Mixture Of Normal Distributions Research Articles

A quantitative prediction method utilizing whole omics data for biosensing

Omics data provide a plethora of quantifiable information that can potentially be used to identify biomarkers targeting the physiological processes and ecological phenomena of organisms. However, omics data have not been fully utilized because current prediction methods in biomarker construction are susceptible to data multidimensionality and noise. We developed OmicSense, a quantitative prediction method that uses a mixture of Gaussian distributions as the probability distribution, yielding the most likely objective variable predicted for each biomarker. Our benchmark test using a transcriptome dataset revealed that OmicSense achieves accurate and robust prediction against background noise without overfitting. Weighted gene co-expression network analysis revealed that OmicSense preferentially utilized hub nodes of the network, indicating the interpretability of the method. Application of OmicSense to single-cell transcriptome, metabolome, and microbiome datasets confirmed high prediction performance (r > 0.8), suggesting applicability to diverse scientific fields. Given the recent rapidly expanding availability of omics data, the developed prediction tool OmicSense, can accelerate the use of omics data as a “biosensor” based on an assemblage of potential biomarkers.

Analyzing Travel and Emission Characteristics of Hazardous Material Transportation Trucks Using BeiDou Satellite Navigation System Data

Road hazardous material transportation plays a critical role in road traffic management. Due to the dangerous nature of the cargo, hazardous material transportation trucks (HMTTs) have different route selection and driving characteristics compared to traditional freight trucks. These differences lead to unique travel and emission patterns, which in turn affect traffic management strategies and emission control measures. However, existing research predominantly focuses on safety aspects related to individual vehicle behavior, with limited exploration of the broader travel and emission characteristics of HMTTs. To bridge this gap, this study develops a comprehensive framework for analyzing the travel patterns and emissions of HMTTs. The methodology begins by applying a Gaussian mixture distribution model to identify vehicle stop points, eliminating biases associated with subjective settings. Origin–destination (OD) pairs are then determined through stop time clustering, followed by the extraction of travel characteristics using non-negative matrix factorization. Emissions are subsequently calculated based on the identified trip data. The relationship between emissions and land use characteristics is further analyzed using geographically weighted regression (GWR). Crucially, this study leverages data from the BeiDou Satellite Navigation System, focusing on HMTTs operating within Shanghai. The processed data reveal three distinct travel modes of HMTTs, categorized by spatiotemporal patterns: Daytime—Surrounding cities, Early morning—In-city, and Midnight—Scattered. Moreover, unlike other road vehicles, HMTT emissions are heavily influenced by industrial and company-related points of interest (POIs). These findings highlight the significant role of BeiDou Satellite Navigation System data in optimizing HMTT management strategies to reduce emissions and improve overall safety.

Gradual Domain Adaptation via Normalizing Flows.

Standard domain adaptation methods do not work well when a large gap exists between the source and target domains. Gradual domain adaptation is one of the approaches used to address the problem. It involves leveraging the intermediate domain, which gradually shifts from the source domain to the target domain. In previous work, it is assumed that the number of intermediate domains is large and the distance between adjacent domains is small; hence, the gradual domain adaptation algorithm, involving self-training with unlabeled data sets, is applicable. In practice, however, gradual self-training will fail because the number of intermediate domains is limited and the distance between adjacent domains is large. We propose the use of normalizing flows to deal with this problem while maintaining the framework of unsupervised domain adaptation. The proposed method learns a transformation from the distribution of the target domains to the gaussian mixture distribution via the source domain. We evaluate our proposed method by experiments using real-world data sets and confirm that it mitigates the problem we have explained and improves the classification performance.

