Bivariate Gaussian Mixture Model Research Articles

On Expectation-Maximization Algorithm with Split and Merge in R

This paper presents a comprehensive study on the implementation of the Expectation-Maximization (EM) algorithm for a 4-component bivariate Gaussian Mixture Model (GMM) with a focus on incorporating the split and merge techniques. The bivariate GMM is widely utilized in various fields, such as pattern recognition, image processing, and data clustering, due to its flexibility in capturing complex data distributions. Our proposed implementation leverages the versatility of the R programming language to create a robust and efficient framework for modeling and optimizing the parameters of the 4-component GMM. The EM algorithm is employed as a powerful tool for iteratively estimating the model parameters, ensuring convergence to a local maximum of the likelihood function. To enhance the model’s flexibility, we introduce the split and merge strategies, enabling the algorithm to adapt to diverse data structures and efficiently manage the complexity of the mixture components. The split operation allows for the subdivision of components when needed, while the merge operation facilitates the combination of components that represent similar patterns. This adaptive approach contributes to the model’s ability to capture intricate data patterns and improve convergence during the optimization process. The implementation is validated through extensive experimentation on synthetic, demonstrating its effectiveness in accurately estimating the parameters of the 4-component bivariate GMM. The proposed methodology proves to be a valuable addition to the existing tools available for GMM based modeling, providing researchers and practitioners with a flexible and powerful framework for analyzing complex data structures in diverse applications

Image Segmentation Based on G.O.A for Finding Deformities in Medical and Aura Images

Objectives: The research aims to develop the segmentation model to identify the deformity in the medical images as accurately as possible and plan for better medical treatment. The study is extended to identify the disease before its appearance in the human body through human aura images to support aura imaging in medical diagnosis. Methods: The study used a brain image from the UCI data set and Aura images from the Biowell data set to identify the disease. The segmentation model Bivariate Gaussian Mixture Model (B.G.M.M) was developed. Model parameters are derived using the Expectation Maximization (E.M) Algorithm. The Grasshopper optimization Algorithm (G.O.A) extracts optimal features from the images. The chosen feature is fed as input to the classification model B.G.M.M. Segmentation accuracy is measured using the quality metrics. Findings: The developed approach shows 97% accuracy in identifying the damaged tissues in MRI images and high-intensity energy zones in the aura images, indicating the potential for deformities. Novelty: This study significantly contributes to the field by offering novel solutions for precise and comprehensive image analysis in medical and aura imaging contexts. Keywords: G.O.A, segmentation, G.M.M, E.M, quality metrics, deformity identification, Hue and saturation

Bayesian learning of Gaussian mixture model for calculating debris flow exceedance probability

ABSTRACT Probabilistic modelling of debris flow data provides useful information for quantitative risk assessment, such as exceedance probabilities (EPs) of debris flow quantities. This task can defy many classical statistical models because debris flow data are frequently collected over years or even decades, and the nonuniformity of the nature and the complex physical mechanism of debris flows lead to multimodal distribution characteristics of observational data. This paper proposes a Bayesian framework for learning Gaussian mixture model (GMM) of debris flow quantities (e.g. total discharge Q total and maximum impact pressure P max) and calculating their EPs for risk-informed decision making. GMM provides great flexibility to fit observation data, but are intrinsically unidentifiable due to the label switching. These computational difficulties are addressed using Random Gibbs Sampling and Bridge Sampling in the proposed framework, allowing incorporating the statistical uncertainty in GMM parameters into EP estimation. Equations are derived for the proposed approach and are illustrated using Q total and P max data at Jiangjia Ravine, China. Results show that the proposed approach identifies a bivariate GMM of Q total and P max reflecting the multimodal characteristics of the observed data and quantifies the statistical uncertainty of GMM parameters. Incorporating the statistical uncertainty into EP estimation provides robust estimates.

A Novel Methodology for Disease Identification Using Metaheuristic Algorithm and Aura Image

Every human has a specific Aura. Every organism in the human body emits energy comprising of Ultra Violet radiation, thermal radiation, and electromagnetic radiation. These energy levels help to underline the physical health inside the human body. In general, these energy levels are called Aura. In order to capture the energy levels, specific cameras like Kirlian are used. These cameras try to capture the energy distribution and map them to the individual organs of the human body. In this article, we present a methodology using Image processing techniques, where Bivariate Gaussian Mixture Model (BGMM) is considered as a classifier to identify the diseases in humans based on the energy distribution. In this article, we have considered five different categories of diseased organs that are identified based on the energy distribution. The preprocessing is subjected to the morphological technique and Particle Swarm Optimization (PSO) algorithm is considered for feature extraction. The segmentation process is carried out using the feature extracted and training is carried out using the BGMM classifier. The result obtained is summarized using various other methods like Support Vector Machine (SVM), Artificial Neural Network (ANN), and Multiclass SVM (MSVM). The results showcase that the proposed methodology exhibits recognition accuracy at 90%.

