Statistical Hurdle Models for Single Cell Gene Expression: Differential Expression and Graphical Modeling

Thesis (Ph.D.)--University of Washington, 2016-06

Zero-non-zero patterned vector error correction modeling for I(2) cointegrated time series with applications in testing PPP and stock market relationships

Vector error-correction models (VECMs) have become increasingly important in their application to financial markets. Standard full-order VECM models assume non-zero entries in all their coefficient matrices. However, applications of VECM models to financial market data have revealed that zero entries are often a necessary part of efficient modelling. In such cases, the use of full-order VECM models may lead to incorrect inferences. Specifically, if indirect causality or Granger non-causality exists among the variables, the use of over-parameterised full-order VECM models may weaken the power of statistical inference. In this paper, it is argued that the zero–non-zero (ZNZ) patterned VECM is a more straightforward and effective means of testing for both indirect causality and Granger non-causality. For a ZNZ patterned VECM framework for time series of integrated order two, we provide a new algorithm to select cointegrating and loading vectors that can contain zero entries. Two case studies are used to demonstrate the usefulness of the algorithm in tests of purchasing power parity and a three-variable system involving the stock market.

Modeling diffraction and imaging of laser beams by the mode-expansion method

We present a new method of modeling imaging of laser beams in the presence of diffraction. Our method is based on the concept of first orthogonally expanding the resultant diffraction field (that would have otherwise been obtained by the laborious application of the Huygens diffraction principle) and then representing it by an effective multimodal laser beam with different beam parameters. We show not only that the process of obtaining the new beam parameters is straightforward but also that it permits a different interpretation of the diffraction-caused focal shift in laser beams. All of the criteria that we have used to determine the minimum number of higher-order modes needed to accurately represent the diffraction field show that the mode-expansion method is numerically efficient. Finally, the characteristics of the mode-expansion method are such that it allows modeling of a vast array of diffraction problems, regardless of the characteristics of the incident laser beam, the diffracting element, or the observation plane. (C) 2005 Optical Society of America.

Forecasting Helicoverpa populations in Australia: A comparison of regression based models and a bio-climatic based modeling approach

Long-term forecasts of pest pressure are central to the effective management of many agricultural insect pests. In the eastern cropping regions of Australia, serious infestations of Helicoverpa punctigera (Wallengren) and H. armigera (Hübner)(Lepidoptera: Noctuidae) are experienced annually. Regression analyses of a long series of light-trap catches of adult moths were used to describe the seasonal dynamics of both species. The size of the spring generation in eastern cropping zones could be related to rainfall in putative source areas in inland Australia. Subsequent generations could be related to the abundance of various crops in agricultural areas, rainfall and the magnitude of the spring population peak. As rainfall figured prominently as a predictor variable, and can itself be predicted using the Southern Oscillation Index (SOI), trap catches were also related to this variable. The geographic distribution of each species was modelled in relation to climate and CLIMEX was used to predict temporal variation in abundance at given putative source sites in inland Australia using historical meteorological data. These predictions were then correlated with subsequent pest abundance data in a major cropping region. The regression-based and bioclimatic-based approaches to predicting pest abundance are compared and their utility in predicting and interpreting pest dynamics are discussed.

An equivalent distributed magnetic current based FDTD method for the calculation of e-fields induced by gradient coils in MRI

This paper evaluates a low-frequency FDTD method applied to the problem of induced E-fields/eddy currents in the human body resulting from the pulsed magnetic field gradients in MRI. In this algorithm, a distributed equivalent magnetic current (DEMC) is proposed as the electromagnetic source and is obtained by quasistatic calculation of the empty coil's vector potential or measurements therein. This technique circumvents the discretizing of complicated gradient coil geometries into a mesh of Yee cells, and thereby enables any type of gradient coil modeling or other complex low frequency sources. The proposed method has been verified against an example with an analytical solution. Results are presented showing the spatial distribution of gradient-induced electric fields in a multilayered spherical phantom model and a complete body model.