Statistical Analysis of Intermediate Frequency Underwater Acoustic Communication Channel Characteristics in Deep-Sea Sound Channel Axis

Based on experimental data from the deep-sea sound channel axis in the Western Pacific, the statistical distribution law of cluster structure and channel delay spread characteristics are analyzed for three typical receiving depths near the sound channel axis in this paper. A ray theory-based underwater acoustic channel model is used to explain the variations in channel parameters over time and the receiving depth. The results indicate that the underwater acoustic communication channel at the channel axis depth over a 20-km range exhibits a clustered structure that depends on the emission angles of sound rays. For the amplitude characteristics, the amplitude of each cluster follows an inverse Gaussian distribution, with the maximum average amplitude observed when the receiver and transmitter depths are similar. The amplitude of each cluster fluctuation decreases as the receiving depth increases. Regarding delay spread characteristics, the delay spread of each cluster, as well as the maximum and root mean square delay spread of the channel, conform to a Gaussian mixture distribution. The mean and fluctuation of the delay spread parameters increase with the receiving depth. Variations in the cluster structure and channel delay spread characteristics above are primarily attributed to the time-varying sound speed along the propagation paths of sound rays emitted at small upward angles.

POLYGAUSSIAN DESCRIPTION OF PROBABILITY DISTRIBUTIONS OF PROCESSES GENERATED BY A NONLINEAR LORENZ SYSTEM IMPLEMENTED IN FIXED-POINT NUMBERS

The article is aimed at analyzing the probability distributions of pseudorandom processes generated on the basis of the Lorentz system solution in fixed-point numbers. Numerical solution of the Lo-renz system by the Euler method in single- and double-precision floating-point numbers with a lim-ited number capacity can lead to a breakdown in signal generation. The use of fixed-point numbers contributes to the reduction of computational complexity in the digital implementation of such sys-tems, which ultimately leads to the simplification of their practical implementation on modern chips of programmable logic. This allows to use resources in more efficient way, and increases productiv-ity in the creation and maintenance of such digital systems. The study of approximation of Lorenz system signals using mixtures of Gaussian distributions is of great importance for predictive analyt-ics and stability of the system. Elimination of signal generation failure also contributes to the for-mation of stable modes of generation of chaotic signals with required statistical characteristics.

A robust Bayesian latent position approach for community detection in networks with continuous attributes

The increasing prevalence of multiplex networks has spurred a critical need to take into account potential dependencies across different layers, especially when the goal is community detection, which is a fundamental learning task in network analysis. We propose a full Bayesian mixture model for community detection in both single-layer and multi-layer networks. A key feature of our model is the joint modeling of the nodal attributes that often come with the network data as a spatial process over the latent space. In addition, our model for multi-layer networks allows layers to have different strengths of dependency in the unique latent position structure and assumes that the probability of a relation between two actors (in a layer) depends on the distances between their latent positions (multiplied by a layer-specific factor) and the difference between their nodal attributes. Under our prior specifications, the actors' positions in the latent space arise from a finite mixture of Gaussian distributions, each corresponding to a cluster. Simulated examples show that our model outperforms existing benchmark models and exhibits significantly greater robustness when handling datasets with missing values. The model is also applied to a real-world three-layer network of employees in a law firm.

Multi-level semantics probability embedding for image–text matching

The requirement of image–text matching is to retrieve matching images or texts based on textual or visual queries. However, image–text matching is inherently a many-to-many problem, as an image can correspond to multiple levels of visual semantic scenes, which can be described by different texts. Similarly, textual descriptions can be visualized through multiple visual scenes. This leads to ambiguity in the matching between images and texts. To better capture these matching relationships, we employ graph convolutional networks to extract multi-level semantic information for image–text pairs, and construct Gaussian distribution representations for image and text instead of conventional point representations. Furthermore, we introduce a inter-modal mixture of Gaussian distribution to constrain the matching relationships between image–text pairs, which ensures more precise distribution representations in a shared space and strengthens the correlation between cross-modal. We conducted experiments on Flickr30K and MS-COCO, which are two widely used datasets, demonstrates the superior performance of our approach.

Efficient training of energy-based models using Jarzynski equality**This article is an updated version of: Carbone D, Hua M, Coste S and Vanden-Eijnden E 2023 Efficient training of energy-based models using Jarzynski equality Advances in Neural Information Processing Systems vol 36, ed A Oh, T Naumann, A Globerson, K Saenko, M Hardt and S Levine (Curran Associates, Inc.) pp 52583–614.

