Bayesian Approach: An Alternative to Periodogram and Time Axes Estimation for Known and Unknown White Noise

Abstract

This study describes the Bayesian approach as an alternative approach for estimating time axes parameters and the periodogram (power spectrum) associated with sinusoidal model when the white noise (sigma) is known or unknown. The conventional method of estimating the time axes parameters and the periodogram has been via the Schuster method that relies solely on Maximum Likelihood Estimation (MLE). The Bayesian alternative approach proposed in this work, on the other hand, adopted the Maximum a Posteriori (MAP) via the Markov Chain Monte Carlo (MCMC) in order to checkmate the problem of re-parameterization and overparameterization associated with MLE in the conventional practice. The rates of heartbeat variability at exactly an hour and two hours after birth of one thousand eight hundred (1800) newly born babies in a state hospital were recorded and subjected to both the Bayesian approach and Schuster approach for inferences. The periodogram estimates, exactly an hour and two hours of after birth, were estimated to be 0.7395 and 0.7549, respectively -and it was deduced that rates of heartbeat (frequency) variability moderated and stabilized the pulse among the babies after two hours of birth. In addition, MAP mean estimates of the parameters approximately equals to the true mean of estimates when round up to curb the problem of re-parameterization and over-parameterization that do affect Schuster method via MLE.Index Terms: Bayesian, Maximum A Posteriori (MAP), Markov Chain Monte Carlo (MCMC), Maximum Likelihood Estimation (MLE), and Periodograms.

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