2018:
848:
5875:
2770:
may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.
2480:
The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.
6254:
6194:
6184:
5801:
5502:
5361:
4981:
2317:
1724:
144:
5603:
5346:
5275:
3151:
6462:
6239:
6234:
6179:
6115:
5880:
5658:
5555:
5140:
4447:
1432:
944:
62:
6059:
5867:
6373:
6149:
5968:
5750:
5703:
5572:
5548:
5528:
5371:
5245:
5125:
4962:
4936:
3357:
with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.
2682:
6378:
5321:
6162:
6123:
5997:
5834:
5678:
5623:
5521:
5485:
5356:
5037:
4815:
4780:
Krieger Lassen, N. C.; Juul Jensen, D.; Conradsen, K. (1994). "On the statistical analysis of orientation data".
6064:
5852:
5618:
5577:
5492:
5446:
5386:
5351:
5240:
5135:
5085:
1928:
1205:
5177:
1346:
1385:
6363:
6305:
5976:
5763:
5673:
5628:
5613:
5533:
5431:
5381:
5376:
5157:
3373:
4322:
3927:
6229:
6088:
5984:
5791:
5235:
5215:
5120:
2741:
2045:
684:
356:
This concept can be extended to the multivariate context by an extension of the simple sum to a number of
74:
6353:
6310:
6154:
5829:
5683:
5663:
5560:
5130:
4354:
3111:{\displaystyle {\overline {S}}(z)={\sqrt {\ln(1/{\overline {R}}^{2})}}={\sqrt {-2\ln({\overline {R}})}}}
138:
4628:"Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold"
6403:
6398:
6393:
6388:
6325:
6295:
6174:
5817:
5708:
5608:
5311:
5270:
5265:
5162:
4841:
4789:
4487:
4452:
4295:
4268:
4241:
2090:
180:
46:
2323:
801:{\displaystyle f(\theta ;\mu ,\kappa )={\frac {e^{\kappa \cos(\theta -\mu )}}{2\pi I_{0}(\kappa )}}}
6337:
5862:
5842:
5812:
5786:
5740:
5668:
5480:
5416:
4702:
2629:
1202:
where ÎŒ and Ï are the mean and standard deviation of the unwrapped distribution, respectively and
82:
6368:
5857:
5638:
5633:
5538:
5475:
5470:
5326:
5316:
5200:
4762:
4657:
4608:
2486:
1691:
2359:
2290:
2522:
912:
183:
around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable
6266:
5693:
5436:
5366:
5331:
5280:
5041:
5015:
4987:
4966:
4940:
4914:
4869:
4649:
4600:
4515:
2767:
2049:
4198:
3350:{\displaystyle {\overline {\delta }}={\frac {1-{{\overline {R}}_{2}}}{2{\overline {R}}^{2}}}}
2247:
1671:
5441:
5115:
4859:
4849:
4797:
4754:
4718:
4684:
4639:
4592:
4505:
4495:
4429:
2779:
2737:
2079:
150:
86:
4218:
2267:
811:
17:
5007:
3122:
2129:
2083:
1992:
1336:{\displaystyle \vartheta (\theta ,\tau )=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}}
836:
66:
54:
4845:
4793:
4561:
4491:
2234:{\displaystyle m_{n}=\operatorname {E} (z^{n})=\int _{\Gamma }P(\theta )z^{n}\,d\theta }
5514:
4910:
4864:
4829:
4510:
4475:
2133:
1972:
1238:
695:
655:
359:
126:
113:
4573:
Fisher, RA (1953) Dispersion on a sphere. Proc. Roy. Soc. London Ser. A., 217, 295â305
4364:
6446:
6137:
5885:
5172:
4661:
4626:
Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava (September 2020).
4596:
4423:
2753:
4766:
4612:
2144:
The raw vector (or trigonometric) moments of a circular distribution are defined as
5054:
2109: = 4, the Bingham distribution is a distribution over the space of unit
2101: â 1)-dimensional sphere with the antipodes identified. For example, if
4739:
4500:
2117:). Since a versor corresponds to a rotation matrix, the Bingham distribution for
4816:
Using the FisherâBingham distribution in stochastic models for protein structure
2744:
may be defined for both the population and a sample drawn from that population.
2110:
2025:
50:
4758:
347:{\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}.}
4801:
4688:
42:
4723:
4706:
4653:
5034:
4854:
105:
81:. More generally, directional statistics deals with observations on compact
4873:
4740:"Fitting mixtures of Kent distributions to aid in joint set identification"
4604:
4519:
2470:{\displaystyle {\overline {m}}_{n}={\frac {1}{N}}\sum _{i=1}^{N}z_{i}^{n}.}
2040:
Three points sets sampled from different Kent distributions on the sphere.
3779:{\displaystyle {\overline {z}}={\overline {R}}e^{i{\overline {\theta }}}}
2053:
645:{\displaystyle \mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}}
4357:
may be applied to the distribution of the sample means. (main article:
2849:{\displaystyle {\overline {\operatorname {Var} (z)}}=1-{\overline {R}}}
2125:
121:
97:
4644:
4627:
92:
2763:
2114:
101:
4885:
4883:
3491:{\displaystyle {\overline {z}}={\frac {1}{N}}\sum _{n=1}^{N}z_{n}}
2064:
675:
The following sections show some relevant circular distributions.
255:{\displaystyle \theta =x_{w}=x{\bmod {2}}\pi \ \ \in (-\pi ,\pi ]}
117:
91:
3553:{\displaystyle {\overline {z}}={\overline {C}}+i{\overline {S}}}
5058:
2623:
In addition, the lengths of the higher moments are defined as:
4830:"A generative, probabilistic model of local protein structure"
2997:{\displaystyle S(z)={\sqrt {\ln(1/R^{2})}}={\sqrt {-2\ln(R)}}}
2082:, and can be used to construct probability distributions over
4419:
For cyclic data â (e.g., is it uniformly distributed) :
2706:
213:
4469:
4467:
100:
can be parameterized as a sequence of points on the unit
4474:
Hamelryck, Thomas; Kent, John T.; Krogh, Anders (2006).
2613:{\displaystyle \theta _{n}=\operatorname {Arg} (m_{n}).}
1925:
where the value of the summand is taken to be zero when
2728:. The lengths of all moments will lie between 0 and 1.
2679:
while the angular parts of the higher moments are just
4534:
Directional features in online handwriting recognition
2796:. For the sample the circular variance is defined as:
4889:
4707:"An Antipodally Symmetric Distribution on the Sphere"
4367:
4325:
4298:
4271:
4244:
4221:
4201:
3968:
3930:
3798:
3732:
3572:
3510:
3433:
3376:
3280:
3221:
3154:
3125:
3009:
2916:
2862:
2802:
2685:
2632:
2571:
2525:
2489:
2397:
2362:
2326:
2293:
2270:
2250:
2153:
1995:
1975:
1931:
1739:
1694:
1674:
1447:
1388:
1349:
1247:
1208:
959:
915:
863:
814:
708:
658:
572:
382:
376:
sums that cover all dimensions in the feature space:
362:
268:
189:
153:
4544:
4542:
120:, so that for example 180 degrees is not a sensible
6346:
6304:
6205:
6041:
6019:
6010:
5894:
5729:
5405:
5302:
5293:
5186:
5106:
5097:
4238:and the integral is subject to the constraint that
4399:
4338:
4311:
4284:
4257:
4230:
4207:
4184:
3943:
3913:
3778:
3712:
3552:
3490:
3412:
3349:
3267:
3197:
3140:
3110:
2996:
2892:
2848:
2720:
2668:
2612:
2554:
2508:
2469:
2377:
2348:
2308:
2279:
2256:
2233:
2001:
1981:
1961:
1917:
1707:
1680:
1660:
1416:
1374:
1335:
1229:
1194:
927:
899:
827:
800:
664:
644:
558:
368:
346:
254:
171:
3268:{\displaystyle \delta ={\frac {1-R_{2}}{2R^{2}}}}
3198:{\displaystyle S(z)^{2}=2\operatorname {Var} (z)}
2784:The most common measures of circular spread are:
2097:dimensions, or equivalently, over points on the (
699:relationship to the wrapped normal distribution.
129:in molecules, orientations, rotations and so on.
4359:Central limit theorem for directional statistics
4834:Proceedings of the National Academy of Sciences
4738:Peel, D.; Whiten, WJ.; McLachlan, GJ. (2001).
900:{\displaystyle U(\theta )={\frac {1}{2\pi }}.}
853:The probability density function (pdf) of the
5070:
4562:The FisherâBingham distribution on the sphere
2721:{\displaystyle (n\theta _{n}){\bmod {2}}\pi }
2032:Distributions on higher-dimensional manifolds
8:
4980:Jammalamadaka, S. Rao; Sengupta, A. (2001).
4361:). It can be shown that the distribution of
2124:These distributions are for example used in
2893:{\displaystyle \operatorname {Var} (z)=1-R}
6016:
5299:
5103:
5077:
5063:
5055:
4675:Downs (1972). "Orientational statistics".
