SechDistribution
✖
SechDistribution
SechDistribution[μ,σ]
represents the hyperbolic secant distribution with location parameter μ and scale parameter σ.
represents the hyperbolic secant distribution with location parameter 0 and scale parameter 1.
Details
- The probability density for value in a hyperbolic secant distribution is proportional to .
- SechDistribution allows μ to be any real number and σ to be any positive real number.
- SechDistribution allows μ and σ to be any quantities of the same unit dimensions. »
- SechDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- SechDistribution[μ,σ] represents a continuous statistical distribution defined and supported on the set of real numbers and parametrized by the real number μ (called a "location parameter") and by the positive real number σ (called the "scale parameter") that together determine the overall behavior of its probability density function (PDF). In general, the PDF of a hyperbolic secant distribution is bell-shaped and unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of μ and σ. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The zero-parameter form SechDistribution[] is equivalent to SechDistribution[0,1] and is sometimes referred to as the standard hyperbolic secant distribution.
- Comprehensive study of the hyperbolic secant distribution seems to begin in the 1950s, when it was isolated as a special case of the logistic distribution (LogisticDistribution) as part of J. Talacko's study of so-called Perks' distributions. The distribution has a number of statistical uses, including in modeling, regression, and inference, and since its inception the hyperbolic secant distribution has been used to model a number of phenomena including income distribution, optics polarization in telecommunications, and star counts in certain galactic structure models.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a hyperbolic secant distribution. Distributed[x,SechDistribution[μ,σ]], written more concisely as xSechDistribution[μ,σ], can be used to assert that a random variable x is distributed according to a hyperbolic secant distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for hyperbolic secant distributions may be given using PDF[SechDistribution[μ,σ],x] and CDF[SechDistribution[μ,σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a hyperbolic secant distribution, EstimatedDistribution to estimate a hyperbolic secant parametric distribution from given data, and FindDistributionParameters to fit data to a hyperbolic secant distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic hyperbolic secant distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic hyperbolic secant distribution.
- TransformedDistribution can be used to represent a transformed hyperbolic secant distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a hyperbolic secant distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving hyperbolic secant distributions.
- SechDistribution is related to a number of other distributions. Qualitatively, SechDistribution is similar to NormalDistribution and LogisticDistribution in that all three are bell-shaped and that each of SechDistribution[μ,σ], NormalDistribution[μ,σ], and LogisticDistribution[ σ/π] has Mean μ and Variance σ2. SechDistribution can be obtained as a transformation (TransformedDistribution) of CauchyDistribution and is a special case of MeixnerDistribution in the sense that the PDF of SechDistribution[μ,σ] is precisely the same as that of MeixnerDistribution[2σ,0,μ,1/2]. SechDistribution is also related to GompertzMakehamDistribution, FrechetDistribution, ParetoDistribution, WeibullDistribution, MaxStableDistribution, and MinStableDistribution.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
https://wolfram.com/xid/01yk2b93w5m-zi0rd
https://wolfram.com/xid/01yk2b93w5m-mzshmb
Cumulative distribution function:
https://wolfram.com/xid/01yk2b93w5m-esjle3
https://wolfram.com/xid/01yk2b93w5m-v9jcr
https://wolfram.com/xid/01yk2b93w5m-hrnyqn
https://wolfram.com/xid/01yk2b93w5m-cqhec3
https://wolfram.com/xid/01yk2b93w5m-ntsoke
Scope (7)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a hyperbolic secant distribution:
https://wolfram.com/xid/01yk2b93w5m-qhtk5j
Compare its histogram to the PDF:
https://wolfram.com/xid/01yk2b93w5m-03mwaz
Distribution parameters estimation:
https://wolfram.com/xid/01yk2b93w5m-45b7g2
Estimate the distribution parameters from sample data:
https://wolfram.com/xid/01yk2b93w5m-epi747
Compare a density histogram of the sample with the PDF of the estimated distribution:
https://wolfram.com/xid/01yk2b93w5m-f8ui5o
Skewness and kurtosis are constant:
https://wolfram.com/xid/01yk2b93w5m-8r705s
https://wolfram.com/xid/01yk2b93w5m-87tck4
Different moments with closed forms as functions of parameters:
https://wolfram.com/xid/01yk2b93w5m-js043h
