Automatic clustering method of multivariate data using Gaussian Dirichlet process mixture model

Hidden Markov Random Field (HMRF) models are important statistical tools for image segmentation. However, such models cannot automatically determine the number of clusters. In many practical applications, to determine the number of clusters in advance is impossible. The Dirichlet process mixture (DPM) models are nonparametric Bayesian models which can automatically determine the number of clusters for the clustering problem. A new HMRF model which combines the DPM model with Student's t-distribution is proposed in this paper. Compared to the Gaussian distribution, the Student's t-distribution with heavier-tailed can have more applications. This model can automatically determine the appropriate clustering number from the observed data. In addition, an efficient variational Bayesian inference algorithm is derived to the proposed model. Experimental results show that the proposed model outperforms other DPM-based models in image segmentation.

This dissertation focuses on the development of multiple imputation models and algorithms for high-dimensional data with variable selection structures. Leveraging on the multivariate linear mixed-effects model with missing responses for clustered data, we incorporate the variable selection routines using spike-and-slab priors within the Bayesian variable selection framework. Specific choice of these priors allow us to "force'' variables of importance (e.g. design variables or variables known to play role in missingness mechanism) into the imputation models. Our ultimate goal is to improve computational speed by removing unnecessary variables. Markov chain Monte Carlo techniques have been designed to sample from the implied posterior distributions for model unknowns as well as missing data. A computationally efficient alternative, namely, variational inference algorithms have also been developed to overcome the computational burden of Markov chain Monte Carlo for sampling the posterior predictive distribution missing data. The continuous imputation model and algorithm can be easily modified through a calibration-based rounding strategy for multiple imputation of categorical variables. We also compare two imputation strategies, i.e., the sequential imputation and the joint modeling imputation, has been made in the presence of variable selection structures. We design a joint modeling based imputation framework using a nonparametric Bayesian Dirichlet process mixture model together with a sparse precision matrix model. Two computational algorithms have been carefully designed: the Gibbs sampler and the variational Bayesian inference algorithm. The proposed methodology can be applied to both continuous and mixed continuous and categorical missing data.

Hidden Markov random field (HMRF) models are widely used for image segmentation, as they appear naturally in problems where a spatially constrained clustering scheme is asked for. A major limitation of HMRF models concerns the automatic selection of the proper number of their states, i.e., the number of region clusters derived by the image segmentation procedure. Existing methods, including likelihood- or entropy-based criteria, and reversible Markov chain Monte Carlo methods, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (DP, infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori; infinite mixture models based on the original DP or spatially constrained variants of it have been applied in unsupervised image segmentation applications showing promising results. Under this motivation, to resolve the aforementioned issues of HMRF models, in this paper, we introduce a nonparametric Bayesian formulation for the HMRF model, the infinite HMRF model, formulated on the basis of a joint Dirichlet process mixture (DPM) and Markov random field (MRF) construction. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally demonstrate its advantages over competing methodologies.

Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers

Robots had a great impact on the manufacturing industry ever since the early seventies when companies such as KUKA and ABB started deploying their first industrial robots. These robots merely performed very specific tasks in specific ways within well-defined environments. Still, they proved to be very useful as they could exceed human performance at these tasks. However, in order to enable robots to enter our daily life, they need to become more versatile and need to operate in much less structured environments. This thesis is partly devoted to stretching these limitations by means of learning, namely imitation learning (IL) and inverse reinforcement learning (IRL). Reinforcement learning (RL) is a powerful approach to enable robots to solve a task in an unknown environment. The practitioner describes a desired behavior by specifying a reward function and the robot autonomously interacts with the environment in order to find a control policy that generates high accumulated reward. However, RL is not suitable for teaching new tasks by non-experts because specifying appropriate reward functions can be difficult. Demonstrating the desired behavior is often easier for non-experts. Imitation learning can be used in order to enable the robot to reproduce the demonstrations. However, without explicitly inferring and modeling the intentions of the demonstrations, it can become difficult to solve the task for unseen situations. Inverse reinforcement learning (IRL) therefore aims to infer a reward function from the demonstrations, such that optimizing this reward function yields the desired behavior even for different situations. This thesis introduces a unifying approach to solve the inverse reinforcement learning problem in the same way as the reinforcement learning problem. This is achieved by framing both problems as information projection problems, i.e., we strive to minimize the relative entropy between a probabilistic model of the robot behavior and a given desired distribution. Furthermore, a trust region on the robot behavior is used to stabilize the optimization. For inverse reinforcement learning, the desired distribution is implicitly given by the expert demonstrations. The resulting optimization can be efficiently solved using state-of-the-art reinforcement learning methods. For reinforcement learning, the log-likelihood of the desired distribution is given by the reward function. The resulting optimization problem corresponds to a standard reinforcement learning formulation, except for an additional objective of maximizing the entropy of the robot behavior. This entropy objective adds little overhead to the optimization, but can lead to better exploration and more diversified policies. Trust-region I-projections are not only useful for training robots, but can also be applied to other machine learning problems. I-projections are typically used for variational inference, in order to approximate an intractable distribution by a simpler model. However, the resulting optimization problems are usually optimized based on stochastic gradient descent which often suffers from high variance in the gradient estimates. As trust-region I-projections where shown to be effective for reinforcement learning and inverse reinforcement learning, this thesis also explores their use for variational inference. More specifically, trust-region I-projections are investigated for the problem of approximating an intractable distribution by a Gaussian mixture model (GMM) with an adaptive number of components. GMMs are highly desirable for variational inference because they can yield arbitrary accurate approximations while inference from GMMs is still relatively cheap. In order to make learning the GMM feasible, we derive a lower bound that enables us to decompose the objective function. The optimization can then be performed by iteratively updating individual components using a technique from reinforcement learning. The resulting method is capable of learning approximations of significantly higher quality than existing variational inference methods. Due to the similarity of the underlying optimization problems, the insights gained from our variational inference method are also useful for IL and IRL. Namely, a similar lower bound can be applied also for the I-projection formulation of imitation learning. However, whereas for variational inference the lower bound serves to decompose the objective function, for imitation learning it allows us to provide a reward signal to the robot that does not depend on its behavior. Compared to reward functions that are relative to the current behavior of the robot---which are typical for popular adversarial methods---behavior-independent reward functions have the advantages that we can show convergence even for greedy optimization. Furthermore, behavior-independent reward functions solve the inverse reinforcement learning problem, thereby closing the gap between imitation learning and IRL. However, algorithms derived from our non-adversarial formulation are actually very similar to existing AIL methods, and we can even show that adversarial inverse reinforcement learning (AIRL) is indeed an instance of our formulation. AIRL was derived from an adversarial formulation, and we point out several problems of that derivation. In contrast, we show that AIRL can be straightforwardly derived from out non-adversarial formulation. Furthermore, we demonstrate that the non-adversarial formulation can be also used to derive novel algorithms by presenting a non-adversarial method for offline imitation learning.

Dirichlet process mixture models (or mixture of Dirichlet process [MDP]) are Bayesian non‐parametric mixture models that can solve the problem of determining the number of components in mixture models by assuming infinitely many components. We propose a new approach for MDP by using a matrix‐generalized half‐ distribution as a non‐informative prior for the covariance in the base distribution. This new approach aims to improve the performance of MDP in cases where conjugacy constraints cause underperformance of current priors, such as Wishart distribution or inverse‐Wishart. The effectiveness of this approach is demonstrated through simulations and by comparing its performance on the Old Faithful geyser dataset with existing MDP specifications.

