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https://github.com/JuliaStats/StatsFuns.jl

Mathematical functions related to statistics.
https://github.com/JuliaStats/StatsFuns.jl

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Mathematical functions related to statistics.

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# StatsFuns.jl



Mathematical functions related to statistics.



[![CI](https://github.com/JuliaStats/StatsFuns.jl/actions/workflows/ci.yml/badge.svg?branch=master)](https://github.com/JuliaStats/StatsFuns.jl/actions/workflows/ci.yml?query=branch%3Amaster)

[![codecov](https://codecov.io/gh/JuliaStats/StatsFuns.jl/branch/master/graph/badge.svg?token=CV9hGTr6cW)](https://codecov.io/gh/JuliaStats/StatsFuns.jl)



This package provides a collection of mathematical constants and numerical functions for statistical computing.



## Constants



```julia

twoπ,       # 2π

fourπ,      # 4π

halfπ,      # π / 2

quartπ,     # π / 4

invπ,       # 1 / π

twoinvπ,    # 2 / π

fourinvπ,   # 4 / π

inv2π,      # 1 / (2π)

inv4π,      # 1 / (4π)

sqrt2,      # √2

sqrt3,      # √3

sqrtπ,      # √π

sqrt2π,     # √2π

sqrt4π,     # √4π

sqrthalfπ,  # √(π / 2)

invsqrt2,   # 1 / √2

invsqrt2π,  # 1 / √2π

loghalf,    # log(1 / 2)

logtwo,     # log(2)

logπ,       # log(π)

log2π,      # log(2π)

log4π,      # log(4π)

```



## Basic Functions



```julia

# basicfuns

xlogx,          # x * log(x), or 0 when x is zero

xlogy,          # x * log(y), or 0 when x is zero

xlog1py,        # x * log(1 + y) for x > 0, or 0 when x == 0

logistic,       # 1 / (1 + exp(-x))

logit,          # log(x / (1 - x))

log1psq,        # log(1 + x^2)

log1pexp,       # log(1 + exp(x))

log1mexp,       # log(1 - exp(x))

log2mexp,       # log(2 - exp(x))

logexpm1,       # log(exp(x) - 1)

softplus,       # alias of log1pexp

invsoftplus,    # alias of logexpm1

log1pmx,        # log(1 + x) - x

logmxp1,        # log(x) - x + 1

logaddexp,      # log(exp(x) + exp(y))

logsubexp,      # log(abs(exp(x) - exp(y)))

logsumexp,      # log(sum(exp(x)))

softmax,        # exp(x_i) / sum(exp(x)), for i

softmax!,       # inplace softmax



# misc

logmvgamma,     # logarithm of multivariate gamma function

lstirling_asym

