644 lines
18 KiB
Python
644 lines
18 KiB
Python
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"""
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.. _statsrefmanual:
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==========================================
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Statistical functions (:mod:`scipy.stats`)
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==========================================
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.. currentmodule:: scipy.stats
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This module contains a large number of probability distributions,
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summary and frequency statistics, correlation functions and statistical
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tests, masked statistics, kernel density estimation, quasi-Monte Carlo
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functionality, and more.
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Statistics is a very large area, and there are topics that are out of scope
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for SciPy and are covered by other packages. Some of the most important ones
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are:
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- `statsmodels <https://www.statsmodels.org/stable/index.html>`__:
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regression, linear models, time series analysis, extensions to topics
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also covered by ``scipy.stats``.
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- `Pandas <https://pandas.pydata.org/>`__: tabular data, time series
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functionality, interfaces to other statistical languages.
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- `PyMC <https://docs.pymc.io/>`__: Bayesian statistical
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modeling, probabilistic machine learning.
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- `scikit-learn <https://scikit-learn.org/>`__: classification, regression,
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model selection.
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- `Seaborn <https://seaborn.pydata.org/>`__: statistical data visualization.
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- `rpy2 <https://rpy2.github.io/>`__: Python to R bridge.
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Probability distributions
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=========================
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Each univariate distribution is an instance of a subclass of `rv_continuous`
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(`rv_discrete` for discrete distributions):
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.. autosummary::
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:toctree: generated/
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rv_continuous
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rv_discrete
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rv_histogram
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Continuous distributions
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------------------------
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.. autosummary::
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:toctree: generated/
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alpha -- Alpha
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anglit -- Anglit
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arcsine -- Arcsine
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argus -- Argus
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beta -- Beta
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betaprime -- Beta Prime
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bradford -- Bradford
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burr -- Burr (Type III)
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burr12 -- Burr (Type XII)
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cauchy -- Cauchy
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chi -- Chi
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chi2 -- Chi-squared
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cosine -- Cosine
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crystalball -- Crystalball
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dgamma -- Double Gamma
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dweibull -- Double Weibull
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erlang -- Erlang
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expon -- Exponential
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exponnorm -- Exponentially Modified Normal
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exponweib -- Exponentiated Weibull
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exponpow -- Exponential Power
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f -- F (Snecdor F)
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fatiguelife -- Fatigue Life (Birnbaum-Saunders)
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fisk -- Fisk
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foldcauchy -- Folded Cauchy
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foldnorm -- Folded Normal
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genlogistic -- Generalized Logistic
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gennorm -- Generalized normal
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genpareto -- Generalized Pareto
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genexpon -- Generalized Exponential
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genextreme -- Generalized Extreme Value
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gausshyper -- Gauss Hypergeometric
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gamma -- Gamma
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gengamma -- Generalized gamma
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genhalflogistic -- Generalized Half Logistic
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genhyperbolic -- Generalized Hyperbolic
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geninvgauss -- Generalized Inverse Gaussian
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gibrat -- Gibrat
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gompertz -- Gompertz (Truncated Gumbel)
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gumbel_r -- Right Sided Gumbel, Log-Weibull, Fisher-Tippett, Extreme Value Type I
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gumbel_l -- Left Sided Gumbel, etc.
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halfcauchy -- Half Cauchy
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halflogistic -- Half Logistic
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halfnorm -- Half Normal
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halfgennorm -- Generalized Half Normal
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hypsecant -- Hyperbolic Secant
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invgamma -- Inverse Gamma
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invgauss -- Inverse Gaussian
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invweibull -- Inverse Weibull
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jf_skew_t -- Jones and Faddy Skew-T
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johnsonsb -- Johnson SB
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johnsonsu -- Johnson SU
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kappa4 -- Kappa 4 parameter
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kappa3 -- Kappa 3 parameter
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ksone -- Distribution of Kolmogorov-Smirnov one-sided test statistic
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kstwo -- Distribution of Kolmogorov-Smirnov two-sided test statistic
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kstwobign -- Limiting Distribution of scaled Kolmogorov-Smirnov two-sided test statistic.
