Wavelet-Variance-Based Estimation for Composite Stochastic Processes

Abstract

This article presents a new estimation method for the parameters of a time series model. We consider here composite Gaussian processes that are the sum of independent Gaussian processes which, in turn, explain an important aspect of the time series, as is the case in engineering and natural sciences. The proposed estimation method offers an alternative to classical estimation based on the likelihood, that is straightforward to implement and often the only feasible estimation method with complex models. The estimator furnishes results as the optimization of a criterion based on a standardized distance between the sample wavelet variances (WV) estimates and the model-based WV. Indeed, the WV provides a decomposition of the variance process through different scales, so that they contain the information about different features of the stochastic model. We derive the asymptotic properties of the proposed estimator for inference and perform a simulation study to compare our estimator to the MLE and the LSE with different models. We also set sufficient conditions on composite models for our estimator to be consistent, that are easy to verify. We use the new estimator to estimate the stochastic error's parameters of the sum of three first order Gauss-Markov processes by means of a sample of over 800,000 issued from gyroscopes that compose inertial navigation systems. Supplementary materials for this article are available online.

Keywords: Allan variance; Kalman filter; Signal processing; Time series.

Figure 1.

Relative RMSE based on 1000 simulations from AR(1) processes with ρ ranging from 0.8 to 1 and with residual variance σ² = 4, of the LSE relative to the GMWM estimator of ρ (“□”) and of σ² (“o”).

Figure 2.

Comparison between a reference trajectory (black dotted line) issued from a mapping flight in which a GPS outage was introduced, with estimated error model based on the AV (light-gray line), the KF-Self-Tunning (dark gray line), and the GMWM (black dashed line).

Figure 3.

Gyroscope-observed error process (top panel) and graphical comparison (log-log scale) between the Haar WV (line “o”) computed from the observed signal and the analytical signal using the estimated parameters of, respectively, the sum of three GM processes (line “▴”), the sum of two GM processes (line “•”), and one GM process (line “▪”).