| """ |
| .. _statsrefmanual: |
| |
| ========================================== |
| Statistical functions (:mod:`scipy.stats`) |
| ========================================== |
| |
| .. currentmodule:: scipy.stats |
| |
| This module contains a large number of probability distributions, |
| summary and frequency statistics, correlation functions and statistical |
| tests, masked statistics, kernel density estimation, quasi-Monte Carlo |
| functionality, and more. |
| |
| Statistics is a very large area, and there are topics that are out of scope |
| for SciPy and are covered by other packages. Some of the most important ones |
| are: |
| |
| - `statsmodels <https://www.statsmodels.org/stable/index.html>`__: |
| regression, linear models, time series analysis, extensions to topics |
| also covered by ``scipy.stats``. |
| - `Pandas <https://pandas.pydata.org/>`__: tabular data, time series |
| functionality, interfaces to other statistical languages. |
| - `PyMC <https://docs.pymc.io/>`__: Bayesian statistical |
| modeling, probabilistic machine learning. |
| - `scikit-learn <https://scikit-learn.org/>`__: classification, regression, |
| model selection. |
| - `Seaborn <https://seaborn.pydata.org/>`__: statistical data visualization. |
| - `rpy2 <https://rpy2.github.io/>`__: Python to R bridge. |
| |
| |
| Probability distributions |
| ========================= |
| |
| Each univariate distribution is an instance of a subclass of `rv_continuous` |
| (`rv_discrete` for discrete distributions): |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| rv_continuous |
| rv_discrete |
| rv_histogram |
| |
| Continuous distributions |
| ------------------------ |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| alpha -- Alpha |
| anglit -- Anglit |
| arcsine -- Arcsine |
| argus -- Argus |
| beta -- Beta |
| betaprime -- Beta Prime |
| bradford -- Bradford |
| burr -- Burr (Type III) |
| burr12 -- Burr (Type XII) |
| cauchy -- Cauchy |
| chi -- Chi |
| chi2 -- Chi-squared |
| cosine -- Cosine |
| crystalball -- Crystalball |
| dgamma -- Double Gamma |
| dpareto_lognorm -- Double Pareto Lognormal |
| dweibull -- Double Weibull |
| erlang -- Erlang |
| expon -- Exponential |
| exponnorm -- Exponentially Modified Normal |
| exponweib -- Exponentiated Weibull |
| exponpow -- Exponential Power |
| f -- F (Snecdor F) |
| fatiguelife -- Fatigue Life (Birnbaum-Saunders) |
| fisk -- Fisk |
| foldcauchy -- Folded Cauchy |
| foldnorm -- Folded Normal |
| genlogistic -- Generalized Logistic |
| gennorm -- Generalized normal |
| genpareto -- Generalized Pareto |
| genexpon -- Generalized Exponential |
| genextreme -- Generalized Extreme Value |
| gausshyper -- Gauss Hypergeometric |
| gamma -- Gamma |
| gengamma -- Generalized gamma |
| genhalflogistic -- Generalized Half Logistic |
| genhyperbolic -- Generalized Hyperbolic |
| geninvgauss -- Generalized Inverse Gaussian |
| gibrat -- Gibrat |
| gompertz -- Gompertz (Truncated Gumbel) |
| gumbel_r -- Right Sided Gumbel, Log-Weibull, Fisher-Tippett, Extreme Value Type I |
| gumbel_l -- Left Sided Gumbel, etc. |
| halfcauchy -- Half Cauchy |
| halflogistic -- Half Logistic |
| halfnorm -- Half Normal |
| halfgennorm -- Generalized Half Normal |
| hypsecant -- Hyperbolic Secant |
| invgamma -- Inverse Gamma |
| invgauss -- Inverse Gaussian |
| invweibull -- Inverse Weibull |
| irwinhall -- Irwin-Hall |
| jf_skew_t -- Jones and Faddy Skew-T |
| johnsonsb -- Johnson SB |
| johnsonsu -- Johnson SU |
| kappa4 -- Kappa 4 parameter |
| kappa3 -- Kappa 3 parameter |
| ksone -- Distribution of Kolmogorov-Smirnov one-sided test statistic |
| kstwo -- Distribution of Kolmogorov-Smirnov two-sided test statistic |
| kstwobign -- Limiting Distribution of scaled Kolmogorov-Smirnov two-sided test statistic. |
| landau -- Landau |
| laplace -- Laplace |
| laplace_asymmetric -- Asymmetric Laplace |
| levy -- Levy |
| levy_l |
| levy_stable |
| logistic -- Logistic |
| loggamma -- Log-Gamma |
| loglaplace -- Log-Laplace (Log Double Exponential) |
| lognorm -- Log-Normal |
| loguniform -- Log-Uniform |
| lomax -- Lomax (Pareto of the second kind) |
| maxwell -- Maxwell |
| mielke -- Mielke's Beta-Kappa |
| moyal -- Moyal |
| nakagami -- Nakagami |
| ncx2 -- Non-central chi-squared |
| ncf -- Non-central F |
| nct -- Non-central Student's T |
| norm -- Normal (Gaussian) |
