Covariance estimation and Graphical Modelling
I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions.
Suppose, $X_i overset{iid}{sim} N_p(mu,Sigma)$, $I=1, dots, n$. Define $S=frac{1}{n}sum (X_i-bar{X})(X_i-bar{X})'$. Then almost in every literature, they claim $S$ is really a 'bad estimator' (or performs poorly) in case of large $p$ small $n$. However I understand $S$ is always non-negative definite and symmetric and will be positive definite (w.p. 1) if $ngeq p+1$. Also $hat{Sigma}_{mle}=S$. My question is
1) Why S performs poorly when $p>n$?
2) If $Sigma$ is sparse why do we use different approach to estimate it (methodology involving Graphical representation of $Q=Sigma^{-1}$) ?
BTW I am reading Graphical Models by S. Lauritzen, Gaussian Markov Random Fields by Rue and Held. Please fell free to suggest any other books, materials etc. That you think will help me understand.
graph-theory normal-distribution statistical-inference machine-learning estimation-theory
edited Nov 23 at 13:33
pointguard0
1,312821
asked Sep 20 '15 at 22:40
Bob
283
add a comment |
I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions.
Suppose, $X_i overset{iid}{sim} N_p(mu,Sigma)$, $I=1, dots, n$. Define $S=frac{1}{n}sum (X_i-bar{X})(X_i-bar{X})'$. Then almost in every literature, they claim $S$ is really a 'bad estimator' (or performs poorly) in case of large $p$ small $n$. However I understand $S$ is always non-negative definite and symmetric and will be positive definite (w.p. 1) if $ngeq p+1$. Also $hat{Sigma}_{mle}=S$. My question is
1) Why S performs poorly when $p>n$?
2) If $Sigma$ is sparse why do we use different approach to estimate it (methodology involving Graphical representation of $Q=Sigma^{-1}$) ?
BTW I am reading Graphical Models by S. Lauritzen, Gaussian Markov Random Fields by Rue and Held. Please fell free to suggest any other books, materials etc. That you think will help me understand.
graph-theory normal-distribution statistical-inference machine-learning estimation-theory
edited Nov 23 at 13:33
pointguard0
1,312821
asked Sep 20 '15 at 22:40
Bob
283
add a comment |
I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions.
Suppose, $X_i overset{iid}{sim} N_p(mu,Sigma)$, $I=1, dots, n$. Define $S=frac{1}{n}sum (X_i-bar{X})(X_i-bar{X})'$. Then almost in every literature, they claim $S$ is really a 'bad estimator' (or performs poorly) in case of large $p$ small $n$. However I understand $S$ is always non-negative definite and symmetric and will be positive definite (w.p. 1) if $ngeq p+1$. Also $hat{Sigma}_{mle}=S$. My question is
1) Why S performs poorly when $p>n$?
2) If $Sigma$ is sparse why do we use different approach to estimate it (methodology involving Graphical representation of $Q=Sigma^{-1}$) ?
BTW I am reading Graphical Models by S. Lauritzen, Gaussian Markov Random Fields by Rue and Held. Please fell free to suggest any other books, materials etc. That you think will help me understand.
graph-theory normal-distribution statistical-inference machine-learning estimation-theory
edited Nov 23 at 13:33
pointguard0
1,312821
asked Sep 20 '15 at 22:40
Bob
283
I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions.
Suppose, $X_i overset{iid}{sim} N_p(mu,Sigma)$, $I=1, dots, n$. Define $S=frac{1}{n}sum (X_i-bar{X})(X_i-bar{X})'$. Then almost in every literature, they claim $S$ is really a 'bad estimator' (or performs poorly) in case of large $p$ small $n$. However I understand $S$ is always non-negative definite and symmetric and will be positive definite (w.p. 1) if $ngeq p+1$. Also $hat{Sigma}_{mle}=S$. My question is
1) Why S performs poorly when $p>n$?
2) If $Sigma$ is sparse why do we use different approach to estimate it (methodology involving Graphical representation of $Q=Sigma^{-1}$) ?
BTW I am reading Graphical Models by S. Lauritzen, Gaussian Markov Random Fields by Rue and Held. Please fell free to suggest any other books, materials etc. That you think will help me understand.
graph-theory normal-distribution statistical-inference machine-learning estimation-theory
graph-theory normal-distribution statistical-inference machine-learning estimation-theory
edited Nov 23 at 13:33
pointguard0
1,312821
asked Sep 20 '15 at 22:40
Bob
283
edited Nov 23 at 13:33
pointguard0
1,312821
asked Sep 20 '15 at 22:40
Bob
283
edited Nov 23 at 13:33
pointguard0
1,312821
edited Nov 23 at 13:33
pointguard0
1,312821
edited Nov 23 at 13:33
pointguard0
1,312821
1,312821
asked Sep 20 '15 at 22:40
Bob
283
asked Sep 20 '15 at 22:40
Bob
283
asked Sep 20 '15 at 22:40
Bob
283
283
add a comment |
add a comment |
1 Answer
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First, I would like to mention that $S$ is always non-negative definite no matter what $p$ and $n$ are. To claim positive definiteness it's not enough to have $n ge p + 1$.
