# NMST539 | Lab Session 3 (Individual work)

## Multivariate Normal Distribution

### LS 2017 | Instad of Thursday 07/03/17

###### Rmd file (UTF8 coding)

A user-friendly interface (one of many): RStudio.

Manuals and introduction into R (in Czech or English):

• Bína, V., Komárek, A. a Komárková, L.: Jak na jazyk R. (PDF súbor)
• Komárek, A.: Základy práce s R. (PDF súbor)
• Kulich, M.: Velmi stručný úvod do R. (PDF súbor)
• De Vries, A. a Meys, J.: R for Dummies. (ISBN-13: 978-1119055808)

#### Conditional Normal Distribution

Let us consider a two-dimensional normal distribution of some random vector $$\Big(\begin{array}{x}X_{1}\\X_{2}\end{array}\Big)$$. The distribution takes the following (general) form:

$$\Big(\begin{array}{x}X_{1}\\X_{2}\end{array}\Big) \sim N_{2}\left(\boldsymbol{\mu} = \Big(\begin{array}{c} \mu_{1} \\ \mu_{2}\end{array}\Big), \Sigma = \left( \begin{array}{cc} \sigma_{1}^{2} & \sigma_{12} \\\sigma_{21} & \sigma_{2}^{2} \end{array} \right) \right)$$,

where $$\boldsymbol{\mu} \in \mathbb{R}^2$$ is the vector of the expected values and $$\Sigma$$ is the variance-covariance matrix, which is a positive definite and symmetric. The correspoding density function (of the two dimensional normal distrubution) is given by the expression

$$\large{f(\boldsymbol{x}) = \frac{1}{2 \pi |\Sigma|^{1/2}} exp\Big\{ -\frac{1}{2} (\boldsymbol{x} - \boldsymbol{\mu})^{\top} \Sigma^{-1} (\boldsymbol{x} - \boldsymbol{\mu}) \Big\},}$$

for $$\boldsymbol{x} = (x_{1}, x_{2})^{\top} \in \mathbb{R}^{2}$$.

This density can be used to derive the marginal distrubution of random variables $$X_{1}$$ and $$X_{2}$$ or the conditional distrubution of $$X_{1}$$ given $$X_{2}$$ (or $$X_{2}$$ given $$X_{1}$$ respectively). In the following we will do both.

For the marginal density of $$X_{1}$$ we have $$f(x_{1}) = \int_{\mathbb{R}} f(x_{1}, x_{2}) \mbox{d}x_{2}$$ and analogously also for the marginal density of $$X_{2}$$ (where we need to integrate the join density wrt the first covariate).

For a simple example with a two dimensional normal distribution the conditional distribution distribution of $$X_{2}$$ given $$X_{1} = x_{1}$$ is normal again and it holds that

$$(X_{2} | X_{1 } = x_{1}) \sim N\Big(\mu_{2} + \frac{\sigma_{21}(x_1 - \mu_1)}{\sigma_{1}^2}, \sigma_{2}^2 - \frac{\sigma_{12}\sigma_{21}}{\sigma_{1}^2}\Big).$$

Now we can apply the formulas given above to obtain the marginal and conditional distributions. Firstly, the library mvtnorm needs to be installed on R. The library is loaded in by running the command:

library("mvtnorm")

Let us consider a simple example with two dimensional normal distribution with the zero mean vector $$\boldsymbol{\mu} = (0,0)^\top$$ and the variance-covariance matrix $$\Sigma = \left( \begin{array}{cc} 1 & 0.8 \\0.8& 1\end{array} \right)$$. We would like to calculate the conditional distribution of $$X_{2}$$ given $$X_{1} = 0.7$$.

Sigma <- matrix(c(1,.8,.8,1), nrow=2) ## variance-covariance matrix

x <- seq(-3,3,0.01)
contour(x,x,outer(x,x,function(x,y){dmvnorm(cbind(x,y),sigma=Sigma)}))

abline(v=.7, lwd=2, lty=2, col = "red")
text(0.75, -2, labels=expression(x[1]==0.7), col = "red", pos = 4)

### conditional distribution of X2 | X1 = 0.7
y <- dnorm(x, mean =  0.8 * 0.7, sd = sqrt(1 - 0.8^2))
lines(y-abs(min(x)),x,lty=2,lwd=2, col = "red")

### marginals
m1 <- m2 <- dnorm(x, 0, 1)
lines(x, m1 - abs(min(x)), lty = 1, lwd = 1)
lines(m2 - abs(min(x)), x, lty = 1, lwd = 1)

And we can add some other conditional distribution of (e.g. for $$X_{2} | X_{1} = -1$$) by adding additional lines of an R code:

contour(x,x,outer(x,x,function(x,y){dmvnorm(cbind(x,y),sigma=Sigma)}))
abline(v=-1, lwd=2, lty=2, col = "blue")

### conditional distribution of X2 | X1 = - 1
y2 <- dnorm(x, mean = 0.8 * (- 1), sd = sqrt(1 - 0.8^2))
lines(-y2 + max(x),x,lty=2,lwd=2, col = "blue")