Improving the predictability of time series by chaotic parameter estimation

Limited literature regarding parameter estimation of dynamic systems has been identified as the central-most reason for not having parametric bounds in chaotic time series. However, literature suggests that a chaotic system displays a sensitive dependence on initial conditions, and our study reveals that the behavior of chaotic system: is also sensitive to changes in parameter values. Therefore, parameter estimation technique could make it possible to establish parametric bounds on a nonlinear dynamic system underlying a given time series, which in turn can improve predictability. By extracting the relationship between parametric bounds and predictability, we implemented chaos-based models for improving prediction in time series. ^ This study describes work done to establish bounds on a set of unknown parameters. Our research results reveal that by establishing parametric bounds, it is possible to improve the predictability of any time series, although the dynamics or the mathematical model of that series is not known apriori. In our attempt to improve the predictability of various time series, we have established the bounds for a set of unknown parameters. These are: (i) the embedding dimension to unfold a set of observation in the phase space, (ii) the time delay to use for a series, (iii) the number of neighborhood points to use for avoiding detection of false neighborhood and, (iv) the local polynomial to build numerical interpolation functions from one region to another. Using these bounds, we are able to get better predictability in chaotic time series than previously reported. In addition, the developments of this dissertation can establish a theoretical framework to investigate predictability in time series from the system-dynamics point of view. ^ In closing, our procedure significantly reduces the computer resource usage, as the search method is refined and efficient. Finally, the uniqueness of our method lies in its ability to extract chaotic dynamics inherent in non-linear time series by observing its values. ^

Factors associated with prompt difficulty in automated essay scoring

This study explores factors related to the prompt difficulty in Automated Essay Scoring. The sample was composed of 6,924 students. For each student, there were 1-4 essays, across 20 different writing prompts, for a total of 20,243 essays. E-rater® v.2 essay scoring engine developed by the Educational Testing Service was used to score the essays. The scoring engine employs a statistical model that incorporates 10 predictors associated with writing characteristics of which 8 were used. The Rasch partial credit analysis was applied to the scores to determine the difficulty levels of prompts. In addition, the scores were used as outcomes in the series of hierarchical linear models (HLM) in which students and prompts constituted the cross-classification levels. This methodology was used to explore the partitioning of the essay score variance.^ The results indicated significant differences in prompt difficulty levels due to genre. Descriptive prompts, as a group, were found to be more difficult than the persuasive prompts. In addition, the essay score variance was partitioned between students and prompts. The amount of the essay score variance that lies between prompts was found to be relatively small (4 to 7 percent). When the essay-level, student-level-and prompt-level predictors were included in the model, it was able to explain almost all variance that lies between prompts. Since in most high-stakes writing assessments only 1-2 prompts per students are used, the essay score variance that lies between prompts represents an undesirable or "noise" variation. Identifying factors associated with this "noise" variance may prove to be important for prompt writing and for constructing Automated Essay Scoring mechanisms for weighting prompt difficulty when assigning essay score.^

Data to Decision in a Dynamic Ocean: Robust Species Distribution Models and Spatial Decision Frameworks

Human use of the oceans is increasingly in conflict with conservation of endangered species. Methods for managing the spatial and temporal placement of industries such as military, fishing, transportation and offshore energy, have historically been post hoc; i.e. the time and place of human activity is often already determined before assessment of environmental impacts. In this dissertation, I build robust species distribution models in two case study areas, US Atlantic (Best et al. 2012) and British Columbia (Best et al. 2015), predicting presence and abundance respectively, from scientific surveys. These models are then applied to novel decision frameworks for preemptively suggesting optimal placement of human activities in space and time to minimize ecological impacts: siting for offshore wind energy development, and routing ships to minimize risk of striking whales. Both decision frameworks relate the tradeoff between conservation risk and industry profit with synchronized variable and map views as online spatial decision support systems.

For siting offshore wind energy development (OWED) in the U.S. Atlantic (chapter 4), bird density maps are combined across species with weights of OWED sensitivity to collision and displacement and 10 km2 sites are compared against OWED profitability based on average annual wind speed at 90m hub heights and distance to transmission grid. A spatial decision support system enables toggling between the map and tradeoff plot views by site. A selected site can be inspected for sensitivity to a cetaceans throughout the year, so as to capture months of the year which minimize episodic impacts of pre-operational activities such as seismic airgun surveying and pile driving.

