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Register a new userRegression analysis with missing data and unknown colored noise: application to the MICROSCOPE space mission
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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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After performing highly sensitive acceleration measurements during two years of drag-free flight around the Earth, MICROSCOPE provided the best constraint on the Weak Equivalence Principle (WEP) to date. Beside being a technological challenge, this experiment required a specialised data analysis pipeline to look for a potential small signal buried in the noise, possibly plagued by instrumental defects, missing data and glitches. This paper describes the frequency-domain iterative least-square technique that we developed for MICROSCOPE. In particular, using numerical simulations, we prove that our estimator is unbiased and provides correct error bars. This paper therefore justifies the robustness of the WEP measurements given by MICROSCOPE.
The MICROSCOPE space mission, launched on April 25, 2016, aims to test the weak equivalence principle (WEP) with a 10^-15 precision. To reach this performance requires an accurate and robust data analysis method, especially since the possible WEP violation signal will be dominated by a strongly colored noise. An important complication is brought by the fact that some values will be missing -therefore, the measured time series will not be strictly regularly sampled. Those missing values induce a spectral leakage that significantly increases the noise in Fourier space, where the WEP violation signal is looked for, thereby complicating scientific returns. Recently, we developed an inpainting algorithm to correct the MICROSCOPE data for missing values. This code has been integrated in the official MICROSCOPE data processing pipeline because it enables us to significantly measure an equivalence principle violation (EPV) signal in a model-independent way, in the inertial satellite configuration. In this work, we present several improvements to the method that may allow us now to reach the MICROSCOPE requirements for both inertial and spin satellite configurations. The main improvement has been obtained using a prior on the power spectrum of the colored-noise that can be directly derived from the incomplete data. We show that after reconstructing missing values with this new algorithm, a least-squares fit may allow us to significantly measure an EPV signal with a 0.96x10^-15 precision in the inertial mode and 1.2x10^-15 precision in the spin mode. Although, the inpainting method presented in this paper has been optimized to the MICROSCOPE data, it remains sufficiently general to be used in the general context of missing data in time series dominated by an unknown colored-noise. The improved inpainting software, called ICON, is freely available at http://www.cosmostat.org/software/icon.
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 instruments 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 {it inpainting} algorithm to mock MICROSCOPE data; {it 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 {it 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 {it inpainting}, a Least-Square fit may allow us to significantly measure a $1.1times10^{-15}$ Equivalence Principle violation signal, which is sufficiently close to the MICROSCOPE requirements to implement {it inpainting} in the official MICROSCOPE data processing and analysis pipeline. Together with the previously published KARMA method, {it inpainting} will then allow us to independently characterize and cross-check an Equivalence Principle violation signal detection down to the $10^{-15}$ level.
The MICROSCOPE mission aimed to test the Weak Equivalence Principle (WEP) to a precision of $10^{-15}$. The WEP states that two bodies fall at the same rate on a gravitational field independently of their mass or composition. In MICROSCOPE, two masses of different compositions (titanium and platinum alloys) are placed on a quasi-circular trajectory around the Earth. They are the test-masses of a double accelerometer. The measurement of their accelerations is used to extract a potential WEP violation that would occur at a frequency defined by the motion and attitude of the satellite around the Earth. This paper details the major drivers of the mission leading to the specification of the major subsystems (satellite, ground segment, instrument, orbit...). Building upon the measurement equation, we derive the objective of the test in statistical and systematic error allocation and provide the missions expected error budget.
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