702:The pdf of the von Mises distribution is:
5014:(2nd ed.). John Wiley and Sons Ltd.
4863:
4853:
4722:
4643:
4509:
4499:
4384:
4371:
4366:
4326:
4324:
4299:
4297:
4272:
4270:
4245:
4243:
4220:
4200:
4171:
4163:
4154:
4133:
4122:
4112:
4099:
4082:
4078:
4068:
4064:
4051:
4038:
4019:
4015:
4005:
4001:
3988:
3975:
3967:
3931:
3929:
3895:
3890:
3880:
3858:
3853:
3845:
3835:
3825:
3815:
3812:
3799:
3797:
3764:
3760:
3746:
3733:
3731:
3701:
3682:
3671:
3657:
3644:
3639:
3630:
3611:
3600:
3586:
3573:
3571:
3540:
3524:
3511:
3509:
3482:
3472:
3461:
3447:
3434:
3432:
3402:
3394:
3381:
3375:
3338:
3328:
3315:
3305:
3303:
3294:
3281:
3279:
3256:
3241:
3228:
3220:
3168:
3153:
3124:
3093:
3076:
3062:
3052:
3046:
3032:
3010:
3008:
2969:
2955:
2946:
2932:
2915:
2861:
2836:
2803:
2801:
2709:
2705:
2696:
2684:
2661:
2655:
2646:
2637:
2631:
2598:
2576:
2570:
2547:
2541:
2532:
2524:
2500:
2488:
2458:
2453:
2443:
2432:
2418:
2409:
2399:
2396:
2361:
2337:
2325:
2292:
2269:
2249:
2224:
2218:
2196:
2180:
2158:
2152:
1994:
1974:
1962:{\displaystyle \theta +2\pi n-\mu \leq 0}
1930:
1902:
1898:
1832:
1825:
1819:
1818:
1801:
1795:
1781:
1744:
1738:
1699:
1693:
1673:
1646:
1601:
1600:
1599:
1584:
1569:
1559:
1525:
1509:
1503:
1489:
1467:
1446:
1399:
1387:
1360:
1348:
1325:
1320:
1310:
1300:
1287:
1273:
1246:
1230:{\displaystyle \vartheta (\theta ,\tau )}
1207:
1170:
1160:
1134:
1111:
1095:
1080:
1046:
1030:
1016:
999:
990:
958:
914:
879:
862:
819:
813:
780:
742:
736:
707:
657:
635:
634:
579:
574:
571:
546:
541:
534:
509:
504:
497:
479:
472:
466:
450:
445:
432:
416:
411:
396:
387:
381:
361:
315:
309:
295:
273:
267:
216:
212:
200:
188:
152:
4415:Goodness of fit and significance testing
2388:Sample moments are analogously defined:
2035:
1375:{\displaystyle w\equiv e^{i\pi \theta }}
125:month, year, etc.), compass directions,
4463:
1417:{\displaystyle q\equiv e^{i\pi \tau }.}
480:
397:
104:. Shown are two views of the spherical
4959:Statistical Analysis of Spherical Data
4548:
4292:are constant, or, alternatively, that
3413:{\displaystyle z_{n}=e^{i\theta _{n}}}
2900:Both will have values between 0 and 1.
2044:There also exist distributions on the
636:
4933:Statistical Analysis of Circular Data
4339:{\displaystyle {\overline {\theta }}}
3944:{\displaystyle {\overline {\theta }}}
7:
6427:
3924:The distribution of the mean angle (
2076:matrix von MisesâFisher distribution
4957:; Lewis, T.; Embleton, BJJ (1993).
4411:in the limit of large sample size.
6468:Types of probability distributions
4814:Kent, J.T., Hamelryck, T. (2005).
4202:
4113:
4100:
2320:of the circular distribution, and
2251:
2197:
2167:
1796:
1791:
1504:
1499:
1288:
1283:
1031:
1026:
467:
462:
433:
428:
310:
305:
25:
4890:Jammalamadaka & Sengupta 2001
6426:
6417:
6416:
4986:. New Jersey: World Scientific.
4597:10.1111/j.1541-0420.2006.00682.x
4443:Circular correlation coefficient
2069:bivariate von Mises distribution
575:
542:
505:
5029:Ley, C.; Verdebout, T. (2017).
4899:Books on directional statistics
4312:{\displaystyle {\overline {R}}}
4285:{\displaystyle {\overline {C}}}
4258:{\displaystyle {\overline {S}}}
4215:is over any interval of length
2762:When data is concentrated, the
2732:Measures of location and spread
2093:is a distribution over axes in
679:von Mises circular distribution
4907:Circular statistics in biology
4564:. J Royal Stat Soc, 44, 71â80.
4394:
4368:
4160:
4147:
4061:
4035:
3998:
3972:
3905:
3877:
3707:
3694:
3636:
3623:
3192:
3186:
3165:
3158:
3135:
3129:
3103:
3090:
3068:
3040:
3026:
3020:
2989:
2983:
2961:
2940:
2926:
2920:
2875:
2869:
2818:
2812:
2702:
2686:
2662:
2647:
2604:
2591:
2548:
2533:
2372:
2366:
2349:{\displaystyle z=e^{i\theta }}
2303:
2297:
2211:
2205:
2186:
2173:
1895:
1870:
1864:
1840:
1771:
1753:
1652:
1633:
1575:
1566:
1534:
1518:
1479:
1454:
1307:
1293:
1263:
1251:
1224:
1212:
1077:
1052:
984:
966:
873:
867:
792:
786:
764:
752:
730:
712:
631:
588:
552:
476:
401:
393:
337:
319:
285:
279:
249:
234:
166:
160:
116:and 360 degrees are identical
1:
5031:Modern Directional Statistics
4983:Topics in Circular Statistics
4432:for possibly multimodal data.
4409:bivariate normal distribution
2669:{\displaystyle R_{n}=|m_{n}|}
2061:von MisesâFisher distribution
2019:Projected normal distribution
2013:Projected normal distribution
909:It can also be thought of as
855:circular uniform distribution
849:Circular uniform distribution
843:Circular uniform distribution
4501:10.1371/journal.pcbi.0020131
4389:
4376:
4331:
4304:
4277:
4250:
4087:
4073:
4056:
4043:
4024:
4010:
3993:
3980:
3936:
3900:
3885:
3863:
3840:
3820:
3804:
3769:
3751:
3738:
3649:
3578:
3545:
3529:
3516:
3439:
3333:
3310:
3286:
3098:
3057:
3015:
2841:
2822:
2404:
672:-th Euclidean basis vector.
145:probability density function
4448:Complex normal distribution
2907:circular standard deviation
2509:{\displaystyle \rho =m_{1}}
2009:is the location parameter.
1708:{\displaystyle \theta _{0}}
1439:wrapped Cauchy distribution
1433:Wrapped Cauchy distribution
1427:Wrapped Cauchy distribution
951:wrapped normal distribution
945:Wrapped normal distribution
939:Wrapped normal distribution
27:Subdiscipline of statistics
18:Circular standard deviation
6484:
6250:Wrapped asymmetric Laplace
5221:Extended negative binomial
5038:Taylor & Francis Group
4963:Cambridge University Press
4937:Cambridge University Press
4759:10.1198/016214501750332974
4480:PLOS Computational Biology
3501:which may be expressed as
2777:
2751:
2378:{\displaystyle P(\theta )}
2309:{\displaystyle P(\theta )}
2264:is any interval of length
2016:
1722:
1430:
942:
846:
682:
136:
41:) is the subdiscipline of
6412:
5909:Generalized extreme value
5689:Relativistic BreitâWigner
5086:Probability distributions
4802:10.1107/S010876739400437X
4536:, Pattern Recognition, 39
2555:{\displaystyle R=|m_{1}|}
2078:is a distribution on the
1731:wrapped LĂ©vy distribution
1725:Wrapped LĂ©vy distribution
1719:Wrapped LĂ©vy distribution
928:{\displaystyle \kappa =0}
3362:Distribution of the mean
1989:is the scale factor and
1688:is the scale factor and
935:of the von Mises above.
5904:Generalized chi-squared
5848:Normal-inverse Gaussian
4905:Batschelet, E. (1981).
4855:10.1073/pnas.0801715105
4689:10.1093/biomet/59.3.665
4208:{\displaystyle \Gamma }
2856:and for the population
2257:{\displaystyle \Gamma }
1681:{\displaystyle \gamma }
96:The overall shape of a
6458:Statistical data types
6453:Directional statistics
6216:Univariate (circular)
5777:Generalized hyperbolic
5206:ConwayâMaxwellâPoisson
5196:Beta negative binomial
5012:Directional Statistics
4724:10.1214/aos/1176342874
4532:Bahlmann, C., (2006),
4426:for a unimodal cluster
4401:
4340:
4313:
4286:
4259:
4232:
4209:
4186:
4138:
3945:
3915:
3780:
3723:or, alternatively as:
3714:
3687:
3616:
3554:
3492:
3477:
3414:
3351:
3269:
3199:
3142:
3112:
2998:
2894:
2850:
2742:statistical dispersion
2722:
2670:
2614:
2556:
2510:
2471:
2448:
2379:
2350:
2310:
2281:
2258:
2235:
2046:two-dimensional sphere
2041:
2003:
1983:
1963:
1919:
1800:
1715:is the peak position.