Routing ships to avoid whale strikes (chapter 5) can be similarly viewed as a tradeoff, but is a different problem spatially. A cumulative cost surface is generated from density surface maps and conservation status of cetaceans, before applying as a resistance surface to calculate least-cost routes between start and end locations, i.e. ports and entrance locations to study areas. Varying a multiplier to the cost surface enables calculation of multiple routes with different costs to conservation of cetaceans versus cost to transportation industry, measured as distance. Similar to the siting chapter, a spatial decisions support system enables toggling between the map and tradeoff plot view of proposed routes. The user can also input arbitrary start and end locations to calculate the tradeoff on the fly.

Essential to the input of these decision frameworks are distributions of the species. The two preceding chapters comprise species distribution models from two case study areas, U.S. Atlantic (chapter 2) and British Columbia (chapter 3), predicting presence and density, respectively. Although density is preferred to estimate potential biological removal, per Marine Mammal Protection Act requirements in the U.S., all the necessary parameters, especially distance and angle of observation, are less readily available across publicly mined datasets.

In the case of predicting cetacean presence in the U.S. Atlantic (chapter 2), I extracted datasets from the online OBIS-SEAMAP geo-database, and integrated scientific surveys conducted by ship (n=36) and aircraft (n=16), weighting a Generalized Additive Model by minutes surveyed within space-time grid cells to harmonize effort between the two survey platforms. For each of 16 cetacean species guilds, I predicted the probability of occurrence from static environmental variables (water depth, distance to shore, distance to continental shelf break) and time-varying conditions (monthly sea-surface temperature). To generate maps of presence vs. absence, Receiver Operator Characteristic (ROC) curves were used to define the optimal threshold that minimizes false positive and false negative error rates. I integrated model outputs, including tables (species in guilds, input surveys) and plots (fit of environmental variables, ROC curve), into an online spatial decision support system, allowing for easy navigation of models by taxon, region, season, and data provider.

For predicting cetacean density within the inner waters of British Columbia (chapter 3), I calculated density from systematic, line-transect marine mammal surveys over multiple years and seasons (summer 2004, 2005, 2008, and spring/autumn 2007) conducted by Raincoast Conservation Foundation. Abundance estimates were calculated using two different methods: Conventional Distance Sampling (CDS) and Density Surface Modelling (DSM). CDS generates a single density estimate for each stratum, whereas DSM explicitly models spatial variation and offers potential for greater precision by incorporating environmental predictors. Although DSM yields a more relevant product for the purposes of marine spatial planning, CDS has proven to be useful in cases where there are fewer observations available for seasonal and inter-annual comparison, particularly for the scarcely observed elephant seal. Abundance estimates are provided on a stratum-specific basis. Steller sea lions and harbour seals are further differentiated by ‘hauled out’ and ‘in water’. This analysis updates previous estimates (Williams & Thomas 2007) by including additional years of effort, providing greater spatial precision with the DSM method over CDS, novel reporting for spring and autumn seasons (rather than summer alone), and providing new abundance estimates for Steller sea lion and northern elephant seal. In addition to providing a baseline of marine mammal abundance and distribution, against which future changes can be compared, this information offers the opportunity to assess the risks posed to marine mammals by existing and emerging threats, such as fisheries bycatch, ship strikes, and increased oil spill and ocean noise issues associated with increases of container ship and oil tanker traffic in British Columbia’s continental shelf waters.

Starting with marine animal observations at specific coordinates and times, I combine these data with environmental data, often satellite derived, to produce seascape predictions generalizable in space and time. These habitat-based models enable prediction of encounter rates and, in the case of density surface models, abundance that can then be applied to management scenarios. Specific human activities, OWED and shipping, are then compared within a tradeoff decision support framework, enabling interchangeable map and tradeoff plot views. These products make complex processes transparent for gaming conservation, industry and stakeholders towards optimal marine spatial management, fundamental to the tenets of marine spatial planning, ecosystem-based management and dynamic ocean management.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Radiotherapy Treatment Assessment using DCE-MRI

Abstract

The goal of modern radiotherapy is to precisely deliver a prescribed radiation dose to delineated target volumes that contain a significant amount of tumor cells while sparing the surrounding healthy tissues/organs. Precise delineation of treatment and avoidance volumes is the key for the precision radiation therapy. In recent years, considerable clinical and research efforts have been devoted to integrate MRI into radiotherapy workflow motivated by the superior soft tissue contrast and functional imaging possibility. Dynamic contrast-enhanced MRI (DCE-MRI) is a noninvasive technique that measures properties of tissue microvasculature. Its sensitivity to radiation-induced vascular pharmacokinetic (PK) changes has been preliminary demonstrated. In spite of its great potential, two major challenges have limited DCE-MRI’s clinical application in radiotherapy assessment: the technical limitations of accurate DCE-MRI imaging implementation and the need of novel DCE-MRI data analysis methods for richer functional heterogeneity information.

