(phc@SSKrSSKJr SSKrSSKJrJr SSKJ r J r Sr "SS5r SSS\RS .S jjr\"S S 5r\ "S 5SSSSSSS.Sj5rSrSSjr\R&\0rSrSrSrS SjrSrSrS\lSr\\\\S.rSrg)!N) namedtuple)optimizestats)check_random_state_transition_to_rngc2[R"U5nUSUS:aSUS[U5S3n[U5e[ USUS5[ USUS54nU(aE[R "US5[R"US5:aSUS3n[U5eU(dUSUS:aSUS3n[U5e[R"[R"U55(dSUS US 3n[U5eU$) zCIntersection of user-defined bounds and distribution PDF/PMF domainrzThere are no values for `z` on the interval .z!There are no integer values for `zY` on the interval defined by the user-provided bounds and the domain of the distribution.z.The intersection of user-provided bounds for `z\` and the domain of the distribution is not finite. Please provide finite bounds for shape `z` in `bounds`.) np atleast_1dlist ValueErrormaxminceilfloorallisfinite)name user_bounds shape_domainintegralmessageboundss C/var/www/html/venv/lib/python3.13/site-packages/scipy/stats/_fit.py_combine_boundsrs4-- ,K1~ A&.tf4F;'(+!!+a.,q/2+a.,q/24FRWWVAY'"((6!9*==6tf=**!! 6!9vay0.tf5**!! 66"++f% & &CD6J77;fNL!! Mch\rSrSrSrSrSrSSjrSSS.S jjrS r S r S r S r SSjr SrSrg) FitResult)aResult of fitting a discrete or continuous distribution to data Attributes ---------- params : namedtuple A namedtuple containing the maximum likelihood estimates of the shape parameters, location, and (if applicable) scale of the distribution. success : bool or None Whether the optimizer considered the optimization to terminate successfully or not. message : str or None Any status message provided by the optimizer. cXlX lX0l[USS5=(d [USS5UlUR c/OUR R S5nU(d[SUSS/-5nO[SUS/-5nU"UR6Ul UR(a7[R"UR55(dSUl SUl[US S5Ul [US S5Ulg) Npmfpdf, FitParamslocscaleFzOOptimization converged to parameter values that are inconsistent with the data.successr)_dist_datadiscretegetattrpxfshapessplitrxparamsr(r rnllfr)selfdistdatar+res shape_namesr%s r__init__FitResult.__init__:s   4-Kud1K KK/bT[[5F5Ft5L "; ug>N0NOI"; ug0EFI'  ;;r{{499;77CK=CKsIt4 sIt4 rc /SQn[[[U55S-nSRUVs/sH:n[ X5cMUR U5S-[ [ X55-PM< sn5$s snf)N)r1r(rr  z: )rmaplenjoinr,rjustrepr)r3keysmkeys r__repr__FitResult.__repr__Psx/ C ! #yy%)M%)cWT-?I#))A,-WT5G0HH%)MN NMs A8-A8NcxUbUO URnUbUO URnURRXS9$)aSNegative log-likelihood function Evaluates the negative of the log-likelihood function of the provided data at the provided parameters. Parameters ---------- params : tuple, optional The shape parameters, location, and (if applicable) scale of the distribution as a single tuple. Default is the maximum likelihood estimates (``self.params``). data : array_like, optional The data for which the log-likelihood function is to be evaluated. Default is the data to which the distribution was fit. Returns ------- nllf : float The negative of the log-likelihood function. )thetar0)r1r*r)nnlf)r3r1r5s rr2FitResult.nllfVs9,"-4;;'tTZZzzV44rhist) plot_typecSSKnURURURUR URS.nUR 5U;a'S[UR553n[U5eXbR 5nUcSSK J n UR5n[R"UR5n U"XS9$![anSn[U5UeSnAff=f)a|Visually compare the data against the fitted distribution. Available only if `matplotlib` is installed. Parameters ---------- ax : `matplotlib.axes.Axes` Axes object to draw the plot onto, otherwise uses the current Axes. plot_type : {"hist", "qq", "pp", "cdf"} Type of plot to draw. Options include: - "hist": Superposes the PDF/PMF of the fitted distribution over a normalized histogram of the data. - "qq": Scatter plot of theoretical quantiles against the empirical quantiles. Specifically, the x-coordinates are the values of the fitted distribution PPF evaluated at the percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is the number of data points, and the y-coordinates are the sorted data points. - "pp": Scatter plot of theoretical percentiles against the observed percentiles. Specifically, the x-coordinates are the percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is the number of data points, and the y-coordinates are the values of the fitted distribution CDF evaluated at the sorted data points. - "cdf": Superposes the CDF of the fitted distribution over the empirical CDF. Specifically, the x-coordinates of the empirical CDF are the sorted data points, and the y-coordinates are the percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is the number of data points. Returns ------- ax : `matplotlib.axes.Axes` The matplotlib Axes object on which the plot was drawn. Examples -------- >>> import numpy as np >>> from scipy import stats >>> import matplotlib.pyplot as plt # matplotlib must be installed >>> rng = np.random.default_rng() >>> data = stats.nbinom(5, 0.5).rvs(size=1000, random_state=rng) >>> bounds = [(0, 30), (0, 1)] >>> res = stats.fit(stats.nbinom, data, bounds) >>> ax = res.plot() # save matplotlib Axes object The `matplotlib.axes.Axes` object can be used to customize the plot. See `matplotlib.axes.Axes` documentation for details. >>> ax.set_xlabel('number of trials') # customize axis label >>> ax.get_children()[0].set_linewidth(5) # customize line widths >>> ax.legend() >>> plt.show() rNz2matplotlib must be installed to use method `plot`.) histogramqqppcdfrJz`plot_type` must be one of )ax fit_params) matplotlibModuleNotFoundError _hist_plot_qq_plot_pp_plot _cdf_plotlowersetrArmatplotlib.pyplotpyplotgcar r r1) r3rQrKrSexcrplotsplotpltrRs rr`FitResult.plotpsp 8  #ooT]]}}T^^* ?? E )3C 4E3FGGW% %__&' : +B]]4;;/ r11%# 8JG%g.C 7 8sC C)C$$C)cSSKJn URR"U6n[R "US5(aUSO[ UR5n[R "US5(aUSO[UR5nUR(aSOSnUR(a[R"XVS-5nUR"U/UQ76n URUSSSU SSSS S 9 [S US S S9n URRU"S S95 UR!S5 UR#S5 Oj[R$"XVS5nUR"U/UQ76n UR'XSSS S 9 [S SSS S9n UR!S5 UR#S5 [)UR5S:dUR(a UR*"UR4SS0U D6 O;UR'UR[R,"UR5SSS S 9 UR/SURR0SUS35 UR2"UR556 U$)Nr) MaxNLocatorr PMFPDFzFitted Distribution PMFC0)labelcolorTleftC1)densitybinsalignrk)integerk--zFitted Distribution PDF2midr0rjzHistogram of Data*Data Fitted $\tt $ z and Histogram)matplotlib.tickerrdr)supportr rrr*rr+aranger-vlinesdictxaxisset_major_locator set_xlabel set_ylabellinspacer`r=rJ zeros_like set_titlerlegendget_legend_handles_labels) r3rQrRrdr|lbubr-r0yoptionss rrUFitResult._hist_plots1**$$j1;;wqz22WQZDJJ;;wqz22WQZDJJ}}e% == "1f%A(Z(A IIafa3B/H   "4avTJG HH & &{4'@ A MM#  MM%  BC(A(Z(A GGA$&?tG L4bTJG MM#  MM% tzz?R 4== GGDJJ E&9 EW E GGDJJ djj 93   . TZZ__$5RuNKL 2//12 rc [R"UR5nUR[ UR55nU(a$SnSnUR R "U/UQ76nUn O#SnSnUnUR R"U/UQ76n URXSSU3SSS 9 UR5n UR5n [U S U S 5[U SU S5/n U(d[U S S 5[U SS54n UR(aRU(aK[U S 5[U SS-5p[R"X5nURXS S S SSSS9 GOUR(aU(dU unnUR R "U6unn[U[R""U5(aS OS5n[U[R""U5(aSOS5nUR R "UU//UQ76up[R"U S- US-5nUR R"U/UQ76nUR%XUSS S SSS9 OURXSS S SSS9 UR'U 5 UR)U 5 UR+SUR R,SU35 UR/SU35 UR1SUR R,SUS35 UR2"UR556 UR7S5 U$)N QuantileszQ-Q PercentileszP-Pr zFitted Distribution rir rjrkzorderro Referencerrg?noneT)rjrkalphamarkerfacecolorclip_ongMbP?g+?-)rjrkrrryz$ Theoretical zData rzz Plotequal)r sortr*_plotting_positionsr=r)ppfrPr`get_xlimget_ylimrrr+intr}r|rstepset_xlimset_ylimrrrrrr set_aspect)r3rQrRrNr5psqprKr0rxlimylimlimq_minq_maxq_idealp_minp_maxabqss r_qp_plotFitResult._qp_plotswwtzz"  % %c$**o 6 BI r/J/AABIA t1j1A c#7 {!C1  &{{}{{}47DG$c$q'47&;<c!fa.#c!fa.0C ==Rs1v;CF1H 5ii-G GGGcC  F ]]2LE5::%%z2DAqBKKNNq=EBKKNNq?E::>>5%.F:FLE57E!G,B0Z0B GGBC{#T   " GGCcCt   " C C djjoo%6nRDIJ bTl# TZZ__$5R {%HI 2//12 g rc *UR"SSS0UD6$)NrNTrr3kwargss rrVFitResult._qq_plots}}////rc *UR"SSS0UD6$)NrNFrrrs rrWFitResult._pp_plots}}0000rcV[R"SUS-5nX2- US-SU-- - $)Nr rg)r r})r3nrrrs rrFitResult._plotting_positions!s/ IIa1 A! $$rc [R"UR5nUR[ UR55n[ [R "U55S:aSOSnUR (aSOSnURX4USSSS 9 UR5n[R"/UQS P76nURR"U/UQ76n URXS S S S 9 URU5 URSS 5 URU5 UR!S5 UR#SURR$S35 UR'5upUR)U SSS2U SSS25 U$)Nrtr rrr0z Empirical CDFrmrri,zFitted Distribution CDFrir CDFryz$ and Empirical CDFrh)r rr*rr=uniquer+rrrr)rPr`rrrrrrrr) r3rQrRr5ecdflsxlabelrqtcdfhandleslabelss rrXFitResult._cdf_plot&s;wwtzz"''DJJ84)B.TC 3 BoT!L{{} KK # #s #zz~~a-*- 8QO D Aq f e TZZ__$55HIJ668 '$B$-". r)r*r)r+rr1r-r(NNN)g?)__name__ __module__ __qualname____firstlineno____doc__r8rDr2r`rUrrVrWrrX__static_attributes__rrrrr)sF 5,N 54L2L2\!F8t01% rrmle)guessmethod optimizerc^UnUn[TS5(aSSS.nSn O%[TS5(aSS0nSn O S n [U 5eTR5n [ R "U5nURS :wa Sn [U 5e[ R"UR[ R5(a/[ R"[ R"U55(d Sn [U 5e[U 5n X(aS OS- nU Vs/sHoRPM nnSRU5nSRUS U5nUc0n[U[ 5(aUR#U5 Un[ R$"U S45n['U 5H;nU URnUR)US 5nUcU UR*nUUU'M= U(a(S[-U53n [.R0"U [2SS9 GO[ R "U[4S9nUR6S:Xa[ R$"S5nURS:wdUR8S S:wa Sn [U 5eUR8SU:aSUSUS3n [U 5eUR8SU :aSU SUS3n [U 5e[ R$"U S45n[;UR=55UUS &UUS [U5&Un/n['U 5HTnU URnUUnU UR*nU UR>n[AUUUU5nURCU5 MV [ R "U5nU Vs/sHoR>PM nnUcS nGO[U[ 5(a[EX5VVs0sH'unnUR[ RF"U5_M) nnn[-U5[-U5- n U (aSU 3n [.R0"U [2SS9 UR#U5 Sn [ R "U Vs/sHnUURPM sn[4S9nOSn [ R "U[4S9nURS :wa [U 5eUR8SU:aS US!US3n [U 5eUR8SU :aS"U S#US3n [U 5e[ RF"US S$9nUUS [U5&UGbURI5n![ RJ"U!U5U!U'[ RL"U!U:g5Sn"U"H<nS%U URS&UUS'U!US3n [.R0"U [2SS9 M> [ RN"U!US S 2S4US S 2S 45n#[ RL"U#U!:g5Sn$U$H<nS%U URS(U!US'U#US3n [.R0"U [2SS9 M> U#nOS nU4U4S)jjn%U4U4S*jjn&U%U&S+.n'U'URQ5n([ RR"S,S,S-9 0n)UbUU)S.'[ RT"U5(aUU)S/'UbUU)S0'U"U(40U)D6n*S S S 5 [WTXW*5$![a!n S TRS 3n [U 5U eS n A ff=fs snf![an Sn [U 5U eS n A ff=fs snfs snnfs snf![an [U 5U eS n A ff=f![an [U 5U eS n A ff=f!,(df  N=f)1a%Fit a discrete or continuous distribution to data Given a distribution, data, and bounds on the parameters of the distribution, return maximum likelihood estimates of the parameters. Parameters ---------- dist : `scipy.stats.rv_continuous` or `scipy.stats.rv_discrete` The object representing the distribution to be fit to the data. data : 1D array_like The data to which the distribution is to be fit. If the data contain any of ``np.nan``, ``np.inf``, or -``np.inf``, the fit method will raise a ``ValueError``. bounds : dict or sequence of tuples, optional If a dictionary, each key is the name of a parameter of the distribution, and the corresponding value is a tuple containing the lower and upper bound on that parameter. If the distribution is defined only for a finite range of values of that parameter, no entry for that parameter is required; e.g., some distributions have parameters which must be on the interval [0, 1]. Bounds for parameters location (``loc``) and scale (``scale``) are optional; by default, they are fixed to 0 and 1, respectively. If a sequence, element *i* is a tuple containing the lower and