ó î&]\c@`s-ddlmZmZmZddlZddlmZmZddlm Z m Z ddl m Z ddl mZddlmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZddlZddl m!Z!m"Z"d d l#m$Z$d gZ%d e&fd „ƒYZ'dS( i(tdivisiontprint_functiontabsolute_importN(tcallablet string_types(tlinalgtspecial(t logsumexp(tcov(t atleast_2dtreshapetzerostnewaxistdottexptpitsqrttraveltpowert atleast_1dtsqueezetsumt transposetones(tchoicetmultivariate_normali(tmvnt gaussian_kdecB`sÅeZdZddd„Zd„ZeZd„Zd„Zdd„Z d„Z dd„Z d„Z d „Z e Zd e_dd „Zd „Zd „Zd„Zed„ƒZed„ƒZRS(s´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`, 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() cC`sýt|ƒ|_|jjdks0tdƒ‚n|jj\|_|_|dk rét|ƒj t ƒ|_ |j t |j ƒ_ |j jdkr¥tdƒ‚nt|j ƒ|jkrÌtdƒ‚ndt |j dƒ|_n|jd|ƒdS(Nis.`dataset` input should have multiple elements.s*`weights` input should be one-dimensional.s%`weights` input should be of length nit bw_method(R tdatasettsizet ValueErrortshapetdtntNoneRtastypetfloatt_weightsRtweightstndimtlent_nefft set_bandwidth(tselfRRR'((s.lib/python2.7/site-packages/scipy/stats/kde.pyt__init__Às c C`sÙt|ƒ}|j\}}||jkrˆ|dkrf||jkrft||jdfƒ}d}qˆd||jf}t|ƒ‚nt|fdtƒ}tj|j ƒ}t ||j ƒ}t ||ƒ}||j krOxßt |j ƒD]Y} |dd…| tf|} t| | ddƒd} ||j| t| ƒ7}qïWnrxot |ƒD]a} ||dd…| tf} t| | ddƒd} tt| ƒ|jddƒ|| >> 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() tscottt silvermans use constantc`sˆS(N(((R(s.lib/python2.7/site-packages/scipy/stats/kde.pytsc`s ˆjˆƒS(N(t _bw_method((R,(s.lib/python2.7/site-packages/scipy/stats/kde.pyRmssC`bw_method` should be 'scott', 'silverman', a scalar or a callable.N( R#Ritcovariance_factorRjRBtisscalart isinstanceRRnRRt_compute_covariance(R,RR6((RR,s.lib/python2.7/site-packages/scipy/stats/kde.pyR+às+       c C`s¿|jƒ|_t|dƒsctt|jdddtd|jƒƒ|_t j |jƒ|_ n|j|jd|_ |j |jd|_ tt jdt|j ƒƒ|j|_dS(scComputes the covariance matrix for each Gaussian kernel using covariance_factor(). t _data_inv_covtrowvaritbiastaweightsiN(RotfactorthasattrR RRtFalseR't_data_covarianceRtinvRsR?R1RtdetRR"R3(R,((s.lib/python2.7/site-packages/scipy/stats/kde.pyRrscC`s |j|ƒS(s× Evaluate the estimated pdf on a provided set of points. 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