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The analysis of physical measurements often copes with highly correlated noises and interruptions caused by outliers, saturation events or transmission losses. We assess the impact of missing data on the performance of linear regression analysis involving the fit of modeled or measured time series. We show that data gaps can significantly alter the precision of the regression parameter estimation in the presence of colored noise, due to the frequency leakage of the noise power. We present a regression method which cancels this effect and estimates the parameters of interest with a precision comparable to the complete data case, even if the noise power spectral density (PSD) is not known a priori. The method is based on an autoregressive (AR) fit of the noise, which allows us to build an approximate generalized least squares estimator approaching the minimal variance bound. The method, which can be applied to any similar data processing, is tested on simulated measurements of the MICROSCOPE space mission, whose goal is to test the Weak Equivalence Principle (WEP) with a precision of 10 −15 . In this particular context the signal of interest is the WEP violation signal expected to be found around a well defined frequency. We test our method with different gap patterns and noise of known PSD and find that the results agree with the mission requirements, decreasing the uncertainty by a factor 60 with respect to ordinary least squares methods. We show that it also provides a test of significance to assess the uncertainty of the measurement.

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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.

Missing data are a common problem in experimental and observational physics. They can be caused by various sources, either an instrument's saturation, or a contamination from an external event, or a data loss. In particular, they can have a disastrous effect when one is seeking to characterize a colored-noise-dominated signal in Fourier space, since they create a spectral leakage that can artificially increase the noise. It is therefore important to either take them into account or to correct for them prior to e.g. a Least-Square fit of the signal to be characterized. In this paper, we present an application of the inpainting algorithm to mock MICROSCOPE data; inpainting is based on a sparsity assumption, and has already been used in various astrophysical contexts; MICROSCOPE is a French Space Agency mission, whose launch is expected in 2016, that aims to test the Weak Equivalence Principle down to the 10 −15 level. We then explore the inpainting dependence on the number of gaps and the total fraction of missing values. We show that, in a worst-case scenario, after reconstructing missing values with inpainting, a Least-Square fit may allow us to significantly measure a 1.1 × 10 −15 Equivalence Principle violation signal, which is sufficiently close to the MICROSCOPE requirements to implement inpainting in the official MICROSCOPE data processing and analysis pipeline. Together with the previously published KARMA method, inpainting will then allow us to independently characterize and cross-check an Equivalence Principle violation signal detection down to the 10 −15 level.