| norminvgauss -- Normal Inverse Gaussian |
| pareto -- Pareto |
| pearson3 -- Pearson type III |
| powerlaw -- Power-function |
| powerlognorm -- Power log normal |
| powernorm -- Power normal |
| rdist -- R-distribution |
| rayleigh -- Rayleigh |
| rel_breitwigner -- Relativistic Breit-Wigner |
| rice -- Rice |
| recipinvgauss -- Reciprocal Inverse Gaussian |
| semicircular -- Semicircular |
| skewcauchy -- Skew Cauchy |
| skewnorm -- Skew normal |
| studentized_range -- Studentized Range |
| t -- Student's T |
| trapezoid -- Trapezoidal |
| triang -- Triangular |
| truncexpon -- Truncated Exponential |
| truncnorm -- Truncated Normal |
| truncpareto -- Truncated Pareto |
| truncweibull_min -- Truncated minimum Weibull distribution |
| tukeylambda -- Tukey-Lambda |
| uniform -- Uniform |
| vonmises -- Von-Mises (Circular) |
| vonmises_line -- Von-Mises (Line) |
| wald -- Wald |
| weibull_min -- Minimum Weibull (see Frechet) |
| weibull_max -- Maximum Weibull (see Frechet) |
| wrapcauchy -- Wrapped Cauchy |
| |
| The ``fit`` method of the univariate continuous distributions uses |
| maximum likelihood estimation to fit the distribution to a data set. |
| The ``fit`` method can accept regular data or *censored data*. |
| Censored data is represented with instances of the `CensoredData` |
| class. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| CensoredData |
| |
| |
| Multivariate distributions |
| -------------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| multivariate_normal -- Multivariate normal distribution |
| matrix_normal -- Matrix normal distribution |
| dirichlet -- Dirichlet |
| dirichlet_multinomial -- Dirichlet multinomial distribution |
| wishart -- Wishart |
| invwishart -- Inverse Wishart |
| multinomial -- Multinomial distribution |
| special_ortho_group -- SO(N) group |
| ortho_group -- O(N) group |
| unitary_group -- U(N) group |
| random_correlation -- random correlation matrices |
| multivariate_t -- Multivariate t-distribution |
| multivariate_hypergeom -- Multivariate hypergeometric distribution |
| normal_inverse_gamma -- Normal-inverse-gamma distribution |
| random_table -- Distribution of random tables with given marginals |
| uniform_direction -- Uniform distribution on S(N-1) |
| vonmises_fisher -- Von Mises-Fisher distribution |
| |
| `scipy.stats.multivariate_normal` methods accept instances |
| of the following class to represent the covariance. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| Covariance -- Representation of a covariance matrix |
| |
| |
| Discrete distributions |
| ---------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| bernoulli -- Bernoulli |
| betabinom -- Beta-Binomial |
| betanbinom -- Beta-Negative Binomial |
| binom -- Binomial |
| boltzmann -- Boltzmann (Truncated Discrete Exponential) |
| dlaplace -- Discrete Laplacian |
| geom -- Geometric |
| hypergeom -- Hypergeometric |
| logser -- Logarithmic (Log-Series, Series) |
| nbinom -- Negative Binomial |
| nchypergeom_fisher -- Fisher's Noncentral Hypergeometric |
| nchypergeom_wallenius -- Wallenius's Noncentral Hypergeometric |
| nhypergeom -- Negative Hypergeometric |
| planck -- Planck (Discrete Exponential) |
| poisson -- Poisson |
| poisson_binom -- Poisson Binomial |
| randint -- Discrete Uniform |
| skellam -- Skellam |
| yulesimon -- Yule-Simon |
| zipf -- Zipf (Zeta) |
| zipfian -- Zipfian |
| |
| |
| An overview of statistical functions is given below. Many of these functions |
| have a similar version in `scipy.stats.mstats` which work for masked arrays. |
| |
| Summary statistics |
| ================== |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| describe -- Descriptive statistics |
| gmean -- Geometric mean |
| hmean -- Harmonic mean |
| pmean -- Power mean |
| kurtosis -- Fisher or Pearson kurtosis |
| mode -- Modal value |
| moment -- Central moment |
| lmoment |
| expectile -- Expectile |
| skew -- Skewness |
| kstat -- |
| kstatvar -- |
| tmean -- Truncated arithmetic mean |
| tvar -- Truncated variance |
| tmin -- |
| tmax -- |
| tstd -- |
| tsem -- |
| variation -- Coefficient of variation |
| find_repeats |
| rankdata |
| tiecorrect |
| trim_mean |
| gstd -- Geometric Standard Deviation |
| iqr |
| sem |
| bayes_mvs |
| mvsdist |
| entropy |
| differential_entropy |
| median_abs_deviation |
| |
