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Nonparametric estimation of a periodic function
 Biometrika
, 2000
"... ABSTRACT. Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. It is shown that nonparametric estimators of pe ..."
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ABSTRACT. Motivated by applications to brightness data on periodic variable stars, we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times. It is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known. Also, firstorder properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known. Numerical simulations and applications to real data show the method to work well in practice. KEY WORDS AND PHRASES. frequency estimation, nonparametric regression, semiparametric estimation, NadarayaWatson estimator, MACHO project, variable star data. SHORT TITLE. Estimation of a periodic function
Robust Frequency Estimation Using Elemental Sets ∗
"... The extraction of sinusoidal signals from timeseries 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 frequen ..."
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Cited by 4 (1 self)
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The extraction of sinusoidal signals from timeseries 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 paper 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 which 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 paper includes simulations with one and two sinusoids and two data examples.
Nonparametric Bayesian Mixedeffects Models for Multitask Learning
, 2013
"... In many real world problems we are interested in learning multiple tasks while the training set for each task is quite small. When the different tasks are related, one can learn all tasks simultaneously and aim to get improved predictive performance by taking advantage of the common aspects of all t ..."
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In many real world problems we are interested in learning multiple tasks while the training set for each task is quite small. When the different tasks are related, one can learn all tasks simultaneously and aim to get improved predictive performance by taking advantage of the common aspects of all tasks. This general idea is known as multitask learning and it has been successfully investigated in several technical settings, with applications in many areas. In this thesis we explore a Bayesian realization of this idea especially using Gaussian Processes (GP) where sharing the prior and its parameters among the tasks can be seen to implement multitask learning. Our focus is on the functional mixedeffects model. More specifically, we propose a family of novel Nonparametric Bayesian models, Grouped mixedeffects GP models, where each individual task is given by a fixedeffect, taken from one of a set of unknown groups, plus a random individual effect function that captures variations among individuals. The proposed models provide a unified algorithmic framework to solve time series prediction, clustering and classification.
Period Analysis of Variable Stars: Temporal Dependence and Local Optima
"... A number of methods have been proposed to estimate the period of a variable star; e.g., a recent approach uses smoothing spline regression to fit tentative periodic functions (light curves) and selects the period minimizing a robust goodnessoffit criterion. These methods assume that measurement er ..."
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A number of methods have been proposed to estimate the period of a variable star; e.g., a recent approach uses smoothing spline regression to fit tentative periodic functions (light curves) and selects the period minimizing a robust goodnessoffit criterion. These methods assume that measurement errors vary independently over time. Empirical evidence, however, indicates substantial temporal dependence, possibly related to changes in observing conditions. Dependence complicates the period analysis in several respects: selection of a “best ” period among several local optima, estimation of the light curve, and evaluation of uncertainty about period and light curve estimates. This article presents methods designed to accommodate dependent errors. An analysis of several data sets shows that the proposed approach can produce substantially different and arguably better results compared with other methods.