This study aims at improving current DCE-MRI techniques and developing new DCE-MRI analysis methods for particular radiotherapy assessment. Thus, the study is naturally divided into two parts. The first part focuses on DCE-MRI temporal resolution as one of the key DCE-MRI technical factors, and some improvements regarding DCE-MRI temporal resolution are proposed; the second part explores the potential value of image heterogeneity analysis and multiple PK model combination for therapeutic response assessment, and several novel DCE-MRI data analysis methods are developed.

I. Improvement of DCE-MRI temporal resolution. First, the feasibility of improving DCE-MRI temporal resolution via image undersampling was studied. Specifically, a novel MR image iterative reconstruction algorithm was studied for DCE-MRI reconstruction. This algorithm was built on the recently developed compress sensing (CS) theory. By utilizing a limited k-space acquisition with shorter imaging time, images can be reconstructed in an iterative fashion under the regularization of a newly proposed total generalized variation (TGV) penalty term. In the retrospective study of brain radiosurgery patient DCE-MRI scans under IRB-approval, the clinically obtained image data was selected as reference data, and the simulated accelerated k-space acquisition was generated via undersampling the reference image full k-space with designed sampling grids. Two undersampling strategies were proposed: 1) a radial multi-ray grid with a special angular distribution was adopted to sample each slice of the full k-space; 2) a Cartesian random sampling grid series with spatiotemporal constraints from adjacent frames was adopted to sample the dynamic k-space series at a slice location. Two sets of PK parameters’ maps were generated from the undersampled data and from the fully-sampled data, respectively. Multiple quantitative measurements and statistical studies were performed to evaluate the accuracy of PK maps generated from the undersampled data in reference to the PK maps generated from the fully-sampled data. Results showed that at a simulated acceleration factor of four, PK maps could be faithfully calculated from the DCE images that were reconstructed using undersampled data, and no statistically significant differences were found between the regional PK mean values from undersampled and fully-sampled data sets. DCE-MRI acceleration using the investigated image reconstruction method has been suggested as feasible and promising.

Second, for high temporal resolution DCE-MRI, a new PK model fitting method was developed to solve PK parameters for better calculation accuracy and efficiency. This method is based on a derivative-based deformation of the commonly used Tofts PK model, which is presented as an integrative expression. This method also includes an advanced Kolmogorov-Zurbenko (KZ) filter to remove the potential noise effect in data and solve the PK parameter as a linear problem in matrix format. In the computer simulation study, PK parameters representing typical intracranial values were selected as references to simulated DCE-MRI data for different temporal resolution and different data noise level. Results showed that at both high temporal resolutions (<1s) and clinically feasible temporal resolution (~5s), this new method was able to calculate PK parameters more accurate than the current calculation methods at clinically relevant noise levels; at high temporal resolutions, the calculation efficiency of this new method was superior to current methods in an order of 102. In a retrospective of clinical brain DCE-MRI scans, the PK maps derived from the proposed method were comparable with the results from current methods. Based on these results, it can be concluded that this new method can be used for accurate and efficient PK model fitting for high temporal resolution DCE-MRI.