upper bound on the *i*\ th parameter of the distribution. In this case, bounds for *all* distribution shape parameters must be provided. Optionally, bounds for location and scale may follow the distribution shape parameters. If a shape is to be held fixed (e.g. if it is known), the lower and upper bounds may be equal. If a user-provided lower or upper bound is beyond a bound of the domain for which the distribution is defined, the bound of the distribution's domain will replace the user-provided value. Similarly, parameters which must be integral will be constrained to integral values within the user-provided bounds. guess : dict or array_like, optional If a dictionary, each key is the name of a parameter of the distribution, and the corresponding value is a guess for the value of the parameter. If a sequence, element *i* is a guess for the *i*\ th parameter of the distribution. In this case, guesses for *all* distribution shape parameters must be provided. If `guess` is not provided, guesses for the decision variables will not be passed to the optimizer. If `guess` is provided, guesses for any missing parameters will be set at the mean of the lower and upper bounds. Guesses for parameters which must be integral will be rounded to integral values, and guesses that lie outside the intersection of the user-provided bounds and the domain of the distribution will be clipped. method : {'mle', 'mse'} With ``method="mle"`` (default), the fit is computed by minimizing the negative log-likelihood function. A large, finite penalty (rather than infinite negative log-likelihood) is applied for observations beyond the support of the distribution. With ``method="mse"``, the fit is computed by minimizing the negative log-product spacing function. The same penalty is applied for observations beyond the support. We follow the approach of [1]_, which is generalized for samples with repeated observations. optimizer : callable, optional `optimizer` is a callable that accepts the following positional argument. fun : callable The objective function to be optimized. `fun` accepts one argument ``x``, candidate shape parameters of the distribution, and returns the objective function value given ``x``, `dist`, and the provided `data`. The job of `optimizer` is to find values of the decision variables that minimizes `fun`. `optimizer` must also accept the following keyword argument. bounds : sequence of tuples The bounds on values of the decision variables; each element will be a tuple containing the lower and upper bound on a decision variable. If `guess` is provided, `optimizer` must also accept the following keyword argument. x0 : array_like The guesses for each decision variable. If the distribution has any shape parameters that must be integral or if the distribution is discrete and the location parameter is not fixed, `optimizer` must also accept the following keyword argument. integrality : array_like of bools For each decision variable, True if the decision variable must be constrained to integer values and False if the decision variable is continuous. `optimizer` must return an object, such as an instance of `scipy.optimize.OptimizeResult`, which holds the optimal values of the decision variables in an attribute ``x``. If attributes ``fun``, ``status``, or ``message`` are provided, they will be included in the result object returned by `fit`. Returns ------- result : `~scipy.stats._result_classes.FitResult` An object with the following fields. params : namedtuple A namedtuple containing the maximum likelihood estimates of the shape parameters, location, and (if applicable) scale of the distribution. success : bool or None Whether the optimizer considered the optimization to terminate successfully or not. message : str or None Any status message provided by the optimizer. The object has the following method: nllf(params=None, data=None) By default, the negative log-likelihood function at the fitted `params` for the given `data`. Accepts a tuple containing alternative shapes, location, and scale of the distribution and an array of alternative data. plot(ax=None) Superposes the PDF/PMF of the fitted distribution over a normalized histogram of the data. See Also -------- rv_continuous, rv_discrete Notes ----- Optimization is more likely to converge to the maximum likelihood estimate when the user provides tight bounds containing the maximum likelihood estimate. For example, when fitting a binomial distribution to data, the number of experiments underlying each sample may be known, in which case the corresponding shape parameter ``n`` can be fixed. References ---------- .. [1] Shao, Yongzhao, and Marjorie G. Hahn. "Maximum product of spacings method: a unified formulation with illustration of strong consistency." Illinois Journal of Mathematics 43.3 (1999): 489-499. Examples -------- Suppose we wish to fit a distribution to the following data. