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I am trying to do a synthesis between 2 Fisher matrices, i.e without only considering a simple sum : I am looking for a final Fisher matrix that performs the XC (cross-correlations). One done that, I could get constraint by just inverting it.

Someone suggested me to look at the "pooled covariance matrix estimation".

One advises me to take the average of eigen values to be closed to the analytical synthesis : what is the justification of this weight taken to the value 0.5 as weight into synthesis ?

Could anyone explain me what is a "pooled covariance matrix estimation" and how to apply it in my case ? this would be fine.

EDIT 1:

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows are the same in the 2 matrixes).

Now I would like to make the cross synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from Maximum Likelihood Estimator method) :

$$\dfrac{1}{\sigma_{\hat{\tau}}^{2}}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\quad(1)$$

$\sigma_{\hat{\tau}}$ represents the best estimator representing the combination of a sample1 ($\sigma_1$) and a sample2 ($\sigma_2$).

Now, I would like to do the same but for my 2 Fisher matrixes, i.e from a matricial point of view.

For this, I tried to diagonalize each of these 2 Fisher matrix. Then, I add the 2 diagonal matrix and I have so a global diagonal Fisher matrix, but I don't know how to come back in the space of start (since the diagonalization don't give the same combination of eigen vectors for each matrix).

If I could get back in the same time to starting parameters space, I could do matrix products to get the final Fisher matrix by doing :

$$\text{Fisher$_{\text{final,cross}}$} = P.\text{Fisher$_{\text{diag,global}}$}.P^{-1}\quad(2)$$

with $P$ the passing matrixes (composed of eigen vectors) and I could get directly the covariance matrix by inverting $\text{Fisher}_{\text{final,cross}}$

How can I come back from $(2)$ of the $\text{Fisher$_{\text{diag,global}}$}$ diagonal matrix to starting space, i.e the single parameters ?

My difficulties come from the fact that the diagonlization of the 2 Fisher matrixes will produce different passing matrix $P_1$ and $P_2$, that is to say, different eigen vectors, so a different linear combination of variables between both. I have written the passing matrix $P$ above but it is not defined, I think an expression of $P$ as a function of $P_1$ and $P_2$ passsing matrixes is the key point of my issue.

There is surely a linear algebra property whih could allow to circumvent this issue of taking into account the 2 different linear combination of variables while being able to come back in the space of start, i.e space of single parameters which represent the Fisher matrixes.

If someone could help me to perform this operation, I hope that I have been clear. If you have any questions, don't hesitate, I would be glad to give you more informations.

EDIT 2: I take the following notations :

  1. $D$ diagonal matrix is the sum of the 2 diagonalized matrix $D_1$ and $D_2$ (from Fisher1 and Fisher2 initial matrixes) : $D=D_1+D_2$

  2. $P_1$ and $P_2$ are respectively the "passing" matrixes (from Fisher1 and Fisher2 diagonalization) composed of eigen vectors.

  3. $M_1$ is the Fisher1 matrix and $M_2$ is the Fisher2 matrix.

So, I am looking a way to find an endomorphism $M$ that checks :

$$D=P^-1.M.P\quad(3)$$ where $P$ is the unkonwn "passing" matrix.

So, there are 2 unknown quantities in my issue :

  1. The "passing" matrix, i.e the eigenvectors (I am yet trying to build it from $P_1$ and $P_2$ matrixes).

  2. The $M$ matrix which represents this endomorphism.

You could tell me : "there are an infinity of endomorphism with ortogonal passing matrices $P_1$ and $P_2$". But in this world of unknown quantities, I know however the eigen values of this wanted endomorphism $M$ (they are equal to the diagonal of matrix $D$).

Would anyone help me to find a way to build this $P$ "passing" matrix from $P_1$ and $P_2$ ? As you can see, a simple sum is not enough.

If an exact building of $P$ from $P_1$ and $P_2$ is not possible, is there a way to approximate it ?

ps : the link A Two-Stage Approach to Synthesizing Covariance Matrices in Meta-Analytic Structural Equation Modeling. that @AJKOER gave are pretty hard to extract useful informations from these papers.

EDIT 2: @AJKOER : thanks for your help. Your idea is a good beginning in my research to carry out a full cross-correlation between my 2 Fisher matrixes, at least for the moment rather an approximation. I say that since in a previous study, I have used antoher method which is also an approximation of an exact formulation of cross-correlation (that will be pretty hard if I have understood your opinion).

For example, with your trick, I can get constraints closed to another method that I have test, so we are on the right way.