```



## Distribution-related functions



Do note that this package only provides low-level distribution functions. We recommend using the [Distributions.jl](https://github.com/JuliaStats/Distributions.jl/) package for a more convenient interface.



```julia

# distrs/beta

betapdf,            # pdf of beta distribution

betalogpdf,         # logpdf of beta distribution

betacdf,            # cdf of beta distribution

betaccdf,           # ccdf of beta distribution

betalogcdf,         # logcdf of beta distribution

betalogccdf,        # logccdf of beta distribution

betainvcdf,         # inverse-cdf of beta distribution

betainvccdf,        # inverse-ccdf of beta distribution

betainvlogcdf,      # inverse-logcdf of beta distribution

betainvlogccdf,     # inverse-logccdf of beta distribution



# distrs/binom

binompdf,           # pdf of binomial distribution

binomlogpdf,        # logpdf of binomial distribution

binomcdf,           # cdf of binomial distribution

binomccdf,          # ccdf of binomial distribution

binomlogcdf,        # logcdf of binomial distribution

binomlogccdf,       # logccdf of binomial distribution

binominvcdf,        # inverse-cdf of binomial distribution

binominvccdf,       # inverse-ccdf of binomial distribution

binominvlogcdf,     # inverse-logcdf of binomial distribution

binominvlogccdf,    # inverse-logccdf of binomial distribution



# distrs/chisq

chisqpdf,           # pdf of chi-square distribution

chisqlogpdf,        # logpdf of chi-square distribution

chisqcdf,           # cdf of chi-square distribution

chisqccdf,          # ccdf of chi-square distribution

chisqlogcdf,        # logcdf of chi-square distribution

chisqlogccdf,       # logccdf of chi-square distribution

chisqinvcdf,        # inverse-cdf of chi-square distribution

chisqinvccdf,       # inverse-ccdf of chi-square distribution

chisqinvlogcdf,     # inverse-logcdf of chi-square distribution

chisqinvlogccdf,    # inverse-logccdf of chi-square distribution



# distrs/fdist

fdistpdf,           # pdf of F distribution

fdistlogpdf,        # logpdf of F distribution

fdistcdf,           # cdf of F distribution

fdistccdf,          # ccdf of F distribution

fdistlogcdf,        # logcdf of F distribution

fdistlogccdf,       # logccdf of F distribution

fdistinvcdf,        # inverse-cdf of F distribution

fdistinvccdf,       # inverse-ccdf of F distribution

fdistinvlogcdf,     # inverse-logcdf of F distribution

fdistinvlogccdf,    # inverse-logccdf of F distribution



# distrs/gamma

gammapdf,           # pdf of gamma distribution

gammalogpdf,        # logpdf of gamma distribution

gammacdf,           # cdf of gamma distribution

gammaccdf,          # ccdf of gamma distribution

gammalogcdf,        # logcdf of gamma distribution

gammalogccdf,       # logccdf of gamma distribution

gammainvcdf,        # inverse-cdf of gamma distribution

gammainvccdf,       # inverse-ccdf of gamma distribution

gammainvlogcdf,     # inverse-logcdf of gamma distribution

gammainvlogccdf,    # inverse-logccdf of gamma distribution



# distrs/hyper

hyperpdf,           # pdf of hypergeometric distribution

hyperlogpdf,        # logpdf of hypergeometric distribution

hypercdf,           # cdf of hypergeometric distribution

hyperccdf,          # ccdf of hypergeometric distribution

hyperlogcdf,        # logcdf of hypergeometric distribution

hyperlogccdf,       # logccdf of hypergeometric distribution

hyperinvcdf,        # inverse-cdf of hypergeometric distribution

hyperinvccdf,       # inverse-ccdf of hypergeometric distribution

hyperinvlogcdf,     # inverse-logcdf of hypergeometric distribution

hyperinvlogccdf,    # inverse-logccdf of hypergeometric distribution



# distrs/nbeta

nbetapdf,           # pdf of noncentral beta distribution

nbetalogpdf,        # logpdf of noncentral beta distribution

nbetacdf,           # cdf of noncentral beta distribution

nbetaccdf,          # ccdf of noncentral beta distribution

nbetalogcdf,        # logcdf of noncentral beta distribution

nbetalogccdf,       # logccdf of noncentral beta distribution

nbetainvcdf,        # inverse-cdf of noncentral beta distribution

nbetainvccdf,       # inverse-ccdf of noncentral beta distribution

nbetainvlogcdf,     # inverse-logcdf of noncentral beta distribution

nbetainvlogccdf,    # inverse-logccdf of noncentral beta distribution



# distrs/nbinom

nbinompdf,           # pdf of negative nbinomial distribution

nbinomlogpdf,        # logpdf of negative nbinomial distribution

nbinomcdf,           # cdf of negative nbinomial distribution

