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Visualizing the Multivariate Normal

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Visualizing the Multivariate Normal

 

Spectral Decomposition

P is orthogonal if PT P = 1 and PPT = 1.

Theorem: Let A be symmetric n × n. Then we can write

A = PDPT ,

where D = diag (λ1, . . . , λn) and P is orthogonal. The λs are the eigenvalues of A and ith column of P is an eigenvector corresponding to λi .

Orthogonal matrices represent rotations of the coordinates. Diagonal matrices represent stretchings/shrinkings of coordinates.

Properties

  • The covariance matrix Σ is symmetric and positive definite, so we know from the spectral decomposition theorem that it can be written as

Σ = PΛPT .

  • Λ is the diagonal matrix of the eigenvalues of Σ.
  • P is the matrix whose columns are the orthonormal eigenvectors of Σ (hence V is an orthogonal matrix).

) Geometrically, orthogonal matrices represent rotations.

) Multiplying by P rotates the coordinate axes so that they are parallel to the eigenvectors of Σ.

) Probabilistically, this tells us that the axes of the probability-contour ellipse are parallel to those eigenvectors.

) The radii of those axes are proportional to the square roots of the eigenvalues.

Can we view the det(Σ) as a “variance“?

  •  

Q

 
  • Variance of one-dimensional
  • From the SDT: det(Σ) = i λi .
  • Eigenvalues (λi ) tell us how stretched or compressed the distribution
  • View det(Σ) as stretching/compressing factor for the MVN
  • We will see this from the contour plots

Our focus is visualizing MVN distributions in R.

What is a Contour Plot?

  • Contour plot is a graphical technique for representing a 3-dimensional
  • We plot constant z slices (contours) on a 2-D
  • The contour plot is an alternative to a 3-D surface The contour plot is formed by:
  • Vertical axis: Independent variable
  • Horizontal axis: Independent variable
  • Lines: iso-response

Contour Plot

The lines of the contour plots denote places of equal probability mass for the MVN distribution

  • The lines represent points of both variables that lead to the same height on the z-axis (the height of the surface)
  • These contours can be constructed from the eigenvalues and eigenvectors of the covariance matrix
  • The direction of the ellipse axes are in the direction of the eigenvalues
  • The length of the ellipse axes are proportional to the constant times the eigenvector
  • More specifically

||Σ1/2(X µ)|| = c2

has ellipsoids centered at µ and axes at √(λi vi )

Visualizing the MVN Distribution Using Contour Plots

The next figure below shows a contour plot of the joint pdf of a bivariate normal distribution. Note: we are plotting the theoretical contour plot. This particular distribution has mean

1

 

µ = . 1 Σ

 

(Solid dot), and variance matrix

1 1

 

Σ =. 2 1 Σ

Code to construct plot

0.04

 

0.06

0.08

 

library(mvtnorm)

x.points <- seq(-3,3,length.out=100) y.points <-x.points

z <- matrix(0,nrow=100,ncol=100) mu <- c(1,1)

sigma <- matrix(c(2,1,1,1),nrow=2) for (i in1:100) {

for (j in1:100) {

z[i,j] <- dmvnorm(c(x.points[i],y.points[j]),

mean=mu,sigma=sigma)

}

}

contour(x.points,y.points,z)

Our findings

  • Probability contours are
  • Density changes comparatively slowly along the major axis, and quickly along the minor
  • The two points marked + in the figure have equal geometric distance from µ.
  • But the one to its right lies on a higher probability contour than the one above it, because of the directions of their displacements from the means

Kernel density estimation (KDE)

  • KDE allows us to estimate the density from which each sample was
  • This method (which you will learn about in other classes) allows us to approximate the density using a
  • There are R packages that use kde’s such as density().

What did we learn?

  • The contour plot of X (bivariate density): Color is the probability density at each point (red is low density and white is high density).
  • Contour lines define regions of probability density (from high to low).
  • Single point where the density is highest (in the white region) and the contours are approximately ellipses (which is what you expect from a Gaussian).

 

What can we say in general about the MVN density?

 

  • The spectral decomposition theorem tells us that the contours of the multivariate normal distribution are
  • The axes of the ellipsoids correspond to eigenvectors of the covariance
  • The radii of the ellipsoids are proportional to square roots of the eigenvalues of the covariance
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