II. Development of DCE-MRI analysis methods for therapeutic response assessment. This part aims at methodology developments in two approaches. The first one is to develop model-free analysis method for DCE-MRI functional heterogeneity evaluation. This approach is inspired by the rationale that radiotherapy-induced functional change could be heterogeneous across the treatment area. The first effort was spent on a translational investigation of classic fractal dimension theory for DCE-MRI therapeutic response assessment. In a small-animal anti-angiogenesis drug therapy experiment, the randomly assigned treatment/control groups received multiple fraction treatments with one pre-treatment and multiple post-treatment high spatiotemporal DCE-MRI scans. In the post-treatment scan two weeks after the start, the investigated Rényi dimensions of the classic PK rate constant map demonstrated significant differences between the treatment and the control groups; when Rényi dimensions were adopted for treatment/control group classification, the achieved accuracy was higher than the accuracy from using conventional PK parameter statistics. Following this pilot work, two novel texture analysis methods were proposed. First, a new technique called Gray Level Local Power Matrix (GLLPM) was developed. It intends to solve the lack of temporal information and poor calculation efficiency of the commonly used Gray Level Co-Occurrence Matrix (GLCOM) techniques. In the same small animal experiment, the dynamic curves of Haralick texture features derived from the GLLPM had an overall better performance than the corresponding curves derived from current GLCOM techniques in treatment/control separation and classification. The second developed method is dynamic Fractal Signature Dissimilarity (FSD) analysis. Inspired by the classic fractal dimension theory, this method measures the dynamics of tumor heterogeneity during the contrast agent uptake in a quantitative fashion on DCE images. In the small animal experiment mentioned before, the selected parameters from dynamic FSD analysis showed significant differences between treatment/control groups as early as after 1 treatment fraction; in contrast, metrics from conventional PK analysis showed significant differences only after 3 treatment fractions. When using dynamic FSD parameters, the treatment/control group classification after 1st treatment fraction was improved than using conventional PK statistics. These results suggest the promising application of this novel method for capturing early therapeutic response.

The second approach of developing novel DCE-MRI methods is to combine PK information from multiple PK models. Currently, the classic Tofts model or its alternative version has been widely adopted for DCE-MRI analysis as a gold-standard approach for therapeutic response assessment. Previously, a shutter-speed (SS) model was proposed to incorporate transcytolemmal water exchange effect into contrast agent concentration quantification. In spite of richer biological assumption, its application in therapeutic response assessment is limited. It might be intriguing to combine the information from the SS model and from the classic Tofts model to explore potential new biological information for treatment assessment. The feasibility of this idea was investigated in the same small animal experiment. The SS model was compared against the Tofts model for therapeutic response assessment using PK parameter regional mean value comparison. Based on the modeled transcytolemmal water exchange rate, a biological subvolume was proposed and was automatically identified using histogram analysis. Within the biological subvolume, the PK rate constant derived from the SS model were proved to be superior to the one from Tofts model in treatment/control separation and classification. Furthermore, novel biomarkers were designed to integrate PK rate constants from these two models. When being evaluated in the biological subvolume, this biomarker was able to reflect significant treatment/control difference in both post-treatment evaluation. These results confirm the potential value of SS model as well as its combination with Tofts model for therapeutic response assessment.

In summary, this study addressed two problems of DCE-MRI application in radiotherapy assessment. In the first part, a method of accelerating DCE-MRI acquisition for better temporal resolution was investigated, and a novel PK model fitting algorithm was proposed for high temporal resolution DCE-MRI. In the second part, two model-free texture analysis methods and a multiple-model analysis method were developed for DCE-MRI therapeutic response assessment. The presented works could benefit the future DCE-MRI routine clinical application in radiotherapy assessment.

(Table 2) Lipid fluxes derived from sediment trap samples

Phytoplankton is a sentinel of marine ecosystem change. Composed by many species with different life-history strategies, it rapidly responds to environment changes. An analysis of the abundance of 54 phytoplankton species in Galicia (NW Spain) between 1989 and 2008 to determine the main components of temporal variability in relation to climate and upwelling showed that most of this variability was stochastic, as seasonality and long term trends contributed to relatively small fractions of the series. In general, trends appeared as non linear, and species clustered in 4 groups according to the trend pattern but there was no defined pattern for diatoms, dinoflagellates or other groups. While, in general, total abundance increased, no clear trend was found for 23 species, 14 species decreased, 4 species increased during the early 1990s, and only 13 species showed a general increase through the series. In contrast, series of local environmental conditions (temperature, stratification, nutrients) and climate-related variables (atmospheric pressure indices, upwelling winds) showed a high fraction of their variability in deterministic seasonality and trends. As a result, each species responded independently to environmental and climate variability, measured by generalized additive models. Most species showed a positive relationship with nutrient concentrations but only a few showed a direct relationship with stratification and upwelling. Climate variables had only measurable effects on some species but no common response emerged. Because its adaptation to frequent disturbances, phytoplankton communities in upwelling ecosystems appear less sensitive to changes in regional climate than other communities characterized by short and well defined productive periods.