1709:
1682:
1662:
1508:
1418:
1376:
1337:
1292:
1231:
1196:
1035:
929:
901:
829:
802:
691:von Mises distribution
685:von Mises distribution
666:
646:
560:
471:
437:
370:
348:
314:
256:
173:
172:{\displaystyle \ p(x)}
133:Circular distributions
109:
69:through the origin in
31:Directional statistics
6261:Bivariate (spherical)
5759:Kaniadakis Îș-Gaussian
4402:
4355:central limit theorem
4341:
4314:
4287:
4260:
4233:
4231:{\displaystyle 2\pi }
4210:
4187:
4118:
3951:) for a circular pdf
3946:
3916:
3781:
3715:
3667:
3596:
3555:
3493:
3457:
3415:
3352:
3270:
3200:
3143:
3113:
2999:
2895:
2851:
2752:Further information:
2723:
2671:
2615:
2557:
2511:
2472:
2428:
2380:
2356:. Since the integral
2351:
2311:
2282:
2280:{\displaystyle 2\pi }
2259:
2236:
2039:
2004:
1984:
1964:
1920:
1777:
1710:
1683:
1663:
1485:
1419:
1377:
1338:
1269:
1239:Jacobi theta function
1232:
1197:
1012:
930:
902:
830:
828:{\displaystyle I_{0}}
803:
667:
647:
561:
441:
407:
371:
349:
291:
257:
174:
139:Circular distribution
95:
6326:Dirac delta function
6273:Bivariate (toroidal)
6230:Univariate von Mises
6101:Multivariate Laplace
5993:Shifted log-logistic
5342:Continuous Bernoulli
4453:Wrapped distribution
4365:
4323:
4296:
4269:
4242:
4219:
4199:
3966:
3959:) will be given by:
3928:
3796:
3730:
3570:
3508:
3431:
3374:
3278:
3219:
3152:
3141:{\displaystyle S(z)}
3123:
3007:
2914:
2860:
2800:
2736:Various measures of
2683:
2630:
2569:
2523:
2487:
2395:
2360:
2324:
2291:
2268:
2248:
2151:
2091:Bingham distribution
2002:{\displaystyle \mu }
1993:
1973:
1929:
1737:
1692:
1672:
1445:
1386:
1347:
1245:
1206:
957:
913:
861:
812:
706:
656:
570:
380:
360:
266:
187:
151:
83:Riemannian manifolds
39:spherical statistics
6374:Natural exponential
6279:Bivariate von Mises
6245:Wrapped exponential
6111:Multivariate stable
6106:Multivariate normal
5427:Benktander 2nd kind
5422:Benktander 1st kind
5211:Discrete phase-type
5010:; Jupp, P. (2000).
4846:2008PNAS..105.8932B
4794:1994AcCrA..50..741K
4492:2006PLSCB...2..131H
3212:circular dispersion
2463:
2057:-dimensional sphere
179:on the line can be
35:circular statistics
6463:Statistical theory
6029:Rectified Gaussian
5914:Generalized Pareto
5772:Generalized normal
5644:Matrix-exponential
4747:J. Am. Stat. Assoc
4397:
4336:
4309:
4282:
4255:
4228:
4205:
4182:
3941:
3911:
3776:
3710:
3550:
3488:
3420:the mean value of
3410:
3347:
3265:
3195:
3138:
3108:
2994:
2890:
2846:
2718:
2666:
2610:
2552:
2506:
2467:
2449:
2375:
2346:
2306:
2277:
2254:
2231:
2042:
1999:
1979:
1959:
1915:
1705:
1678:
1658:
1414:
1372:
1333:
1227:
1192:
925:
897:
825:
798:
662:
642:
556:
366:
344:
252:
169:
110:
6440:
6439:
6037:
6036:
6006:
6005:
5897:whose type varies
5843:Normal (Gaussian)
5797:Hyperbolic secant
5746:Exponential power
5649:MaxwellâBoltzmann
5397:Wigner semicircle
5289:
5288:
5261:Parabolic fractal
5251:Negative binomial
5047:978-1-4987-0664-3
4840:(26): 8932â8937.
4645:10.1214/19-BA1176
4632:Bayesian Analysis
4392:
4379:
4334:
4307:
4280:
4253:
4090:
4076:
4059:
4046:
4027:
4013:
3996:
3983:
3939:
3903:
3888:
3866:
3856:
3851:
3843:
3823:
3807:
3772:
3754:
3741:
3665:
3652:
3642:
3594:
3581:
3548:
3532:
3519:
3455:
3442:
3345:
3336:
3313:
3289:
3263:
3106:
3101:
3071:
3060:
3018:
2992:
2964:
2844:
2825:
2792:circular variance
2426:
2407:
2084:rotation matrices
2050:Kent distribution
1982:{\displaystyle c}
1913:
1816:
1815:
1656:
1597:
1579:
1185:
1155:
1124:
1102:
1010:
1007:
892:
796:
665:{\displaystyle k}
369:{\displaystyle F}
230:
227:
156:
16:(Redirected from
6475:
6430:
6429:
6420:
6419:
6359:Compound Poisson
6334:
6322:
6291:von MisesâFisher
6287:
6275:
6263:
6225:Circular uniform
6221:
6141:
6085:
6056:
6017:
5919:MarchenkoâPastur
5782:Geometric stable
5699:Truncated normal
5592:Inverse Gaussian
5498:Hyperexponential
5337:Beta rectangular
5305:bounded interval
5300:
5168:Discrete uniform
5153:Poisson binomial
5104:
5079:
5072:
5065:
5056:
5051:
5025:
5003:
5001:
5000:
4976:
4950:
4924:
4893:
4887:
4878:
4877:
4867:
4857:
4825:
4819:
4812:
4806:
4805:
4782:Acta Crystallogr
4777:
4771:
4770:
4744:
4735:
4729:
4728:
4726:
4717:(6): 1201â1225.
