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machine-learningmathplotstatisticsnormal-distribution

how to plot three or even more dimensional multivariate gaussian distribution

In the study of machine learning and pattern recognition, we know that if a sample i has two dimensional feature like (length, weight), both of length and weight belongs to Gaussian distribution, so we can use a multivariate Gaussian distribution to describe it

it's just a 3d plot looks like this :

enter image description here

where z axis is the possibility , but what if this sample i has three dimensional features, x1, x2 , x3 ....xn or even more, how do we correctly plot it using one plot???

Solution

  • You can use dimensionality reduction methods to visualize higher dimensional data. https://scikit-learn.org/stable/auto_examples/manifold/plot_compare_methods.html#sphx-glr-auto-examples-manifold-plot-compare-methods-py

    1. convert D dimensional data into 2 or 3 dimensional data
    2. plot the transformed data points on 2 or 3 data points depending upon the dimension to which the data was reduced.

    Lets consider an example. Take 10th dimensional Gaussian

    import matplotlib.pyplot as plt
import numpy as np
DIMENSION = 10
mean = np.zeros((DIMENSION,))
cov = np.eye(DIMENSION)
X = np.random.multivariate_normal(mean, cov, 5000)

    Then perform dimensionality reduction (I used PCA, you can choose any other method depending upon the prior knowledge of effectiveness of the algorithm for a particular type of data)

    import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from sklearn.decomposition import PCA

X_2d = PCA(n_components=2).fit_transform(X)
X_3d = PCA(n_components=3).fit_transform(X)

    Then Plot them

    fig = plt.figure(figsize=(12,4))
ax = fig.add_subplot(121, projection='3d')
ax.scatter(X_3d[:,0],X_3d[:,1],X_3d[:,2])
plt.title('3D')
fig.add_subplot(122)
plt.scatter(X_2d[:,0], X_2d[:,1])
plt.title('2D')

    results for 10D gaussian

    You can play with other algos as well. Each offers different kind of advantage. I hope this answers your question.

    Note: In higher dimension, phenomenon like "curse of dimensionality" also comes into play. so accurate projection in lower dimensional may not be possible. Something like why Greenland appears to be of similar size to that of Africa on cartographic map.