The aim of this study was to implement a Dirichlet process mixture (DPM) model for automatic tumor edge identification on (18)F-fluorodeoxyglucose positron emission tomography ((18)F-FDG PET) images by optimizing the parameters on which the algorithm depends, to validate it experimentally, and to test its robustness. The DPM model belongs to the class of the Bayesian nonparametric models and uses the Dirichlet process prior for flexible nonparametric mixture modeling, without any preliminary choice of the number of mixture components. The DPM algorithm implemented in the statistical software package R was used in this work. The contouring accuracy was evaluated on several image data sets: on an IEC phantom (spherical inserts with diameter in the range 10-37 mm) acquired by a Philips Gemini Big Bore PET-CT scanner, using 9 different target-to-background ratios (TBRs) from 2.5 to 70; on a digital phantom simulating spherical/uniform lesions and tumors, irregular in shape and activity; and on 20 clinical cases (10 lung and 10 esophageal cancer patients). The influence of the DPM parameters on contour generation was studied in two steps. In the first one, only the IEC spheres having diameters of 22 and 37 mm and a sphere of the digital phantom (41.6 mm diameter) were studied by varying the main parameters until the diameter of the spheres was obtained within 0.2% of the true value. In the second step, the results obtained for this training set were applied to the entire data set to determine DPM based volumes of all available lesions. These volumes were compared to those obtained by applying already known algorithms (Gaussian mixture model and gradient-based) and to true values, when available. Only one parameter was found able to significantly influence segmentation accuracy (ANOVA test). This parameter was linearly connected to the uptake variance of the tested region of interest (ROI). In the first step of the study, a calibration curve was determined to automatically generate the optimal parameter from the variance of the ROI. This "calibration curve" was then applied to contour the whole data set. The accuracy (mean discrepancy between DPM model-based contours and reference contours) of volume estimation was below (1 ± 7)% on the whole data set (1 SD). The overlap between true and automatically segmented contours, measured by the Dice similarity coefficient, was 0.93 with a SD of 0.03. The proposed DPM model was able to accurately reproduce known volumes of FDG concentration, with high overlap between segmented and true volumes. For all the analyzed inserts of the IEC phantom, the algorithm proved to be robust to variations in radius and in TBR. The main advantage of this algorithm was that no setting of DPM parameters was required in advance, since the proper setting of the only parameter that could significantly influence the segmentation results was automatically related to the uptake variance of the chosen ROI. Furthermore, the algorithm did not need any preliminary choice of the optimum number of classes to describe the ROIs within PET images and no assumption about the shape of the lesion and the uptake heterogeneity of the tracer was required.

In the era of data-driven decision-making, the Gaussian Mixture Model (GMM) stands as a cornerstone in statistical modeling, particularly in clustering and density estimation. The Improved GMM presents a robust solution to a fundamental problem in clustering: the determination of the optimal number of clusters. Unlike its predecessor, it does not rely on a predetermined cluster count but employs model selection criteria, such as the Bayesian Information Criterion (BIC) or Akaike Information Criterion (AIC), to automatically identify the most suitable cluster count for the given data. This inherent adaptability is a hallmark of the Improved GMM, making it a versatile tool in a broad spectrum of applications, from market segmentation to image processing. Furthermore, the Improved GMM revolutionizes parameter estimation and model fitting. It leverages advanced optimization techniques, such as the Expectation-Maximization (EM) algorithm or variational inference, to achieve convergence to more favorable local optima. This results in precise and reliable parameter estimates, including cluster means, covariances, and component weights. The Improved GMM is particularly invaluable when dealing with data of varying complexities, non-standard data distributions, and clusters with differing shapes and orientations. It excels at capturing the nuanced relationships within the data, providing a powerful framework for understanding complex systems. One of the key differentiators of the Improved GMM is its accommodation of full covariance matrices for each component. This feature empowers the model to account for intricate interdependencies between variables, which is essential for modeling real-world data effectively. It is capable of handling data that exhibits non-spherical or irregular cluster shapes, a significant limitation of the traditional GMM.

