- #1
fab13
- 318
- 6
- TL;DR Summary
- I would like to find a way to combine 2 probes in cosmology represented each one by a Fisher matrix. This combination should increase the Figure of Merit (FoM) by a significant gain, that is, doing make cross-correlations between the 2 probes.
This post is slightly different from a previous post sent to mathematical forum : this is because I talk about here the MATLAB function "eig" with 2 arguments but this concerns actually the combination between 2 biased tracers in Cosmology context.
I am looking for a common basis of eigenvectors between 2 Fisher or Covariance matrices A and B. Different algorithms exist to "joint diagonalize" 2 matrices like for example qndiag. They allow to find the same eigen vectors that diagonalize both A and B symmetric matrix.
In Matlab, the command `[V, D] = eig(A, B)` should solve the Generalized Eigenvalue Problem `A*V = B*V*D`. Knowing vectors `V`, this equality is true also for all other vectors under the form `V=VD'` with `D'` a chosen diagonal matrix.
I want to get the same basis for matrices A and B because I want to apply the MLE (Maximum Likelihood Estimator) by doing on diagonalized matrix :
1) For Fisher matrix, the sum coming from MLE :
##\dfrac{1}{\sigma_{\tau}^2}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\quad(1)##
2) Or For Covariance matrix, putting on the diagonal the variances :
##\sigma_{\tau}^2=\bigg(\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\bigg)^{-1}\quad(2)##
Unfortunately, But in both cases, it doesn't increase the FoM (Figure of Merit, equal to ##\dfrac{1}{det(block(2 parameters))}##), I mean that constraints are not better than with a classical synthesis ((where we simply sum sthe 2 Fisher matrices) : as a conclusion, I can't manage to do cross-correlations since I have no gain on constraints.
Moreover, with the first method 1) above, after the building of Fisher matrix and its inversion, I can marginalize over the nuisance parameters and re-invert to get a Fisher matrix where nuisance parameters (shot noise, intrinsic alignement) estimations are encoded into it.
But I can't do the same for method 2). I can fix parameters (remove directly lines/columns) in Fisher matrix but this produces too high FoM (and so too small constraints) since I have less error parameters to estimate.
So, I am looking for a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations between 2 given Fisher matrices (which represent actually 2 different probes into a cosmology context : spectroscopic and photometric probes).
And to carry out it, I would like to know if the Generalized eigenvectors and eigenvalues formulated with the Matlab command `[V, D] = eig(A,B) could help me by taking for example (I am not sure) a special or rather an appropriate diagonal matrix `D'`.Indeed, I didn't yet grasp all the usefulness of using `[V, D] = eig(A, B)` in my case : I only understand that it allows to find ```ectors such that `AV` is "parallel" to `BV` (I mean the both are linked by a diagonal matrix).
I guess you think it is about numerical stuff but the underying issue is about Cosmology and the way to combine 2 probes in this context.
I hope you will understand the formulation of my issue.
Regards
I am looking for a common basis of eigenvectors between 2 Fisher or Covariance matrices A and B. Different algorithms exist to "joint diagonalize" 2 matrices like for example qndiag. They allow to find the same eigen vectors that diagonalize both A and B symmetric matrix.
In Matlab, the command `[V, D] = eig(A, B)` should solve the Generalized Eigenvalue Problem `A*V = B*V*D`. Knowing vectors `V`, this equality is true also for all other vectors under the form `V=VD'` with `D'` a chosen diagonal matrix.
I want to get the same basis for matrices A and B because I want to apply the MLE (Maximum Likelihood Estimator) by doing on diagonalized matrix :
1) For Fisher matrix, the sum coming from MLE :
##\dfrac{1}{\sigma_{\tau}^2}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\quad(1)##
2) Or For Covariance matrix, putting on the diagonal the variances :
##\sigma_{\tau}^2=\bigg(\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\bigg)^{-1}\quad(2)##
Unfortunately, But in both cases, it doesn't increase the FoM (Figure of Merit, equal to ##\dfrac{1}{det(block(2 parameters))}##), I mean that constraints are not better than with a classical synthesis ((where we simply sum sthe 2 Fisher matrices) : as a conclusion, I can't manage to do cross-correlations since I have no gain on constraints.
Moreover, with the first method 1) above, after the building of Fisher matrix and its inversion, I can marginalize over the nuisance parameters and re-invert to get a Fisher matrix where nuisance parameters (shot noise, intrinsic alignement) estimations are encoded into it.
But I can't do the same for method 2). I can fix parameters (remove directly lines/columns) in Fisher matrix but this produces too high FoM (and so too small constraints) since I have less error parameters to estimate.
So, I am looking for a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations between 2 given Fisher matrices (which represent actually 2 different probes into a cosmology context : spectroscopic and photometric probes).
And to carry out it, I would like to know if the Generalized eigenvectors and eigenvalues formulated with the Matlab command `[V, D] = eig(A,B) could help me by taking for example (I am not sure) a special or rather an appropriate diagonal matrix `D'`.Indeed, I didn't yet grasp all the usefulness of using `[V, D] = eig(A, B)` in my case : I only understand that it allows to find ```ectors such that `AV` is "parallel" to `BV` (I mean the both are linked by a diagonal matrix).
I guess you think it is about numerical stuff but the underying issue is about Cosmology and the way to combine 2 probes in this context.
I hope you will understand the formulation of my issue.
Regards