STATISTICAL DATA FUSION WITH DENSITY RATIO MODEL AND EXTENSION TO RESIDUAL COHERENCE

Abstract

Nowadays, the statistical analysis of data from diverse sources has become more prevalent. The Density Ratio Model (DRM) is one of the methods for fusing and analyzing such data. The population distributions of different samples can be estimated basedon fused data, which leads to more precise estimates of the probability distributions. These probability distributions are related by assuming the ratios of their probability density functions (PDFs) follow a parametric form. In the previous works, this parametric form is assumed to be uniform for all ratios. In Chapter 1, an extension is made to allow this parametric form to vary for different ratios. Two methods of determining the parametric form for each ratio are developed based on asymptotic test and Akaike Information Criterion (AIC). This extended DRM is applied to Radon concentration and Pertussis rates to demonstrate the use of this extension in univariate case and multivariate case, respectively.

The above analysis is made possible when data in each sample are independent and identically distributed (IID). However, in many cases, statistical analysis is entailed for time series in which data appear to be sequentially dependent. In Chapter 2, an extension is made for DRM to account for weakly dependent data, which allows us to investigate the structure of multiple time series on the strength of each other. It is shown that the IID assumption can be replaced by proper stationarity, mixing and moment conditions. This extended DRM is applied to the analysis of air quality data which are recorded in chronological order.

As mentioned above, DRM is suitable for the situation where we investigate a single time series based on multiple alternative ones. These time series are assumed to be mutually independent. However, in time series analysis, it is often of interest to detect linear and nonlinear dependence between different time series. In such dependent scenario, coherence is a common tool to measure the linear dependence between two time series, and residual coherence is used to detect a possible quadratic relationship. In Chapter 3, we extend the notion of residual coherence and develop statistical tests for detecting linear and nonlinear associations between time series. These tests are applied to the analysis of brain functional connectivity data.