其實密度估計是一個非常簡單的概念，我們已經熟悉了一種常見的密度估計技術:直方圖。密度估計在無監督學習，特徵工程和數據建模三個領域都有應用。高斯混合模型就是一種流行和有用的密度估計技術和基於近鄰域的方法。高斯混合技術還可用作無監督聚類方案。
直方圖是一種最簡單的數據可視化方法，可以在下圖的左上面板中看到:簡單的一維核密度估計 這個示例使用sklearn.neighbors。
第一個圖顯示了使用直方圖可視化一維中點密度的問題之一。直觀地說，直方圖可以被認爲是一種方案，其中一個單元“塊”被堆放在規則網格的每個點之上。然而，正如最上面的兩個面板所顯示的，這些塊的網格選擇可能導致對密度分佈的基本形狀產生嚴重分歧。如果我們把每個塊放在它所代表的點上，我們就會得到左下角顯示的估計值。這個想法可以推廣到其他核形狀:第一個圖的右下角顯示了在相同分佈上的高斯核密度估計。
Scikit-learn通過sklearn.neighbors使用球樹或KD樹結構實現有效的核密度估計。KernelDensity估計量。可用的內核顯示在本例的第二個圖中。
第三個圖比較了1維100個樣本分佈的核密度估計數。雖然本例使用的是一維分佈，但核密度估計也可以輕鬆有效地擴展到更高的維數。
# Author: Jake Vanderplas <[email protected]> # import numpy as np import matplotlib import matplotlib.pyplot as plt from distutils.version import LooseVersion from scipy.stats import norm from sklearn.neighbors import KernelDensity # `normed` is being deprecated in favor of `density` in histograms if LooseVersion(matplotlib.__version__) >= '2.1': density_param = {'density': True} else: density_param = {'normed': True} # ---------------------------------------------------------------------- # Plot the progression of histograms to kernels np.random.seed(1) N = 20 X = np.concatenate((np.random.normal(0, 1, int(0.3 * N)), np.random.normal(5, 1, int(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 1 ax[0, 0].hist(X[:, 0], bins=bins, fc='#AAAAFF', **density_param) ax[0, 0].text(-3.5, 0.31, "Histogram") # histogram 2 ax[0, 1].hist(X[:, 0], bins=bins + 0.75, fc='#AAAAFF', **density_param) ax[0, 1].text(-3.5, 0.31, "Histogram, bins shifted") # tophat KDE kde = 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 KDE kde = 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.full(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 kernels X_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' % x for 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 example N = 100 np.random.seed(1) X = np.concatenate((np.random.normal(0, 1, int(0.3 * N)), np.random.normal(5, 1, int(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') colors = ['navy', 'cornflowerblue', 'darkorange'] kernels = ['gaussian', 'tophat', 'epanechnikov'] lw = 2 for color, kernel in zip(colors, kernels): 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), color=color, lw=lw, linestyle='-', 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()
下圖展示了一個基於地理空間數據的基於鄰居的查詢示例(特別是核密度估計)，使用基於哈弗斯距離度量(即經緯度上的點的距離)構建的球樹。如果可以的話，本示例使用basemap繪製南美洲的海岸線和國界。
這個示例沒有對數據執行任何學習(請參閱物種分佈建模，瞭解基於此數據集中屬性的分類示例)。簡單地用地理空間座標表示觀測數據點的核密度估計值。
# Author: Jake Vanderplas <[email protected]> # # License: BSD 3 clause import numpy as np import matplotlib.pyplot as plt from sklearn.datasets import fetch_species_distributions from sklearn.neighbors import KernelDensity # if basemap is available, we'll use it. # otherwise, we'll improvise later... try: from mpl_toolkits.basemap import Basemap basemap = True except ImportError: basemap = False def construct_grids(batch): """Construct the map grid from the batch object Parameters ---------- batch : Batch object The object returned by :func:`fetch_species_distributions` Returns ------- (xgrid, ygrid) : 1-D arrays The grid corresponding to the values in batch.coverages """ # x,y coordinates for corner cells xmin = batch.x_left_lower_corner + batch.grid_size xmax = xmin + (batch.Nx * batch.grid_size) ymin = batch.y_left_lower_corner + batch.grid_size ymax = ymin + (batch.Ny * batch.grid_size) # x coordinates of the grid cells xgrid = np.arange(xmin, xmax, batch.grid_size) # y coordinates of the grid cells ygrid = np.arange(ymin, ymax, batch.grid_size) return (xgrid, ygrid) # Get matrices/arrays of species IDs and locations data = fetch_species_distributions() species_names = ['Bradypus Variegatus', 'Microryzomys Minutus'] Xtrain = np.vstack([data['train']['dd lat'], data['train']['dd long']]).T ytrain = np.array([d.decode('ascii').startswith('micro') for d in data['train']['species']], dtype='int') Xtrain *= np.pi / 180. # Convert lat/long to radians # Set up the data grid for the contour plot xgrid, ygrid = construct_grids(data) X, Y = np.meshgrid(xgrid[::5], ygrid[::5][::-1]) land_reference = data.coverages[6][::5, ::5] land_mask = (land_reference > -9999).ravel() xy = np.vstack([Y.ravel(), X.ravel()]).T xy = xy[land_mask] xy *= np.pi / 180. # Plot map of South America with distributions of each species fig = plt.figure() fig.subplots_adjust(left=0.05, right=0.95, wspace=0.05) for i in range(2): plt.subplot(1, 2, i + 1) # construct a kernel density estimate of the distribution print(" - computing KDE in spherical coordinates") kde = KernelDensity(bandwidth=0.04, metric='haversine', kernel='gaussian', algorithm='ball_tree') kde.fit(Xtrain[ytrain == i]) # evaluate only on the land: -9999 indicates ocean Z = np.full(land_mask.shape[0], -9999, dtype='int') Z[land_mask] = np.exp(kde.score_samples(xy)) Z = Z.reshape(X.shape) # plot contours of the density levels = np.linspace(0, Z.max(), 25) plt.contourf(X, Y, Z, levels=levels, cmap=plt.cm.Reds) if basemap: print(" - plot coastlines using basemap") m = Basemap(projection='cyl', llcrnrlat=Y.min(), urcrnrlat=Y.max(), llcrnrlon=X.min(), urcrnrlon=X.max(), resolution='c') m.drawcoastlines() m.drawcountries() else: print(" - plot coastlines from coverage") plt.contour(X, Y, land_reference, levels=[-9998], colors="k", linestyles="solid") plt.xticks([]) plt.yticks([]) plt.title(species_names[i]) plt.show()