Exercise - Multivariate Gaussian

Introduction

To define a univariate gaussian distribution, we only need the variance and the mean. For multivariate gaussian we need the vectorized mean and the covariance matrix. In this notebook you will implement the functions to calculate both, the vectorized mean- and the covariance_matrix.

In two dimensional vector space, the multivariate gaussian is called bivariate gaussian, which will be used throughout the whole notebook, so we are still able to visualize our data.

In order to detect errors in your own code, execute the notebook cells containing assert or assert_almost_equal. These statements raise exceptions, as long as the calculated result is not yet correct.

Requirements

Knowledge

To complete this exercise notebook, you should possess knowledge about the following topics.

  • Univariate gaussian
  • Multivariate gaussian
  • Variance
  • Covariance
  • Covariance matrix

The following material can help you to acquire this knowledge:

  • Gaussian, variance, covariance, covariance matrix:
  • Chapter 3 of the Deep Learning Book [GOO16]
  • Chapter 1 of the book Pattern Recognition and Machine Learning by Christopher M. Bishop [BIS07]
  • Univariate gaussian:
  • Video and the follwoing of Khan Academy [KHA18]
  • Multivariate gaussian:
  • Video PP 6.1 and following in the playlist Probability Primer of the youtube user mathematicalmonk [MAT18]

Python Modules

# External Modules
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm
from scipy.stats import multivariate_normal as mvn
from mpl_toolkits.mplot3d import Axes3D
%%html
# prevents output cells to receive a height limit and a scroll bar
<style>
.output_wrapper, .output {
    height:auto !important;
    max-height:1000px;  /* your desired max-height here */
}
.output_scroll {
    box-shadow:none !important;
    webkit-box-shadow:none !important;
}
</style>

Exercises

From an experiment we obtain a size NN random sample x1,,xN{\bf x}_1, \dots, {\bf x}_N from a population f(x)f(x), where each random sample x=[x1,,xp]{\bf x}=[x_1, \dots, x_p] contains pp variables. Each variable x1,,xpx_1, \dots, x_p is normally distributed in itself and independent from each other (xNxN1{\bf x}_N \perp {\bf x}_{N-1}).

Sidenote:

In most literature, bold variables are used to denote different vectors (examples), whereas non-bold variables are used for the elements of one vector.

Usually the independence sign has two vertical lines, but unfortunately it is not part of the standard LATEX\mathrm{L\!\!^{{}_{A}} \!\!\!\!\!\;\; T\!_{\displaystyle E} \! X} package.

N = 1000
p = 2
X = np.zeros((N,2))

mean_x1 = -1
mean_x2 = 2
std_dev_x1 = np.sqrt(0.6)
std_dev_x2 = np.sqrt(2.0)

X[:,0] = np.random.normal(size=N, loc=mean_x1, scale=std_dev_x1)
X[:,1] = np.random.normal(size=N, loc=mean_x2, scale=std_dev_x2)

The following plot visualizes the drawn sample.

fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(111)
ax.scatter(X[...,0], X[...,1], c='b', marker='x', label="data")
Y_lim = -1,5
X_lim = -4,2

ax.set_xlim(X_lim[0],X_lim[1])
ax.set_ylim(Y_lim[0],Y_lim[1])
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
plt.draw()
plt.show()

Exercise - Mean and Covariance Matrix

Task:

Write a vectorized mean- and the covariance_matrix-function using only the numpy methods/functions/operators: +, -, *, /,np.sum(..), np.dot(..), np.shape(..), X.T)

Hint:

You don't need all of them (depending on your implementation). You can use your implemented mean function inside covariance_matrix.

# Implement this function

def mean(X):
    """
    Calculates the vectorized mean over the first axis
    
    :X: input-matrix.
    :X type: numpy ndarray with n dimensions (n >= 1) of type float.
    
