Gaussian regression and power spectral density estimation with missing data: The MICROSCOPE space mission as a case study

TL;DR

This paper introduces a Gaussian regression method for time series with missing data and unknown PSD, demonstrated on the MICROSCOPE space mission to test the weak equivalence principle with high precision.

Contribution

The paper develops a novel iterative Gaussian regression algorithm that accurately estimates missing data and PSD in stationary time series, applicable to space mission data analysis.

Findings

Method maintains high-precision WEP testing despite data gaps

Provides reliable noise and PSD estimates

Produces consistent reconstructed data for scientific use

Abstract

We present a Gaussian regression method for time series with missing data and stationary residuals of unknown power spectral density (PSD). The missing data are efficiently estimated by their conditional expectation as in universal Kriging, based on the circulant approximation of the complete data covariance. After initialization with an autoregessive fit of the noise, a few iterations of estimation/reconstruction steps are performed until convergence of the regression and PSD estimates, in a way similar to the expectation-conditional-maximization algorithm. The estimation can be performed for an arbitrary PSD provided that it is sufficiently smooth. The algorithm is developed in the framework of the MICROSCOPE space mission whose goal is to test the weak equivalence principle (WEP) with a precision of $10^{-15}$. We show by numerical simulations that the developed method allows us to meet three major requirements: to maintain the targeted precision of the WEP test in spite of the loss of data, to calculate a reliable estimate of this precision and of the noise level, and finally to provide consistent and faithful reconstructed data to the scientific community.