Now, as for your questions:
- When $p > n$ you basically have $frac{p(p+1)}2$ parameters to estimate with $n$ observations only. Imagine, you want to estimate a 3dimensional random vector with only one observation. What would your estimate be? Would that be consistent in any sense?
- There are many different methods for sparse covariance matrix estimation, but the reason people don't use estimator $S$ (mentioned in OPs question) is that the sparsity pattern is assumed unknown, in other words, you don't know which elements of matrix $Sigma$ are exactly $0$ and which are not. One of the ways to handle this is to use Graphical models.
I tried to explain the intuition behind the answers but of course mathematically rigorous explanations also exist.
answered Nov 23 at 13:37
pointguard0
1,312821
add a comment |
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1 Answer
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1 Answer
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votes
First, I would like to mention that $S$ is always non-negative definite no matter what $p$ and $n$ are. To claim positive definiteness it's not enough to have $n ge p + 1$.
Now, as for your questions:
- When $p > n$ you basically have $frac{p(p+1)}2$ parameters to estimate with $n$ observations only. Imagine, you want to estimate a 3dimensional random vector with only one observation. What would your estimate be? Would that be consistent in any sense?
- There are many different methods for sparse covariance matrix estimation, but the reason people don't use estimator $S$ (mentioned in OPs question) is that the sparsity pattern is assumed unknown, in other words, you don't know which elements of matrix $Sigma$ are exactly $0$ and which are not. One of the ways to handle this is to use Graphical models.
I tried to explain the intuition behind the answers but of course mathematically rigorous explanations also exist.
answered Nov 23 at 13:37
pointguard0
1,312821
add a comment |
First, I would like to mention that $S$ is always non-negative definite no matter what $p$ and $n$ are. To claim positive definiteness it's not enough to have $n ge p + 1$.
Now, as for your questions:
- When $p > n$ you basically have $frac{p(p+1)}2$ parameters to estimate with $n$ observations only. Imagine, you want to estimate a 3dimensional random vector with only one observation. What would your estimate be? Would that be consistent in any sense?
- There are many different methods for sparse covariance matrix estimation, but the reason people don't use estimator $S$ (mentioned in OPs question) is that the sparsity pattern is assumed unknown, in other words, you don't know which elements of matrix $Sigma$ are exactly $0$ and which are not. One of the ways to handle this is to use Graphical models.
I tried to explain the intuition behind the answers but of course mathematically rigorous explanations also exist.
answered Nov 23 at 13:37
pointguard0
1,312821
add a comment |
First, I would like to mention that $S$ is always non-negative definite no matter what $p$ and $n$ are. To claim positive definiteness it's not enough to have $n ge p + 1$.
Now, as for your questions:
- When $p > n$ you basically have $frac{p(p+1)}2$ parameters to estimate with $n$ observations only. Imagine, you want to estimate a 3dimensional random vector with only one observation. What would your estimate be? Would that be consistent in any sense?
- There are many different methods for sparse covariance matrix estimation, but the reason people don't use estimator $S$ (mentioned in OPs question) is that the sparsity pattern is assumed unknown, in other words, you don't know which elements of matrix $Sigma$ are exactly $0$ and which are not. One of the ways to handle this is to use Graphical models.
I tried to explain the intuition behind the answers but of course mathematically rigorous explanations also exist.
answered Nov 23 at 13:37
pointguard0
1,312821
First, I would like to mention that $S$ is always non-negative definite no matter what $p$ and $n$ are. To claim positive definiteness it's not enough to have $n ge p + 1$.
Now, as for your questions:
- When $p > n$ you basically have $frac{p(p+1)}2$ parameters to estimate with $n$ observations only. Imagine, you want to estimate a 3dimensional random vector with only one observation. What would your estimate be? Would that be consistent in any sense?
- There are many different methods for sparse covariance matrix estimation, but the reason people don't use estimator $S$ (mentioned in OPs question) is that the sparsity pattern is assumed unknown, in other words, you don't know which elements of matrix $Sigma$ are exactly $0$ and which are not. One of the ways to handle this is to use Graphical models.
I tried to explain the intuition behind the answers but of course mathematically rigorous explanations also exist.
answered Nov 23 at 13:37
pointguard0
1,312821
answered Nov 23 at 13:37
pointguard0
1,312821
answered Nov 23 at 13:37
pointguard0
1,312821
answered Nov 23 at 13:37
pointguard0
1,312821
1,312821
add a comment |
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