Splines and vector splines

The thrust of this report concerns spline theory and some of the background to spline theory and follows the development in (Wahba, 1991). We also review methods for determining hyper-parameters, such as the smoothing parameter, by Generalised Cross Validation. Splines have an advantage over Gaussian Process based procedures in that we can readily impose atmospherically sensible smoothness constraints and maintain computational efficiency. Vector splines enable us to penalise gradients of vorticity and divergence in wind fields. Two similar techniques are summarised and improvements based on robust error functions and restricted numbers of basis functions given. A final, brief discussion of the application of vector splines to the problem of scatterometer data assimilation highlights the problems of ambiguous solutions.

Neural network-based wind vector retrieval from satellite scatterometer data

Obtaining wind vectors over the ocean is important for weather forecasting and ocean modelling. Several satellite systems used operationally by meteorological agencies utilise scatterometers to infer wind vectors over the oceans. In this paper we present the results of using novel neural network based techniques to estimate wind vectors from such data. The problem is partitioned into estimating wind speed and wind direction. Wind speed is modelled using a multi-layer perceptron (MLP) and a sum of squares error function. Wind direction is a periodic variable and a multi-valued function for a given set of inputs; a conventional MLP fails at this task, and so we model the full periodic probability density of direction conditioned on the satellite derived inputs using a Mixture Density Network (MDN) with periodic kernel functions. A committee of the resulting MDNs is shown to improve the results.

Gaussian processes for classification: mean field algorithms:mean-field algorithms

We derive a mean field algorithm for binary classification with Gaussian processes which is based on the TAP approach originally proposed in Statistical Physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler 'naive' mean field theory and support vector machines (SVM) as limiting cases. For both mean field algorithms and support vectors machines, simulation results for three small benchmark data sets are presented. They show 1. that one may get state of the art performance by using the leave-one-out estimator for model selection and 2. the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The latter result is a taken as a strong support for the internal consistency of the mean field approach.

Gaussian processes and SVM: Mean field results and leave-one-out

In this chapter, we elaborate on the well-known relationship between Gaussian processes (GP) and Support Vector Machines (SVM). Secondly, we present approximate solutions for two computational problems arising in GP and SVM. The first one is the calculation of the posterior mean for GP classifiers using a `naive' mean field approach. The second one is a leave-one-out estimator for the generalization error of SVM based on a linear response method. Simulation results on a benchmark dataset show similar performances for the GP mean field algorithm and the SVM algorithm. The approximate leave-one-out estimator is found to be in very good agreement with the exact leave-one-out error.

Gaussian processes:iterative sparse approximations

In recent years there has been an increased interest in applying non-parametric methods to real-world problems. Significant research has been devoted to Gaussian processes (GPs) due to their increased flexibility when compared with parametric models. These methods use Bayesian learning, which generally leads to analytically intractable posteriors. This thesis proposes a two-step solution to construct a probabilistic approximation to the posterior. In the first step we adapt the Bayesian online learning to GPs: the final approximation to the posterior is the result of propagating the first and second moments of intermediate posteriors obtained by combining a new example with the previous approximation. The propagation of em functional forms is solved by showing the existence of a parametrisation to posterior moments that uses combinations of the kernel function at the training points, transforming the Bayesian online learning of functions into a parametric formulation. The drawback is the prohibitive quadratic scaling of the number of parameters with the size of the data, making the method inapplicable to large datasets. The second step solves the problem of the exploding parameter size and makes GPs applicable to arbitrarily large datasets. The approximation is based on a measure of distance between two GPs, the KL-divergence between GPs. This second approximation is with a constrained GP in which only a small subset of the whole training dataset is used to represent the GP. This subset is called the em Basis Vector, or BV set and the resulting GP is a sparse approximation to the true posterior. As this sparsity is based on the KL-minimisation, it is probabilistic and independent of the way the posterior approximation from the first step is obtained. We combine the sparse approximation with an extension to the Bayesian online algorithm that allows multiple iterations for each input and thus approximating a batch solution. The resulting sparse learning algorithm is a generic one: for different problems we only change the likelihood. The algorithm is applied to a variety of problems and we examine its performance both on more classical regression and classification tasks and to the data-assimilation and a simple density estimation problems.