This old known test gives for 1 sigma error the following constraints :

wm +/- 0.0012417036832725796
wb +/- 0.0005995521931530912
w0 +/- 0.020152731097000408
wa +/- 0.07473741589196892
h +/- 0.001003370899834334
ns +/- 0.0018175790165196942
s8 +/- 0.0010147066130214034

and with the method of average eigen values into diagonal Fisher matrix1 matrix2, I get :

wm +/- 0.002003104719056934
wb +/- 0.0006792309014032004
w0 +/- 0.023872510159754036
wa +/- 0.08488679494420406
h +/- 0.0008331587961258862
ns +/- 0.002039401547697202
s8 +/- 0.0019000745102872813

Except for the first and last parameters (wm and s8 parameters), as I already said, contraints are closed. I have also a criterion of precision (determinant of the sub-block (w_0,w_a)) which is pretty similar.

But I think I can do better, what are your suggestions ?

A colleague told me to apply a prior but How could I proceed ?

The method of average on eigen values is already a "prior method", isn't it ?

By the way, what does this average from a Fisher information point of view or physics point of view mean ?

I have to keep on studying the papers given by @AJKOER, I have difficulties to assimilate them and extract a more rigorous method !

EDIT 3: If I perform the average on each Fisher diagonal matrix, so I get better contraints when I get the final Fisher matrix :

$F_{\text{final}}=P_1.D.P_1^{-1} + P_2.D.P_2^{-1}$

with D the diagonal matrix built by : $D=(D_1+D_2)/2$

To avoid confusion with the figure below :

$$M_1=P_1.D.P_1^{-1}$$ that represents GCsp

$$M_2=P_2.D.P_2^{-1}$$ that represents GCph+WL+XC

If I do the same the same method (average on variance) by summing directly the covariance matrices, constraints are very bad (even worse that the best contsraints given by one of two initial covariance matrix).

Is my approach correct from a Fisher's formalism point of view ?

Here below the figure where I plot the FoM (figure of Merit : the higher it is, better constraints are : FoM = 1./np.sqrt(np.det(cov[2:4,2:4])) as a function of contribution $\alpha$ of the first Fisher matrix (the normal case should be, from my opinion, $\alpha=0.5$ corresponding to the average but I have difficulties to quantify another value). I expect roughly a FoM ~ 1400-2000 (from other methods of colleagues).

For this, I have covered all the possible contribution by making vary the "alpha" quantity comprised between 0 and 1 and that represents this relative contribution between the 2 diagonal matrices.

In python, this corresponds to :

for alpha in np.arange(0,1,0.01):
  FISH_eigen_sp_flat = alpha*np.linalg.inv(np.diag(eigen_sp_flat)) + (1-alpha)*np.linalg.inv(np.diag(eigen_xc_flat))
  FISH_eigen_xc_flat = FISH_eigen_sp_flat

Figure of merit with pooled Fisher matrices

How to quantify this weight $\alpha$ of each Fisher diagonal matrices ? On which criterion could I decide a specific value ?

SUGGESTIONS :

  • I could assess this weight $\alpha$ by the ratio between Figure of Merit of the 2 initial Fisher's matrix and take it to $\alpha=FoM_1/FoM_2$ ? This would be a good indicator on the weight on each of the Likelihood associated to each Fisher matrix to combine ?

  • I could simply assess this weight by the ratio of each determinant of the 2 matrices : but the deviation between both is very high, making the ratio very small.

  • I think also for example this weight could be parametrized by others conditions, but I have not enough culture in this field.

IMPORTANT REMARK :

Just a further remark: if I take a new diagonal matrix D = (D1 + D2) / 2, then my Fisher matrix (which is supposed to be cross-correlated) can be written:

$M=P_1.(D_1+D_2)/2.P_1^{-1}+P_2.(D_1+D_2)/2.P_2^{-1}$

$= P_1.D_1/2.P_1^{-1}+P_1.D_1/2.P_1^{-1}+P_1.D_2/2.P_1^{-1}+P_2.D_1/2.P_2^{-1}$

$= M_1/2+M_2/2+P_1.D_2/2.P_1^{-1}+P_2.D_1/2.P_2^{-1}$

We would therefore provide half of the information from probe 1 (M1) and probe 2 (M2) + the other 2 terms supposed to represent the contribution of cross-correlations.

What do you think about ?

Any help is welcome.

PS: I give the links of the 2 Fishers matrices to combine (I mean to cross- combine)

Fisher GCsp Fisher GCph+WL+XC

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