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> dist = stats.nbinom >>> shapes = (5, 0.5) >>> data = dist.rvs(*shapes, size=1000, random_state=rng) Suppose we do not know how the data were generated, but we suspect that it follows a negative binomial distribution with parameters *n* and *p*\. (See `scipy.stats.nbinom`.) We believe that the parameter *n* was fewer than 30, and we know that the parameter *p* must lie on the interval [0, 1]. We record this information in a variable `bounds` and pass this information to `fit`. >>> bounds = [(0, 30), (0, 1)] >>> res = stats.fit(dist, data, bounds) `fit` searches within the user-specified `bounds` for the values that best match the data (in the sense of maximum likelihood estimation). In this case, it found shape values similar to those from which the data were actually generated. >>> res.params FitParams(n=5.0, p=0.5028157644634368, loc=0.0) # may vary We can visualize the results by superposing the probability mass function of the distribution (with the shapes fit to the data) over a normalized histogram of the data. >>> import matplotlib.pyplot as plt # matplotlib must be installed to plot >>> res.plot() >>> plt.show() Note that the estimate for *n* was exactly integral; this is because the domain of the `nbinom` PMF includes only integral *n*, and the `nbinom` object "knows" that. `nbinom` also knows that the shape *p* must be a value between 0 and 1. In such a case - when the domain of the distribution with respect to a parameter is finite - we are not required to specify bounds for the parameter. >>> bounds = {'n': (0, 30)} # omit parameter p using a `dict` >>> res2 = stats.fit(dist, data, bounds) >>> res2.params FitParams(n=5.0, p=0.5016492009232932, loc=0.0) # may vary If we wish to force the distribution to be fit with *n* fixed at 6, we can set both the lower and upper bounds on *n* to 6. Note, however, that the value of the objective function being optimized is typically worse (higher) in this case. >>> bounds = {'n': (6, 6)} # fix parameter `n` >>> res3 = stats.fit(dist, data, bounds) >>> res3.params FitParams(n=6.0, p=0.5486556076755706, loc=0.0) # may vary >>> res3.nllf() > res.nllf() True # may vary Note that the numerical results of the previous examples are typical, but they may vary because the default optimizer used by `fit`, `scipy.optimize.differential_evolution`, is stochastic. However, we can customize the settings used by the optimizer to ensure reproducibility - or even use a different optimizer entirely - using the `optimizer` parameter. >>> from scipy.optimize import differential_evolution >>> rng = np.random.default_rng(767585560716548) >>> def optimizer(fun, bounds, *, integrality): ... return differential_evolution(fun, bounds, strategy='best2bin', ... rng=rng, integrality=integrality) >>> bounds = [(0, 30), (0, 1)] >>> res4 = stats.fit(dist, data, bounds, optimizer=optimizer) >>> res4.params FitParams(n=5.0, p=0.5015183149259951, loc=0.0) r#)rr)r r )r&r'Fr"r&Tz?`dist` must be an instance of `rv_continuous` or `rv_discrete.`zDistribution `z[` is not yet supported by `scipy.stats.fit` because shape information has not been defined.Nr z'`data` must be exactly one-dimensional.z.All elements of `data` must be finite numbers.rgr$zKBounds provided for the following unrecognized parameters will be ignored: ) stackleveldtyper)rrgzEach element of a `bounds` sequence must be a tuple containing two elements: the lower and upper bound of a distribution parameter.zcEach element of `bounds` must be a tuple specifying the lower and upper bounds of a shape parameterz*A `bounds` sequence must contain at least zP elements: tuples specifying the lower and upper bounds of all shape parameters r z.A `bounds` sequence may not contain more than zS elements: tuples specifying the lower and upper bounds of distribution parameters zLGuesses provided for the following unrecognized parameters will be ignored: zLEach element of `guess` must be a scalar guess for a distribution parameter.z)A `guess` sequence must contain at least z@ elements: scalar guesses for the distribution shape parameters z-A `guess` sequence may not contain more than z: elements: scalar guesses for the distribution parameters axiszGuess for parameter `z` rounded from z to z` clipped from c>[R"SSS9 TRX5sSSS5 $!,(df  g=fNignoreinvaliddivide)r errstate_penalized_nnlf free_paramsr5r4s rr2fit..nllfs+ [[( ;'' :< ; ; 2 Ac>[R"SSS9 TRX5sSSS5 $!,(df  g=fr)r r_penalized_nlpsfrs rnlpsffit..nlpsfs+ [[( ;((;< ; ;r)rmserrr integralityx0),hasattrr _param_infoAttributeErrorrr asarrayndim issubdtypernumberrrr=r> isinstancerupdateemptyrangepopdomainrZwarningswarnRuntimeWarningfloatsizeshaper valuesrrappendzipmeancopyroundwherecliprYranyr)+r4r5rrrrr user_guessdefault_boundsr+r param_infoen_paramsn_shapesparam param_list param_namesr7user_bounds_arrayi param_name user_boundvalidated_boundsr param_domainrcombinedr guess_arraybound default_guess unrecognized guess_roundedrounded guess_clippedclippedr2rmethods objectivekwdsr6s+` rfitr'<sFKJtU!'&9 u  '!!)%%'  ::d D yyA~;!! MM$**bii 0 0r{{4())B!!:H1a0H*45****J5))J'K))Jy12K +t$$k*$ HHh]3xA#A++J$T:J!']11 #- a ! 669+6F5GIG MM'>a @ -**[>K1$ hhv.    ![%6%6q%9Q%>IGW% %   Q ( *CH:N99D QHGW% %   Q ( *G"$B%a)GW% %HHh]3'+N,A,A,C'D()$/:+3{+,#K 8_!}!!&q) !!}++ a=,,"4\8L) ZZ( )F2<=*$$*K= J % %-0-DF-D\UERWWU^3-D F:]);; 66B^EG MM'>a @Z(9 -**3=&?3=%'4EJJ&?3=&?FKMK 9 -Jea @  vad|VAqD\J ((=M9:1=A.z!}/A/A.BC''4Q'7&8'*+1.G MM'>a @   $;!%<5)G 'I Xh 7  #DN 66+  "-D   DJ *T* 8 T43 //[ )#DII;/''!q( )&66 -3GW%1 ,  -H>F&? -W%1 , - -W%1 , -h 8 7s[\ ?\\..\3\>(\9 \>] <]: \%\\ \+\&&\+9\>> ] ]] ]7& ]22]7: ^GoodnessOfFitResult) fit_result statisticpvaluenull_distribution random_stateadi') known_paramsrRguessed_paramsr* n_mc_samplesrngc ^^^^^[TXX4XVT5nUu mpppn m[TXU 5nU"U5nT"U6mUU4Sjn[TXU 5m[U5(aUmO [Um[ TSS5nUUU4Sjn[ R "UUUSUSUS9n[R"5nSUl SUl UUl [[TUS U5URURUR 5$) uG Perform a goodness of fit test comparing data to a distribution family. Given a distribution family and data, perform a test of the null hypothesis that the data were drawn from a distribution in that family. Any known parameters of the distribution may be specified. Remaining parameters of the distribution will be fit to the data, and the p-value of the test is computed accordingly. Several statistics for comparing the distribution to data are available. Parameters ---------- dist : `scipy.stats.rv_continuous` The object representing the distribution family under the null hypothesis. data : 1D array_like Finite, uncensored data to be tested. known_params : dict, optional A dictionary containing name-value pairs of known distribution parameters. Monte Carlo samples are randomly drawn from the null-hypothesized distribution with these values of the parameters. Before the statistic is evaluated for the observed `data` and each Monte Carlo sample, only remaining unknown parameters of the null-hypothesized distribution family are fit to the samples; the known parameters are held fixed. If all parameters of the distribution family are known, then the step of fitting the distribution family to each sample is omitted. fit_params : dict, optional A dictionary containing name-value pairs of distribution parameters that have already been fit to the data, e.g. using `scipy.stats.fit` or the ``fit`` method of `dist`. Monte Carlo samples are drawn from the null-hypothesized distribution with these specified values of the parameter. However, these and all other unknown parameters of the null-hypothesized distribution family are always fit to the sample, whether that is the observed `data` or a Monte Carlo sample, before the statistic is evaluated. guessed_params : dict, optional A dictionary containing name-value pairs of distribution parameters which have been guessed. These parameters are always considered as free parameters and are fit both to the provided `data` as well as to the Monte Carlo samples drawn from the null-hypothesized distribution. The purpose of these `guessed_params` is to be used as initial values for the numerical fitting procedure. statistic : {"ad", "ks", "cvm", "filliben"} or callable, optional The statistic used to compare data to a distribution after fitting unknown parameters of the distribution family to the data. The Anderson-Darling ("ad") [1]_, Kolmogorov-Smirnov ("ks") [1]_, Cramer-von Mises ("cvm") [1]_, and Filliben ("filliben") [7]_ statistics are available. Alternatively, a callable with signature ``(dist, data, axis)`` may be supplied to compute the statistic. Here ``dist`` is a frozen distribution object (potentially with array parameters), ``data`` is an array of Monte Carlo samples (of compatible shape), and ``axis`` is the axis of ``data`` along which the statistic must be computed. n_mc_samples : int, default: 9999 The number of Monte Carlo samples drawn from the null hypothesized distribution to form the null distribution of the statistic. The sample size of each is the same as the given `data`. rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a ``Generator``. Returns ------- res : GoodnessOfFitResult An object with the following attributes. fit_result : `~scipy.stats._result_classes.FitResult` An object representing the fit of the provided `dist` to `data`. This object includes the values of distribution family parameters that fully define the null-hypothesized distribution, that is, the distribution from which Monte Carlo samples are drawn. statistic : float The value of the statistic comparing provided `data` to the null-hypothesized distribution. pvalue : float The proportion of elements in the null distribution with statistic values at least as extreme as the statistic value of the provided `data`. null_distribution : ndarray The value of the statistic for each Monte Carlo sample drawn from the null-hypothesized distribution. Notes ----- This is a generalized Monte Carlo goodness-of-fit procedure, special cases of which correspond with various Anderson-Darling tests, Lilliefors' test, etc. The test is described in [2]_, [3]_, and [4]_ as a parametric bootstrap test. This is a Monte Carlo test in which parameters that specify the distribution from which samples are drawn have been estimated from the data. We describe the