Local half-region depth for functional data

In some applications of data mining, clustering analysis of directional data is often involved. In this case, conventional model-based clustering methods are not suitable for fitting the data of such type of structure. Therefore, a Dirichlet Process Mixture Model based on von Mises-Fisher distribution was proposed for the clustering analysis of directional data. The main motivation is that as a non-parametric Bayesian model, The Dirichlet process can automatically determine the complexity of the mixture model when the number of data categories is unknown. We use the accelerated Variational inference algorithm to quickly estimate the parameters involved in the model, which enables the method to be applied in applications with large data scale. The validity of the proposed model was verified by using different scale simulation data and clustering analysis of gene expression data.

Bayesian robust multi-extreme learning machine

Attention Mechanism Based Mixture of Gaussian Processes

Using Gaussian mixture models for clustering is a statistically mature method for clustering in data science with numerous successful applications in science and engineering. The parameters for a Gaussian mixture model are typically estimated from training data using the iterative expectation-maximization algorithm, which requires the number of Gaussian components a priori. In this study, we propose two algorithms rooted in numerical algebraic geometry, namely an area-based algorithm and a local maxima algorithm, to identify the optimal number of components. The area-based algorithm transforms several Gaussian mixture models with varying number of components into sets of equivalent polynomial regression splines. Next, it uses homotopy continuation methods for evaluating the resulting splines to identify the number of components that results in the best fit. The local maxima algorithm forms a set of polynomials by fitting a smoothing spline to a kernel density estimate of the data. Next, it uses numerical algebraic geometry to solve the system of the first derivatives for finding the local maxima of the resulting smoothing spline, which estimates the number of mixture components. The local maxima algorithm also identifies the location of the centers of Gaussian components. Using a real-world case study in automotive manufacturing and multiple simulations, we compare the performance of the proposed algorithms with that of Akaike information criterion (AIC) and Bayesian information criterion (BIC), which are popular methods in the literature. We show the proposed algorithms are more robust than AIC and BIC when the Gaussian assumption is violated.

We present an unsupervised segmentation algorithm combining the mean shift procedure and deterministic annealing expectation maximization (DAEM) called MS-DAEM algorithm. We use the mean shift procedure to determine the number of components in a mixture model and to detect their modes of each mixture component. Next, we have adopted the Gaussian mixture model (GMM) to represent the probability distribution of color feature vectors. A DAEM formula is used to estimate the parameters of the GMM which represents the multi-colored objects statistically. The experimental results show that the mean shift part of the proposed MS-DAEM algorithm is efficient to determine the number of components and initial modes of each component in mixture models. And also it shows that the DAEM part provides a global optimal solution for the parameter estimation in a mixture model and the natural color images are segmented efficiently by using the GMM with components estimated by MS-DAEM algorithm.

In rarefied gas flows, discontinuity phenomena such as velocity slip and temperature jump commonly appear in the gas layer adjacent to a solid boundary. Due to the physical complexity of the interactions at the gas-solid interface, particularly in the case of systems with local nonequilibrium state, boundary models with limited number of parameters cannot completely describe the reflection of gas molecules at the boundary. In this work, the Gaussian mixture (GM) model, which is an unsupervised machine learning technique, is employed to construct a statistical gas-solid surface scattering model based on the collisional data obtained from molecular dynamics (MD) simulations. The GM model is applied to study Couette flow for different inert gases (Ar and He) confined between two parallel infinite gold walls at different temperatures. A direct comparison between the results obtained from the GM model and the Cercignani-Lampis-Lord (CLL) scattering kernel against the MD collisional data in terms of the distribution of the predicted postcollisional velocities, and accommodation coefficients has shown that the results from the GM model are an excellent match with the MD results outperforming the CLL scattering kernel. As an example, for He gas, while the predicted energy accommodation coefficient by the CLL model is more than two times higher than the MD predictions, the value computed by the GM model is in excellent agreement with the MD results. This superior performance of the GM model confirms its high potential to derive a generalized boundary condition in systems encountered with highly nonequilibrium and complex gas flow conditions.