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laplace -- Laplace
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laplace_asymmetric -- Asymmetric Laplace
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levy -- Levy
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levy_l
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levy_stable
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logistic -- Logistic
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loggamma -- Log-Gamma
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loglaplace -- Log-Laplace (Log Double Exponential)
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lognorm -- Log-Normal
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loguniform -- Log-Uniform
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lomax -- Lomax (Pareto of the second kind)
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maxwell -- Maxwell
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mielke -- Mielke's Beta-Kappa
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moyal -- Moyal
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nakagami -- Nakagami
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ncx2 -- Non-central chi-squared
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ncf -- Non-central F
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nct -- Non-central Student's T
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norm -- Normal (Gaussian)
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norminvgauss -- Normal Inverse Gaussian
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pareto -- Pareto
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pearson3 -- Pearson type III
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powerlaw -- Power-function
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powerlognorm -- Power log normal
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powernorm -- Power normal
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rdist -- R-distribution
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rayleigh -- Rayleigh
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rel_breitwigner -- Relativistic Breit-Wigner
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rice -- Rice
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recipinvgauss -- Reciprocal Inverse Gaussian
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semicircular -- Semicircular
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skewcauchy -- Skew Cauchy
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skewnorm -- Skew normal
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studentized_range -- Studentized Range
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t -- Student's T
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trapezoid -- Trapezoidal
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triang -- Triangular
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truncexpon -- Truncated Exponential
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truncnorm -- Truncated Normal
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truncpareto -- Truncated Pareto
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truncweibull_min -- Truncated minimum Weibull distribution
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tukeylambda -- Tukey-Lambda
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uniform -- Uniform
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vonmises -- Von-Mises (Circular)
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vonmises_line -- Von-Mises (Line)
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wald -- Wald
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weibull_min -- Minimum Weibull (see Frechet)
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weibull_max -- Maximum Weibull (see Frechet)
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wrapcauchy -- Wrapped Cauchy
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The ``fit`` method of the univariate continuous distributions uses
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maximum likelihood estimation to fit the distribution to a data set.
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The ``fit`` method can accept regular data or *censored data*.
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Censored data is represented with instances of the `CensoredData`
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class.
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.. autosummary::
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:toctree: generated/
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CensoredData
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Multivariate distributions
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--------------------------
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.. autosummary::
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:toctree: generated/
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multivariate_normal -- Multivariate normal distribution
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matrix_normal -- Matrix normal distribution
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dirichlet -- Dirichlet
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dirichlet_multinomial -- Dirichlet multinomial distribution
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wishart -- Wishart
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invwishart -- Inverse Wishart
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multinomial -- Multinomial distribution
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special_ortho_group -- SO(N) group
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ortho_group -- O(N) group
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unitary_group -- U(N) group
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random_correlation -- random correlation matrices
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multivariate_t -- Multivariate t-distribution
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multivariate_hypergeom -- Multivariate hypergeometric distribution
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random_table -- Distribution of random tables with given marginals
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uniform_direction -- Uniform distribution on S(N-1)
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vonmises_fisher -- Von Mises-Fisher distribution
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`scipy.stats.multivariate_normal` methods accept instances
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of the following class to represent the covariance.
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.. autosummary::
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:toctree: generated/
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Covariance -- Representation of a covariance matrix
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Discrete distributions
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----------------------
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.. autosummary::
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:toctree: generated/
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bernoulli -- Bernoulli
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betabinom -- Beta-Binomial
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betanbinom -- Beta-Negative Binomial
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binom -- Binomial
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boltzmann -- Boltzmann (Truncated Discrete Exponential)
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dlaplace -- Discrete Laplacian
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geom -- Geometric
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hypergeom -- Hypergeometric
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logser -- Logarithmic (Log-Series, Series)
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nbinom -- Negative Binomial
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nchypergeom_fisher -- Fisher's Noncentral Hypergeometric
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nchypergeom_wallenius -- Wallenius's Noncentral Hypergeometric
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nhypergeom -- Negative Hypergeometric
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planck -- Planck (Discrete Exponential)
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poisson -- Poisson
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randint -- Discrete Uniform
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skellam -- Skellam
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yulesimon -- Yule-Simon
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zipf -- Zipf (Zeta)
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zipfian -- Zipfian
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An overview of statistical functions is given below. Many of these functions
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have a similar version in `scipy.stats.mstats` which work for masked arrays.
