On Functional Data Analysis: Methodologies and Applications

In economic analyses, the variables of interest are often functions defined on continua such as time or space, though we may only have access to discrete observations -- such type of variables are said to be ``functional'' (Ramsay, 1982). Traditional economic analyses model discrete observations using discrete methods, which can cause misspecification when the data are driven by functional underlying processes and further lead to inconsistent estimation and invalid inference. This thesis contains three chapters on functional data analysis (FDA), which concerns data that are functional in nature. As a nonparametric method accommodating functional data of different levels of smoothness, not only does FDA recover the functional underlying processes from discrete observations without misspecification, it also allows for analyses of derivatives of the functional data. Specifically, Chapter 1 provides an application of FDA in examining the distribution equality of GDP functions across different versions of the Penn World Tables (PWT). Through our bootstrap-based hypothesis test and applying the properties of the derivatives of functional data, we find no support for the distribution equality hypothesis, indicating that GDP functions in different versions do not share a common underlying distribution. This result suggests a need to use caution in drawing conclusions from a particular PWT version, and conduct appropriate sensitivity analyses to check the robustness of results. In Chapter 2, we utilize a FDA approach to generalize dynamic factor models. The newly proposed generalized functional dynamic factor model adopts two-dimensional loading functions to accommodate possible instability of the loadings and lag effects of the factors nonparametrically. Large sample theories and simulation results are provided. We also present an application of our model using a widely used macroeconomic data set. In Chapter 3, I consider a functional linear regression model with a forward-in-time-only causality from functional predictors onto a functional response. In this chapter, (i) a uniform convergence rate of the estimated functional coefficients is derived depending on the degree of cross-sectional dependence; (ii) asymptotic normality of the estimated coefficients can be obtained under proper conditions, with unknown forms of cross-sectional dependence; (iii) a bootstrap method is proposed for approximating the distribution of the estimated functional coefficients. A simulation analysis is provided to illustrate the estimation and bootstrap procedures and to demonstrate the properties of the estimators

Variational Bayesian analysis of survival data using a log-logistic accelerated failure time model

The log-logistic regression model is one of the most commonly used accelerated failure time (AFT) models in survival analysis, for which statistical inference methods are mainly established under the frequentist framework. Recently, Bayesian inference for log-logistic AFT models using Markov chain Monte Carlo (MCMC) techniques has also been widely developed. In this work, we develop an alternative approach to MCMC methods and infer the parameters of the log-logistic AFT model via a mean-field variational Bayes (VB) algorithm. A piecewise approximation technique is embedded in deriving the VB algorithm to achieve conjugacy. The proposed VB algorithm is evaluated and compared with typical frequentist inferences and MCMC inference using simulated data under various scenarios. A publicly available dataset is employed for illustration. We demonstrate that the proposed VB algorithm can achieve good estimation accuracy and has a lower computational cost compared with MCMC methods

Hydrothermal Dolomite Paleokarst Reservoir Development in Wolonghe Gasfield, Sichuan Basin, Revealed by Seismic Characterization

Hydrothermal dolomite paleokarst reservoir is a type of porous carbonate reservoir, which has a secondary porosity and can store a large amount of oil and gas underground. The reservoir is formed by magnesium-rich hydrothermal fluids during the karstification and later stages of the transformation. Due to the strong heterogeneity and thin thickness of hydrothermal dolomite paleokarst reservoirs, it is a real challenge to characterize the spatial distribution of the reservoirs. In this paper, we studied the hydrothermal dolomite paleokarst reservoir in the Wolonghe gasfield of the eastern Sichuan Basin. First, based on detailed observations of core samples, the characteristics and storage space types of the dolomite reservoir were described. Secondly, the petrophysical parameters of the paleokarst reservoirs were analyzed, and then the indicator factor for the dolomite reservoirs was established. Thirdly, using the time–depth conversion method, the geological characteristics near boreholes were connected with a three-dimensional (3D) seismic dataset. Several petrophysical parameters were predicted by prestack synchronous inversion technology, including the P-wave velocity, S-wave velocity, P-wave impedance, and the hydrothermal dolomite paleokarst reservoir indicator factor. Finally, the hydrothermal dolomite paleokarst reservoirs were quantitatively predicted, and their distribution model was built. The 3D geophysical characterization approach improves our understanding of hydrothermal dolomite paleokarst reservoirs, and can also be applied to other similar heterogeneous reservoirs