nbinomccdf,          # ccdf of negative nbinomial distribution

nbinomlogcdf,        # logcdf of negative nbinomial distribution

nbinomlogccdf,       # logccdf of negative nbinomial distribution

nbinominvcdf,        # inverse-cdf of negative nbinomial distribution

nbinominvccdf,       # inverse-ccdf of negative nbinomial distribution

nbinominvlogcdf,     # inverse-logcdf of negative nbinomial distribution

nbinominvlogccdf,    # inverse-logccdf of negative nbinomial distribution



# distrs/nchisq

nchisqpdf,           # pdf of noncentral chi-square distribution

nchisqlogpdf,        # logpdf of noncentral chi-square distribution

nchisqcdf,           # cdf of noncentral chi-square distribution

nchisqccdf,          # ccdf of noncentral chi-square distribution

nchisqlogcdf,        # logcdf of noncentral chi-square distribution

nchisqlogccdf,       # logccdf of noncentral chi-square distribution

nchisqinvcdf,        # inverse-cdf of noncentral chi-square distribution

nchisqinvccdf,       # inverse-ccdf of noncentral chi-square distribution

nchisqinvlogcdf,     # inverse-logcdf of noncentral chi-square distribution

nchisqinvlogccdf,    # inverse-logccdf of noncentral chi-square distribution



# distrs/nfdist

nfdistpdf,           # pdf of noncentral F distribution

nfdistlogpdf,        # logpdf of noncentral F distribution

nfdistcdf,           # cdf of noncentral F distribution

nfdistccdf,          # ccdf of noncentral F distribution

nfdistlogcdf,        # logcdf of noncentral F distribution

nfdistlogccdf,       # logccdf of noncentral F distribution

nfdistinvcdf,        # inverse-cdf of noncentral F distribution

nfdistinvccdf,       # inverse-ccdf of noncentral F distribution

nfdistinvlogcdf,     # inverse-logcdf of noncentral F distribution

nfdistinvlogccdf,    # inverse-logccdf of noncentral F distribution



# distrs/norm

normpdf,            # pdf of normal distribution

normlogpdf,         # logpdf of normal distribution

normcdf,            # cdf of normal distribution

normccdf,           # ccdf of normal distribution

normlogcdf,         # logcdf of normal distribution

normlogccdf,        # logccdf of normal distribution

norminvcdf,         # inverse-cdf of normal distribution

norminvccdf,        # inverse-ccdf of normal distribution

norminvlogcdf,      # inverse-logcdf of normal distribution

norminvlogccdf,     # inverse-logccdf of normal distribution



# distrs/ntdist

ntdistpdf,           # pdf of noncentral t distribution

ntdistlogpdf,        # logpdf of noncentral t distribution

ntdistcdf,           # cdf of noncentral t distribution

ntdistccdf,          # ccdf of noncentral t distribution

ntdistlogcdf,        # logcdf of noncentral t distribution

ntdistlogccdf,       # logccdf of noncentral t distribution

ntdistinvcdf,        # inverse-cdf of noncentral t distribution

ntdistinvccdf,       # inverse-ccdf of noncentral t distribution

ntdistinvlogcdf,     # inverse-logcdf of noncentral t distribution

ntdistinvlogccdf,    # inverse-logccdf of noncentral t distribution



# distrs/pois

poispdf,           # pdf of Poisson distribution

poislogpdf,        # logpdf of Poisson distribution

poiscdf,           # cdf of Poisson distribution

poisccdf,          # ccdf of Poisson distribution

poislogcdf,        # logcdf of Poisson distribution

poislogccdf,       # logccdf of Poisson distribution

poisinvcdf,        # inverse-cdf of Poisson distribution

poisinvccdf,       # inverse-ccdf of Poisson distribution

poisinvlogcdf,     # inverse-logcdf of Poisson distribution

poisinvlogccdf,    # inverse-logccdf of Poisson distribution



# distrs/tdist

tdistpdf,           # pdf of student's t distribution

tdistlogpdf,        # logpdf of student's t distribution

tdistcdf,           # cdf of student's t distribution

tdistccdf,          # ccdf of student's t distribution

tdistlogcdf,        # logcdf of student's t distribution

tdistlogccdf,       # logccdf of student's t distribution

tdistinvcdf,        # inverse-cdf of student's t distribution

tdistinvccdf,       # inverse-ccdf of student's t distribution

tdistinvlogcdf,     # inverse-logcdf of student's t distribution

tdistinvlogccdf,    # inverse-logccdf of student's t distribution

```



## License



All of the code for this package is released under the MIT/Expat license, but this package depends upon the Rmath.jl package, which currently links to a GPL-licensed library. As such, the package currently constitutes a derived work. Work is underway to remove that dependency.