Last updated: 2021-03-01

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library(mvtnorm)
Warning: package 'mvtnorm' was built under R version 3.6.2

Pre-requisites

You should be familiar with the Multivariate normal distribution, and with the eigen-decomposition for symmetric positive semi-definite (PSD) matrices.

Introduction

Getting an intuition for what the \(p\)-dimensional multivariate normal distribution, \(N_p(\mu,\Sigma)\), “looks like” can be difficult. For \(p=1,2\) things are not too bad: we can directly visualize a univariate normal distribution by plotting its density, and visualize a bivariate normal distribution by plotting a contour plot of the density, or by simulating samples from the distribution and visualizing them using a 2d scatterplot. For example, the following code does this for \(N(0,\Sigma)\) where \[\Sigma = \begin{pmatrix} 1.0 & 0.7 \\ 0.7 & 1.0 \end{pmatrix}\]:

Sigma= cbind(c(1,0.7),c(0.7,1))
X = rmvnorm(1000,c(0,0),Sigma)
plot(X[,1],X[,2],main="Samples from bivariate normal with variance Sigma",asp=1)

But in \(p=100\) dimensions, or even just \(p=4\) dimensions, things become much harder because direct visualization is impractical.

So how can we get intuition about the multivariate normal distribution, \(N_p(\mu,\Sigma)\) when \(p\) is large?

Note first that the mean \(\mu\) is just a vector of \(p\) numbers, and generally causes few problem in interpretation: you can just think of each number as specifying the mean in each of the \(p\) coordinates one at a time.

In contrast, the covariance matrix \(\Sigma\) is a \(p \times p\) matrix that captures potentially more complex patterns, and creates more challenges for intuition. One possible approach is to plot a heatmap of this matrix, and this can certainly be helpful in certain situations. However, this vignette describes a more algebraic approach, based on the eigen-decomposition of \(\Sigma\).

Some linear algebra

Recall that any valid \(p \times p\) covariance matrix \(\Sigma\) must be symmetric and positive semi-definite (PSD). Furthermore, recall that any such PSD matrix must have eigen-decomposition: \[\Sigma = V \Lambda V'\] where:

  • \(\Lambda\) is a \(K \times K\) diagonal matrix with the non-zero eigenvalues of \(\Sigma\), \(\lambda_1,\dots,\lambda_K\) say, on the diagonal (\(K \leq p\) is the rank of \(\Sigma\)).

  • \(V\) is a \(p \times K\) orthonormal matrix (\(V'V=I_K\)), whose columns \(v_1,\dots,v_K\) are the normalized eigenvectors of \(\Sigma\) corresponding to the non-zero eigenvalues.

Recall also that if \(Z \sim N_p(0, I_p)\) and \(A\) is any \(n \times p\) matrix then \(\mu + AZ \sim N(\mu, AA')\).

Now apply this last result with \(A= V \Lambda^{0.5}\) where \(\Lambda^{0.5}\) is the diagonal matrix with \(\lambda_1^{0.5},\dots,\lambda_K^{0.5}\) on the diagonal. We get \[ \mu + V \Lambda^{0.5} Z \sim N_p(\mu, V \Lambda^{0.5} \Lambda^{0.5} V').\] That is, \[\mu + V \Lambda^{0.5} Z \sim N_p(\mu, \Sigma).\] We can write the matrix multiple \(V\Lambda^{0.5} Z\) as a sum to make the structure more obvious: \[\mu + \sum_{k=1}^K \lambda_k^{0.5} z_k v_k \sim N_p(\mu, \Sigma).\] Here \(\mu\) and \(v_1,\dots,v_K\) are all column vector of length \(p\), whereas the \(\lambda_k\) and \(z_k\) are all scalars.

Interpration as a random linear combination of eigenvectors

From this algebra, if \(X \sim N_p(\mu,\Sigma)\), then we can think of \(X\) as being generated by taking the mean \(\mu\), and adding a random linear combination of the eigenvectors of \(\Sigma\). Specifically \[X = \mu + \sum_{k=1}^K b_k v_k,\] where the weights \[b_k=\lambda_k^{0.5} z_k \sim N(0,\lambda_k).\] are independent of one another.

Note that if \(\lambda_k\) is small then \(b_k \approx 0\), so the eigenvectors with small eigenvalues contribute little to \(X\), and we can focus on the eigenvectors with large eigenvalues. Indeed, this approach provides the simplest insights when most of the \(\lambda_k\) are negligible, and only one or two eigenvectors contribute meaningfully to the sum.

Example: rank 1 covariance

To make a simple example, set \(\mu=0\) and assume \(\Sigma\) is a rank 1 matrix. That is, \(\Sigma\) has only one eigenvector: \[\Sigma = \lambda vv'\] for some \(p\)-vector \(v\).

In this case the algebra above gives the representation \(X= b v\) where \(b \sim N(0,\lambda)\). That is \(X\) is simply a multiple of \(v\), where the multiplier is randomly distributed from a univariate normal. Thus in this case the randomness in \(X\) boils down to the randomness in a single random univarate normal, which is easy to visualize.

To give a specific example, suppose that \(v\) is the vector of all 1s \(v=(1,\dots,1)\) and \(\lambda=1\). That is \(\Sigma\) is a matrix of all 1s. Then \(X= (b,b,b,\dots,b)\) where \(b \sim N(0,1)\).

To give another specific example, if \(v=(-1,-1,-1,1,1)\) and \(\lambda=2\) then \(X= (-b,-b,-b,b,b)\) where \(b \sim N(0,2)\).


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] mvtnorm_1.1-1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.6       rstudioapi_0.11  whisker_0.4      knitr_1.29      
 [5] magrittr_1.5     workflowr_1.6.2  R6_2.4.1         rlang_0.4.8     
 [9] stringr_1.4.0    tools_3.6.0      xfun_0.16        git2r_0.27.1    
[13] htmltools_0.5.0  ellipsis_0.3.1   yaml_2.2.1       digest_0.6.27   
[17] rprojroot_1.3-2  tibble_3.0.4     lifecycle_0.2.0  crayon_1.3.4    
[21] later_1.1.0.1    vctrs_0.3.4      fs_1.5.0         promises_1.1.1  
[25] glue_1.4.2       evaluate_0.14    rmarkdown_2.3    stringi_1.4.6   
[29] compiler_3.6.0   pillar_1.4.6     backports_1.1.10 httpuv_1.5.4    
[33] pkgconfig_2.0.3

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