Modeling the duration and size of wildfires using joint mixture models

AbstractFire duration and fire size are key outcomes for quantifying the survivorship of extended attack fires, here considered to be fires with duration greater than 2 days and size greater than 4 ha. Past studies suggest that these key outcomes are correlated. As well, fire behavior, linked to hidden effects, tends to yield that fires arise from different subpopulations. Indeed, it is not unusual for fire behavior to be identified as arising from normal or extreme subpopulations, for example. Here, we embed these two concepts into a new framework for jointly modeling fire duration and fire size. We develop a bivariate finite mixture framework that can be used to model duration and size with four subpopulations of the outcomes whereby duration and size are either normal or extreme. We utilize a shared random effect model as well as a bivariate Gaussian mixture model for such mixture modeling. We also incorporate the effect of explanatory variables associated with each fire event, on the posterior probability of the component that the fire belongs to, through a Dirichlet model. In an analysis of fire outcomes from British Columbia, Canada, we find that the majority of the fires are of normal or extreme magnitude in both outcomes, with strong evidence indicating correlation between duration and size. The effect of fire center, month, and several environmental covariates are identified as key predictors and we are able to determine through these approaches how these covariates differentially affect the four subpopulations.

Gaussian Mixture Models Based on Principal Components and Applications

Data scientists use various machine learning algorithms to discover patterns in large data that can lead to actionable insights. In general, high-dimensional data are reduced by obtaining a set of principal components so as to highlight similarities and differences. In this work, we deal with the reduced data using a bivariate mixture model and learning with a bivariate Gaussian mixture model. We discuss a heuristic for detecting important components by choosing the initial values of location parameters using two different techniques: cluster means, k-means and hierarchical clustering, and default values in the “mixtools” R package. The parameters of the model are obtained via an expectation maximization algorithm. The criteria from Bayesian point are evaluated for both techniques, demonstrating that both techniques are efficient with respect to computation capacity. The effectiveness of the discussed techniques is demonstrated through a simulation study and using real data sets from different fields.

SU22 - BIVARIATE GAUSSIAN MIXTURE MODEL FOR GWAS SUMMARY STATISTICS

Background Identifying shared genetics is important as it uncovers hidden relationship between complex traits and improves our understanding of disease etiology. Today genetic correlation is commonly used as the principal measure that quantifies genetic overlap. Available methods can calculate genetic correlation from raw genotypes (restricted maximum likelihood, polygenic risk scores), from a set of Single-Nucleotide Polymorphisms (SNPs) that pass genome-wide significance threshold (Mendelian Randomization), or utilizing all data from Genome Wide Association Studies (GWASes) including SNPs that do not reach genome-wide significance (Cross-Trait Linkage Disequilibrium (LD) Score Regression). These methods for evaluating genetic overlap are unable to capture a mixture of effect directions across shared genetic variants (i.e., only reporting overall positive, negative or no genetic correlations), while recent analyses suggest that correlation across polygenic traits in the effect size and directionality of a SNP is not the same across all SNPs. Methods Bivariate Gaussian Mixture Model provides a novel way to detect and quantify shared genetics independently from genetic correlation. In the absence of genetic correlation we use another measure of polygenic overlap, expressed as the proportion of SNPs associated with both traits. To estimate this quantity, we model true per-SNP effect sizes as a mixture of four bivariate normal distributions: two causal components specific to each trait, one causal component of SNPs affecting both traits, and a null component of SNPs with no effect on either trait. We interpret the weight of each component in the mixture as the proportion of SNPs in that component. The parameters of the mixture model are estimated from the summary statistics data by direct optimization of the maximum likelihood. Our statistical model relates observed signed test statistics (GWAS z-scores) to the underlying per-SNP effect sizes, incorporating effects of LD structure, minor allele frequency, sample size, cryptic relationships / sample stratification, and sample overlap, to capture all these effects on GWAS z-scores. Results We show in simulations that our model differentiates cases with no polygenic overlap versus cases with significant overlap in the absence of genetic correlation. We show genetic overlap between schizophrenia, waist-hip-ratio and triglycerides, despite the fact that genetic correlation is close to zero. We apply our model to schizophrenia, bipolar disorder and IQ data, and show that in the presence of genetic correlation our model reports consistent results with those from cross-trait LD score regression. Discussion It appears to be a common practice to interpret the lack of genetic correlation as no relation between traits. However, lack of correlation does not necessarily imply independence - a fact well known from statistics. Thus, we advocate that certain important cases of genetic overlap are not captured by genetic correlation. Our model quantifies genetic overlap between traits both with and without genetic correlation. It controls for the fact that if both traits are highly polygenic then some proportion of SNPs is associated with both traits by chance or due to LD structure. In addition to the insights into genetic architecture our model can improve SNP discovery, by estimating posterior effect size of one trait given observed GWAS z-score in another trait. Further, more accurately estimated effect sizes can be used for improving polygenic risk scores.

