Abstract
The paper shows how the linear regression depends on the selection of the reference frame. The slope of the fitted line and the corresponding Pearson’s correlation coefficient are expressed in terms of the rotation angle. The correlation coefficient is found to be maximized for a certain optimal angle, for which the slope attains a special optimal value. The optimal angle, the value of the optimal slope, and the corresponding maximum correlation coefficient were expressed in terms of the covariance matrix, but also in terms of the values of the slope, derived from the fitting at the nonrotated and right-angle-rotated axes. The potential of the new method is to improve the derived values of the fitting parameters by detecting the optimal rotation angle, that is, the one that maximizes the correlation coefficient. The presented analysis was applied to the linear regression of density and temperature measurements characterizing the proton plasma in the inner heliosheath, the outer region of our heliosphere.
Highlights
I=1 of a statistical model V (X; α) in the i=1 domain X ∈ Dx ⊆ < [1,2,3,4,5] involves finding the optimal parameter value α = α∗ in α ∈ Dα ⊆ < that minimizes the sum of squared residuals, known in physical sciences as total square deviations (TSD) between model and data, TSD(α)2 =
We focus on the optimal fitting value of the slope β, derived from the linear regression of data expressed on the rotated Cartesian axes
While most of the space plasmas exhibit positive correlations between density and temperature, there are several plasmas with negative correlations, consistent with constant or quasi-constant thermal pressure, that were found in heliosheath [10,17,18,19,20,21] and planetary magnetosheaths
Summary
We focus on the optimal fitting value of the slope β, derived from the linear regression of data expressed on the rotated Cartesian axes. The presented analysis was applied to the linear regression of density and temperature measurements characterizing the proton plasma in the inner heliosheath, the outer region of our heliosphere. Those measurements can be fitted with a line of negative slope, which gives the polytropic index of the plasma. The new statistical method improves the applied linear regression and the estimated value of the polytropic index
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