4699:
4693:
4692:
4672:
4666:
4665:
4647:
4623:
4617:
4616:
4580:
4574:
4571:
4565:
4558:
4552:
4546:
4537:
4530:
4524:
4523:
4513:
4503:
4471:
4406:
4404:
4403:
4400:{\displaystyle }
4398:
4393:
4385:
4380:
4372:
4345:
4343:
4342:
4337:
4335:
4327:
4318:
4316:
4315:
4310:
4308:
4300:
4291:
4289:
4288:
4283:
4281:
4273:
4264:
4262:
4261:
4256:
4254:
4246:
4237:
4235:
4234:
4229:
4214:
4212:
4211:
4206:
4191:
4189:
4188:
4183:
4181:
4177:
4176:
4175:
4159:
4158:
4137:
4132:
4117:
4116:
4104:
4103:
4091:
4083:
4077:
4069:
4060:
4052:
4047:
4039:
4028:
4020:
4014:
4006:
3997:
3989:
3984:
3976:
3950:
3948:
3947:
3942:
3940:
3932:
3920:
3918:
3917:
3912:
3904:
3896:
3894:
3889:
3881:
3867:
3859:
3857:
3854:
3852:
3850:
3849:
3844:
3836:
3830:
3829:
3824:
3816:
3813:
3808:
3800:
3785:
3783:
3782:
3777:
3775:
3774:
3773:
3765:
3755:
3747:
3742:
3734:
3719:
3717:
3716:
3711:
3706:
3705:
3686:
3681:
3666:
3658:
3653:
3645:
3643:
3640:
3635:
3634:
3615:
3610:
3595:
3587:
3582:
3574:
3559:
3557:
3556:
3551:
3549:
3541:
3533:
3525:
3520:
3512:
3497:
3495:
3494:
3489:
3487:
3486:
3476:
3471:
3456:
3448:
3443:
3435:
3419:
3417:
3416:
3411:
3409:
3408:
3407:
3406:
3386:
3385:
3356:
3354:
3353:
3348:
3346:
3344:
3343:
3342:
3337:
3329:
3322:
3321:
3320:
3319:
3314:
3306:
3295:
3290:
3282:
3274:
3272:
3271:
3266:
3264:
3262:
3261:
3260:
3247:
3246:
3245:
3229:
3214:
3213:
3204:
3202:
3201:
3196:
3173:
3172:
3147:
3145:
3144:
3139:
3117:
3115:
3114:
3109:
3107:
3102:
3094:
3077:
3072:
3067:
3066:
3061:
3053:
3050:
3033:
3019:
3011:
3003:
3001:
3000:
2995:
2993:
2970:
2965:
2960:
2959:
2950:
2933:
2909:
2908:
2899:
2897:
2896:
2891:
2855:
2853:
2852:
2847:
2845:
2837:
2826:
2821:
2804:
2794:
2793:
2780:Yamartino method
2748:Central tendency
2738:central tendency
2727:
2725:
2724:
2719:
2714:
2713:
2701:
2700:
2675:
2673:
2672:
2667:
2665:
2660:
2659:
2650:
2642:
2641:
2619:
2617:
2616:
2611:
2603:
2602:
2581:
2580:
2561:
2559:
2558:
2553:
2551:
2546:
2545:
2536:
2515:
2513:
2512:
2507:
2505:
2504:
2476:
2474:
2473:
2468:
2462:
2457:
2447:
2442:
2427:
2419:
2414:
2413:
2408:
2400:
2384:
2382:
2381:
2376:
2355:
2353:
2352:
2347:
2345:
2344:
2315:
2313:
2312:
2307:
2286:
2284:
2283:
2278:
2263:
2261:
2260:
2255:
2240:
2238:
2237:
2232:
2223:
2222:
2201:
2200:
2185:
2184:
2163:
2162:
2080:Stiefel manifold
2008:
2006:
2005:
2000:
1988:
1986:
1985:
1980:
1968:
1966:
1965:
1960:
1924:
1922:
1921:
1916:
1914:
1912:
1911:
1910:
1906:
1868:
1867:
1836:
1820:
1817:
1814:
1803:
1802:
1799:
1794:
1752:
1751:
1714:
1712:
1711:
1706:
1704:
1703:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1657:
1655:
1651:
1650:
1613:
1602:
1598:
1596:
1585:
1580:
1578:
1574:
1573:
1564:
1563:
1530:
1529:
1510:
1507:
1502:
1472:
1471:
1423:
1421:
1420:
1415:
1410:
1409:
1381:
1379:
1378:
1373:
1371:
1370:
1342:
1340:
1339:
1334:
1332:
1331:
1330:
1329:
1315:
1314:
1305:
1304:
1291:
1286:
1236:
1234:
1233:
1228:
1201:
1199:
1198:
1193:
1191:
1187:
1186:
1184:
1176:
1175:
1174:
1161:
1156:
1154:
1146:
1135:
1125:
1123:
1112:
1107:
1103:
1101:
1100:
1099:
1086:
1085:
1084:
1047:
1034:
1029:
1011:
1009:
1008:
1000:
991:
934:
932:
931:
926:
906:
904:
903:
898:
893:
891:
880:
835:is the modified
834:
832:
831:
826:
824:
823:
807:
805:
804:
799:
797:
795:
785:
784:
768:
767:
737:
671:
669:
668:
663:
651:
649:
648:
643:
641:
640:
639:
584:
583:
578:
565:
563:
562:
557:
555:
551:
550:
545:
539:
538:
514:
513:
508:
502:
501:
483:
470:
465:
455:
454:
436:
431:
421:
420:
400:
392:
391:
375:
373:
372:
367:
353:
351:
350:
345:
340:
313:
308:
278:
277:
261:
259:
258:
253:
228:
225:
221:
220:
205:
204:
178:
176:
175:
170:
154:
112:The fact that 0
87:Stiefel manifold
45:that deals with
21:
6483:
6482:
6478:
6477:
6476:
6474:
6473:
6472:
6443:
6442:
6441:
6436:
6408:
6384:Maximum entropy
6342:
6330:
6318:
6308:
6300:
6283:
6271:
6259:
6214:
6201:
6138:Matrix-valued:
6135:
6081:
6052:
6044:
6033:
6021:
6012:
6002:
5896:
5890:
5807:
5733:
5731:
5725:
5654:MaxwellâJĂŒttner
5503:Hypoexponential
5409:
5407:
5406:supported on a
5401:
5362:Noncentral beta
5322:BaldingâNichols
5304:
5303:supported on a
5295:
5285:
5188:
5182:
5178:ZipfâMandelbrot
5108:
5099:
5093:
5083:
5048:
5028:
5022:
5006:
4998:
4996:
4994:
4979:
4973:
4953:
4947:
4927:
4921:
4904:
4901:
4896:
4888:
4881:
4827:
4826:
4822:
4813:
4809:
4779:
4778:
4774:
4742:
4737:
4736:
4732:
4701:
4700:
4696:
4674:
4673:
4669:
4625:
4624:
4620:
4582:
4581:
4577:
4572:
4568:
4560:Kent, J (1982)
4559:
4555:
4547:
4540:
4531:
4527:
4473:
4472:
4465:
4461:
4439:
4417:
4363:
4362:
4321:
4320:
4294:
4293:
4267:
4266:
4240:
4239:
4217:
4216:
4197:
4196:
4167:
4150:
4143:
4139:
4108:
4095:
3964:
3963:
3926:
3925:
3855: and
3834:
3814:
3794:
3793:
3756:
3728:
3727:
3697:
3641: and
3626:
3568:
3567:
3506:
3505:
3478:
3429:
3428:
3424:is defined as:
3398:
3390:
3377:
3372:
3371:
3366:Given a set of
3364:
3327:
3323:
3304:
3296:
3276:
3275:
3252:
3248:
3237:
3230:
3217:
3216:
3211:
3210:
3164:
3150:
3149:
3121:
3120:
3051:
3005:
3004:
2951:
2912:
2911:
2906:
2905:
2858:
2857:
2805:
2798:
2797:
2791:
2790:
2782:
2776:
2756:
2750:
2734:
2692:
2681:
2680:
2651:
2633:
2628:
2627:
2594:
2572:
2567:
2566:
2537:
2521:
2520:
2496:
2485:
2484:
2398:
2393:
2392:
2358:
2357:
2333:
2322:
2321:
2289:
2288:
2266:
2265:
2246:
2245:
2214:
2192:
2176:
2154:
2149:
2148:
2142:
2130:crystallography
2034:
2021:
2015:
1991:
1990:
1971:
1970:
1927:
1926:
1894:
1869:
1821:
1807:
1740:
1735:
1734:
1729:The pdf of the
1727:
1721:
1695:
1690:
1689:
1670:
1669:
1642:
1614:
1603:
1589:
1565:
1555:
1521:
1514:
1463:
1443:
1442:
1437:The pdf of the
1435:
1429:
1395:
1384:
1383:
1356:
1345:
1344:
1321:
1316:
1306:
1296:
1243:
1242:
1204:
1203:
1177:
1166:
1162:
1147:
1136:
1133:
1129:
1116:
1091:
1087:
1076:
1048:
1042:
995:
955:
954:
949:The pdf of the
947:
941:
911:
910:
884:
859:
858:
851:
845:
837:Bessel function
815:
810:
809:
776:
769:
738:
704:
703:
687:
681:
654:
653:
630:
573:
568:
567:
540:
530:
503:
493:
446:
412:
383:
378:
377:
358:
357:
269:
264:
263:
196:
185:
184:
149:
148:
141:
135:
127:dihedral angles
55:Euclidean space
28:
23:
22:
15:
12:
11:
5:
6481:
6479:
6471:
6470:
6465:
6460:
6455:
6445:
6444:
6438:
6437:
6435:
6434:
6424:
6413:
6410:
6409:
6407:
6406:
6401:
6396:
6391:
6386:
6381:
6379:Locationâscale
6376:
6371:
6366:
6361:
6356:
6350:
6348:
6344:
6343:
6341:
6340:
6335:
6328:
6323:
6315:
6313:
6302:
6301:
6299:
6298:
6293:
6288:
6281:
6276:
6269:
6264:
6257:
6252:
6247:
6242:
6240:Wrapped Cauchy
6237:
6235:Wrapped normal
6232:
6227:
6222:
6211:
6209:
6203:
6202:
6200:
6199:
6198:
6197:
6192:
6190:Normal-inverse
6187:
6182:
6172:
6171:
6170:
6160:
6152:
6147:
6142:
6133:
6132:
6131:
6121:
6113:
6108:
6103:
6098:
6097:
6096:
6086:
6079:
6078:
6077:
6072:
6062:
6057:
6049:
6047:
6039:
6038:
6035:
6034:
6032:
6031:
6025:
6023:
6014:
6008:
6007:
6004:
6003:
6001:
6000:
5995:
5990:
5982:
5974:
5966:
5957:
5948:
5939:
5930:
5921:
5916:
5911:
5906:
5900:
5898:
5892:
5891:
5889:
5888:
5883:
5881:Variance-gamma
5878:
5873:
5865:
5860:
5855:
5850:
5845:
5840:
5832:
5827:
5826:
5825:
5815:
5810:
5805:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5761:
5756:
5748:
5743:
5737:
5735:
5727:
5726:
5724:
5723:
5721:Wilks's lambda
5718:
5717:
5716:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5666:
5661:
5659:Mittag-Leffler
5656:
5651:
5646:
5641:
5636:
5631:
5626:
5621:
5616:
5611:
5606:
5601:
5600:
5599:
5589:
5580:
5575:
5570:
5569:
5568:
5558:
5556:gamma/Gompertz
5553:
5552:
5551:
5546:
5536:
5531:
5526:
5525:
5524:
5512:
5511:
5510:
5505:
5500:
5490:
5489:
5488:
5478:
5473:
5468:
5467:
5466:
5465:
5464:
5454:
5444:
5439:
5434:
5429:
5424:
5419:
5413:
5411:
5408:semi-infinite
5403:
5402:
5400:
5399:
5394:
5389:
5384:
5379:
5374:
5369:
5364:
5359:
5354:
5349:
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5308:
5306:
5297:
5291:
5290:
5287:
5286:
5284:
5283:
5278:
5273:
5268:
5263:
5258:
5253:
5248:
5243:
5238:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5198:
5192:
5190:
5187:with infinite
5184:
5183:
5181:
5180:
5175:
5170:
5165:
5160:
5155:
5150:
5149:
5148:
5141:Hypergeometric
5138:
5133:
5128:
5123:
5118:
5112:
5110:
5101:
5095:
5094:
5084:
5082:
5081:
5074:
5067:
5059:
5053:
5052:
5046:
5026:
5020:
5004:
4992:
4977:
4971:
4951:
4945:
4925:
4919:
4911:Academic Press
4900:
4897:
4895:
4894:
4879:
4820:
4807:
4788:(6): 741â748.
