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Computing the PDF of a low-rank multivariate normal

0

$begingroup$

I have a question which seems simple, but I would appreciate some comments!

Sometimes models involve a low-rank approximation to a covariance matrix. What confuses me is how you can compute the PDF of these low-rank matrices. As a very simple example, consider the case where:

$textbfx sim mathcalN(pmbmu, Sigma)$

$textbfy = K textbfx$

where $textbfx$ is a vector of length $m$ and $K$ is an $n times m$ matrix, with $n > m$.

Now by using some standard identities, I believe the distribution of $textbfy$ is:

$textbfy sim mathcalN(K pmbmu, K Sigma K^T)$

That's all well and good, but the covariance matrix

$Sigma^* = K Sigma K^T$

is now only of rank $m$, but of size $n times n$. The multivariate normal PDF involves a term:

$p(textbfx|mu, Sigma) propto textrmexp(-frac12 textbfx^T Sigma^-1 textbfx) $

However, while everything here seems coherent, the matrix $Sigma^*$ is not invertible because it is not full rank. I suppose this makes sense; we can't compute the PDF for just any value, because the covariance matrix does not span the space.

I suppose my question is: does this all make sense, and if so, what do people tend to do about this? Does it make sense to use a pseudo-inverse to compute the PDF, and what would that mean?

matrices statistics normal-distribution

share|cite|improve this question

asked Mar 20 at 8:32

noctiluxnoctilux

1305

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Your question is similar to asking what is the density function of a $N(0,0)$ distribution or the density function of a multivariate normal with $Sigma=beginbmatrix 1 & 1 \ 1 & 1 endbmatrix$. In an ordinary sense they do not have a density function
  $endgroup$
  – Henry
  Mar 20 at 8:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  en.wikipedia.org/wiki/….
  $endgroup$
  – StubbornAtom
  Mar 20 at 9:53
  

  
  
  
  

add a comment | 

0

$begingroup$

I have a question which seems simple, but I would appreciate some comments!

Sometimes models involve a low-rank approximation to a covariance matrix. What confuses me is how you can compute the PDF of these low-rank matrices. As a very simple example, consider the case where:

$textbfx sim mathcalN(pmbmu, Sigma)$

$textbfy = K textbfx$

where $textbfx$ is a vector of length $m$ and $K$ is an $n times m$ matrix, with $n > m$.

Now by using some standard identities, I believe the distribution of $textbfy$ is:

$textbfy sim mathcalN(K pmbmu, K Sigma K^T)$

That's all well and good, but the covariance matrix

$Sigma^* = K Sigma K^T$

is now only of rank $m$, but of size $n times n$. The multivariate normal PDF involves a term:

$p(textbfx|mu, Sigma) propto textrmexp(-frac12 textbfx^T Sigma^-1 textbfx) $

However, while everything here seems coherent, the matrix $Sigma^*$ is not invertible because it is not full rank. I suppose this makes sense; we can't compute the PDF for just any value, because the covariance matrix does not span the space.

I suppose my question is: does this all make sense, and if so, what do people tend to do about this? Does it make sense to use a pseudo-inverse to compute the PDF, and what would that mean?

matrices statistics normal-distribution

share|cite|improve this question

asked Mar 20 at 8:32

noctiluxnoctilux

1305

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Your question is similar to asking what is the density function of a $N(0,0)$ distribution or the density function of a multivariate normal with $Sigma=beginbmatrix 1 & 1 \ 1 & 1 endbmatrix$. In an ordinary sense they do not have a density function
  $endgroup$
  – Henry
  Mar 20 at 8:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  en.wikipedia.org/wiki/….
  $endgroup$
  – StubbornAtom
  Mar 20 at 9:53
  

  
  
  
  

add a comment | 

$begingroup$

I have a question which seems simple, but I would appreciate some comments!

Sometimes models involve a low-rank approximation to a covariance matrix. What confuses me is how you can compute the PDF of these low-rank matrices. As a very simple example, consider the case where:

$textbfx sim mathcalN(pmbmu, Sigma)$

$textbfy = K textbfx$

where $textbfx$ is a vector of length $m$ and $K$ is an $n times m$ matrix, with $n > m$.

Now by using some standard identities, I believe the distribution of $textbfy$ is:

$textbfy sim mathcalN(K pmbmu, K Sigma K^T)$

That's all well and good, but the covariance matrix

$Sigma^* = K Sigma K^T$

is now only of rank $m$, but of size $n times n$. The multivariate normal PDF involves a term:

$p(textbfx|mu, Sigma) propto textrmexp(-frac12 textbfx^T Sigma^-1 textbfx) $

However, while everything here seems coherent, the matrix $Sigma^*$ is not invertible because it is not full rank. I suppose this makes sense; we can't compute the PDF for just any value, because the covariance matrix does not span the space.

I suppose my question is: does this all make sense, and if so, what do people tend to do about this? Does it make sense to use a pseudo-inverse to compute the PDF, and what would that mean?

matrices statistics normal-distribution

share|cite|improve this question

asked Mar 20 at 8:32

noctiluxnoctilux

1305

$endgroup$

share|cite|improve this question

asked Mar 20 at 8:32

noctiluxnoctilux

1305

share|cite|improve this question

asked Mar 20 at 8:32

noctiluxnoctilux

1305

asked Mar 20 at 8:32

noctiluxnoctilux

1305

asked Mar 20 at 8:32

noctiluxnoctilux

1305