An Overview of Composite Standard Elastic-Net Distribution Based on Complex Wavelet Coefficients

Abstract

Statistical tools have changed significantly in the past decade; when the maximum likelihood method is usually applied, it provides an inaccurate solution due to its unsuitable properties and causes problems in fit. However, the current Ordinary Least Square (OLS) Model is more reliable in specific situations where the estimation is based on the slope only. On the other hand, some methods depend on the slope, and the intercept is recommended. These methods can be described as thresholding rules. Another often preferred method is the posterior mean (PM) technique. This procedure depends on two parts, the first part is the likelihood, and the other is the prior distribution, where the second part plays a significant role in the estimation. In this article, the standard elastic-net distribution is assumed as the prior part, which consists of two parts, the first being the normal distribution and the second being the double exponential distribution. The reason why this model is used is that wavelet tools have different levels of resolution. Thus, this model may provide a more accurate estimation for the wavelet coefficients, which might be estimated using a normal or double exponential distribution. In the past, some properties of elastic-net penalized were introduced and discussed. However, more properties are introduced for this distribution. In addition, two models based on the elastic-net method are demonstrated, involving the point mass prior. The first model combines normal as likelihood, and elastic-net distributions as prior, while the other combines the double exponential distribution as likelihood with the elastic-net distribution as prior. Moreover, the level-dependent components are estimated at each resolutions. A simulated investigation is studied using the Markov Chain Monte Carlo (MCMC) tool to estimate the underlying features, where real data are involved and modelled using the proposed methods. A stationary wavelet basis is also applied. As a result, the proposed procedure reduces noise levels which may be helpful since noise levels often corrupt real data, usually a significant cause of most numerical estimation problems.