(phbSSKrSSKrSSKJrJr SSKJr SSKJrJ r J r J r J r J r JrJrJrJrJrJrJrJrJrJr SSKrSSKJr SSKJrJr S/r\R>"5r "S S5r!S r"g) N)linalgspecial)check_random_state)asarray atleast_2dreshapezerosnewaxisexppisqrtravelpower atleast_1dsqueezesum transposeonescov)_mvn)gaussian_kernel_estimategaussian_kernel_estimate_log gaussian_kdec\rSrSrSrSSjrSr\rSrSr SSjr S r SS jr S r S r\ rS \lSSjrSr\S5rSrSrSr\S5r\S5rSrg)r+a.Representation of a kernel-density estimate using Gaussian kernels. Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. `gaussian_kde` works for both uni-variate and multi-variate data. It includes automatic bandwidth determination. The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be oversmoothed. Parameters ---------- dataset : array_like Datapoints to estimate from. In case of univariate data this is a 1-D array, otherwise a 2-D array with shape (# of dims, # of data). bw_method : str, scalar or callable, optional The method used to calculate the estimator bandwidth. This can be 'scott', 'silverman', a scalar constant or a callable. If a scalar, this will be used directly as `kde.factor`. If a callable, it should take a `gaussian_kde` instance as only parameter and return a scalar. If None (default), 'scott' is used. See Notes for more details. weights : array_like, optional weights of datapoints. This must be the same shape as dataset. If None (default), the samples are assumed to be equally weighted Attributes ---------- dataset : ndarray The dataset with which `gaussian_kde` was initialized. d : int Number of dimensions. n : int Number of datapoints. neff : int Effective number of datapoints. .. versionadded:: 1.2.0 factor : float The bandwidth factor, obtained from `kde.covariance_factor`. The square of `kde.factor` multiplies the covariance matrix of the data in the kde estimation. covariance : ndarray The covariance matrix of `dataset`, scaled by the calculated bandwidth (`kde.factor`). inv_cov : ndarray The inverse of `covariance`. Methods ------- evaluate __call__ integrate_gaussian integrate_box_1d integrate_box integrate_kde pdf logpdf resample set_bandwidth covariance_factor Notes ----- Bandwidth selection strongly influences the estimate obtained from the KDE (much more so than the actual shape of the kernel). Bandwidth selection can be done by a "rule of thumb", by cross-validation, by "plug-in methods" or by other means; see [3]_, [4]_ for reviews. `gaussian_kde` uses a rule of thumb, the default is Scott's Rule. Scott's Rule [1]_, implemented as `scotts_factor`, is:: n**(-1./(d+4)), with ``n`` the number of data points and ``d`` the number of dimensions. In the case of unequally weighted points, `scotts_factor` becomes:: neff**(-1./(d+4)), with ``neff`` the effective number of datapoints. Silverman's Rule [2]_, implemented as `silverman_factor`, is:: (n * (d + 2) / 4.)**(-1. / (d + 4)). or in the case of unequally weighted points:: (neff * (d + 2) / 4.)**(-1. / (d + 4)). Good general descriptions of kernel density estimation can be found in [1]_ and [2]_, the mathematics for this multi-dimensional implementation can be found in [1]_. With a set of weighted samples, the effective number of datapoints ``neff`` is defined by:: neff = sum(weights)^2 / sum(weights^2) as detailed in [5]_. `gaussian_kde` does not currently support data that lies in a lower-dimensional subspace of the space in which it is expressed. For such data, consider performing principal component analysis / dimensionality