logo Use CA10RAM to get 10%* Discount.
Order Nowlogo
(5/5)

Visualizing the Multivariate Normal

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

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
(5/5)
Attachments:

Related Questions

. Introgramming & Unix Fall 2018, CRN 44882, Oakland University Homework Assignment 6 - Using Arrays and Functions in C

DescriptionIn this final assignment, the students will demonstrate their ability to apply two ma

. The standard path finding involves finding the (shortest) path from an origin to a destination, typically on a map. This is an

Path finding involves finding a path from A to B. Typically we want the path to have certain properties,such as being the shortest or to avoid going t

. Develop a program to emulate a purchase transaction at a retail store. This program will have two classes, a LineItem class and a Transaction class. The LineItem class will represent an individual

Develop a program to emulate a purchase transaction at a retail store. Thisprogram will have two classes, a LineItem class and a Transaction class. Th

. SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 1 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

. Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

Ask This Question To Be Solved By Our ExpertsGet A+ Grade Solution Guaranteed

expert
Um e HaniScience

664 Answers

Hire Me
expert
Muhammad Ali HaiderFinance

888 Answers

Hire Me
expert
Husnain SaeedComputer science

845 Answers

Hire Me
expert
Atharva PatilComputer science

747 Answers

Hire Me

Get Free Quote!

379 Experts Online