test using "Monte Carlo" rather than "parametric bootstrap" throughout to avoid confusion with the more familiar nonparametric bootstrap, and describe how the test is performed below. *Traditional goodness of fit tests* Traditionally, critical values corresponding with a fixed set of significance levels are pre-calculated using Monte Carlo methods. Users perform the test by calculating the value of the test statistic only for their observed `data` and comparing this value to tabulated critical values. This practice is not very flexible, as tables are not available for all distributions and combinations of known and unknown parameter values. Also, results can be inaccurate when critical values are interpolated from limited tabulated data to correspond with the user's sample size and fitted parameter values. To overcome these shortcomings, this function allows the user to perform the Monte Carlo trials adapted to their particular data. *Algorithmic overview* In brief, this routine executes the following steps: 1. Fit unknown parameters to the given `data`, thereby forming the "null-hypothesized" distribution, and compute the statistic of this pair of data and distribution. 2. Draw random samples from this null-hypothesized distribution. 3. Fit the unknown parameters to each random sample. 4. Calculate the statistic between each sample and the distribution that has been fit to the sample. 5. Compare the value of the statistic corresponding with `data` from (1) against the values of the statistic corresponding with the random samples from (4). The p-value is the proportion of samples with a statistic value greater than or equal to the statistic of the observed data. In more detail, the steps are as follows. First, any unknown parameters of the distribution family specified by `dist` are fit to the provided `data` using maximum likelihood estimation. (One exception is the normal distribution with unknown location and scale: we use the bias-corrected standard deviation ``np.std(data, ddof=1)`` for the scale as recommended in [1]_.) These values of the parameters specify a particular member of the distribution family referred to as the "null-hypothesized distribution", that is, the distribution from which the data were sampled under the null hypothesis. The `statistic`, which compares data to a distribution, is computed between `data` and the null-hypothesized distribution. Next, many (specifically `n_mc_samples`) new samples, each containing the same number of observations as `data`, are drawn from the null-hypothesized distribution. All unknown parameters of the distribution family `dist` are fit to *each resample*, and the `statistic` is computed between each sample and its corresponding fitted distribution. These values of the statistic form the Monte Carlo null distribution (not to be confused with the "null-hypothesized distribution" above). The p-value of the test is the proportion of statistic values in the Monte Carlo null distribution that are at least as extreme as the statistic value of the provided `data`. More precisely, the p-value is given by .. math:: p = \frac{b + 1} {m + 1} where :math:`b` is the number of statistic values in the Monte Carlo null distribution that are greater than or equal to the statistic value calculated for `data`, and :math:`m` is the number of elements in the Monte Carlo null distribution (`n_mc_samples`). The addition of :math:`1` to the numerator and denominator can be thought of as including the value of the statistic corresponding with `data` in the null distribution, but a more formal explanation is given in [5]_. *Limitations* The test can be very slow for some distribution families because unknown parameters of the distribution family must be fit to each of the Monte Carlo samples, and for most distributions in SciPy, distribution fitting performed via numerical optimization. *Anti-Pattern* For this reason, it may be tempting to treat parameters of the distribution pre-fit to `data` (by the user) as though they were `known_params`, as specification of all parameters of the distribution precludes the need to fit the distribution to each Monte Carlo sample. (This is essentially how the original Kilmogorov-Smirnov test is performed.) Although such a test can provide evidence against the null hypothesis, the test is conservative in the sense that small p-values will tend to (greatly) *overestimate* the probability of making a type I error (that is, rejecting the null hypothesis although it is true), and the power of the test is low (that is, it is less likely to reject the null hypothesis even when the null hypothesis is false). This is because the Monte Carlo samples are less likely to agree with the null-hypothesized distribution as well as `data`. This tends to increase the values of the statistic recorded in the null distribution, so that a larger number of them exceed the value of statistic for `data`, thereby inflating the p-value. References ---------- .. [1] M. A. Stephens (1974). "EDF Statistics for Goodness