I am trying to use SciPy's gaussian_kde function to estimate the density of multivariate data. In my code below I sample a 3D multivariate normal and fit the kernel density but I'm not sure how to evaluate my fit.
import numpy as np from scipy import stats mu = np.array([1, 10, 20]) sigma = np.matrix([[4, 10, 0], [10, 25, 0], [0, 0, 100]]) data = np.random.multivariate_normal(mu, sigma, 1000) values = data.T kernel = stats.gaussian_kde(values)
I saw this but not sure how to extend it to 3D.
Also not sure how do I even begin to evaluate the fitted density? How do I visualize this?
There are several ways you might visualize the results in 3D.
The easiest is to evaluate the gaussian KDE at the points that you used to generate it, and then color the points by the density estimate.
For example:
import numpy as np from scipy import stats import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D mu=np.array([1,10,20]) sigma=np.matrix([[4,10,0],[10,25,0],[0,0,100]]) data=np.random.multivariate_normal(mu,sigma,1000) values = data.T kde = stats.gaussian_kde(values) density = kde(values) fig, ax = plt.subplots(subplot_kw=dict(projection='3d')) x, y, z = values ax.scatter(x, y, z, c=density) plt.show()
If you had a more complex (i.e. not all lying in a plane) distribution, then you might want to evaluate the KDE on a regular 3D grid and visualize isosurfaces (3D contours) of the volume. It's easiest to use Mayavi for the visualiztion:
import numpy as np from scipy import stats from mayavi import mlab mu=np.array([1,10,20]) # Let's change this so that the points won't all lie in a plane... sigma=np.matrix([[20,10,10], [10,25,1], [10,1,50]]) data=np.random.multivariate_normal(mu,sigma,1000) values = data.T kde = stats.gaussian_kde(values) # Create a regular 3D grid with 50 points in each dimension xmin, ymin, zmin = data.min(axis=0) xmax, ymax, zmax = data.max(axis=0) xi, yi, zi = np.mgrid[xmin:xmax:50j, ymin:ymax:50j, zmin:zmax:50j] # Evaluate the KDE on a regular grid... coords = np.vstack([item.ravel() for item in [xi, yi, zi]]) density = kde(coords).reshape(xi.shape) # Visualize the density estimate as isosurfaces mlab.contour3d(xi, yi, zi, density, opacity=0.5) mlab.axes() mlab.show()