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Summary statistics
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==================
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.. autosummary::
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:toctree: generated/
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describe -- Descriptive statistics
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gmean -- Geometric mean
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hmean -- Harmonic mean
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pmean -- Power mean
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kurtosis -- Fisher or Pearson kurtosis
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mode -- Modal value
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moment -- Central moment
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expectile -- Expectile
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skew -- Skewness
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kstat --
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kstatvar --
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tmean -- Truncated arithmetic mean
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tvar -- Truncated variance
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tmin --
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tmax --
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tstd --
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tsem --
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variation -- Coefficient of variation
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find_repeats
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rankdata
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tiecorrect
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trim_mean
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gstd -- Geometric Standard Deviation
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iqr
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sem
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bayes_mvs
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mvsdist
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entropy
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differential_entropy
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median_abs_deviation
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Frequency statistics
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====================
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.. autosummary::
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:toctree: generated/
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cumfreq
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percentileofscore
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scoreatpercentile
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relfreq
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.. autosummary::
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:toctree: generated/
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binned_statistic -- Compute a binned statistic for a set of data.
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binned_statistic_2d -- Compute a 2-D binned statistic for a set of data.
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binned_statistic_dd -- Compute a d-D binned statistic for a set of data.
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Hypothesis Tests and related functions
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======================================
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SciPy has many functions for performing hypothesis tests that return a
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test statistic and a p-value, and several of them return confidence intervals
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and/or other related information.
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The headings below are based on common uses of the functions within, but due to
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the wide variety of statistical procedures, any attempt at coarse-grained
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categorization will be imperfect. Also, note that tests within the same heading
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are not interchangeable in general (e.g. many have different distributional
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assumptions).
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One Sample Tests / Paired Sample Tests
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--------------------------------------
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One sample tests are typically used to assess whether a single sample was
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drawn from a specified distribution or a distribution with specified properties
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(e.g. zero mean).
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.. autosummary::
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:toctree: generated/
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ttest_1samp
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binomtest
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quantile_test
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skewtest
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kurtosistest
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normaltest
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jarque_bera
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shapiro
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anderson
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cramervonmises
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ks_1samp
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goodness_of_fit
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chisquare
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power_divergence
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Paired sample tests are often used to assess whether two samples were drawn
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from the same distribution; they differ from the independent sample tests below
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in that each observation in one sample is treated as paired with a
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closely-related observation in the other sample (e.g. when environmental
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factors are controlled between observations within a pair but not among pairs).
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They can also be interpreted or used as one-sample tests (e.g. tests on the
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mean or median of *differences* between paired observations).
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.. autosummary::
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:toctree: generated/
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ttest_rel
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wilcoxon
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Association/Correlation Tests
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-----------------------------
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These tests are often used to assess whether there is a relationship (e.g.
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linear) between paired observations in multiple samples or among the
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coordinates of multivariate observations.
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.. autosummary::
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:toctree: generated/
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linregress
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pearsonr
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spearmanr
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pointbiserialr
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kendalltau
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weightedtau
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somersd
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siegelslopes
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theilslopes
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page_trend_test
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multiscale_graphcorr
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These association tests and are to work with samples in the form of contingency
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tables. Supporting functions are available in `scipy.stats.contingency`.
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.. autosummary::
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:toctree: generated/
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chi2_contingency
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fisher_exact
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barnard_exact
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boschloo_exact
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Independent Sample Tests
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------------------------
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Independent sample tests are typically used to assess whether multiple samples
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were independently drawn from the same distribution or different distributions
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with a shared property (e.g. equal means).
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Some tests are specifically for comparing two samples.