reduction and using `gaussian_kde` with the transformed data. References ---------- .. [1] D.W. Scott, "Multivariate Density Estimation: Theory, Practice, and Visualization", John Wiley & Sons, New York, Chicester, 1992. .. [2] B.W. Silverman, "Density Estimation for Statistics and Data Analysis", Vol. 26, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1986. .. [3] B.A. Turlach, "Bandwidth Selection in Kernel Density Estimation: A Review", CORE and Institut de Statistique, Vol. 19, pp. 1-33, 1993. .. [4] D.M. Bashtannyk and R.J. Hyndman, "Bandwidth selection for kernel conditional density estimation", Computational Statistics & Data Analysis, Vol. 36, pp. 279-298, 2001. .. [5] Gray P. G., 1969, Journal of the Royal Statistical Society. Series A (General), 132, 272 Examples -------- Generate some random two-dimensional data: >>> import numpy as np >>> from scipy import stats >>> def measure(n): ... "Measurement model, return two coupled measurements." ... m1 = np.random.normal(size=n) ... m2 = np.random.normal(scale=0.5, size=n) ... return m1+m2, m1-m2 >>> m1, m2 = measure(2000) >>> xmin = m1.min() >>> xmax = m1.max() >>> ymin = m2.min() >>> ymax = m2.max() Perform a kernel density estimate on the data: >>> X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j] >>> positions = np.vstack([X.ravel(), Y.ravel()]) >>> values = np.vstack([m1, m2]) >>> kernel = stats.gaussian_kde(values) >>> Z = np.reshape(kernel(positions).T, X.shape) Plot the results: >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r, ... extent=[xmin, xmax, ymin, ymax]) >>> ax.plot(m1, m2, 'k.', markersize=2) >>> ax.set_xlim([xmin, xmax]) >>> ax.set_ylim([ymin, ymax]) >>> plt.show() Nc [[U55UlURRS:d [ S5eURR uUlUlUb[U5R[5Ul U=R[UR5-sl URRS:wa [ S5e[UR5UR:wa [ S5eS[URS-5- UlUR UR:a Sn[ U5eUR#US9 g![$R&anSn[$R&"U5UeSnAff=f) Nrz.`dataset` input should have multiple elements.z*`weights` input should be one-dimensional.z%`weights` input should be of length na1Number of dimensions is greater than number of samples. This results in a singular data covariance matrix, which cannot be treated using the algorithms implemented in `gaussian_kde`. Note that `gaussian_kde` interprets each *column* of `dataset` to be a point; consider transposing the input to `dataset`. bw_methodabThe data appears to lie in a lower-dimensional subspace of the space in which it is expressed. This has resulted in a singular data covariance matrix, which cannot be treated using the algorithms implemented in `gaussian_kde`. Consider performing principal component analysis / dimensionality reduction and using `gaussian_kde` with the transformed data.)rrdatasetsize ValueErrorshapednrastypefloat_weightsrweightsndimlen_neff set_bandwidthr LinAlgError)selfr!r r*msges C/var/www/html/venv/lib/python3.13/site-packages/scipy/stats/_kde.py__init__gaussian_kde.__init__sA!''"23 ||  1$MN N++  &w/66u=DM MMS/ /M||  A% !MNN4==!TVV+ !HII3t}}a/00DJ 66DFF?-C S/ ! 1     3!! 