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kde.cpython-35.pyc
�B�W3E � @ s d d l m Z m Z m Z d d l Z d d l m Z m Z d d l m Z m Z d d l m Z m Z m Z m Z m Z m Z m Z m Z m Z m Z m Z m Z m Z m Z d d l Z d d l m Z m Z d d l m Z d g Z Gd d � d e! � Z" d S) � )�division�print_function�absolute_importN)�callable� string_types)�linalg�special)� atleast_2d�reshape�zeros�newaxis�dot�exp�pi�sqrt�ravel�power� atleast_1d�squeeze�sum� transpose)�randint�multivariate_normal� )�mvn� gaussian_kdec @ s� e Z d Z d Z d d d � Z d d � Z e Z d d � Z d d � Z d d d � Z d d � Z d d d � Z d d � Z d d � Z e Z d e _ d d d � Z d d � Z d d � Z d d � Z d S) 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. Attributes ---------- dataset : ndarray The dataset with which `gaussian_kde` was initialized. d : int Number of dimensions. n : int Number of datapoints. 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. Silverman's Rule [2]_, implemented as `silverman_factor`, is:: (n * (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]_. 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. 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() Nc C sY t | � | _ | j j d k s- t d � � | j j \ | _ | _ | j d | � d S)Nr z.`dataset` input should have multiple elements.� bw_method)r �dataset�size� ValueError�shape�d�n� set_bandwidth)�selfr r � r% �/kde.py�__init__� s zgaussian_kde.__init__c C s� t | � } | j \ } } | | j k r� | d k rf | | j k rf t | | j d f � } d } n d | | j f } t | � � t | f d t �} | | j k r#x� t | j � D]c } | j d d � | t f | } t | j | � } t | | d d �d } | t | � } q� Wn� x} t | � D]o } | j | d d � | t f } t | j | � } t | | d d �d } t t | � d d �| | <q0W| | j } | S)a Evaluate 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. r z2points have dimension %s, dataset has dimension %sZdtypeN�axisr g @)r r r! r r r �floatr" �ranger r r �inv_covr r � _norm_factor) r$ Zpointsr! �m�msg�result�i�diff�tdiffZenergyr% r% r& �evaluate� s. ! zgaussian_kde.evaluatec C sE t t | � � } t | � } | j | j f k rF t d | j � � | j | j | j f k rt t d | j � � | d d � t f } | j | } t j | � } | j | } t j | | � } t j t j | d � � } t d t | j d d � | } t | | d d �d } t t | � d d �| | j } | 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. z mean does not have dimension %sz%covariance does not have dimension %sNr � g @r( )r r r r r! r r � covariancer � cho_factorr � cho_solve�np�prod�diagonalr r r r r" ) r$ �mean�cov�sum_cov� sum_cov_cholr1 r2 �sqrt_det� norm_const�energiesr/ r% r% r& �integrate_gaussian� s "$z gaussian_kde.integrate_gaussianc C s� | j d k r t d � � t t | j � � d } t | | j | � } t | | j | � } t j t j | � t j | � � } | S)a� Computes the integral of a 1D pdf between two bounds. Parameters ---------- low : scalar Lower bound of integration. high : scalar Upper bound of integration. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDE is over more than one dimension. r z'integrate_box_1d() only handles 1D pdfsr ) r! r r r r5 r r8 r; r Zndtr)r$ �low�highZstdevZnormalized_lowZnormalized_high�valuer% r% r& �integrate_box_1d s z gaussian_kde.integrate_box_1dc C sp | d k r d | i } n i } t j | | | j | j | � \ } } | rl d | j d } t j | � | S)a� Computes 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. N�maxptsz5An integral in mvn.mvnun requires more points than %si� )r Zmvnunr r5 r! �warnings�warn)r$ Z low_boundsZ high_boundsrG Z extra_kwdsrE Zinformr. r% r% r& � integrate_box< s zgaussian_kde.integrate_boxc C sV | j | j k r t d � � | j | j k r? | } | } n | } | } | j | j } t j | � } d } x� t | j � D]x } | j d d � | t f } | j | } t j | | � } t | | d d �d } | t t | � d d �7} q� Wt j t j | d � � } t d t | j d d � | } | | | j | j } | 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$KDEs are not the same dimensionalityg Nr( r g @r4 )r! r r" r5 r r6 r* r r r7 r r r8 r9 r: r r r )r$ �otherZsmallZlarger= r> r/ r0 r; r1 r2 rA r? r@ r% r% r& � integrate_kde\ s( ! "zgaussian_kde.integrate_kdec C s{ | d k r | j } t t t | j f t � | j d | �� } t d | j d | �} | j d d � | f } | | S)at 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 underlying dataset. Returns ------- resample : (self.d, `size`) ndarray The sampled dataset. Nr r ) r" r r r r! r) r5 r r )r$ r Znorm�indicesZmeansr% r% r& �resample� s zgaussian_kde.resamplec C s t | j d | j d � S)Ng �?� g �)r r" r! )r$ r% r% r& � scotts_factor� s zgaussian_kde.scotts_factorc C s* t | j | j d d d | j d � S)Ng @g @g �?rO g �)r r" r! )r$ r% r% r& �silverman_factor� s z gaussian_kde.silverman_factora0 Computes 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`.c s� � d k r n� � d k r* � j � _ n� � d k rE � j � _ n t j � � r� t � t � r� d � _ � f d d � � _ n? t � � r� � � _ � f d d � � _ n d } t | � � � j � d S) a6 Compute 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 constantc s � S)Nr% r% )r r% r& �<lambda>� s z,gaussian_kde.set_bandwidth.<locals>.<lambda>c s � j � � S)N)� _bw_methodr% )r$ r% r& rR � s zC`bw_method` should be 'scott', 'silverman', a scalar or a callable.) rP �covariance_factorrQ r8 Zisscalar� isinstancer rS r r �_compute_covariance)r$ r r. r% )r r$ r& r# � s + zgaussian_kde.set_bandwidthc C s� | j � | _ t | d � sZ t t j | j d d d d �� | _ t j | j � | _ | j | j d | _ | j | j d | _ t t j d t | j � � | j | _ d S)zcComputes the covariance matrix for each Gaussian kernel using covariance_factor(). � _data_inv_covZrowvarr ZbiasFr4 N)rT Zfactor�hasattrr r8 r< r Z_data_covariancer �invrW r5 r+ r Zdetr r" r, )r$ r% r% r& rV � s z gaussian_kde._compute_covariancec C s | j | � 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. )r3 )r$ �xr% r% r& �pdf s zgaussian_kde.pdfc C s t j | j | � � S)z� Evaluate the log of the estimated pdf on a provided set of points. Notes ----- See `gaussian_kde.evaluate` for more details; this method simply returns ``np.log(gaussian_kde.evaluate(x))``. )r8 �logr3 )r$ rZ r% r% r& �logpdf s zgaussian_kde.logpdf)�__name__� __module__� __qualname__�__doc__r' r3 �__call__rB rF rJ rL rN rP rQ rT r# rV r[ r] r% r% r% r& r ) s" { 6 5 " 2 > )#Z __future__r r r rH Zscipy._lib.sixr r Zscipyr r Znumpyr r r r r r r r r r r r r r r8 Z numpy.randomr r � r �__all__�objectr r% r% r% r& �<module> s ^
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