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.. autosummary::
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:toctree: generated/
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ttest_ind_from_stats
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poisson_means_test
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ttest_ind
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mannwhitneyu
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bws_test
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ranksums
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brunnermunzel
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mood
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ansari
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cramervonmises_2samp
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epps_singleton_2samp
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ks_2samp
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kstest
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Others are generalized to multiple samples.
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.. autosummary::
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||
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:toctree: generated/
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||
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f_oneway
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||
|
|
tukey_hsd
|
||
|
|
dunnett
|
||
|
|
kruskal
|
||
|
|
alexandergovern
|
||
|
|
fligner
|
||
|
|
levene
|
||
|
|
bartlett
|
||
|
|
median_test
|
||
|
|
friedmanchisquare
|
||
|
|
anderson_ksamp
|
||
|
|
|
||
|
|
Resampling and Monte Carlo Methods
|
||
|
|
----------------------------------
|
||
|
|
The following functions can reproduce the p-value and confidence interval
|
||
|
|
results of most of the functions above, and often produce accurate results in a
|
||
|
|
wider variety of conditions. They can also be used to perform hypothesis tests
|
||
|
|
and generate confidence intervals for custom statistics. This flexibility comes
|
||
|
|
at the cost of greater computational requirements and stochastic results.
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
monte_carlo_test
|
||
|
|
permutation_test
|
||
|
|
bootstrap
|
||
|
|
|
||
|
|
Instances of the following object can be passed into some hypothesis test
|
||
|
|
functions to perform a resampling or Monte Carlo version of the hypothesis
|
||
|
|
test.
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
MonteCarloMethod
|
||
|
|
PermutationMethod
|
||
|
|
BootstrapMethod
|
||
|
|
|
||
|
|
Multiple Hypothesis Testing and Meta-Analysis
|
||
|
|
---------------------------------------------
|
||
|
|
These functions are for assessing the results of individual tests as a whole.
|
||
|
|
Functions for performing specific multiple hypothesis tests (e.g. post hoc
|
||
|
|
tests) are listed above.
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
combine_pvalues
|
||
|
|
false_discovery_control
|
||
|
|
|
||
|
|
|
||
|
|
The following functions are related to the tests above but do not belong in the
|
||
|
|
above categories.
|
||
|
|
|
||
|
|
Quasi-Monte Carlo
|
||
|
|
=================
|
||
|
|
|
||
|
|
.. toctree::
|
||
|
|
:maxdepth: 4
|
||
|
|
|
||
|
|
stats.qmc
|
||
|
|
|
||
|
|
Contingency Tables
|
||
|
|
==================
|
||
|
|
|
||
|
|
.. toctree::
|
||
|
|
:maxdepth: 4
|
||
|
|
|
||
|
|
stats.contingency
|
||
|
|
|
||
|
|
Masked statistics functions
|
||
|
|
===========================
|
||
|
|
|
||
|
|
.. toctree::
|
||
|
|
|
||
|
|
stats.mstats
|
||
|
|
|
||
|
|
|
||
|
|
Other statistical functionality
|
||
|
|
===============================
|
||
|
|
|
||
|
|
Transformations
|
||
|
|
---------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
boxcox
|
||
|
|
boxcox_normmax
|
||
|
|
boxcox_llf
|
||
|
|
yeojohnson
|
||
|
|
yeojohnson_normmax