1?C$$S)q 0 1s EF /FF c[[U55nURup#X R:waLUS:Xa)X0R:Xa[ XRS45nSnOSUSUR3n[ U5e[ URU5upV[U"URRURSS2S4URURU5nUSS2S4$)aEvaluate the estimated pdf on a set of points. Parameters ---------- points : (# of dimensions, # of points)-array Alternatively, a (# of dimensions,) vector can be passed in and treated as a single point. Returns ------- values : (# of points,)-array The values at each point. Raises ------ ValueError : if the dimensionality of the input points is different than the dimensionality of the KDE. rpoints have dimension , dataset has dimension Nr) rrr$r%rr#_get_output_dtype covariancerr!Tr*cho_cov)r0pointsr%mr1 output_dtypespecresults r3evaluategaussian_kde.evaluates(GFO,|| ;Av!vv+ &&!5/s3004x9 o%.tG )$/ LLNNDLLD1 HHdllL2ad|c[[U55n[U5nURUR4:wa[ SUR35eURURUR4:wa[ SUR35eUSS2[ 4nURU-n[R"U5nURU- n[R"XE5n[R"[R"US55n[S[ -URSS- 5U-n[#XV-SS9S- n [#[%U *5UR&-SS9U- n U $)a Multiply estimated density by a multivariate Gaussian and integrate over the whole space. Parameters ---------- mean : aray_like A 1-D array, specifying the mean of the Gaussian. cov : array_like A 2-D array, specifying the covariance matrix of the Gaussian. Returns ------- result : scalar The value of the integral. Raises ------ ValueError If the mean or covariance of the input Gaussian differs from the KDE's dimensionality. zmean does not have dimension z#covariance does not have dimension Nrr@axis)rrrr$r%r#r r:r cho_factorr! cho_solvenpproddiagonalrr rr r*) r0meanrsum_cov sum_cov_choldifftdiffsqrt_det norm_constenergiesrAs r3integrate_gaussiangaussian_kde.integrate_gaussians80'$-(o ::$&& "t|| _5 ^4567 rDc*UbSU0nO0n[ [R"XURURUR 40UD6upVSSS5 W(a'SUR S-3n[R"USS9 W$!,(df  N>=f)aComputes the integral of a pdf over a rectangular interval. Parameters ---------- low_bounds : array_like A 1-D array containing the lower bounds of integration. high_bounds : array_like A 1-D array containing the upper bounds of integration. maxpts : int, optional The maximum number of points to use for integration. Returns ------- value : scalar The result of the integral. Nmaxptsz4An integral in _mvn.mvnun requires more points than ir) stacklevel) MVN_LOCKrmvnun_weightedr!r*r:r%warningswarn)r0 low_bounds high_boundsrc extra_kwdsr_informr1s r3 integrate_boxgaussian_kde.integrate_boxos$  "F+JJ  // 04 dll04OCMOME HRVXC MM#! , Xs ;B BcURUR:wa [S5eURUR:aUnUnOUnUnURUR-n[R "U5nSn[ UR5HnURSS2U[4nURU- n [R"XY5n [X-SS9S- n U[[U *5UR-SS9URU-- nM [R"[R"US55n [!S["-UR$SS- 5U -n Xm-nU$)a' Computes the integral of the product of this kernel density estimate with another. Parameters ---------- other : gaussian_kde instance The other kde. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDEs have different dimensionality. z$KDEs are not the same dimensionalitygNrrGrFr)r%r#r&r:rrIranger!r rJrr r*rKrLrMrr r$)r0othersmalllargerOrPrAirNrQrRrUrSrTs r3 integrate_kdegaussian_kde.integrate_kdesC* 77dff CD D 77TVV EEEE""U%5%55((1 uwwA==Aw/D==4'D$$\8E4D)$/ 99 466)U #T__4:  %%dff4<<%H QZ(|rDcN[URSURS-- 5$)zGCompute Scott's factor. Returns ------- s : float Scott's factor. rrzr%r0s r3 scotts_factorgaussian_kde.scotts_factors!TYYTVVAX//rDct[URURS--S- SURS-- 5$)zSCompute the Silverman factor. Returns ------- s : float The silverman factor. rFg@rrrrs r3silverman_factorgaussian_kde.silverman_factors3TYYs +C/dffQh@@rDa0Computes the coefficient (`kde.factor`) that multiplies the data covariance matrix to obtain the kernel covariance matrix. The default is `scotts_factor`. A subclass can overwrite this method to provide a different method, or set it through a call to `kde.set_bandwidth`.cv^^TcOTS:XaTRTlOTS:XaTRTlOs[R"T5(a([ T[ 5(dSTlU4SjTlO0[T5(aTTlU4SjTlO Sn[U5eTR5 g)a8Compute the estimator bandwidth with given method. The new bandwidth calculated after a call to `set_bandwidth` is used for subsequent evaluations of the estimated density. Parameters ---------- bw_method : str, scalar or callable, optional The method used to calculate the estimator bandwidth. This can be 'scott', 'silverman', a scalar