    :return: vectorized mean of X.
    .r type: numpy ndarray with n-1 dimensions of type float.
    """
    raise NotImplementedError
# Your implementations should pass this test:

mu = mean(X)
np.testing.assert_almost_equal(mu, X.mean(axis=0))
def covariance_matrix(X):
    """
    Calculates the covariance matrix of X
    
    :X: input-matrix.
    :X type: numpy ndarray with shape (n,m) of type float.
    
    :return: covariance matrix of X.
    .r type: numpy ndarray with shape (m,m) of type float.
    """
    raise NotImplementedError
# Your implementations should pass this test:

cov = covariance_matrix(X)
cov_np = np.cov(X, rowvar=0)
np.testing.assert_array_almost_equal(cov, cov_np)

Plots

To examine the underlying distribution of our drawn sample, we could, e.g. plot all the individual dimensions as histrograms:

fig = plt.figure(figsize=(12,5), dpi=80)
bins = 15

ax = fig.add_subplot(121)
ax.hist(X[:,0], bins=bins)
ax.set_xlabel(r'$x_1$', fontsize=20)
ax.set_ylabel(r'$p$', fontsize=20)

ax2 = fig.add_subplot(122)
ax2.hist(X[:,1], bins=bins)
ax2.set_xlabel(r'$x_2$', fontsize=20)
ax2.set_ylabel(r'$p$', fontsize=20)

plt.draw()
plt.show()

Since the sample we use is only two dimensional, we can even visualize them together:

%matplotlib notebook

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
hist, xedges, yedges = np.histogram2d(X[:,0], X[:,1], bins=10, range=[[X_lim[0], X_lim[1]], [Y_lim[0], Y_lim[1]]])

xpos, ypos = np.meshgrid(xedges[:-1], yedges[:-1])
xpos = xpos.flatten('F')
ypos = ypos.flatten('F')
zpos = np.zeros_like(xpos)

dx = 0.5 * np.ones_like(zpos)
dy = dx.copy()
dz = hist.flatten()

ax.bar3d(xpos, ypos, zpos, dx, dy, dz, color='cyan', zsort='average')
ax.view_init(45, 45) # 90,90 = topdown

plt.draw()
plt.show()

Considdering the plots of the individual distributions and the joint distribution it is not far-fetched to assume a bivariate normal distribution.

Using the mean and the covariance matrix of the sample, it is possible to estimate the underlying gaussian probability density function:

%matplotlib notebook

M=60
A = np.linspace(X_lim[0],X_lim[1], M)
B = np.linspace(Y_lim[0],Y_lim[1], M)
A_, B_ = np.meshgrid(A, B)
xy = np.column_stack([A_.flat, B_.flat])

# density values at the grid points
Z = mvn.pdf(xy, mu, cov).reshape(A_.shape)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_xlabel(r'$x_1$', fontsize=20)
ax.set_ylabel(r'$x_2$', fontsize=20)
ax.set_zlabel(r'$p$', fontsize=20)
ax.plot_surface(A_,B_,Z,cmap=cm.coolwarm)
ax.view_init(45, 45) # 90,90 = topdown
plt.draw()
plt.show()

A way to visualize both, the drawn sample and the estimated bivariate gaussian is using a 2D-scatter-plot together with a contour-plot. In a contour-plot all data points on a line with the same colour have the same value in the third dimension:

%matplotlib inline

# arbitrary contour levels
max = 0.12
contour_level = [max * 1/3, max * 2/3, max]
fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(111)

ax.contour(A, B, Z, levels = contour_level)
ax.scatter(X[...,0], X[...,1], c='b', marker='x', label="data")
ax.set_xlim(X_lim[0],X_lim[1])
ax.set_ylim(Y_lim[0],Y_lim[1])
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
plt.draw()
plt.show()

Exercise - Joint Normality

Task:

If x1,,xpx_1, \dots, x_p are all normally distributed, does this always imply that they are jointly normally distributed, i.e. does the pair (x1,,xp)(x_1, \dots, x_p) have a multivariate normal distribution?

Literature

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise - Multivariate Gaussian
by Christian Herta, Klaus Strohmenger
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Christian Herta, Klaus Strohmenger

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.