The reconstruction of dark energy with Ridge Regression Approach

It may be determined by non-parametric method if the dark energy evolves with time. We propose a method of combining PCA and biased estimation on the basis of ridge regression analysis to reconstruct parameters, meanwhile we present an interesting principal component selection criterion to avoid the arbitrariness of principal component selections, and use numerical integral by Lagrange interpolation to linearize the luminosity distance integral formula in nearly flat space to avoid instability of derivative for functional data. We get the preliminary test results that shows if $\Delta \overline w (z) = \overline {\left| {1 + w(z)} \right|} <=0.05$ included $w(z) = -1$, the probability of making a type I error for ${w_{recon}} \ne -1$ is almost zero ($1\%$) in the test; otherwise, if $\Delta \overline w (z) = \overline {\left| {1 + w(z)} \right|} > 0.05$, the probability of making a type I error for ${w_{recon}} = -1$ is not more than $10\%$. Finally, we use JLA sample to reconstruct $w(z)$, and the results reject ${\rm{w(z) }} \ne {\rm{ - 1}}$, which is agreement with $\Lambda CDM$ model.