| Frequency statistics |
| ==================== |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| cumfreq |
| percentileofscore |
| scoreatpercentile |
| relfreq |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| binned_statistic -- Compute a binned statistic for a set of data. |
| binned_statistic_2d -- Compute a 2-D binned statistic for a set of data. |
| binned_statistic_dd -- Compute a d-D binned statistic for a set of data. |
| |
| .. _hypotests: |
| |
| Hypothesis Tests and related functions |
| ====================================== |
| SciPy has many functions for performing hypothesis tests that return a |
| test statistic and a p-value, and several of them return confidence intervals |
| and/or other related information. |
| |
| The headings below are based on common uses of the functions within, but due to |
| the wide variety of statistical procedures, any attempt at coarse-grained |
| categorization will be imperfect. Also, note that tests within the same heading |
| are not interchangeable in general (e.g. many have different distributional |
| assumptions). |
| |
| One Sample Tests / Paired Sample Tests |
| -------------------------------------- |
| One sample tests are typically used to assess whether a single sample was |
| drawn from a specified distribution or a distribution with specified properties |
| (e.g. zero mean). |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| ttest_1samp |
| binomtest |
| quantile_test |
| skewtest |
| kurtosistest |
| normaltest |
| jarque_bera |
| shapiro |
| anderson |
| cramervonmises |
| ks_1samp |
| goodness_of_fit |
| chisquare |
| power_divergence |
| |
| Paired sample tests are often used to assess whether two samples were drawn |
| from the same distribution; they differ from the independent sample tests below |
| in that each observation in one sample is treated as paired with a |
| closely-related observation in the other sample (e.g. when environmental |
| factors are controlled between observations within a pair but not among pairs). |
| They can also be interpreted or used as one-sample tests (e.g. tests on the |
| mean or median of *differences* between paired observations). |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| ttest_rel |
| wilcoxon |
| |
| Association/Correlation Tests |
| ----------------------------- |
| |
| These tests are often used to assess whether there is a relationship (e.g. |
| linear) between paired observations in multiple samples or among the |
| coordinates of multivariate observations. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| linregress |
| pearsonr |
| spearmanr |
| pointbiserialr |
| kendalltau |
| chatterjeexi |
| weightedtau |
| somersd |
| siegelslopes |
| theilslopes |
| page_trend_test |
| multiscale_graphcorr |
| |
| These association tests and are to work with samples in the form of contingency |
| tables. Supporting functions are available in `scipy.stats.contingency`. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| chi2_contingency |
| fisher_exact |
| barnard_exact |
| boschloo_exact |
| |
| Independent Sample Tests |
| ------------------------ |
| Independent sample tests are typically used to assess whether multiple samples |
| were independently drawn from the same distribution or different distributions |
| with a shared property (e.g. equal means). |
| |
| Some tests are specifically for comparing two samples. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| ttest_ind_from_stats |
| poisson_means_test |
| ttest_ind |
| mannwhitneyu |
| bws_test |
| ranksums |
| brunnermunzel |
| mood |
| ansari |
| cramervonmises_2samp |
| epps_singleton_2samp |
| ks_2samp |
| kstest |
| |
| Others are generalized to multiple samples. |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| f_oneway |
| 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 |
| power |
| |
| 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. |
| |
| Random Variables |
| ================ |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| make_distribution |
| Normal |
| Uniform |
| Mixture |
| order_statistic |
| truncate |
| abs |
| exp |
| log |
| |
| 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 |
| |
| 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 |
| |
| """ |
|
|
| 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 |
|
|