https://wolfram.com/xid/01yk2b93w5m-rx074o
https://wolfram.com/xid/01yk2b93w5m-pknsqa
Closed form for symbolic order:
https://wolfram.com/xid/01yk2b93w5m-fjybja
https://wolfram.com/xid/01yk2b93w5m-zg9ct4
https://wolfram.com/xid/01yk2b93w5m-9gzmth
Closed form for symbolic order:
https://wolfram.com/xid/01yk2b93w5m-psguco
https://wolfram.com/xid/01yk2b93w5m-03niay
https://wolfram.com/xid/01yk2b93w5m-dsa3mu
https://wolfram.com/xid/01yk2b93w5m-18qzy9
https://wolfram.com/xid/01yk2b93w5m-7nob1q
https://wolfram.com/xid/01yk2b93w5m-sfhmv2
https://wolfram.com/xid/01yk2b93w5m-z9uscl
Consistent use of Quantity in parameters yields QuantityDistribution:
https://wolfram.com/xid/01yk2b93w5m-dptrrh
https://wolfram.com/xid/01yk2b93w5m-opezt3
Applications (2)Sample problems that can be solved with this function
SechDistribution is an envelope of a soliton wave. Find the full width at half maximum:
https://wolfram.com/xid/01yk2b93w5m-wc126
https://wolfram.com/xid/01yk2b93w5m-bhm9pf
https://wolfram.com/xid/01yk2b93w5m-k7epk
The arc tangent hyperbolic of the interclass correlation coefficient between twins' characteristics that follow a binormal distribution with equal means and equal variances follows hyperbolic secant distribution:
https://wolfram.com/xid/01yk2b93w5m-4la2o
The interclass correlation coefficient is independent of mean and variance , so define the arc tangent hyperbolic of the coefficient for and :
https://wolfram.com/xid/01yk2b93w5m-fo4l33
Draw a sample and test the hypothesis that it could have been drawn from SechDistribution:
https://wolfram.com/xid/01yk2b93w5m-kgg7wo
https://wolfram.com/xid/01yk2b93w5m-qz4ry
Confirm goodness of fit with QuantilePlot:
https://wolfram.com/xid/01yk2b93w5m-mtoc4w
Properties & Relations (6)Properties of the function, and connections to other functions
Sech distribution is closed under translation and scaling by a positive factor:
https://wolfram.com/xid/01yk2b93w5m-dlbe01
Relationships to other distributions:
SechDistribution is a special case of MeixnerDistribution:
https://wolfram.com/xid/01yk2b93w5m-bijwrs
https://wolfram.com/xid/01yk2b93w5m-keecw5
https://wolfram.com/xid/01yk2b93w5m-cvtks1
SechDistribution can be obtained by functional transformation from CauchyDistribution:
https://wolfram.com/xid/01yk2b93w5m-b27gmk
https://wolfram.com/xid/01yk2b93w5m-oa9dz
https://wolfram.com/xid/01yk2b93w5m-dyfxt
Compare with expressions for sech distribution:
https://wolfram.com/xid/01yk2b93w5m-iso3in
https://wolfram.com/xid/01yk2b93w5m-rhs6g
The PDF of SechDistribution is bell-shaped, similar to NormalDistribution and LogisticDistribution:
https://wolfram.com/xid/01yk2b93w5m-b3eu5d
https://wolfram.com/xid/01yk2b93w5m-tyc43v
The tails of SechDistribution are heavier than those of NormalDistribution or LogisticDistribution:
https://wolfram.com/xid/01yk2b93w5m-mclmuy
Mean and variance of all three distributions are equal:
https://wolfram.com/xid/01yk2b93w5m-t4bll1
https://wolfram.com/xid/01yk2b93w5m-9rp0pn
SechDistribution is more leptokurtic than LogisticDistribution:
https://wolfram.com/xid/01yk2b93w5m-spwmlq
Hyperbolic secant distribution mimics LogisticDistribution:
https://wolfram.com/xid/01yk2b93w5m-cxoi2f
Fit the hyperbolic secant distribution to data:
https://wolfram.com/xid/01yk2b93w5m-flpor
Compare the histogram to the PDF of the estimated distribution:
https://wolfram.com/xid/01yk2b93w5m-jbr8w7
Fit the logistic distribution to data:
https://wolfram.com/xid/01yk2b93w5m-cxhd00
https://wolfram.com/xid/01yk2b93w5m-2drv7c
DistributionFitTest rejects the hypothesis that data was drawn from the hyperbolic secant distribution:
https://wolfram.com/xid/01yk2b93w5m-dft1e6
Wolfram Research (2010), SechDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SechDistribution.html (updated 2016).
Text
Wolfram Research (2010), SechDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SechDistribution.html (updated 2016).
Wolfram Research (2010), SechDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SechDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "SechDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/SechDistribution.html.
Wolfram Language. 2010. "SechDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/SechDistribution.html.
APA
Wolfram Language. (2010). SechDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SechDistribution.html
Wolfram Language. (2010). SechDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SechDistribution.html
BibTeX
@misc{reference.wolfram_2024_sechdistribution, author="Wolfram Research", title="{SechDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/SechDistribution.html}", note=[Accessed: 12-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_sechdistribution, organization={Wolfram Research}, title={SechDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/SechDistribution.html}, note=[Accessed: 12-January-2025
]}