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Khalil Arya, FaridTemporal and Spatial Analysis of Water Quality Time Series
Doctor of Philosophy, University of Akron, 2015, Civil Engineering
The models that are able to appropriately study the temporal and spatial dependence structure of water quality and hydrological time series are the essential tools to evaluate the future state of water availability, pollution loading, and best watershed management options. The mass-balance model is one of the current approaches for modeling water quality. However, it has the following limitations: extensive data for inputs, limited effectiveness for many water quality parameters, and ineffectiveness for long-term forecasts. To address the above limitations, statistical and stochastic models, such as classical ARIMA, ANN, and TFN modeling approaches have also been applied to investigate water quality and hydrological time series. However, they also have the some limitations, i.e., Gaussian process and/or linear dependence. Thereby, this study proposes to investigate the water quality and hydrological time series with the use of the following methodologies: (1) applying the order series transformation method to fulfill the assumptions and to address the limitations of the classic (F)AR(I)MA time series modeling approach; (2) applying the copula theory to investigate the spatial dependence pattern for water quality and hydrological time series at the different locations within the same watershed; (3) investigating the temporal dependence for the observed sequences with the use of copula-based Markov process to address the limitations existed in the classic Markov process. To valid the proposed approaches, three watersheds (i.e., Stillaguamish and Snohomish watersheds in Washington: Forest watershed, Chattahoochee River Watershed in Georgia: Urban Watershed, and Cuyahoga River Watershed in Ohio: watershed with mixed LULC) are selected as case-studies. The findings of this study showed: (1) the order series transformation may successfully transform the heavy-skewed and/or fat-tailed univariate time series to Gaussian process to fulfill the assumptions of (F)AR(I)models; (2) the length of records should be considered in evaluating the Hurst phenomenon, if it is existed; (3) the copula theory is an efficient tool in modeling the spatial dependence pattern of water quality and hydrological time series by relaxing the assumptions of classic multivariate analysis; (4) the copula-based Markov processes are able to successfully model the temporal dependence of the observed water quality and hydrological time series by relaxing the assumptions of classic Markov process modeling approach; (5) the estimated VaRs may provide valuable information for risk analysis and management; and (6) the proposed procedures can be adopted by other watersheds for watershed management and decision making.

Committee:

Lan Zhang, Dr (Advisor)

Subjects:

Civil Engineering; Environmental Engineering; Water Resource Management

Keywords:

Univariate time series analysis; Copula; Copula-based Markov process; Risk Analysis; Water quality

Xu, ZhiguangModeling Non-Gaussian Time-correlated Data Using Nonparametric Bayesian Method
Doctor of Philosophy, The Ohio State University, 2014, Statistics
This dissertation proposes nonparametric Bayesian methods to study a large class of non-Gaussian time-correlated data, including non-Gaussian time series and non-Gaussian longitudinal datasets. When a time series is noticeably non-Gaussian, classical methods with Gaussian innovations will yield poor fits and forecasts, but the joint distribution of a non-Gaussian time series is often difficult to specify. To overcome this difficulty, we propose the copula-transformed AR (CTAR) model. This model utilizes the copula method to determine the joint distribution of the observed series by separating the marginal distribution from the serial dependence. In implementation, we model the observed series as a nonlinear, nonparametric transformation from a latent Gaussian series. The marginal distribution of the observed series follows a nonparametric Bayesian prior distribution having large support, and therefore any non-Gaussian distribution can be well approximated. The dependence structure of the observed series is characterized indirectly through the latent Gaussian time series, so that we can borrow some classic Gaussian time series modeling methods to model the serial dependence. We also extend the proposed nonparametric Bayesian copula methods to model stationary time series with changing conditional volatility by developing copula-transformed AR-GARCH (CTAR-GARCH) model, which describes the observed series as a nonlinear, nonparametric transformation from an AR-GARCH latent series. We conduct simulations and show the CTAR and CTAR-GARCH models' advantages in capturing non-Gaussian marginal and predictive distributions. We also fit the CTAR-GARCH models to stock index return series and conclude that they yield better predictions than the classical AR-GARCH models with Gaussian innovation. We further extend our models to the non-Gaussian longitudinal analysis setting. We model an observed within-subject response series as a transformation from a latent Gaussian series. The latent series specifies the within-subject dependence structure and the transformation function specifies marginal distribution of response variable. Similar to CTAR models, a marginal distribution of the response variable has a nonparametric Bayesian prior distribution and is therefore flexible in shape. We conduct simulations and study a 100km-race real dataset where the response variable is noticeably non-Gaussian. The data analysis demonstrates the advantage of copula-transformed models' performance in model fitting and prediction compared with the Gaussian-based models when the data is truly non-Gaussian and when the mean function is correctly specified. We also study the situations where the mean function shifts in the out-of-sample data. We find that the model's predictive performance for individuals is impacted by the shifts. The copula-transformed models are more sensitive to the shift than the Gaussian-based models. We also study the predictive performance of the contrasts. The models' predictive performance remains fairly robust to the shifts, and the copula-transformed models outperform the Gaussian-based models in contrast predictions. The proposed method can be extended in many directions, including using other transformation functions (e.g., a transformation using Polya tree prior).