of Fit and Some Comparisons." Journal of the American Statistical Association, Vol. 69, pp. 730-737. .. [2] W. Stute, W. G. Manteiga, and M. P. Quindimil (1993). "Bootstrap based goodness-of-fit-tests." Metrika 40.1: 243-256. .. [3] C. Genest, & B Rémillard. (2008). "Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models." Annales de l'IHP Probabilités et statistiques. Vol. 44. No. 6. .. [4] I. Kojadinovic and J. Yan (2012). "Goodness-of-fit testing based on a weighted bootstrap: A fast large-sample alternative to the parametric bootstrap." Canadian Journal of Statistics 40.3: 480-500. .. [5] B. Phipson and G. K. Smyth (2010). "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1. .. [6] H. W. Lilliefors (1967). "On the Kolmogorov-Smirnov test for normality with mean and variance unknown." Journal of the American statistical Association 62.318: 399-402. .. [7] Filliben, James J. "The probability plot correlation coefficient test for normality." Technometrics 17.1 (1975): 111-117. Examples -------- A well-known test of the null hypothesis that data were drawn from a given distribution is the Kolmogorov-Smirnov (KS) test, available in SciPy as `scipy.stats.ks_1samp`. Suppose we wish to test whether the following data: >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> x = stats.uniform.rvs(size=75, random_state=rng) were sampled from a normal distribution. To perform a KS test, the empirical distribution function of the observed data will be compared against the (theoretical) cumulative distribution function of a normal distribution. Of course, to do this, the normal distribution under the null hypothesis must be fully specified. This is commonly done by first fitting the ``loc`` and ``scale`` parameters of the distribution to the observed data, then performing the test. >>> loc, scale = np.mean(x), np.std(x, ddof=1) >>> cdf = stats.norm(loc, scale).cdf >>> stats.ks_1samp(x, cdf) KstestResult(statistic=0.1119257570456813, pvalue=0.2827756409939257, statistic_location=0.7751845155861765, statistic_sign=-1) An advantage of the KS-test is that the p-value - the probability of obtaining a value of the test statistic under the null hypothesis as extreme as the value obtained from the observed data - can be calculated exactly and efficiently. `goodness_of_fit` can only approximate these results. >>> known_params = {'loc': loc, 'scale': scale} >>> res = stats.goodness_of_fit(stats.norm, x, known_params=known_params, ... statistic='ks', rng=rng) >>> res.statistic, res.pvalue (0.1119257570456813, 0.2788) The statistic matches exactly, but the p-value is estimated by forming a "Monte Carlo null distribution", that is, by explicitly drawing random samples from `scipy.stats.norm` with the provided parameters and calculating the stastic for each. The fraction of these statistic values at least as extreme as ``res.statistic`` approximates the exact p-value calculated by `scipy.stats.ks_1samp`. However, in many cases, we would prefer to test only that the data were sampled from one of *any* member of the normal distribution family, not specifically from the normal distribution with the location and scale fitted to the observed sample. In this case, Lilliefors [6]_ argued that the KS test is far too conservative (that is, the p-value overstates the actual probability of rejecting a true null hypothesis) and thus lacks power - the ability to reject the null hypothesis when the null hypothesis is actually false. Indeed, our p-value above is approximately 0.28, which is far too large to reject the null hypothesis at any common significance level. Consider why this might be. Note that in the KS test above, the statistic always compares data against the CDF of a normal distribution fitted to the *observed data*. This tends to reduce the value of the statistic for the observed data, but it is "unfair" when computing the statistic for other samples, such as those we randomly draw to form the Monte Carlo null distribution. It is easy to correct for this: whenever we compute the KS statistic of a sample, we use the CDF of a normal distribution fitted to *that sample*. The null distribution in this case has not been calculated exactly and is tyically approximated using Monte Carlo methods as described above. This is where `goodness_of_fit` excels. >>> res = stats.goodness_of_fit(stats.norm, x, statistic='ks', ... rng=rng) >>> res.statistic, res.pvalue (0.1119257570456813, 0.0196) Indeed, this p-value is much smaller, and small enough to (correctly) reject the null hypothesis at common significance levels, including 5% and 2.5%. However, the KS statistic is not very sensitive to all deviations from normality. The original advantage of the KS statistic was the ability to compute the null distribution theoretically, but a more sensitive statistic - resulting in a higher test power - can be used now that we can approximate the null distribution computationally. The Anderson-Darling statistic [1]_ tends to be more sensitive, and critical values of the this statistic have been tabulated for various significance levels and sample sizes using Monte Carlo methods. >>> res = stats.anderson(x, 'norm') >>> print(res.statistic) 1.2139573337497467 >>> print(res.critical_values) [0.549 0.625 0.75 0.875 1.041] >>> print(res.significance_level) [15. 