4772:
4753:(453): 56â63.
4730:
4694:
4683:(3): 665â676.
4667:
4638:(3): 871â908.
4618:
4591:(2): 505â512.
4575:
4566:
4553:
4538:
4525:
4462:
4460:
4457:
4456:
4455:
4450:
4445:
4438:
4435:
4434:
4433:
4427:
4416:
4413:
4396:
4391:
4388:
4383:
4378:
4375:
4370:
4346:are constant.
4333:
4330:
4306:
4303:
4279:
4276:
4252:
4249:
4227:
4224:
4204:
4193:
4192:
4180:
4174:
4170:
4166:
4162:
4157:
4153:
4149:
4146:
4142:
4136:
4131:
4128:
4125:
4121:
4115:
4111:
4107:
4102:
4098:
4094:
4089:
4086:
4081:
4075:
4072:
4067:
4063:
4058:
4055:
4050:
4045:
4042:
4037:
4034:
4031:
4026:
4023:
4018:
4012:
4009:
4004:
4000:
3995:
3992:
3987:
3982:
3979:
3974:
3971:
3938:
3935:
3922:
3921:
3910:
3907:
3902:
3899:
3893:
3887:
3884:
3879:
3876:
3873:
3870:
3865:
3862:
3848:
3842:
3839:
3833:
3828:
3822:
3819:
3811:
3806:
3803:
3787:
3786:
3771:
3768:
3763:
3759:
3753:
3750:
3745:
3740:
3737:
3721:
3720:
3709:
3704:
3700:
3696:
3693:
3690:
3685:
3680:
3677:
3674:
3670:
3664:
3661:
3656:
3651:
3648:
3638:
3633:
3629:
3625:
3622:
3619:
3614:
3609:
3606:
3603:
3599:
3593:
3590:
3585:
3580:
3577:
3561:
3560:
3547:
3544:
3539:
3536:
3531:
3528:
3523:
3518:
3515:
3499:
3498:
3485:
3481:
3475:
3470:
3467:
3464:
3460:
3454:
3451:
3446:
3441:
3438:
3405:
3401:
3397:
3393:
3389:
3384:
3380:
3363:
3360:
3359:
3358:
3341:
3335:
3332:
3326:
3318:
3312:
3309:
3302:
3299:
3293:
3288:
3285:
3259:
3255:
3251:
3244:
3240:
3236:
3233:
3227:
3224:
3206:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3171:
3167:
3163:
3160:
3157:
3137:
3134:
3131:
3128:
3105:
3100:
3097:
3092:
3089:
3086:
3083:
3080:
3075:
3070:
3065:
3059:
3056:
3049:
3045:
3042:
3039:
3036:
3031:
3028:
3025:
3022:
3017:
3014:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2968:
2963:
2958:
2954:
2949:
2945:
2942:
2939:
2936:
2931:
2928:
2925:
2922:
2919:
2901:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2843:
2840:
2835:
2832:
2829:
2824:
2820:
2817:
2814:
2811:
2808:
2775:
2772:
2749:
2746:
2733:
2730:
2717:
2712:
2708:
2704:
2699:
2695:
2691:
2688:
2677:
2676:
2664:
2658:
2654:
2649:
2645:
2640:
2636:
2621:
2620:
2609:
2606:
2601:
2597:
2593:
2590:
2587:
2584:
2579:
2575:
2563:
2562:
2550:
2544:
2540:
2535:
2531:
2528:
2517:
2516:
2503:
2499:
2495:
2492:
2478:
2477:
2466:
2461:
2456:
2452:
2446:
2441:
2438:
2435:
2431:
2425:
2422:
2417:
2412:
2406:
2403:
2374:
2371:
2368:
2365:
2343:
2340:
2336:
2332:
2329:
2305:
2302:
2299:
2296:
2276:
2273:
2253:
2242:
2241:
2230:
2227:
2221:
2217:
2213:
2210:
2207:
2204:
2199:
2195:
2191:
2188:
2183:
2179:
2175:
2172:
2169:
2166:
2161:
2157:
2141:
2138:
2134:bioinformatics
2033:
2030:
2024:symmetric nor
2017:Main article:
2014:
2011:
1998:
1978:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1909:
1905:
1901:
1897:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1835:
1831:
1828:
1824:
1813:
1810:
1806:
1798:
1793:
1790:
1787:
1784:
1780:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1750:
1747:
1743:
1723:Main article:
1720:
1717:
1702:
1698:
1677:
1654:
1649:
1645:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1612:
1609:
1606:
1595:
1592:
1588:
1583:
1577:
1572:
1568:
1562:
1558:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1528:
1524:
1520:
1517:
1513:
1506:
1501:
1498:
1495:
1492:
1488:
1484:
1481:
1478:
1475:
1470:
1466:
1462:
1459:
1456:
1453:
1450:
1431:Main article:
1428:
1425:
1413:
1408:
1405:
1402:
1398:
1394:
1391:
1369:
1366:
1363:
1359:
1355:
1352:
1328:
1324:
1319:
1313:
1309:
1303:
1299:
1295:
1290:
1285:
1282:
1279:
1276:
1272:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1226:
1223:
1220:
1217:
1214:
1211:
1190:
1183:
1180:
1173:
1169:
1165:
1159:
1153:
1150:
1145:
1142:
1139:
1132:
1128:
1122:
1119:
1115:
1110:
1106:
1098:
1094:
1090:
1083:
1079:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1045:
1041:
1038:
1033:
1028:
1025:
1022:
1019:
1015:
1006:
1003:
998:
994:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
943:Main article:
940:
937:
924:
921:
918:
896:
890:
887:
883:
878:
875:
872:
869:
866:
847:Main article:
844:
841:
822:
818:
794:
791:
788:
783:
779:
775:
772:
766:
763:
760:
757:
754:
751:
748:
745:
741:
735:
732:
729:
726:
723:
720:
717:
714:
711:
696:wrapped normal
683:Main article:
680:
677:
661:
638:
633:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
582:
577:
554:
549:
544:
537:
533:
529:
526:
523:
520:
517:
512:
507:
500:
496:
492:
489:
486:
482:
478:
475:
469:
464:
461:
458:
453:
449:
444:
440:
435:
430:
427:
424:
419:
415:
410:
406:
403:
399:
395:
390:
386:
365:
343:
339:
336:
333:
330:
327:
324:
321:
318:
312:
307:
304:
301:
298:
294:
290:
287:
284:
281:
276:
272:
251:
248:
245:
242:
239:
236:
233:
224:
219:
215:
211:
208:
203:
199:
195:
192:
168:
165:
162:
159:
137:Main article:
134:
131:
85:including the
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6480:
6469:
6466:
6464:
6461:
6459:
6456:
6454:
6451:
6450:
6448:
6433:
6425:
6423:
6415:
6414:
6411:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6351:
6349:
6345:
6339:
6336:
6333:
6329:
6327:
6324:
6321:
6317:
6316:
6314:
6312:
6307:
6303:
6297:
6294:
6292:
6289:
6286:
6282:
6280:
6277:
6274:
6270:
6268:
6265:
6262:
6258:
6256:
6253:
6251:
6248:
6246:
6243:
6241:
6238:
6236:
6233:
6231:
6228:
6226:
6223:
6220:
6219:
6213:
6212:
6210:
6208:
6204:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6178:
6177:
6176:
6173:
6169:
6166:
6165:
6164:
6161:
6159:
6158:
6153:
6151:
6150:Matrix normal
6148:
6146:
6143:
6140:
6139:
6134:
6130:
6127:
6126:
6125:
6122:
6120:
6119:
6116:Multivariate
6114:
6112:
6109:
6107:
6104:
6102:
6099:
6095:
6092:
6091:
6090:
6087:
6084:
6080:
6076:
6073:
6071:
6068:
6067:
6066:
6063:
6061:
6058:
6055:
6051:
6050:
6048:
6046:
6043:Multivariate
6040:
6030:
6027:
6026:
6024:
6018:
6015:
6009:
5999:
5996:
5994:
5991:
5989:
5987:
5983:
5981:
5979:
5975:
5973:
5971:
5967:
5965:
5963:
5958:
5956:
5954:
5949:
5947:
5945:
5940:
5938:
5936:
5931:
5929:
5927:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5905:
5902:
5901:
5899:
5895:with support
5893:
5887:
5884:
5882:
5879:
5877:
5874:
5872:
5871:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5838:
5833:
5831:
5828:
5824:
5821:
5820:
5819:
5816:
5814:
5811:
5809:
5808:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5767:
5762:
5760:
5757:
5755:
5754:
5749:
5747:
5744:
5742:
5739:
5738:
5736:
5732:on the whole
5728:
5722:
5719:
5715:
5712:
5711:
5710:
5707:
5705:
5704:type-2 Gumbel
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5598:
5595:
5594:
5593:
5590:
5588:
5586:
5581:
5579:
5576:
5574:
5573:Half-logistic
5571:
5567:
5564:
5563:
5562:
5559:
5557:
5554:
5550:
5547:
5545:
5542:
5541:
5540:
5537:
5535:
5532:
5530:
5529:Folded normal
5527:
5523:
5520:
5519:
5518:
5517:
5513:
5509:
5506:
5504:
5501:
5499:
5496:
5495:
5494:
5491:
5487:
5484:
5483:
5482:
5479:
5477:
5474:
5472:
5469:
5463:
5460:
5459:
5458:
5455:
5453:
5450:
5449:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5414:
5412:
5404:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5372:Raised cosine
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5309:
5307:
5301:
5298:
5292:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5246:Mixed Poisson
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5193:
5191:
5185:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5159:
5156:
5154:
5151:
5147:
5144:
5143:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5126:Beta-binomial
5124:
5122:
5119:
5117:
5114:
5113:
5111:
5105:
5102:
5096:
5091:
5087:
5080:
5075:
5073:
5068:
5066:
5061:
5060:
5057:
5049:
5043:
5039:
5036:
5032:
5027:
5023:
5021:0-471-95333-4
5017:
5013:
5009:
5008:Mardia, K. V.