constant or a callable. If a scalar, this will be used directly as `kde.factor`. If a callable, it should take a `gaussian_kde` instance as only parameter and return a scalar. If None (default), nothing happens; the current `kde.covariance_factor` method is kept. Notes ----- .. versionadded:: 0.11 Examples -------- >>> import numpy as np >>> import scipy.stats as stats >>> x1 = np.array([-7, -5, 1, 4, 5.]) >>> kde = stats.gaussian_kde(x1) >>> xs = np.linspace(-10, 10, num=50) >>> y1 = kde(xs) >>> kde.set_bandwidth(bw_method='silverman') >>> y2 = kde(xs) >>> kde.set_bandwidth(bw_method=kde.factor / 3.) >>> y3 = kde(xs) >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.plot(x1, np.full(x1.shape, 1 / (4. * x1.size)), 'bo', ... label='Data points (rescaled)') >>> ax.plot(xs, y1, label='Scott (default)') >>> ax.plot(xs, y2, label='Silverman') >>> ax.plot(xs, y3, label='Const (1/3 * Silverman)') >>> ax.legend() >>> plt.show() Nscott silvermanz use constantc>T$Nrsr3,gaussian_kde.set_bandwidth..5sYrDc&>TRT5$r) _bw_methodrsr3rr8sT__T-BrDzC`bw_method` should be 'scott', 'silverman', a scalar or a callable.) rcovariance_factorrrKisscalar isinstancestrrcallabler#_compute_covariance)r0r r1s`` r3r.gaussian_kde.set_bandwidthsX    ' !%)%7%7D " + %%)%:%:D " [[ # #Jy#,F,F,DO%6D " i 'DO%BD "#CS/ !   "rDc |UR5Ul[US5(dR[[ UR SSUR S95Ul[R"URSS9Ul URURS--Ul URUR-R[R5UlS[R "[R""UR[R$"S[&-5-55R)5-Ulg) zSComputes the covariance matrix for each Gaussian kernel using covariance_factor(). _data_cho_covrFrowvarbiasaweightsT)lowerrN)rfactorhasattrrrr!r*_data_covariancercholeskyrr:r'rKfloat64r<logdiagr r rlog_detrs r3r gaussian_kde._compute_covariance@s,,. t_--$.s4<<498< 0F%GD !"(1F1F7;"=D //$++q.@**T[[8@@L  *,''!B$-)8!9::=#%@ rDc UR5Ul[[URSSUR S95Ul[R"UR 5URS-- $)NrFrr) rrrrr!r*rrinvrs r3inv_covgaussian_kde.inv_covRs],,. *3t||A05 ,N!Ozz$//04;;>AArDc$URU5$)z Evaluate the estimated pdf on a provided set of points. Notes ----- This is an alias for `gaussian_kde.evaluate`. See the ``evaluate`` docstring for more details. )rB)r0xs r3pdfgaussian_kde.pdf^s}}QrDc[U5nURup4X0R:waLUS:Xa)X@R:Xa[X RS45nSnOSUSUR3n[ U5e[ UR U5upg[U"URRURSS2S4URURU5nUSS2S4$)zD Evaluate the log of the estimated pdf on a provided set of points. rr7r8Nr) rr$r%rr#r9r:rr!r;r*r<) r0rr=r%r>r1r?r@rAs r3logpdfgaussian_kde.logpdfjsA|| ;Av!vv+ &&!5/s3004x9 o%.tG -d3 LLNNDLLD1 HHdllL2ad|rDcf[R"U5n[R"UR[R5(d Sn[ U5e[ UR5nUR5nXBUS:-X"S:'[ [R"U55[ U5:wa Sn[ U5eUS:X$:-n[R"U5(aSXVSUS3n[ U5eURUnURn[XpR5US9$)aReturn a marginal KDE distribution Parameters ---------- dimensions : int or 1-d array_like The dimensions of the multivariate distribution corresponding with the marginal variables, that is, the indices of the dimensions that are being retained. The other dimensions are marginalized out. Returns ------- marginal_kde : gaussian_kde An object representing the marginal distribution. Notes ----- .. versionadded:: 1.10.0 zaElements of `dimensions` must be integers - the indices of the marginal variables being retained.rz,All elements of `dimensions` must be unique.z Dimensions z# are invalid for a distribution in z dimensions.)r r*)rKr issubdtypedtypeintegerr#r,r!copyuniqueanyr*rr) r0 dimensionsdimsr1r& original_dims i_invalidr!r*s r3marginalgaussian_kde.marginals *}}Z(}}TZZ44?CS/ !   $(^+AX ryy 3t9 ,ACS/ !AX$), 66)   !9 :;,,-3l>*5Lxx %..H1}   Q   X    . ;H:F rD)# threadingrgscipyrrscipy._lib._utilrnumpyrrrr r r r r rrrrrrrrrKr_statsrr__all__Lockrerr9rrDr3rsc*"/J   >> V V rrD