B &]\Q@sddlmZmZmZddlZddlmZmZddlm Z m Z ddl m Z ddl mZddlmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZddlZddl m!Z!m"Z"d d l#m$Z$d gZ%Gd d d e&Z'dS) )divisionprint_functionabsolute_importN)callable string_types)linalgspecial) logsumexp)cov) atleast_2dreshapezerosnewaxisdotexppisqrtravelpower atleast_1dsqueezesum transposeones)choicemultivariate_normal)mvn gaussian_kdec@seZdZdZd"ddZddZeZddZd d Zd#d d Z d dZ d$ddZ ddZ ddZ e Zde_d%ddZddZddZddZeddZed d!ZdS)&raRepresentation 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`, with which the covariance matrix is multiplied. 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]_. 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: >>> 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() NcCst||_|jjdkstd|jj\|_|_|dk rt|t |_ |j t |j _ |j j dkrntdt|j |jkrtddt |j d|_|j|ddS)Nrz.`dataset` input should have multiple elements.z*`weights` input should be one-dimensional.z%`weights` input should be of length n) bw_method)r datasetsize ValueErrorshapednrZastypefloat_weightsrweightsndimlen_neff set_bandwidth)selfr!r r)r/.lib/python3.7/site-packages/scipy/stats/kde.py__init__s   zgaussian_kde.__init__c CsXt|}|j\}}||jkrZ|dkrD||jkrDt||jdf}d}nd||jf}t|t|ftd}t|j }t ||j }t ||}||j krxt |j D]F} |dd| tf|} t| | ddd} ||j| t| 7}qWnXxVt |D]J} ||dd| tf} t| | ddd} tt| |jdd|| <qW||j |j}|S)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 %s)dtypeNr)axisg@)r r$r%r r#r r'rZcholeskyinv_covrr!r&rangerrr)r _norm_factor) r.pointsr%mmsgresultZ whiteningZscaled_datasetZ scaled_pointsidiffenergyr/r/r0evaluates0        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 %sNrrg@)r3)rrr r$r%r#r covariancer cho_factorr! cho_solvenpproddiagonalrrrrr)) r.meanr sum_cov sum_cov_cholr<tdiffsqrt_det norm_constenergiesr:r/r/r0integrate_gaussian s      zgaussian_kde.integrate_gaussiancCsl|jdkrtdtt|jd}t||j|}t||j|}t|jt |t |}|S)a Computes the integral of a 1D pdf between two bounds. Parameters ---------- low : scalar Lower bound of integration. high : scalar Upper bound of integration. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDE is over more than one dimension. rz'integrate_box_1d() only handles 1D pdfsr) r%r#rrr?r!rBrr)rZndtr)r.ZlowZhighZstdevZnormalized_lowZnormalized_highvaluer/r/r0integrate_box_1dAs zgaussian_kde.integrate_box_1dcCsV|dk rd|i}ni}tj|||j|j|jf|\}}|rRd|jd}t||S)aComputes the integral of a pdf over a rectangular interval. Parameters ---------- low_bounds : array_like A 1-D array containing the lower bounds of integration. high_bounds : array_like A 1-D array containing the upper bounds of integration. maxpts : int, optional The maximum number of points to use for integration. Returns ------- value : scalar The result of the integral. Nmaxptsz5An integral in mvn.mvnun requires more points than %si)rZmvnun_weightedr!r)r?r%warningswarn)r.Z low_boundsZ high_boundsrOZ extra_kwdsrMZinformr9r/r/r0 integrate_boxds   zgaussian_kde.integrate_boxcCs|j|jkrtd|j|jkr*|}|}n|}|}|j|j}t|}d}xvt|jD]h}|jdd|tf}|j|} t || } t | | ddd} |t t | |j dd|j |7}qXWt t |d} tdt|jdd| } || }|S)a Computes the integral of the product of this kernel density estimate with another. Parameters ---------- other : gaussian_kde instance The other kde. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDEs have different dimensionality. z$KDEs are not the same dimensionalitygNr)r3g@r)r%r#r&r?rr@r5r!rrArrr)rBrCrDrrr$)r.otherZsmallZlargerFrGr:r;rEr<rHrKrIrJr/r/r0 integrate_kdes(      *zgaussian_kde.integrate_kdecCs\|dkrt|j}ttt|jft|j|d}t|j ||j d}|j dd|f}||S)a Randomly sample a dataset from the estimated pdf. Parameters ---------- size : int, optional The number of samples to draw. If not provided, then the size is the same as the effective number of samples in the underlying dataset. Returns ------- resample : (self.d, `size`) ndarray The sampled dataset. N)r")r"p) intneffrrr r%r'r?rr&r)r!)r.r"ZnormindicesZmeansr/r/r0resamples zgaussian_kde.resamplecCst|jd|jdS)Ng)rrWr%)r.r/r/r0 scotts_factorszgaussian_kde.scotts_factorcCs$t|j|jddd|jdS)Ng@g@grZ)rrWr%)r.r/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) a6Compute 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 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.ones(x1.shape) / (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() NZscottZ silvermanz use constantcsS)Nr/r/)r r/r0sz,gaussian_kde.set_bandwidth..cs S)N) _bw_methodr/)r.r/r0r]szC`bw_method` should be 'scott', 'silverman', a scalar or a callable.) r[covariance_factorr\rBZisscalar isinstancerr^rr#_compute_covariance)r.r r9r/)r r.r0r-s+  zgaussian_kde.set_bandwidthcCs||_t|ds)r.xr/r/r0pdf.s zgaussian_kde.pdfc Csht|}|j\}}||jkrZ|dkrD||jkrDt||jdf}d}nd||jf}t|t|ftd}||jkrt|j|ftd}xNt|jD]@}|j dd|t f|} t |j | } t | | ddd||<qWt| |j||j|jdd}nlxjt|D]^}|j |dd|t f} t |j | } t | | ddd}t| |j|j|jd ||<qW|S) zT Evaluate the log of the estimated pdf on a provided set of points. rz2points have dimension %s, dataset has dimension %s)r2Nr)r3g@)br3)rg)r r$r%r r#r r'r&r5r!rrr4rr r)r6) r.rer7r%r8r9r:r=r;r<rHr/r/r0logpdf:s4        zgaussian_kde.logpdfcCs4y|jStk r.t|j|j|_|jSXdS)N)r(AttributeErrorrr&)r.r/r/r0r)cs zgaussian_kde.weightscCs6y|jStk r0dt|jd|_|jSXdS)Nrr)r,rirr))r.r/r/r0rWks zgaussian_kde.neff)NN)N)N)N)__name__ __module__ __qualname____doc__r1r>__call__rLrNrRrTrYr[r\r_r-rarfrhpropertyr)rWr/r/r/r0r+s( 85# !2  > ) )(Z __future__rrrrPZscipy._lib.sixrrZscipyrrZ scipy.specialr Zscipy._lib._numpy_compatr Znumpyr r r rrrrrrrrrrrrrBZ numpy.randomrrr__all__objectrr/r/r/r0s  D