# Forecast in cosmology survey : loss or gain of information with cross-correlations

In the context of a forecast in cosmology, I am currently working on Fisher's formalism which is part of a more general theory, that of information. My problem applies to estimating cosmological parameters from input data with the Fisher formalism and recipes to build a Fisher matrix. The context is Astrophysics but it can apply to many subjects.

Here is a summary: First I generate with CAMB code matter power spetrum that allow me to compute observable power spectrum. This way, I can build my Fisher matrix with others following input data. Then, the other input data are 4 columns of data, the first representing the redshift of the galaxies (i.e their distance) and then the other 3 each corresponding to the bias (ie roughly to the uncertainty about their position) of a given galaxy type: there are therefore 3 types of given galaxies and one value per redshift (8 redshifts in all). So I have a table of 8x3 values of bias.

1) First part : Now, I'm trying to cross data to try to extract additional information because for example, for the first type of galaxy, I have only the first 2 biases that are non-zero (I mean for First 2 redshifts), and for the third type, I have 6 different values ​​of 0 for the 6 redshifts above the 2 previous ones.

Here the file of biases for the 3 populations (b1,b2,b3) as a function of redshift (first column) :

# z                    b1                b2              b3
1.7500000000e-01 1.1133849956e+00 0.0000000000e+00 0.0000000000e+00
4.2500000000e-01 1.7983127401e+00 0.0000000000e+00 0.0000000000e+00
6.5000000000e-01 0.0000000000e+00 1.4469899900e+00 7.1498329000e-01
8.5000000000e-01 0.0000000000e+00 1.4194157200e+00 7.0135835000e-01
1.0500000000e+00 0.0000000000e+00 1.4006739400e+00 6.9209771000e-01
1.2500000000e+00 0.0000000000e+00 0.0000000000e+00 6.8562140000e-01
1.4500000000e+00 0.0000000000e+00 0.0000000000e+00 6.8097541000e-01
1.6500000000e+00 0.0000000000e+00 0.0000000000e+00 6.7756594000e-01


My teacher suggested to me to merge the first column (corresponding to the first type of galaxy) with the third one (corresponding to the third type of population of galaxies), so as to obtain a single vector with only values ​​for the non-zero bias). This way, I simulate a "single population" processing since one wants to avoid including zero values.

From a statistical point of view (maybe on entropy level or quantity of information level), will there be a loss or a gain of information if I do this fusion of the 2 columns ? The problem seems rather complex because everything depends on the value of the data.

2) Second part : Another point of view suggested by my teacher: if I take a sample and I cut it in 2 parts, if I cross-correlate data between the 2 subsets obtained, will I win or lose information from a statisticl point of view, i.e at the level of the accuracy of parameters that I will extract from the cross-correlation between the 2 subsets. (in my case, i.e galaxies biaises and Fisher formalism, I mean the constraints that I get after having built my Fisher matrix and invert it) ?

He thinks that at first sight, I can not lose information (which seems intuitive because cutting a sample in 2 is not a loss of info per se) but he says that everything depends on whether I know or not precisely the ratio of the biases between the 2 subsamples : I did not quite understand this notion of ratio between theses 2 biases of subsets.

I am therefore looking for information on this problem, maybe on this forum, statisticians will be able to help me in this technique of cross-correlations and the fact of knowing or not if one gains or one loses some information by bringing together several sources of information.

I think that the gain or the loss of info will be a function of the redundancy of the data (we speak of entropy of Shannon I think, don't we ?).

3) Third part : I could also cross data between overlapped data for 2 columns of data (2 values ​​for each redshift) but here I think it's another problem from a statistical point of view: by the way, I'm talking about at the beginning of data crossing with the merging of 2 vectors but the "cross-correlation" is rather defined in the case of overlapped values, right?

However, in both cases, we cross data, in some way.

For the moment, in my algorithm, I process the first 2 values ​​of the 1st type of population, the 3 others overlapped between the second and the 3rd type, and the last 3 of the 3rd type of population, which is 8 values at total (I mean 8 redshifts): so there are 2 "auto-spectrum" and 1 overlapped spectrum.

My measure of the information gain that I have been talking about since the beginning of this post is done with the computation of constraints by inverting the Fisher matrix, which gives me the covariance matrix and therefore the variance and the correlation of the parameters that I want to estimate: the smaller the standard deviations, the higher the information gain. That's how this forecast is carried out.

Your advices or suggestions on the issue are precious and will help me better understand the logic of this "data cross-correlation" matter on the 3 points I have cited.

Any help is welcome.

Regards

• Anyone could give me some clues or idea to see clearer the problematic ? Thanks
– user87745
Nov 20, 2019 at 17:11
• Something I did not get, if the bias correspond to one galaxy type, why did you have galaxy with two types and therefore two bias? Nov 21, 2019 at 15:47
• Could you be more explicit about what you call "cross data"? Nov 21, 2019 at 16:12
• So you have on dataset per redshift and galaxy type (except those without value), and you extract information from those dataset and now you want to extract information from the combined set of dataset, right? Nov 21, 2019 at 16:26
• Nov 21, 2019 at 16:43

At least for the first question, if I understand correctly what you are doing, the data you have are actually $$b(z,i)$$ so the biais as a function of the redshift $$z$$ and the type of the galaxy $$i$$. Therefore the the fact that the bias is zero is then an information and removing this value from the dataset will remove information on your system because you will lose information about the underlying type of galaxy. However as a zero value of a bias does not seems physical, I guess that indicate the absence of the type of galaxy at this redshift.