5005:
4995:
4993:981-02-3778-2
4989:
4985:
4984:
4978:
4974:
4972:0-521-45699-1
4968:
4964:
4960:
4956:
4955:Fisher, N. I.
4952:
4948:
4946:0-521-35018-2
4942:
4938:
4934:
4930:
4929:Fisher, N. I.
4926:
4922:
4920:0-12-081050-6
4916:
4912:
4908:
4903:
4902:
4898:
4891:
4886:
4884:
4880:
4875:
4871:
4866:
4861:
4856:
4851:
4847:
4843:
4839:
4835:
4831:
4824:
4821:
4817:
4811:
4808:
4803:
4799:
4795:
4791:
4787:
4783:
4776:
4773:
4768:
4764:
4760:
4756:
4752:
4748:
4741:
4734:
4731:
4725:
4720:
4716:
4712:
4708:
4704:
4698:
4695:
4690:
4686:
4682:
4678:
4671:
4668:
4663:
4659:
4655:
4651:
4646:
4641:
4637:
4633:
4629:
4622:
4619:
4614:
4610:
4606:
4602:
4598:
4594:
4590:
4586:
4579:
4576:
4570:
4567:
4563:
4557:
4554:
4550:
4545:
4543:
4539:
4535:
4529:
4526:
4521:
4517:
4512:
4507:
4502:
4497:
4493:
4489:
4485:
4481:
4477:
4470:
4468:
4464:
4458:
4454:
4451:
4449:
4446:
4444:
4441:
4440:
4436:
4431:
4430:Kuiper's test
4428:
4425:
4424:Rayleigh test
4422:
4421:
4420:
4414:
4412:
4410:
4407:approaches a
4386:
4381:
4373:
4360:
4356:
4351:
4347:
4328:
4301:
4274:
4247:
4225:
4222:
4178:
4172:
4168:
4164:
4155:
4151:
4144:
4140:
4134:
4129:
4126:
4123:
4119:
4109:
4105:
4096:
4092:
4084:
4079:
4070:
4065:
4053:
4048:
4040:
4032:
4029:
4021:
4016:
4007:
4002:
3990:
3985:
3977:
3969:
3962:
3961:
3960:
3958:
3954:
3933:
3908:
3897:
3891:
3882:
3874:
3871:
3868:
3860:
3846:
3837:
3831:
3826:
3817:
3809:
3801:
3792:
3791:
3790:
3766:
3761:
3757:
3748:
3743:
3735:
3726:
3725:
3724:
3702:
3698:
3691:
3688:
3683:
3678:
3675:
3672:
3668:
3662:
3659:
3654:
3646:
3631:
3627:
3620:
3617:
3612:
3607:
3604:
3601:
3597:
3591:
3588:
3583:
3575:
3566:
3565:
3564:
3542:
3537:
3534:
3526:
3521:
3513:
3504:
3503:
3502:
3483:
3479:
3473:
3468:
3465:
3462:
3458:
3452:
3449:
3444:
3436:
3427:
3426:
3425:
3423:
3403:
3399:
3395:
3391:
3387:
3382:
3378:
3370:measurements
3369:
3361:
3339:
3330:
3324:
3316:
3307:
3300:
3297:
3291:
3283:
3257:
3253:
3249:
3242:
3238:
3234:
3231:
3225:
3222:
3215:
3207:
3189:
3183:
3180:
3177:
3174:
3169:
3161:
3155:
3132:
3126:
3095:
3087:
3084:
3081:
3078:
3073:
3063:
3054:
3047:
3043:
3037:
3034:
3029:
3023:
3012:
2986:
2980:
2977:
2974:
2971:
2966:
2956:
2952:
2947:
2943:
2937:
2934:
2929:
2923:
2917:
2910:
2902:
2887:
2884:
2881:
2878:
2872:
2866:
2863:
2838:
2833:
2830:
2827:
2815:
2809:
2806:
2795:
2787:
2786:
2785:
2781:
2773:
2771:
2769:
2765:
2760:
2755:
2754:Circular mean
2747:
2745:
2743:
2739:
2731:
2729:
2715:
2710:
2697:
2693:
2689:
2656:
2652:
2643:
2638:
2634:
2626:
2625:
2624:
2607:
2599:
2595:
2588:
2585:
2582:
2577:
2573:
2565:
2564:
2542:
2538:
2529:
2526:
2519:
2518:
2501:
2497:
2493:
2490:
2483:
2482:
2481:
2464:
2459:
2454:
2450:
2444:
2439:
2436:
2433:
2429:
2423:
2420:
2415:
2410:
2401:
2391:
2390:
2389:
2386:
2369:
2363:
2341:
2338:
2334:
2330:
2327:
2319:
2300:
2294:
2274:
2271:
2228:
2225:
2219:
2215:
2208:
2202:
2193:
2189:
2181:
2177:
2170:
2164:
2159:
2155:
2147:
2146:
2145:
2139:
2137:
2135:
2131:
2127:
2122:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2087:
2085:
2081:
2077:
2072:
2070:
2066:
2062:
2058:
2056:
2051:
2048:(such as the
2047:
2038:
2031:
2029:
2027:
2020:
2012:
2010:
1996:
1976:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1907:
1903:
1899:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1837:
1833:
1829:
1826:
1822:
1811:
1808:
1804:
1788:
1785:
1782:
1778:
1774:
1768:
1765:
1762:
1759:
1756:
1748:
1745:
1741:
1732:
1726:
1718:
1716:
1700:
1696:
1675:
1647:
1643:
1639:
1636:
1630:
1627:
1624:
1621:
1618:
1615:
1610:
1607:
1604:
1593:
1590:
1586:
1581:
1570:
1560:
1556:
1552:
1549:
1546:
1543:
1540:
1537:
1531:
1526:
1522:
1515:
1511:
1496:
1493:
1490:
1486:
1482:
1476:
1473:
1468:
1464:
1460:
1457:
1451:
1448:
1440:
1434:
1426:
1424:
1411:
1406:
1403:
1400:
1396:
1392:
1389:
1367:
1364:
1361:
1357:
1353:
1350:
1326:
1322:
1317:
1311:
1301:
1297:
1280:
1277:
1274:
1270:
1266:
1260:
1257:
1254:
1248:
1240:
1221:
1218:
1215:
1209:
1188:
1181:
1178:
1171:
1167:
1163:
1157:
1151:
1148:
1143:
1140:
1137:
1130:
1126:
1120:
1117:
1113:
1108:
1104:
1096:
1092:
1088:
1081:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1049:
1043:
1039:
1036:
1023:
1020:
1017:
1013:
1004:
1001:
996:
992:
987:
981:
978:
975:
972:
969:
963:
960:
952:
946:
938:
936:
922:
919:
916:
907:
894:
888:
885:
881:
876:
870:
864:
856:
850:
842:
840:
838:
820:
816:
789:
781:
777:
773:
770:
761:
758:
755:
749:
746:
743:
739:
733:
727:
724:
721:
718:
715:
709:
700:
697:
692:
686:
678:
676:
673:
659:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
585:
580:
547:
535:
531:
527:
524:
521:
518:
515:
510:
498:
494:
490:
487:
484:
473:
459:
456:
451:
447:
442:
438:
425:
422:
417:
413:
408:
404:
388:
384:
363:
354:
341:
334:
331:
328:
325:
322:
316:
302:
299:
296:
292:
288:
282:
274:
270:
246:
243:
240:
237:
231:
222:
217:
209:
206:
201:
197:
193:
190:
182:
163:
157:
146:
140:
132:
130:
128:
123:
119:
115:
107:
103:
99:
94:
90:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
36:
32:
19:
6331:
6319:
6285:Multivariate
6284:
6272:
6260:
6255:Wrapped LĂ©vy
6217:
6215:
6206:
6163:Matrix gamma
6156:
6136:
6124:Normal-gamma
6117:
6083:Continuous:
6082:
6053:
5998:Tukey lambda
5985:
5977:
5972:-exponential
5969:
5961:
5952:
5943:
5934:
5928:-exponential
5925:
5869:
5836:
5803:
5765:
5752:
5679:Poly-Weibull
5624:Log-logistic
5584:
5583:Hotelling's
5515:
5357:Logit-normal
5231:GaussâKuzmin
5226:FloryâSchulz
5107:with finite
5030:
5011:
4997:. Retrieved
4982:
4958:
4932:
4906:
4837:
4833:
4823:
4810:
4785:
4781:
4775:
4750:
4746:
4733:
4714:
4710:
4697:
4680:
4676:
4670:
4635:
4631:
4621:
4588:
4584:
4578:
4569:
4556:
4528:
4483:
4479:
4418:
4352:
4348:
4194:
3956:
3952:
3923:
3788:
3722:
3562:
3500:
3421:
3367:
3365:
3209:
2904:
2789:
2783:
2761:
2757:
2735:
2678:
2622:
2479:
2387:
2243:
2143:
2123:
2118:
2106:
2102:
2098:
2094:
2088:
2073:
2054:
2043:
2022:
1730:
1728:
1438:
1436:
950:
948:
908:
857:is given by
854:
852:
839:of order 0.