A Hybrid Approach for Identification of Manhole and Staircase to Assist Visually Challenged

Recognition of an object is a bare minimum restraint for an individual in order to sort out or classify the type of the object. This situation becomes a tricky experience with respect to the blind persons; therefore, to assist visually challenged persons, in particular, while recognizing the staircases and manholes, a prototype of mobility recognition is presented using the feature vector identification and sensor computed processor Arduino chips. This prototype provides more sovereignty to the sightless people while walking on the roads and helps to pass through on their own without any backing. This prototype is developed using Arduino kit along with feature detection module and helps the visually challenged in reaching their destinations with ease. A low weight stick is built to facilitate the visually challenged people toward effective recognition of the obstacles. In order to recognize the manholes, the chip is programmed and embedded in the stick that also holds the code for detection of the staircases based on a bivariate Gaussian mixture model; speeded up robust features algorithm is considered for extraction of features. The developed model shows an accuracy of around 90% for manhole detection and 88% for staircase detection.

3DSpectra: A 3-dimensional quantification algorithm for LC-MS labeled profile data.

Quantitative proteomics is pivotal for many systems biology related fields, such as biomarker discovery. The quantification quality provided by the adopted software is crucial for the success of protein differential expression studies. To determine the reliability of a quantitative computational method, we suggest evaluating performance parameters like accuracy and precision of the quantifications, robustness to outliers and proteome coverage. A quantitative comparison of these parameters is highly desirable since it enables to benchmark software performance. We applied this strategy to 3DSpectra, a 3-dimensional approach to spectra analysis for MS1 peptide quantification. It distinguishes peptide peaks from spurious peaks interfering in the survey scan. 3DSpectra was compared to ASAPRatio in terms of quantification quality performance parameters and showed an overall improvement.

20th century intraseasonal Asian monsoon dynamics viewed from Isomap

Abstract. The Asian summer monsoon is a high-dimensional and highly nonlinear phenomenon involving considerable moisture transport towards land from the ocean, and is critical for the whole region. We have used daily ECMWF reanalysis (ERA-40) sea-level pressure (SLP) anomalies on the seasonal cycle, over the region 50–145° E, 20° S–35° N, to study the nonlinearity of the Asian monsoon using Isomap. We have focused on the two-dimensional embedding of the SLP anomalies for ease of interpretation. Unlike the unimodality obtained from tests performed in empirical orthogonal function space, the probability density function, within the two-dimensional Isomap space, turns out to be bimodal. But a clustering procedure applied to the SLP data reveals support for three clusters, which are identified using a three-component bivariate Gaussian mixture model. The modes are found to appear similar to active and break phases of the monsoon over South Asia in addition to a third phase, which shows active conditions over the western North Pacific. Using the low-level wind field anomalies, the active phase over South Asia is found to be characterised by a strengthening and an eastward extension of the Somali jet. However during the break phase, the Somali jet is weakened near southern India, while the monsoon trough in northern India also weakens. Interpretation is aided using the APHRODITE gridded land precipitation product for monsoon Asia. The effect of large-scale seasonal mean monsoon and lower boundary forcing, in the form of ENSO, is also investigated and discussed. The outcome here is that ENSO is shown to perturb the intraseasonal regimes, in agreement with conceptual ideas.

Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection

It is known that the empirical orthogonal function method is unable to detect possible nonlinear structure in climate data. Here, isometric feature mapping (Isomap), as a tool for nonlinear dimensionality reduction, is applied to 1958–2001 ERA‐40 sea‐level pressure anomalies to study nonlinearity of the Asian summer monsoon intraseasonal variability. Using the leading two Isomap time series, the probability density function is shown to be bimodal. A two‐dimensional bivariate Gaussian mixture model is then applied to identify the monsoon phases, the obtained regimes representing enhanced and suppressed phases, respectively. The relationship with the large‐scale seasonal mean monsoon indicates that the frequency of monsoon regime occurrence is significantly perturbed in agreement with conceptual ideas, with preference for enhanced convection on intraseasonal time scales during large‐scale strong monsoons. Trend analysis suggests a shift in concentration of monsoon convection, with less emphasis on South Asia and more on the East China Sea.

