Trouble shooting for covariance fitting in highly correlated data

We report a possible solution to the trouble that the covariance fitting fails when the data is highly correlated and the covariance matrix has small eigenvalues. As an example, we choose the data analysis of highly correlated $B_K$ data on the basis of the SU(2) staggered chiral perturbation theory. Basically, the essence of the problem is that we do not have an accurate fitting function so that we cannot fit the highly correlated and precise data. When some eigenvalues of the covariance matrix are small, even a tiny error of fitting function can produce large chi-square and spoil the fitting procedure. We have applied a number of prescriptions available in the market such as diagonal approximation and cutoff method. In addition, we present a new method, the eigenmode shift method which fine-tunes the fitting function while keeping the covariance matrix untouched.

💡 Research Summary

The paper addresses a persistent problem in lattice QCD data analysis: standard covariance‑matrix fitting often fails when the data are highly correlated and the covariance matrix possesses very small eigenvalues. Using the calculation of the neutral kaon mixing parameter (B_K) as a concrete example, the authors demonstrate that the failure is not merely a statistical fluke but stems from a combination of (i) an imperfect fitting function derived from staggered chiral perturbation theory (SChPT) and (ii) the amplification of any residual fitting‑function error by the inverse of the smallest eigenvalues of the covariance matrix.

The authors begin by reviewing the multivariate normal model for the sample means (\bar y_i) and the associated chi‑square‑like statistic
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