10. 5. 2.5 1. ] Here, the observed value of the statistic exceeds the critical value corresponding with a 1% significance level. This tells us that the p-value of the observed data is less than 1%, but what is it? We could interpolate from these (already-interpolated) values, but `goodness_of_fit` can estimate it directly. >>> res = stats.goodness_of_fit(stats.norm, x, statistic='ad', ... rng=rng) >>> res.statistic, res.pvalue (1.2139573337497467, 0.0034) A further advantage is that use of `goodness_of_fit` is not limited to a particular set of distributions or conditions on which parameters are known versus which must be estimated from data. Instead, `goodness_of_fit` can estimate p-values relatively quickly for any distribution with a sufficiently fast and reliable ``fit`` method. For instance, here we perform a goodness of fit test using the Cramer-von Mises statistic against the Rayleigh distribution with known location and unknown scale. >>> rng = np.random.default_rng() >>> x = stats.chi(df=2.2, loc=0, scale=2).rvs(size=1000, random_state=rng) >>> res = stats.goodness_of_fit(stats.rayleigh, x, statistic='cvm', ... known_params={'loc': 0}, rng=rng) This executes fairly quickly, but to check the reliability of the ``fit`` method, we should inspect the fit result. >>> res.fit_result # location is as specified, and scale is reasonable params: FitParams(loc=0.0, scale=2.1026719844231243) success: True message: 'The fit was performed successfully.' >>> import matplotlib.pyplot as plt # matplotlib must be installed to plot >>> res.fit_result.plot() >>> plt.show() If the distribution is not fit to the observed data as well as possible, the test may not control the type I error rate, that is, the chance of rejecting the null hypothesis even when it is true. We should also look for extreme outliers in the null distribution that may be caused by unreliable fitting. These do not necessarily invalidate the result, but they tend to reduce the test's power. >>> _, ax = plt.subplots() >>> ax.hist(np.log10(res.null_distribution)) >>> ax.set_xlabel("log10 of CVM statistic under the null hypothesis") >>> ax.set_ylabel("Frequency") >>> ax.set_title("Histogram of the Monte Carlo null distribution") >>> plt.show() This plot seems reassuring. If ``fit`` method is working reliably, and if the distribution of the test statistic is not particularly sensitive to the values of the fitted parameters, then the p-value provided by `goodness_of_fit` is expected to be a good approximation. >>> res.statistic, res.pvalue (0.2231991510248692, 0.0525) c$>TRUTS9$)N)rr-)rvs)rnhd_distr2s rr5goodness_of_fit..rvsgs||C|88r alternativegreatercZ>[R"XS5nT"U5nT"U6nT"X0SS9$)Nrhr)r moveaxis)r5rrfd_valsrfd_dist compare_funr4fit_funs r statistic_fun&goodness_of_fit..statistic_funrs2{{4r*4=?833rTrh) vectorized n_resamplesrr8z#The fit was performed successfully.F)_gof_iv _get_fit_funcallable _compare_dictr,rmonte_carlo_testrOptimizeResultr(rr0r(rr*r+r,)r4r5r/rRr0r*r1r2argsfixed_nhd_paramsfixed_rfd_paramsguessed_nhd_paramsguessed_rfd_paramsn_mc_samples_int nhd_fit_funnhd_valsr5r8r@r6opt_resr>r?r6s` ` @@@rgoodness_of_fitrSsn 4ZC 1D>B;T4#3$4ctT/1K4 HXH94;KLG  #I. +}i@K4  sMd-9-8 :C%%'GGO;GOGI ytUGD"}}cjj"44 66rc^^^^ ^ ^ TRc/OTRRS5nUSS/-nUVs/sHnSU-PM snm [T 5RT5(+nUVs/sHoT;dM TR US5PM snm U(a UU 4Sjn U $T[ ;a UU4Sjn U $UUUU 4Sjm U 4Sjn U $s snfs snf) Nr$r&r'fc:>TVs/sHnTUPM sn$s snfrr)r5r fixed_params fparam_namess rr?_get_fit_fun..fit_funs 3?@<4L&<@ @@sc>[T"U40TD6n[R"[R"U65nURS:aUS[R 4nU$)Nr .) _fit_funsr rbroadcast_arraysrnewaxis)r5r1r4rWs rr?rYsQt_T:\:FZZ 3 3V <=F{{QRZZ0Mrc4>TR"U/TQ70TDTD6$r)r')r5r4rWr0guessed_shapess r fit_fun_1d _get_fit_fun..fit_fun_1ds,88D,>,^,*, ,rc>[R"TSUS9nURS:aURS[R4nU$)Nrh)rarrr .)r apply_along_axisrTr])r5r1r`s rr?rYs=(("$GF{{Q#rzz/2Mr)r.r/rZ differencerr[) r4r5r0rWr7rr all_fixedr0r?r`rXr_s ` `` @@@rrErEs +"1B1B41HK 00K)45CH5L %00>>I*C*!>.A2n((D1*CN A* N%   " N , ,  N76CsC 1 C>CcUnUnUc1Uc.[R"USS9n[R"USSS9nX44$Uc[R"USS9nX44$Uc([R"X- S-RSS95nX44$)Nrhrr )ddofrrg)r rstdsqrt)r5flocfscaler&r's r _fit_normrns C E {u}ggd$t!"- : ggd$ : $*q..B.78 :rc "[R"USS9nURSn[R"SUS-5nSU-S- U- UR U5UR USSSS245--n[R "USS9nU*U- $)Nrhrr rg.)r rrr}logcdflogsfsum)r4r5rr0rrSiSs r_anderson_darlingrus 2A 2A !QqSA A#'1 AAc4R4iL)AA BB rA 26MrczURSn[R"SUS-5U- U- RSS9$)Nrhg?r rrr r}rcdfvalsrs r_compute_dplusrzs= bA IIc1q5 !A % / 4 4" 4 ==rctURSnU[R"SU5U- - RSS9$)Nrhgrrwrxs r_compute_dminusr|s9 bA biiQ') ) . .B . 77rc[R"XS9nURU5n[R"XBS5n[ U5n[ U5n[R "XV5$)Nrrh)r rrPr;rzr|maximum)r4r5rr0ryDplusDminuss r_kolmogorov_smirnovrsP  AhhqkGkk',G 7 #E W %F ::e $$rcURSSS9nURSSS9n[R"X- X- -SS9n[R"[R"X- S-SS9[R"X- S-SS9-5nXE- $)NrhT)rkeepdimsrrg)rr rrrk)XMXmMmnumdens r_corrrs} R$ 'B R$ 'B &&!&QV$2 .C ''"&&!&12.! 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