Image Segmentation and Retrievals based on Finite Doubly Truncated Bivariate Gaussian Mixture Model and KMeans

A new Image Segmentation method based on Finite Doubly Truncated Bivariate Gaussian Mixture Model is proposed in this paper. The Truncated Bivariate Gaussian Distribution includes several of the skewed and asymmetric distributions as particular cases with finite range. This distribution also includes the Gaussian distribution as a limiting case. We use Expectation maximization (EM) algorithm to estimate the model parameters of the image data and the number of mixture components is estimated by using K-means Clustering algorithm. The K-means clustering algorithm is also utilized for developing the initial estimates of the EM algorithm. The segmentation is carried out by clustering of feature vector into appropriate component according to the Maximum Likelihood Estimation criteria. The advantage of our method lies its efficiency on initialization of the model parameters and segmenting the images in a totally unsupervised manner. The performance of the proposed algorithm is studied by computing the segmentation performance measures like, PRI, GCE and VOI. The ability of this method for image retrieval is demonstrated by computing the image quality metrics for six images namely OSTRICH, POT, TOWER, BEARS, DEER and BIRD. The experimental results show that this method outperforms the existing model based image segmentation methods.

Efficient discovery of abundant post-translational modifications and spectral pairs using peptide mass and retention time differences

BackgroundPeptide identification via tandem mass spectrometry is the basic task of current proteomics research. Due to the complexity of mass spectra, the majority of mass spectra cannot be interpreted at present. The existence of unexpected or unknown protein post-translational modifications is a major reason.ResultsThis paper describes an efficient and sequence database-independent approach to detecting abundant post-translational modifications in high-accuracy peptide mass spectra. The approach is based on the observation that the spectra of a modified peptide and its unmodified counterpart are correlated with each other in their peptide masses and retention time. Frequently occurring peptide mass differences in a data set imply possible modifications, while small and consistent retention time differences provide orthogonal supporting evidence. We propose to use a bivariate Gaussian mixture model to discriminate modification-related spectral pairs from random ones. Due to the use of two-dimensional information, accurate modification masses and confident spectral pairs can be determined as well as the quantitative influences of modifications on peptide retention time.ConclusionExperiments on two glycoprotein data sets demonstrate that our method can effectively detect abundant modifications and spectral pairs. By including the discovered modifications into database search or by propagating peptide assignments between paired spectra, an average of 10% more spectra are interpreted.

Correlating Two Continuous Variables Subject to Detection Limits in the Context of Mixture Distributions

Summary In individuals who are infected with human immunodeficiency virus (HIV), distributions of quantitative HIV ribonucleic acid measurements may be highly left censored with an extra spike below the limit of detection LD of the assay. A two-component mixture model with the lower component entirely supported on [0, LD] is recommended to model the extra spike in univariate analysis better. Let LD1 and LD2 be the limits of detection for the two HIV viral load measurements. When estimating the correlation coefficient between two different measures of viral load obtained from each of a sample of patients, a bivariate Gaussian mixture model is recommended to model the extra spike on [0, LD1] and [0, LD2] better when the proportion below LD is incompatible with the left-hand tail of a bivariate Gaussian distribution. When the proportion of both variables falling below LD is very large, the parameters of the lower component may not be estimable since almost all observations from the lower component are falling below LD. A partial solution is to assume that the lower component’s entire support is on [0, LD1]×[0, LD2]. Maximum likelihood is used to estimate the parameters of the lower and higher components. To evaluate whether there is a lower component, we apply a Monte Carlo approach to assess the p-value of the likelihood ratio test and two information criteria: a bootstrap-based information criterion and a cross-validation-based information criterion. We provide simulation results to evaluate the performance and compare it with two ad hoc estimators and a single-component bivariate Gaussian likelihood estimator. These methods are applied to the data from a cohort study of HIV-infected men in Rio de Janeiro, Brazil, and the data from the Women’s Interagency HIV oral study. These results emphasize the need for caution when estimating correlation coefficients from data with a large proportion of non-detectable values when the proportion below LD is incompatible with the left-hand tail of a bivariate Gaussian distribution.