Committee:

Steven MacEachern (Advisor); Xinyi Xu (Advisor); Mario Peruggia (Committee Member)

Subjects:

Statistics

Keywords:

time series, longitudinal data, copula, nonparametric Bayesian method

Sucheston, Lara E.STATISTICAL METHODS FOR THE GENETIC ANALYSIS OF DEVELOPMENTAL DISORDERS
Doctor of Philosophy, Case Western Reserve University, 2007, Epidemiology and Biostatistics
This dissertation focuses on approaches to the genetic analysis of longitudinal measures of developmental disorders (DD) with specific application to a longitudinal pedigree study of children ascertained on the basis of a Speech Sound Disorder (SSD). Analysis of this longitudinal cohort is complicated by non-normal trait distributions and a potentially non-linear cognitive developmental trajectory. Prior to developing a longitudinal model I measured the power of the SSD dataset to correctly detect linkage of a quantitative trait to a genetic marker. Assuming that the function describing the genetic effect across time is correctly specified the power of the SSD data set is .18 at a .01 level of signficance. Additional data collection is planned and by doubling the sample size (from 200 to 400 sibling pairs) and number of measurement points (from 2 to 4) the power increases to .83 for the same significance level. It is therefore reasonable to develop a longitudinal approach for use at a later date. As an alternative to the longitudinal analysis, multivariate dependence functions, called copulas, are used to develop a cross-sectional model to test for polygenic*age interaction. These functions separate a multivariate joint distribution into two parts: one describing the interdependency of the probabilities (correlation), the other describing the distribution of the margins (the phenotypes). Using these functions for analysis simultaneously addresses both the non-normality problem, as the margins can be modeled with a wide variety of parametric probability distributions and the developmental trajectory question, as we incorporate age into the analysis through the use of a correlation function, the parameter estimate of which can be tested for significance using a chi-square test statistic. Four of the 13 SSD test measures showed nominal p-values less than .05. While at the broadest level the 4 tests measure different cognitive skills, short term memory plays an important role in each of these tests. This provides preliminary evidence that the genetic contribution to phenotypic variance of tasks involving memory is not stationary in children ages 6 to 18.

Committee:

Sudha Iyengar (Advisor)

Subjects:

Statistics

Keywords:

speech sound disorder; longitudinal model; gene age interaction; longitudinal copula model

Takahashi, SonokoThe Interrogative Marker KA in Japanese
Master of Arts, The Ohio State University, 1995, East Asian Languages and Literatures

Committee:

Mineharu Nakayama (Advisor)

Keywords:

polite; Non-Copula; KA; Predicates; Question Particles; desu; SENTENCES

Longla, MartialModeling dependence and limit theorems for Copula-based Markov chains
PhD, University of Cincinnati, 2013, Arts and Sciences: Mathematical Sciences
This dissertation is concerned with the notion of copula and its importance in modeling and estimation. We use the theory of copulas to assess dependence properties of stationary Markov chains and convergence to the Brownian motion. In the introductory chapter we overview the theory of copulas and their relationship with dependence coefficients for Markov chains. In Chapter 1, we investigate the rates of convergence to zero of the dependence coefficients of copula-based Markov chains. We first review some theoretical results, then improve them and propose an estimate of the maximal correlation coefficient between consecutive states. We also comment on the relationship between geometric ergodicity and exponential ρ-mixing for reversible Markov chains. We show that convex combinations of (absolutely regular) geometrically ergodic stationary reversible Markov chains are (absolutely regular) geometrically ergodic and exponential ρ-mixing. Moreover, we show that this result holds if only one of the summands is geometrically ergodic. Most of the results are based on our observation that if the absolutely continuous part of the copula has a density bounded away from 0 on a set of measure 1, then it generates absolutely regular stationary Markov chains. Many other striking results are provided on this topic in subsequent sections of Chapter 1. We also use small sets to investigate β-mixing rates for the Frechet and Mardia families of copulas. We provide new copula families with functions as parameters. We derive the copula for the general Metropolis-Hastings algorithm and use it to apply our results to this class of processes. In Chapter 2, we provide a background survey on functional central limit theorem for stationary Markov chains with a general state space. We emphasize the relationship between the dependence coefficients studied in Chapter 1 and convergence of normalized partial sums of the chain to a standard normal random variable. We present results showing that in most of the cases we consider, the invariance principle holds. In Chapter 3, we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion. The maximal inequalities are based on a new forward-backward martingale decomposition with a triangular array of differences.

