Timeline for Covariance Linear Shrinkage Estimator : Implied Data
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2021 at 21:42 | answer | added | seanv507 | timeline score: 0 | |
| Jan 13, 2021 at 22:01 | comment | added | whuber♦ | That means you're really asking about the correlation matrix. | |
| Jan 13, 2021 at 21:38 | comment | added | quantguy | @whuber. Frobenius norm is one way to do this. To address scaling issues, you could normalize the data ${\bf X}$ first. | |
| Jan 13, 2021 at 21:33 | comment | added | quantguy | @seanv507, the linear shrinkage estimator is defined as ${\bf \hat\Sigma}=\delta {\bf S} + (1-\delta) {\bf T}$ where $\bf T$ is a "shrinkage target" and $\delta$ is a scalar found by a simple optimization. The simplest target (and often the most efficient) is the identity matrix. The shrinkage introduces bias, but reduces overall estimation error. | |
| Jan 13, 2021 at 19:14 | comment | added | seanv507 | Can you define the linear shrinkage operator in the question. I would believe you can, just as you do for Principal components analysis. I would suggest expressing X in terms of SVD and take it from there... | |
| Jan 13, 2021 at 19:13 | comment | added | whuber♦ | That sounds interesting, but first you need to explain what you mean by "as close as possible:" what metric do you have in mind? A potential problem is that typically the columns of a data matrix measure incommensurable things--height, weight, counts, etc.--but "close" means you have to condense all those differences into a single number. How? | |
| Jan 13, 2021 at 18:19 | comment | added | quantguy | Not exactly, we have some information on the process to obtain the "valid" covariance matrix : the data $\bf X$ used to obtain it. The question relates to the existence of $\bf Y$, but also how to obtain it so that it is as close as possible to $\bf X$. Do you have any ideas on how to do this ? | |
| Jan 13, 2021 at 17:02 | comment | added | whuber♦ | Thank you. Doesn't this question amount to asking "given a valid covariance matrix and a dataset size $T,$ do there exist data of length $T$ for which it is the (empirical) covariance matrix?" | |
| Jan 13, 2021 at 15:59 | history | edited | quantguy | CC BY-SA 4.0 |
added 77 characters in body
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| Jan 13, 2021 at 15:54 | comment | added | quantguy | Thanks for the comment, I will clarify the question | |
| Jan 13, 2021 at 15:47 | comment | added | whuber♦ | Some things about this question aren't quite right. First, $S$ is not the covariance matrix because the columns have not been centered, so are you actually asking about this sums-of-products matrix or do you truly intend to ask about the covariance matrix? Second, could you clearly indicate what "estimated by linear shrinkage" is? There may be several ways to interpret that. Third, could you articulate some sense of "closest"? How would you measure or quantify that? | |
| Jan 13, 2021 at 15:38 | history | asked | quantguy | CC BY-SA 4.0 |