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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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dpareto_lognorm -- Double Pareto Lognormal |
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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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irwinhall -- Irwin-Hall |
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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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landau -- Landau |
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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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normal_inverse_gamma -- Normal-inverse-gamma 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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poisson_binom -- Poisson Binomial |
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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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lmoment |
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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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.. _hypotests: |
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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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chatterjeexi |
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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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:toctree: generated/ |
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f_oneway |
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tukey_hsd |
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dunnett |
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kruskal |
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alexandergovern |
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fligner |
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levene |
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bartlett |
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median_test |
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friedmanchisquare |
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anderson_ksamp |
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Resampling and Monte Carlo Methods |
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---------------------------------- |
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The following functions can reproduce the p-value and confidence interval |
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results of most of the functions above, and often produce accurate results in a |
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wider variety of conditions. They can also be used to perform hypothesis tests |
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and generate confidence intervals for custom statistics. This flexibility comes |
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at the cost of greater computational requirements and stochastic results. |
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.. autosummary:: |
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:toctree: generated/ |
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monte_carlo_test |
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permutation_test |
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bootstrap |
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power |
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Instances of the following object can be passed into some hypothesis test |
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functions to perform a resampling or Monte Carlo version of the hypothesis |
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test. |
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.. autosummary:: |
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:toctree: generated/ |
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MonteCarloMethod |
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PermutationMethod |
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BootstrapMethod |
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Multiple Hypothesis Testing and Meta-Analysis |
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--------------------------------------------- |
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These functions are for assessing the results of individual tests as a whole. |
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Functions for performing specific multiple hypothesis tests (e.g. post hoc |
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tests) are listed above. |
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.. autosummary:: |
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:toctree: generated/ |
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combine_pvalues |
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false_discovery_control |
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The following functions are related to the tests above but do not belong in the |
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above categories. |
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Random Variables |
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================ |
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.. autosummary:: |
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:toctree: generated/ |
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make_distribution |
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Normal |
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Uniform |
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Mixture |
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order_statistic |
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truncate |
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abs |
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exp |
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log |
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Quasi-Monte Carlo |
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================= |
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.. toctree:: |
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:maxdepth: 4 |
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stats.qmc |
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Contingency Tables |
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================== |
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.. toctree:: |
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:maxdepth: 4 |
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stats.contingency |
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Masked statistics functions |
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=========================== |
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.. toctree:: |
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stats.mstats |
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Other statistical functionality |
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=============================== |
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Transformations |
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--------------- |
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.. autosummary:: |
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:toctree: generated/ |
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boxcox |
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boxcox_normmax |
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boxcox_llf |
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yeojohnson |
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yeojohnson_normmax |
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yeojohnson_llf |
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obrientransform |
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sigmaclip |
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trimboth |
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trim1 |
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zmap |
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zscore |
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gzscore |
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Statistical distances |
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--------------------- |
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.. autosummary:: |
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:toctree: generated/ |
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wasserstein_distance |
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wasserstein_distance_nd |
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energy_distance |
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Sampling |
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-------- |
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.. toctree:: |
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:maxdepth: 4 |
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stats.sampling |
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Fitting / Survival Analysis |
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--------------------------- |
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.. autosummary:: |
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:toctree: generated/ |
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fit |
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ecdf |
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logrank |
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|
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Directional statistical functions |
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--------------------------------- |
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|
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.. autosummary:: |
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:toctree: generated/ |
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|
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directional_stats |
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circmean |
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circvar |
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circstd |
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|
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|
Sensitivity Analysis |
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|
-------------------- |
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|
|
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|
.. autosummary:: |
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|
:toctree: generated/ |
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|
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sobol_indices |
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|
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|
Plot-tests |
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|
---------- |
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|
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|
.. autosummary:: |
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:toctree: generated/ |
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|
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|
ppcc_max |
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|
ppcc_plot |
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|
probplot |
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|
boxcox_normplot |
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|
yeojohnson_normplot |
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|
|
|
|
Univariate and multivariate kernel density estimation |
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|
----------------------------------------------------- |
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|
|
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|
.. autosummary:: |
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|
:toctree: generated/ |
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|
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|
gaussian_kde |
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|
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|
Warnings / Errors used in :mod:`scipy.stats` |
|
|
-------------------------------------------- |
|
|
|
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|
.. autosummary:: |
|
|
:toctree: generated/ |
|
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|
|
|
DegenerateDataWarning |
|
|
ConstantInputWarning |
|
|
NearConstantInputWarning |
|
|
FitError |
|
|
|
|
|
Result classes used in :mod:`scipy.stats` |
|
|
----------------------------------------- |
|
|
|
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|
.. 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 |
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|
|
|
|
stats._result_classes |
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|
""" |
|
|
|
|
|
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, power, |
|
|
MonteCarloMethod, PermutationMethod, BootstrapMethod) |
|
|
from ._entropy import * |
|
|
from ._hypotests import * |
|
|
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 * |
|
|
from ._distribution_infrastructure import ( |
|
|
make_distribution, Mixture, order_statistic, truncate, exp, log, abs |
|
|
) |
|
|
from ._new_distributions import Normal, Uniform |
|
|
from ._mgc import multiscale_graphcorr |
|
|
from ._correlation import chatterjeexi |
|
|
|
|
|
|
|
|
|
|
|
from . import ( |
|
|
biasedurn, kde, morestats, mstats_basic, mstats_extras, mvn, stats |
|
|
) |
|
|
|
|
|
|
|
|
__all__ = [s for s in dir() if not s.startswith("_")] |
|
|
|
|
|
from scipy._lib._testutils import PytestTester |
|
|
test = PytestTester(__name__) |
|
|
del PytestTester |
|
|
|