701:
690:
688:
674:
355:
142:
111:
78:
70:
58:
51:unit vectors
38:
34:
30:
29:
6369:Exponential
6218:directional
6207:Directional
6094:Generalized
6065:Multinomial
6020:continuous-
5960:Kaniadakis
5951:Kaniadakis
5942:Kaniadakis
5933:Kaniadakis
5924:Kaniadakis
5876:TracyâWidom
5853:Skew normal
5835:Noncentral
5619:Log-Laplace
5597:Generalized
5578:Half-normal
5544:Generalized
5508:Logarithmic
5493:Exponential
5447:Chi-squared
5387:U-quadratic
5352:Kumaraswamy
5294:Continuous
5241:Logarithmic
5136:Categorical
4703:Bingham, C.
4549:Fisher 1993
4486:(9): e131.
2111:quaternions
6447:Categories
6364:Elliptical
6320:Degenerate
6306:Degenerate
6054:Discrete:
6013:univariate
5868:Student's
5823:Asymmetric
5802:Johnson's
5730:supported
5674:Phase-type
5629:Log-normal
5614:Log-Cauchy
5604:Kolmogorov
5522:Noncentral
5452:Noncentral
5432:Beta prime
5382:Triangular
5377:Reciprocal
5347:IrwinâHall
5296:univariate
5276:YuleâSimon
5158:Rademacher
5100:univariate
4999:2011-05-15
4909:. London:
4677:Biometrika
4585:Biometrics
4459:References
3148:, we have
2778:See also:
2774:Dispersion
47:directions
43:statistics
6089:Dirichlet
6070:Dirichlet
5980:-Gaussian
5955:-Logistic
5792:Holtsmark
5764:Gaussian
5751:Fisher's
5734:real line
5236:Geometric
5216:Delaporte
5121:Bernoulli
5098:Discrete
5035:CRC Press
4711:Ann. Stat
4662:209974627
4654:1936-0975
4390:¯
4377:¯
4332:¯
4329:θ
4305:¯
4278:¯
4251:¯
4226:π
4203:Γ
4169:θ
4152:θ
4120:∏
4114:Γ
4110:∫
4106:⋯
4101:Γ
4097:∫
4088:¯
4085:θ
4074:¯
4057:¯
4054:θ
4044:¯
4025:¯
4011:¯
3994:¯
3981:¯
3937:¯
3934:θ
3901:¯
3886:¯
3875:
3864:¯
3861:θ
3841:¯
3821:¯
3805:¯
3770:¯
3767:θ
3752:¯
3739:¯
3699:θ
3692:
3669:∑
3650:¯
3628:θ
3621:
3598:∑
3579:¯
3546:¯
3530:¯
3517:¯
3459:∑
3440:¯
3400:θ
3334:¯
3311:¯
3301:−
3287:¯
3284:δ
3235:−
3223:δ
3184:
3099:¯
3088:
3079:−
3058:¯
3038:
3016:¯
2981:
2972:−
2938:
2885:−
2867:
2842:¯
2834:−
2823:¯
2810:
2716:π
2694:θ
2589:
2574:θ
2491:ρ
2430:∑
2405:¯
2370:θ
2342:θ
2301:θ
2275:π
2252:Γ
2229:θ
2209:θ
2198:Γ
2194:∫
2171:
2063:) or the
1997:μ
1954:≤
1951:μ
1948:−
1942:π
1933:θ
1892:μ
1889:−
1883:π
1874:θ
1862:μ
1859:−
1853:π
1844:θ
1827:−
1812:π
1797:∞
1792:∞
1789:−
1779:∑
1763:μ
1757:θ
1733:(WL) is:
1697:θ
1676:γ
1644:θ
1640:−
1637:θ
1631:
1625:−
1622:γ
1619:
1611:γ
1608:
1594:π
1557:θ
1553:−
1547:π
1538:θ
1523:γ
1516:π
1512:γ
1505:∞
1500:∞
1497:−
1487:∑
1477:γ
1465:θ
1458:θ
1441:(WC) is:
1407:τ
1404:π
1393:≡
1368:θ
1365:π
1354:≡
1289:∞
1284:∞
1281:−
1271:∑
1261:τ
1255:θ
1249:ϑ
1222:τ
1216:θ
1210:ϑ
1182:π
1168:σ
1152:π
1144:μ
1141:−
1138:θ
1127:ϑ
1121:π
1093:σ
1071:π
1065:−
1062:μ
1059:−
1056:θ
1050:−
1040:
1032:∞
1027:∞
1024:−
1014:∑
1005:π
997:σ
982:σ
976:μ
970:θ
953:(WN) is:
917:κ
889:π
871:θ
790:κ
774:π
762:μ
759:−
756:θ
750:
744:κ
728:κ
722:μ
716:θ
622:…
598:…
528:π
519:⋯
491:π
481:θ
468:∞
463:∞
460:−
443:∑
439:⋯
434:∞
429:∞
426:−
409:∑
398:θ
332:π
323:θ
311:∞
306:∞
303:−
293:∑
283:θ
247:π
241:π
238:−
232:∈
223:π
191:θ
181:"wrapped"
106:histogram
75:rotations
6422:Category
6354:Circular
6347:Families
6332:Singular
6311:singular
6075:Negative
6022:discrete
5988:-Weibull
5946:-Weibull
5830:Logistic
5714:Discrete
5684:Rayleigh
5664:Nakagami
5587:-squared
5561:Gompertz
5410:interval
5146:Negative
5131:Binomial
4931:(1993).
4874:18579771
4767:11667311
4705:(1974).
