Pic2, Kernel Density Estimates

Kernel function estimation

Demo I

 

import sys,re,osimport numpy as npfrom scipy import stats import matplotlib.pylab as plt if __name__ == ‘__main__‘:    # random data    grade = [np.random.rand(100) * 100]    fig = plt.figure()        # KDE    ax1 = fig.add_subplot(211)    ind = np.arange(0.,100.,1)    gkde = stats.kde.gaussian_kde(grade, bw_method = ‘scott‘)    ax1.plot(ind, gkde(ind), label=‘Gods\‘ Grade‘, color="g")    ax1.set_title(‘Kernel Density Estimation‘)    ax1.legend()        # hisogram    ax2 = fig.add_subplot(212)    ax2.hist(grade, 100, range = (0,100), normed = True)    plt.show()

 

Demo II

Demo in scikit-learn

Http://scikit-learn.org/stable/auto_examples/neighbors/plot_kde_1d.html

 

# -*- coding: utf-8 -*-"""Created on Wed Oct 22 20:38:13 2014@author: dell"""# Author: Jake Vanderplas <[email protected]>#import numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import normfrom sklearn.neighbors import KernelDensity#----------------------------------------------------------------------# Plot the progression of histograms to kernelsnp.random.seed(1)N = 20X = np.concatenate((np.random.normal(0, 1, 0.3 * N),                    np.random.normal(5, 1, 0.7 * N)))[:, np.newaxis]X_plot = np.linspace(-5, 10, 1000)[:, np.newaxis]bins = np.linspace(-5, 10, 10)fig, ax = plt.subplots(2, 2, sharex=True, sharey=True)fig.subplots_adjust(hspace=0.05, wspace=0.05)# histogram 1ax[0, 0].hist(X[:, 0], bins=bins, fc=‘#AAAAFF‘, normed=True)ax[0, 0].text(-3.5, 0.31, "Histogram")# histogram 2ax[0, 1].hist(X[:, 0], bins=bins + 0.75, fc=‘#AAAAFF‘, normed=True)ax[0, 1].text(-3.5, 0.31, "Histogram, bins shifted")# tophat KDEkde = KernelDensity(kernel=‘tophat‘, bandwidth=0.75).fit(X)log_dens = kde.score_samples(X_plot)ax[1, 0].fill(X_plot[:, 0], np.exp(log_dens), fc=‘#AAAAFF‘)ax[1, 0].text(-3.5, 0.31, "Tophat Kernel Density")# Gaussian KDEkde = KernelDensity(kernel=‘gaussian‘, bandwidth=0.75).fit(X)log_dens = kde.score_samples(X_plot)ax[1, 1].fill(X_plot[:, 0], np.exp(log_dens), fc=‘#AAAAFF‘)ax[1, 1].text(-3.5, 0.31, "Gaussian Kernel Density")for axi in ax.ravel():    axi.plot(X[:, 0], np.zeros(X.shape[0]) - 0.01, ‘+k‘)    axi.set_xlim(-4, 9)    axi.set_ylim(-0.02, 0.34)for axi in ax[:, 0]:    axi.set_ylabel(‘Normalized Density‘)for axi in ax[1, :]:    axi.set_xlabel(‘x‘)#----------------------------------------------------------------------# Plot all available kernelsX_plot = np.linspace(-6, 6, 1000)[:, None]X_src = np.zeros((1, 1))fig, ax = plt.subplots(2, 3, sharex=True, sharey=True)fig.subplots_adjust(left=0.05, right=0.95, hspace=0.05, wspace=0.05)def format_func(x, loc):    if x == 0:        return ‘0‘    elif x == 1:        return ‘h‘    elif x == -1:        return ‘-h‘    else:        return ‘%ih‘ % xfor i, kernel in enumerate([‘gaussian‘, ‘tophat‘, ‘epanechnikov‘,                            ‘exponential‘, ‘linear‘, ‘cosine‘]):    axi = ax.ravel()[i]    log_dens = KernelDensity(kernel=kernel).fit(X_src).score_samples(X_plot)    axi.fill(X_plot[:, 0], np.exp(log_dens), ‘-k‘, fc=‘#AAAAFF‘)    axi.text(-2.6, 0.95, kernel)    axi.xaxis.set_major_formatter(plt.FuncFormatter(format_func))    axi.xaxis.set_major_locator(plt.MultipleLocator(1))    axi.yaxis.set_major_locator(plt.NullLocator())    axi.set_ylim(0, 1.05)    axi.set_xlim(-2.9, 2.9)ax[0, 1].set_title(‘Available Kernels‘)#----------------------------------------------------------------------# Plot a 1D density exampleN = 100np.random.seed(1)X = np.concatenate((np.random.normal(0, 1, 0.3 * N),                    np.random.normal(5, 1, 0.7 * N)))[:, np.newaxis]X_plot = np.linspace(-5, 10, 1000)[:, np.newaxis]true_dens = (0.3 * norm(0, 1).pdf(X_plot[:, 0])             + 0.7 * norm(5, 1).pdf(X_plot[:, 0]))fig, ax = plt.subplots()ax.fill(X_plot[:, 0], true_dens, fc=‘black‘, alpha=0.2,        label=‘input distribution‘)for kernel in [‘gaussian‘, ‘tophat‘, ‘epanechnikov‘]:    kde = KernelDensity(kernel=kernel, bandwidth=0.5).fit(X)    log_dens = kde.score_samples(X_plot)    ax.plot(X_plot[:, 0], np.exp(log_dens), ‘-‘,            label="kernel = ‘{0}‘".format(kernel))ax.text(6, 0.38, "N={0} points".format(N))ax.legend(loc=‘upper left‘)ax.plot(X[:, 0], -0.005 - 0.01 * np.random.random(X.shape[0]), ‘+k‘)ax.set_xlim(-4, 9)ax.set_ylim(-0.02, 0.4)plt.show()

 

Pic2, Kernel Density Estimates