Robust Frequency Estimation Using Elemental Sets

The extraction of sinusoidal signals from time-series data is a classic problem of ongoing interest in the statistics and signal processing literatures. Obtaining least squares estimates is difficult because the sum of squares has local minima O(1/n) apart in the frequencies. In practice the frequencies are often estimated using ad hoc and inefficient methods. Problems of data quality have received little attention. An elemental set is a subset of the data containing the minimum number of points such that the unknown parameters in the model can be identified. This article shows that, using a variant of the classical method of Prony, parameter estimates for a sum of sinusoids can be obtained algebraically from an elemental set. Elemental set methods are used to construct finite algorithm estimators that approximately minimize the least squares, least trimmed sum of squares, or least median of squares criteria. The elemental set estimators prove able in simulations to resolve the frequencies to the correct local minima of the objective functions. When used as the first stage of an MM estimator, the constructed estimators based on the trimmed sum of squares and least median of squares criteria produce final estimators which have high breakdown properties and which are simultaneously efficient when no outliers are present. The approach can also be applied to sums of exponentials, and sums of damped sinusoids. The article includes simulations with one and two sinusoids and two data examples.