4613:14293602
4605:17688502
4520:17002495
4437:See also
2026:unimodal
6432:Commons
6404:Wrapped
6399:Tweedie
6394:Pearson
6389:Mixture
6296:Bingham
6195:Complex
6185:Inverse
6175:Wishart
6168:Inverse
6155:Matrix
6129:Inverse
6045:(joint)
5964:-Erlang
5818:Laplace
5709:Weibull
5566:Shifted
5549:Inverse
5534:Fréchet
5457:Inverse
5392:Uniform
5312:Arcsine
5271:Skellam
5266:Poisson
5189:support
5163:Soliton
5116:Benford
5109:support
4865:2440424
4842:Bibcode
4790:Bibcode
4511:1570370
4488:Bibcode
2316:is the
2140:Moments
2126:geology
2115:versors
2052:), the
1237:is the
652:is the
114:degrees
98:protein
6338:Cantor
6180:Normal
6011:Mixed
5937:-Gamma
5863:Stable
5813:Landau
5787:Gumbel
5741:Cauchy
5669:Pareto
5481:Erlang
5462:Scaled
5417:Benini
5256:Panjer
5044:
5018:
4990:
4969:
4943:
4917:
4872:
4862:
4765:
4660:
4652:
4611:
4603:
4518:
4508:
4195:where
3957:θ
3872:arctan
3789:where
3563:where
2764:median
2244:where
1668:where
1343:where
808:where
566:where
229:
226:
155:
147:(pdf)
118:angles
102:sphere
33:(also
6060:Ewens
5886:Voigt
5858:Slash
5639:Lomax
5634:Log-t
5539:Gamma
5486:Hyper
5476:Davis
5471:Dagum
5327:Bates
5317:ARGUS
5201:Borel
4763:S2CID
4743:(PDF)
4658:S2CID
4609:S2CID
2067:(the
2065:torus
2059:(the
73:) or
67:lines
6309:and
6267:Kent
5694:Rice
5609:LĂ©vy
5437:Burr
5367:PERT
5332:Beta
5281:Zeta
5173:Zipf
5090:list
5042:ISBN
5016:ISBN
4988:ISBN
4967:ISBN
4941:ISBN
4915:ISBN
4870:PMID
4650:ISSN
4601:PMID
4516:PMID
4353:The
4319:and
4265:and
3208:The
2903:The
2788:The
2768:mode
2766:and
2740:and
2132:and
2089:The
2074:The
1616:cosh
1605:sinh
1382:and
689:The
143:Any
122:mean
63:axes
6145:LKJ
5442:Chi
4860:PMC
4850:doi
4838:105
4798:doi
4786:A50
4755:doi
4719:doi
4685:doi
4640:doi
4593:doi
4506:PMC
4496:doi
3689:sin
3618:cos
3181:Var
2864:Var
2807:Var
2707:mod
2586:Arg
2318:PDF
2071:).
1628:cos
1037:exp
747:cos
262:is
214:mod
77:in
61:),
53:in
37:or
6449::
5040:.
5033:.
4965:.
4961:.
4939:.
4935:.
4913:.
4882:^
4868:.
4858:.
4848:.
4836:.
4832:.
4796:.
4784:.
4761:.
4751:96
4749:.
4745:.
4713:.
4709:.
4681:59
4679:.
4656:.
4648:.
4636:15
4634:.
4630:.
4607:.
4599:.
4589:63
4587:.
4541:^
4514:.
4504:.
4494:.
4482:.
4478:.
4466:^
3085:ln
3035:ln
2978:ln
2935:ln
2287:,
2136:.
2128:,
2086:.
2028:.
1969:,
1241::
89:.
57:,
6157:t
6118:t
5986:q
5978:q
5970:q
5962:Îș
5953:Îș
5944:Îș
5935:Îș
5926:Îș
5870:t
5837:t
5806:U
5804:S
5766:q
5753:z
5585:T
5516:F
5092:)
5088:(
5078:e
5071:t
5064:v
5050:.
5024:.
5002:.
4975:.
4949:.
4923:.
4892:.
4876:.
4852::
4844::
4804:.
4800::
4792::
4769:.
4757::
4727:.
4721::
4715:2
4691:.
4687::
4664:.
4642::
4615:.
4595::
4551:.
4522:.
4498::
4490::
4484:2
4395:]
4387:S
4382:,
4374:C
4369:[
4302:R
4275:C
4248:S
4223:2
4179:]
4173:n
4165:d
4161:)
4156:n
4148:(
4145:P
4141:[
4135:N
4130:1
4127:=
4124:n
4093:=
4080:d
4071:R
4066:d
4062:)
4049:,
4041:R
4036:(
4033:P
4030:=
4022:S
4017:d
4008:C
4003:d
3999:)
3991:S
3986:,
3978:C
3973:(
3970:P
3955:(
3953:P
3909:.
3906:)
3898:C
3892:/
3883:S
3878:(
3869:=
3847:2
3838:S
3832:+
3827:2
3818:C
3810:=
3802:R
3762:i
3758:e
3749:R
3744:=
3736:z
3708:)
3703:n
3695:(
3684:N
3679:1
3676:=
3673:n
3663:N
3660:1
3655:=
3647:S
3637:)
3632:n
3624:(
3613:N
3608:1
3605:=
3602:n
3592:N
3589:1
3584:=
3576:C
3543:S
3538:i
3535:+
3527:C
3522:=
3514:z
3484:n
3480:z
3474:N
3469:1
3466:=
3463:n
3453:N
3450:1
3445:=
3437:z
3422:z
3404:n
3396:i
3392:e
3388:=
3383:n
3379:z
3368:N
3340:2
3331:R
3325:2
3317:2
3308:R
3298:1
3292:=
3258:2
3254:R
3250:2
3243:2
3239:R
3232:1
3226:=
3205:.
3193:)
3190:z
3187:(
3178:2
3175:=
3170:2
3166:)
3162:z
3159:(
3156:S
3136:)
3133:z
3130:(
3127:S
3104:)
3096:R
3091:(
3082:2
3074:=
3069:)
3064:2
3055:R
3048:/
3044:1
3041:(
3030:=
3027:)
3024:z
3021:(
3013:S
2990:)
2987:R
2984:(
2975:2
2967:=
2962:)
2957:2
2953:R
2948:/
2944:1
2941:(
2930:=
2927:)
2924:z
2921:(
2918:S
2888:R
2882:1
2879:=
2876:)
2873:z
2870:(
2839:R
2831:1
2828:=
2819:)
2816:z
2813:(
2711:2
2703:)
2698:n
2690:n
2687:(
2663:|
2657:n
2653:m
2648:|
2644:=
2639:n
2635:R
2608:.
2605:)
2600:n
2596:m
2592:(
2583:=
2578:n
2549:|
2543:1
2539:m
2534:|
2530:=
2527:R
2502:1
2498:m
2494:=
2465:.
2460:n
2455:i
2451:z
2445:N
2440:1
2437:=
2434:i
2424:N
2421:1
2416:=
2411:n
2402:m
2373:)
2367:(
2364:P
2339:i
2335:e
2331:=
2328:z
2304:)
2298:(
2295:P
2272:2
2226:d
2220:n
2216:z
2212:)
2206:(
2203:P
2190:=
2187:)
2182:n
2178:z
2174:(
2168:E
2165:=
2160:n
2156:m
2119:N
2113:(
2107:N
2103:N
2099:N
2095:N
2055:N
1977:c
1957:0
1945:n
1939:2
1936:+
1908:2
1904:/
1900:3
1896:)
1886:n
1880:2
1877:+
1871:(
1865:)
1856:n
1850:2
1847:+
1841:(
1838:2
1834:/
1830:c
1823:e
1809:2
1805:c
1786:=
1783:n
1775:=
1772:)
1769:c
1766:,
1760:;
1754:(
1749:L
1746:W
1742:f
1701:0
1653:)
1648:0
1634:(
1591:2
1587:1
1582:=
1576:)
1571:2
1567:)
1561:0
1550:n
1544:2
1541:+
1535:(
1532:+
1527:2
1519:(
1494:=
1491:n
1483:=
1480:)
1474:,
1469:0
1461:;
1455:(
1452:C
1449:W
1412:.
1401:i
1397:e
1390:q
1362:i
1358:e
1351:w
1327:2
1323:n
1318:q
1312:n
1308:)
1302:2
1298:w
1294:(
1278:=
1275:n
1267:=
1264:)
1258:,
1252:(
1225:)
1219:,
1213:(
1189:)
1179:2
1172:2
1164:i
1158:,
1149:2
1131:(
1118:2
1114:1
1109:=
1105:]
1097:2
1089:2
1082:2
1078:)
1074:k
1068:2
1053:(
1044:[
1021:=
1018:k
1002:2
993:1
988:=
985:)
979:,
973:;
967:(
964:N
961:W
923:0
920:=
895:.
886:2
882:1
877:=
874:)
868:(
865:U
821:0
817:I
793:)
787:(
782:0
778:I
771:2
765:)
753:(
740:e
734:=
731:)
725:,
719:;
713:(
710:f
660:k
637:T
632:)
628:0
625:,
619:,
616:0
613:,
610:1
607:,
604:0
601:,
595:,
592:0
589:(
586:=
581:k
576:e
553:)
548:F
543:e
536:F
532:k
525:2
522:+
516:+
511:1
506:e
499:1
495:k
488:2
485:+
477:(
474:p
457:=
452:F
448:k
423:=
418:1
414:k
405:=
402:)
394:(
389:w
385:p
364:F
342:.
338:)
335:k
329:2
326:+
320:(
317:p
300:=
297:k
289:=
286:)
280:(
275:w
271:p
250:]
244:,
235:(
218:2
210:x
207:=
202:w
198:x
194:=
167:)
164:x
161:(
158:p
79:R
71:R
65:(
59:R
49:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
