U |ea@sddlZddlmZmZddlmZddlmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZddlZddlmZddlmZmZdgZGd ddZd d ZdS) N)linalgspecial)check_random_state)asarray atleast_2dreshapezerosnewaxisexppisqrtravelpower atleast_1dsqueezesum transposeonescov)_mvn)gaussian_kernel_estimategaussian_kernel_estimate_log gaussian_kdec@seZdZdZd&ddZddZeZddZd d Zd'd d Z d dZ d(ddZ ddZ ddZ e Zde_d)ddZddZeddZddZddZd d!Zed"d#Zed$d%ZdS)*ra&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 principle 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 Cstt||_|jjdks"td|jj\|_|_|dk rt| t |_ |j t |j _ |j jdkrrtdt|j |jkrtddt |j d|_|j|jkrd}t|z|j|dWn4tjk r}zd}t||W5d}~XYnXdS) 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 principle component analysis / dimensionality reduction and using `gaussian_kde` with the transformed data.)rrdatasetsize ValueErrorshapednrastypefloat_weightsrweightsndimlen_neff set_bandwidthr LinAlgError)selfrrr&msger/M/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/stats/_kde.py__init__s(   zgaussian_kde.__init__cCstt|}|j\}}||jkr^|dkrH||jkrHt||jdf}d}nd||jf}t|t|j|\}}t||j j |j dddf|j |j |}|dddfS)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. rz2points have dimension %s, dataset has dimension %sNr) rrr r!rr_get_output_dtype covariancerrTr&cho_cov)r,pointsr!mr- output_dtypespecresultr/r/r0evaluates&   zgaussian_kde.evaluatec Cstt|}t|}|j|jfkr0td|j|j|j|jfkrPtd|j|ddtf}|j|}t |}|j |}t ||}t t |d}tdt|jdd|}t||ddd} tt| |jdd|} | S)aW 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 %sz%covariance does not have dimension %sNrr@axis)rrrr r!rr r3r cho_factorr cho_solvenpproddiagonalrr rr r&) r,meanrsum_cov sum_cov_choldifftdiffsqrt_det norm_constenergiesr:r/r/r0integrate_gaussians      zgaussian_kde.integrate_gaussiancCsl|jdkrtdtt|jd}t||j|}t||j|}t|jt |t |}|S)a Computes the integral of a 1D pdf between two bounds. Parameters ---------- low : scalar Lower bound of integration. high : scalar Upper bound of integration. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDE is over more than one dimension. rz'integrate_box_1d() only handles 1D pdfsr) r!rr r r3rrArr&rndtr)r,lowhighstdevZnormalized_lowZnormalized_highvaluer/r/r0integrate_box_1dIs zgaussian_kde.integrate_box_1dcCsV|dk rd|i}ni}tj|||j|j|jf|\}}|rRd|jd}t||S)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. Nmaxptsz6An integral in _mvn.mvnun requires more points than %si)rZmvnun_weightedrr&r3r!warningswarn)r,Z low_boundsZ high_boundsrSZ extra_kwdsrQZinformr-r/r/r0 integrate_boxls   zgaussian_kde.integrate_boxcCs|j|jkrtd|j|jkr*|}|}n|}|}|j|j}t|}d}t|jD]h}|jdd|tf}|j|} t || } t | | ddd} |t t | |j dd|j |7}qVt t |d} tdt|jdd| } || }|S)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 dimensionalitygNrr=r<r)r!rr"r3rr?rangerr r@rr r&rArBrCrr r )r,othersmalllargerErFr:irDrGrHrKrIrJr/r/r0 integrate_kdes(      (zgaussian_kde.integrate_kdecCsh|dkrt|j}t|}t|jt|jft|j|d}|j |j ||j d}|j dd|f}||S)aARandomly sample a dataset from the estimated pdf. Parameters ---------- size : int, optional The number of samples to draw. If not provided, then the size is the same as the effective number of samples in the underlying dataset. seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- resample : (self.d, `size`) ndarray The sampled dataset. N)r)rp) intneffrrmultivariate_normalrr!r$r3choicer"r&r)r,rseed random_statenormindicesmeansr/r/r0resamples  zgaussian_kde.resamplecCst|jd|jdS)zoCompute Scott's factor. Returns ------- s : float Scott's factor. rr_r!r,r/r/r0 scotts_factorszgaussian_kde.scotts_factorcCs$t|j|jddd|jdS)z{Compute the Silverman factor. Returns ------- s : float The silverman factor. r<g@rhrirjrkr/r/r0silverman_factorszgaussian_kde.silverman_factora0Computes 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`.csdkr nxdkrj_nfdkr.j_nTtrXttsXd_fdd_n*trv_fdd_n d}t | dS) aXCompute 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() NscottZ silvermanz use constantcsSNr/r/rr/r02z,gaussian_kde.set_bandwidth..cs Sro) _bw_methodr/rkr/r0rp5rqzC`bw_method` should be 'scott', 'silverman', a scalar or a callable.) rlcovariance_factorrmrAisscalar isinstancestrrrcallabler_compute_covariance)r,rr-r/)rr,r0r*s,  zgaussian_kde.set_bandwidthc Cs||_t|ds@tt|jdd|jd|_tj |jdd|_ |j|jd|_ |j |j t j|_dt t |jt dt|_dS) zcComputes the covariance matrix for each Gaussian kernel using covariance_factor(). _data_cho_covrFrowvarbiasaweightsT)lowerrN)rsfactorhasattrrrrr&_data_covariancercholeskyryr3r#rAfloat64r5logdiagr r rZlog_detrkr/r/r0rx=s     z gaussian_kde._compute_covariancecCs:||_tt|jdd|jd|_t|j|jdS)NrFrzr) rsrrrrr&rrinvrkr/r/r0inv_covOs    zgaussian_kde.inv_covcCs ||S)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. )r;)r,xr/r/r0pdf[s zgaussian_kde.pdfc Cst|}|j\}}||jkr^|dkrD||jkrDt||jdf}d}nd|d|j}t|t|j|\}}t||jj |j dddf|j |j |}|dddfS)zT Evaluate the log of the estimated pdf on a provided set of points. rzpoints have dimension z, dataset has dimension Nr) rr r!rrr2r3rrr4r&r5) r,rr6r!r7r-r8r9r:r/r/r0logpdfgs"  zgaussian_kde.logpdfc Cst|}t|jtjs&d}t|t|j}|}|||dk||dk<tt |t|krrd}t||dk||kB}t |rd||d|d}t||j|}|j }t || |dS)a)Return 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.)rr&)rAr issubdtypedtypeintegerrr(rcopyuniqueanyr&rrs) r, dimensionsdimsr-r"Z original_dimsZ i_invalidrr&r/r/r0marginals&     zgaussian_kde.marginalcCs:z|jWStk r4t|j|j|_|jYSXdSro)r%AttributeErrorrr"rkr/r/r0r&s zgaussian_kde.weightscCs<z|jWStk r6dt|jd|_|jYSXdS)Nrr)r)rrr&rkr/r/r0r_s zgaussian_kde.neff)NN)N)NN)N)__name__ __module__ __qualname____doc__r1r;__call__rLrRrVr\rgrlrmrsr*rxpropertyrrrrr&r_r/r/r/r0r(s2 &(5# !2 #   ?  1 cCs\t||}t|j}|dkr&d}n.|dkr4d}n |dkrBd}nt|d|||fS)z Calculates the output dtype and the "spec" (=C type name). This was necessary in order to deal with the fused types in the Cython routine `gaussian_kernel_estimate`. See gh-10824 for details. rir$double) z long doublez has unexpected item size: )rA common_typeritemsizer)r3r6r8rr9r/r/r0r2s   r2) rTscipyrrscipy._lib._utilrnumpyrrrrr r r r r rrrrrrrrAr_statsrr__all__rr2r/r/r/r0s H