Shrinkage priors on complex-valued circular-symmetric autoregressive processes

Abstract

We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximal likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler parameter spaces, utilizing a positive continuous eigenfunction of the Laplace-Beltrami operator with a negative eigenvalue. We present numerical experiments on a complex-valued stationary autoregressive model of order 1.√ −1 φ Z equals the distribution of Z. We focus on complex-valued circular-symmetric discrete Gaussian processes, which are defined as complex-valued processes whose finite dimensional marginal distributions are complex normal distributions. The precise definitions of a complex normal distribution and a complex-valued Gaussian process are given in Section 2.The circular-symmetry of complex normal distributions with mean 0 ∈ C is of great importance in practical applications [13,12]. Complex processes are commonly used for directional processes, such as wind, radar, and sonar signals. Also, complex-valued representations are widely used in diverse fields, such as electronics, physics, and biomedicine.