5.5. CR data with covariance¶
With USINE, it is possible to use a covariance matrix of errors on selected CR data.
5.5.1. Context¶
| Error types in publications: | |||||||||||||||||||||||||||||||||||||||||||
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For past measurements, at best, the overall uncertainties (statistical and systematics combined) were provided. In more recent data, at least statistical and systematic uncertainties are provided separately (format of data gathered in CRDB). Recent experiments (e.g., AMS02) go further and provide also systematics broken down to various contributions (see, e.g., Table 5.1).
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| What about the covariance: | |||||||||||||||||||||||||||||||||||||||||||
However, this may not be sufficient for the model analysis of high-precision data. The next step is the information encoded in the covariance matrix, which gives the error correlations between different energies for all types of errors in the instrument. |
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5.5.2. Covariance definition and \(\chi^2\)¶
Given data at \(n_E\) energies, the covariance matrix is a symmetric \(n_E \times n_E\) matrix containing the covariance of all pairs of data point (see more, e.g., in wikipedia). For \(n_\alpha\) different types of systematic uncertainties, we have
\({\rm Cov}_{ij}^\alpha = \rho_{ij}^\alpha \times \sigma_i^\alpha \times \sigma_j^\alpha\),
so that
\(\chi^2_{\rm cov} = \sum_{i,j \in ={0\dots n_E}} ({\rm data}_i-{\rm model}_i) \left(\sum_{\alpha}{\rm Cov}^{\alpha}\right)^{-1}_{ij} ({\rm data}_j-{\rm model}_j)\),
with
- \(\alpha\in\{{\rm STAT}, {\rm BACK}\dots\}\) for data in Table 5.1, with \(i\) and \(j \in \{R_1, R_2,\dots R_n\}\)
- \(\sigma_i^\alpha\) the relative uncertainty (for error type \(\alpha\) at rigidity \(R_i\))
- \(\rho_{ij}\) the correlation coefficient (e.g., 0=no correl., 1=full correl.)
Note
\({\rm Cov}_{ij}^\alpha\) coefficients are symmetric (Gaussian errors), with \(\rho_{ii}^\alpha=1\) (by construction), and \(\rho_{i\neq j}^{\rm STAT}=0\) (by definition).
5.5.3. Covariance matrix from relative errors¶
Actually, USINE relies on the covariance matrix of relative errors \({\cal C}_{\rm rel}^\alpha\), and as discussed in Derome et al. (2009), this matrix be related to the covariance \({\cal C}_\alpha\) required for the above equation in two different ways:
\(({\cal C}_{\rm model}^\alpha)_{ij} = ({\cal C}^\alpha_{\rm rel})_{ij} \times y^{\rm model}_i \times y^{\rm model}_j\quad\quad\) or \(\quad\quad({\cal C}_{\rm data}^\alpha)_{ij} = ({\cal C}^\alpha_{\rm rel})_{ij} \times y^{\rm data}_i \times y^{\rm data}_j\)
The initialisation keyword IsModelOrDataForRelCov allows to select one of the two options (see initialisation parameter in UsineFit@TOAData). In particular, for the case of a global normalisation factor in the data (or equivalently, an infinite correlation length for the systematics), \({\cal C}_{\rm model}\) should be preferred over \({\cal C}_{\rm data}\) not to bias the reconstructed model parameters.
5.5.4. Relative covariance file format¶
In USINE, a file for the covariance matrix (of data relative errors) is uniquely associated to a measured quantity from an experiment for its data taking period. The user must provide a covariance matrix for each measured quantity selected as such in the minimisation procedure (see the initialisation parameter UsineFit@TOAData).
Shown below is an excerpt of the file inputs/CRDATA_COVARIANCE/cov_AMS02_201105201605__BC_R.dat, a toy-model for the covariance of AMS02 B/C(R) data for the data taking period 2011/05-2016/05 (courtesy of L. Derome).
##########################################################################
# N.B.: line starting with # is omitted
#
# Content:
# --------
# Covariance matrices associated to Exp_Qty_Etype given in file name.
# cov_ij^alpha = rho_ij^alpha * sigma_i^alpha * sigma_j^alpha
# with
# alpha: Types of errors for which we have a covariance
# i,j: Energy points (as many as data points nE)
# sigma_i^alpha: Relative uncertainty at E_i for error alpha
# rho_ij^alpha: Correlation coeff. (e.g., 0/1 = not/fully correlated)
#
# Note that:
# - covariance matrices are symmetric by construction (Gaussian errors)
# - rho_ii=1 by construction => sigma_i can be recovered from diagonal
#
#
# File format:
# ------------
# ErrorType: name_alpha1
# cov_11_alpha1 cov_12_alpha1 ... cov_1nE_alpha1
# cov_21_alpha1 cov_22_alpha1 ... cov_2nE_alpha1
# ... ... ... ...
# cov_nE1_alpha1 cov_nE12_alpha1 ... cov_nEnE_alpha1
# ###
# .
# .
# .
# ErrorType: name_alphaN
# cov_11_alphaN cov_12_alphaN ... cov_1nE_alphaN
# cov_21_alphaN cov_22_alphaN ... cov_2nE_alphaN
# ... ... ... ...
# cov_nE1_alphaN cov_nE1_alphaN ... cov_nEnE_alphaN
##########################################################################
####
ErrorType: stat
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