Committee:

Magda Peligrad, Ph.D. (Committee Chair); Wlodzimierz Bryc, Ph.D. (Committee Member); Jeesen Chen, Ph.D. (Committee Member); Joanna Mitro, Ph.D. (Committee Member); Dan Ralescu, Ph.D. (Committee Member); Siva Sivaganesan, Ph.D. (Committee Member)

Subjects:

Mathematics

Keywords:

Markov chains;Copula and dependence coefficients;Central Limit theorems;Reversible Markov processes;Ergodicity;Invariance principle;

Gyamfi, MichaelModelling The Financial Market Using Copula
Master of Science, University of Akron, 2017, Applied Mathematics
This project is to track the differences and the movements between the Actual and theoretical future prices using Copula. Standard & Poor's 500 Index (SPX) and 10-year treasury bond yield rate was downloaded from Yahoo! website and SPX future prices were downloaded from Moore Research Centre website and their observations from January 2, 2001 to May 27, 2016 were used for this analysis. Log-returns of the future prices were taken to model and analyse the direct movements of the future prices. The distributions of the marginals and the best family of copula was selected and simulated. We compared the copula method to the classical method after 2000 simulation. A high level of mis-pricing in the future price which corresponds to the period 2008-2009 was observed. This observed mis-pricing could be as a result of relative over-reaction of the Financial market compared to future market. Inverse relationship between the performance of SPX and the volatility of future prices was observed. Standardised Student's t-distribution was concluded to be the marginal distribution using the maximum likelihood method to estimate their distribution parameters. Student t-Copula was concluded to be the best family of copula to measure the dependence. In further studies, modelling the risk associated with futures stock price and pricing with copula based simulation will be a major red flag to be addressed.

Committee:

Nao Mimoto, Dr (Advisor); Patrick Wilber, Dr (Other); Kevin Kreider, Dr (Other)

Subjects:

Applied Mathematics

Keywords:

Copula, Theoretical Future Price and Actual Future Price

Yu, LiTau-Path Test - A Nonparametric Test For Testing Unspecified Subpopulation Monotone Association
Doctor of Philosophy, The Ohio State University, 2009, Statistics
In data mining and other settings, there is sometimes a need to identify relationships between variables when the relationship may hold only over a subset of the observations available. For example, expression of a particular gene may cause resistance to an anticancer drug, but only over certain types of cancer cell-lines. It may not be known in advance which types of cancer cell-lines (e.g., estrogen-regulated, newly differentiated, central nervous system) employ such a method of resistance. This situation differs from the usual setting in which partial correlations are estimated conditional on a known selection, such as the value of another variable. For any pair of variables of interest, the goal is to test if these are associated in some unspecified subpopulation that is represented by a subsample of the data we have available. Nothing in the literature deals directly with this problem. We have tried several parametric and non-parametric approaches, and for both inferential and computational reasons have chosen to present a procedure based on a sequential development of Kendall's tau measure of monotone association. The sequence is achieved by reordering observations so that the sample tau coefficients for the first k of the n observations form a monotone decreasing path, ending at Kendall's tau coefficient. Boundaries are constructed so that 95% of the paths remain within the boundaries under the null hypothesis of independence. A boundary crossing at any point k is evidence of a stronger than expected association amongst a subpopulation represented by the k observations involved. The method is used to screen for association between gene expression and compound activity amongst types of cancer cell-lines in the NCI-60 database. We prove that a particular method of reordering the observations is optimal against any other ordering for simultaneously identifying the highest Kendall's tau association in subsets of size k (k = 2,...,n). Furthermore, assuming a subpopulation of k, we present a way of quantifying how likely any observation is to be in that subpopulation. From the statistical model point of view, first we show that the semi-parametric bivariate Mallow's model provides a good tool to model the paired empirical ranking through Kendall's distance for bivariate samples from parametric copula models. Then the Mallow's model can also model the bivariate samples from special three/two components (Frechet-Hoeffding upper bound copula, positively associated copula and independence copula) copula models. At last, a generalize permutation governed n-stage bivariate Mallow's model is proposed to model n independent bivariate samples from n copulas from the same family but with n different association parameters. It is shown that both the power of the Tau-path test in detecting subpopulation association and the ability of the method to identify the associated subpopulation does not depend on the copula models for positively associated copula in the two-copula mixture (positively associated copula and independent copula) case. In the n-copula mixture case, the tau-path method can be applied to put the n observations in an order close to the order of the strength of association for their parent copula models when there are a reasonably proportion of these copulas with moderate or strong association.

Committee:

Joseph Verducci (Committee Chair); Douglas E. Critchlow (Other); Shili Lin (Other)

Subjects:

Statistics

Keywords:

Concordance matrix; copula; drug assay; microarray; mixture;permutation; quassinoids; bivariate; Mallow's model; multistage