|
||
|
|
yeojohnson_llf
|
||
|
|
obrientransform
|
||
|
|
sigmaclip
|
||
|
|
trimboth
|
||
|
|
trim1
|
||
|
|
zmap
|
||
|
|
zscore
|
||
|
|
gzscore
|
||
|
|
|
||
|
|
Statistical distances
|
||
|
|
---------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
wasserstein_distance
|
||
|
|
wasserstein_distance_nd
|
||
|
|
energy_distance
|
||
|
|
|
||
|
|
Sampling
|
||
|
|
--------
|
||
|
|
|
||
|
|
.. toctree::
|
||
|
|
:maxdepth: 4
|
||
|
|
|
||
|
|
stats.sampling
|
||
|
|
|
||
|
|
Random variate generation / CDF Inversion
|
||
|
|
-----------------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
rvs_ratio_uniforms
|
||
|
|
|
||
|
|
Fitting / Survival Analysis
|
||
|
|
---------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
fit
|
||
|
|
ecdf
|
||
|
|
logrank
|
||
|
|
|
||
|
|
Directional statistical functions
|
||
|
|
---------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
directional_stats
|
||
|
|
circmean
|
||
|
|
circvar
|
||
|
|
circstd
|
||
|
|
|
||
|
|
Sensitivity Analysis
|
||
|
|
--------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
sobol_indices
|
||
|
|
|
||
|
|
Plot-tests
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
ppcc_max
|
||
|
|
ppcc_plot
|
||
|
|
probplot
|
||
|
|
boxcox_normplot
|
||
|
|
yeojohnson_normplot
|
||
|
|
|
||
|
|
Univariate and multivariate kernel density estimation
|
||
|
|
-----------------------------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
gaussian_kde
|
||
|
|
|
||
|
|
Warnings / Errors used in :mod:`scipy.stats`
|
||
|
|
--------------------------------------------
|
||
|
|
|
||
|
|
.. autosummary::
|
||
|
|
:toctree: generated/
|
||
|
|
|
||
|
|
DegenerateDataWarning
|
||
|
|
ConstantInputWarning
|
||
|
|
NearConstantInputWarning
|
||
|
|
FitError
|
||
|
|
|
||
|
|
Result classes used in :mod:`scipy.stats`
|
||
|
|
-----------------------------------------
|
||
|
|
|
||
|
|
.. warning::
|
||
|
|
|
||
|
|
These classes are private, but they are included here because instances
|
||
|
|
of them are returned by other statistical functions. User import and
|
||
|
|
instantiation is not supported.
|
||
|
|
|
||
|
|
.. toctree::
|
||
|
|
:maxdepth: 2
|
||
|
|
|
||
|
|
stats._result_classes
|
||
|
|
|
||
|
|
""" # noqa: E501
|
||
|
|
|
||
|
|
from ._warnings_errors import (ConstantInputWarning, NearConstantInputWarning,
|
||
|
|
DegenerateDataWarning, FitError)
|
||
|
|
from ._stats_py import *
|
||
|
|
from ._variation import variation
|
||
|
|
from .distributions import *
|
||
|
|
from ._morestats import *
|
||
|
|
from ._multicomp import *
|
||
|
|
from ._binomtest import binomtest
|
||
|
|
from ._binned_statistic import *
|
||
|
|
from ._kde import gaussian_kde
|
||
|
|
from . import mstats
|
||
|
|
from . import qmc
|
||
|
|
from ._multivariate import *
|
||
|
|
from . import contingency
|
||
|
|
from .contingency import chi2_contingency
|
||
|
|
from ._censored_data import CensoredData
|
||
|
|
from ._resampling import (bootstrap, monte_carlo_test, permutation_test,
|
||
|
|
MonteCarloMethod, PermutationMethod, BootstrapMethod)
|
||
|
|
from ._entropy import *
|
||
|
|
from ._hypotests import *
|
||
|
|
from ._rvs_sampling import rvs_ratio_uniforms
|
||
|
|
from ._page_trend_test import page_trend_test
|
||
|
|
from ._mannwhitneyu import mannwhitneyu
|
||
|
|
from ._bws_test import bws_test
|
||
|
|
from ._fit import fit, goodness_of_fit
|
||
|
|
from ._covariance import Covariance
|
||
|
|
from ._sensitivity_analysis import *
|
||
|
|
from ._survival import *
|
||
|
|
|
||
|
|
# Deprecated namespaces, to be removed in v2.0.0
|
||
|
|
from . import (
|
||
|
|
biasedurn, kde, morestats, mstats_basic, mstats_extras, mvn, stats
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
__all__ = [s for s in dir() if not s.startswith("_")] # Remove dunders.
|
||
|
|
|
||
|
|
from scipy._lib._testutils import PytestTester
|
||
|
|
test = PytestTester(__name__)
|
||
|
|
del PytestTester
|