Abstract Energy-based models (EBMs) are generative models inspired by statistical physics with a wide range of applications in unsupervised learning. Their performance is well measured by the cross-entropy (CE) of the model distribution relative to the data distribution. Using the CE as the objective for training is, however, challenging because the computation of its gradient with respect to the model parameters requires sampling of the model distribution. Here, we show how the results for nonequilibrium thermodynamics based on the Jarzynski equality together with tools from sequential Monte Carlo sampling can be used to perform this computation efficiently and avoid the uncontrolled approximations made using the standard contrastive divergence algorithm. Specifically, we introduce a modification of the unadjusted Langevin algorithm (ULA), in which each walker acquires a weight that enables the estimation of the gradient of the CE at any step during gradient descent, thereby bypassing sampling biases induced by slow mixing of the ULA. We illustrate these results with numerical experiments on Gaussian mixture distributions as well as the MNIST and CIFAR-10 datasets. We show that the proposed approach outperforms methods based on the contrastive divergence algorithm in all the considered situations.

A comparative study of different deep learning methods for time-series probabilistic residential load power forecasting

The widespread adoption of nonlinear power electronic devices in residential settings has significantly increased the stochasticity and uncertainty of power systems. The original load power data, characterized by numerous irregular, random, and probabilistic components, adversely impacts the predictive performance of deep learning techniques, particularly neural networks. To address this challenge, this paper proposes a time-series probabilistic load power prediction technique based on the mature neural network point prediction technique, i.e., decomposing the load power data into deterministic and stochastic components. The deterministic component is predicted using deep learning neural network technology, the stochastic component is fitted with Gaussian mixture distribution model and the parameters are fitted using great expectation algorithm, after which the stochastic component prediction data is obtained using the stochastic component generation method. Using a mature neural network point prediction technique, the study evaluates six different deep learning methods to forecast residential load power. By comparing the prediction errors of these methods, the optimal model is identified, leading to a substantial improvement in prediction accuracy.

Visual object detection model integrating deep active learning

The perception of surrounding objects by vehicle autonomous driving is an important means to ensure traffic safety. Object detection models based on deep learning are widely used, but they require a large amount of labeled data for training. This paper proposes an active visual object detection model that uses Gaussian mixture distribution to estimate the uncertainty of unlabeled images, reducing the dependence of model training on labeled data. First, a mixed density network is used as the detection head, and the image features extracted by the deep neural network are used as input to estimate the probability distribution of the classification and positioning of the target prediction box. Secondly, the classification score value of the target prediction box is mapped to the probability space, and the classification uncertainty of the target is calculated using the edge uncertainty; the positioning uncertainty of the target is measured using the prediction box positioning variance. Finally, the most unstable samples are selected for labeling. Compared with other typical active learning sampling strategies on the VOC dataset, the proposed method achieves the best performance, and the 54%data annotation amount can reach the performance of YOLOX supervised learning 98.8%, saving nearly 45% of the data annotation amount.

Enhancing DHA supplementation adherence: A Bayesian approach with finite mixture models and irregular interim schedules in adaptive trial designs.

Docosahexaenoic acid (DHA) supplementation has proven beneficial in reducing preterm births. However, the challenge lies in addressing nonadherence to prescribed supplementation regimens-a hurdle that significantly impacts clinical trial outcomes. Conventional methods of adherence estimation, such as pill counts and questionnaires, usually fall short when estimating adherence within a specific dosage group. Thus, we propose a Bayesian finite mixture model to estimate adherence among women with low baseline red blood cell phospholipid DHA levels (<6%) receiving higher DHA doses. In our model, adherence is defined as the proportion of participants classified into one of the two distinct components in a normal mixture distribution. Subsequently, based on the estimands from the adherence model, we introduce a novel Bayesian adaptive trial design. Unlike conventional adaptive trials that employ regularly spaced interim schedules, the novelty of our proposed trial design lies in its adaptability to adherence percentages across the treatment arm through irregular interims. The irregular interims in the proposed trial are based on the effect size estimation informed by the finite mixture model. In summary, this study presents innovative methods for leveraging the capabilities of Bayesian finite mixture models in adherence analysis and the design of adaptive clinical trials.

Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits

The training of a parameterized model largely depends on the landscape of the underlying loss function. In particular, vanishing gradients are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways. However, a caveat of most existing gradient bound results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on loss and gradient concentration for a large class of parameterized quantum circuits and arbitrary observables, which are significantly stronger than prior work. Moreover, we show that these bounds, as well as the variance of the loss itself, can be estimated efficiently and classically-providing practical tools to study the loss landscapes of VQA models, including verifying whether or not a circuit/observable induces barren plateaus. In particular, our results can readily be leveraged to rule out barren plateaus for a realistic class of ansätze and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs). We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. This implies that qGANs with appropriately chosen generators do not suffer from barren plateaus even at scale-making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training.

Deciphering gene expression patterns using large-scale transcriptomic data and its applications.

Gene expression varies stochastically across genders, racial groups, and health statuses. Deciphering these patterns is crucial for identifying informative genes, classifying samples, and understanding diseases like cancer. This study analyzes 11,252 bulk RNA-seq samples to explore expression patterns of 19,156 genes, including 10,512 cancer tissue samples and 740 normal samples. Additionally, 4,884 single-cell RNA-seq samples are examined. Statistical analysis using 16 probability distributions shows that normal samples display a wider range of distributions compared to cancer samples. Cancer samples tend to favor asymmetric distributions such as generalized extreme value, logarithmic normal, and Gaussian mixture distributions. In contrast, certain genes in normal samples exhibit symmetric distributions. Remarkably, more than 95.5% of genes exhibit non-normal distributions, which challenges traditional assumptions. Furthermore, distributions differ significantly between bulk and single-cell RNA-seq data. Many cancer driver genes exhibit distinct distribution patterns across sample types, suggesting potential for gene selection and classification based on distribution characteristics. A novel skewness-based metric is proposed to quantify distribution variation across datasets, showing genes with significant skewness differences have biological relevance. Finally, an improved naïve Bayes method incorporating gene-specific distributions demonstrates superior performance in simulations over traditional methods. This work enhances understanding of gene expression and its application in omics-based gene selection and sample classification.

Mix-DDPM: Enhancing Diffusion Models through Fitting Mixture Noise with Global Stochastic Offset

Denoising diffusion probabilistic models (DDPM) have shown impressive performance in various domains as a class of deep generative models. In this article, we introduce the mixture noise-based DDPM (Mix-DDPM), which considers the Markov diffusion posterior as a Gaussian mixture model. Specifically, Mix-DDPM randomly selects a Gaussian component and then adds the chosen Gaussian noise, which can be demonstrated as a more efficient way to perturb the signals into a simple known distribution. We further define the reverse probabilistic model as a parameterized Gaussian mixture kernel. Due to the intractability in calculating the KL divergence between Gaussian mixture models, we derive a variational bound to maximize the likelihood, offering a concise formulation for optimizing the denoising model and valuable insights for designing the sampling strategies. Our theoretical derivation highlights that Mix-DDPM need only shift image which requires the inclusion of a global stochastic offset in both the diffusion and reverse processes , which can be efficiently implemented with just several lines of code. The global stochastic offset effectively fits a Gaussian mixture distribution enhancing the degrees of freedom of the entire diffusion model. Furthermore, we present three streamlined sampling strategies that interface with diverse fast dedicated solvers for diffusion ordinary differential equations, boosting the efficacy of image representation in the sampling phase and alleviating the issue of slow generation speed, thereby enhancing both efficiency and accuracy. Extensive experiments on benchmark datasets demonstrate the effectiveness of Mix-DDPM and its superiority over the original DDPM.

Domain knowledge-driven encoder–decoder for nasopharyngeal carcinoma segmentation

Accurate image segmentation is vital for the diagnosis and treatment of nasopharyngeal carcinoma (NPC). Although deep neural networks have shown promising performance in NPC segmentation, they depend on large-scale pixel-level annotation datasets for purely data-driven model training, which poses practical limitations. To address this challenge, this paper explores the domain knowledge, namely the expert knowledge from radiologists, and proposes a domain knowledge-driven encoder–decoder architecture. Specifically, the domain knowledge on image spatial information is formulated as Gaussian mixture distribution and then transformed into an optimal transport-based expert-prior regularization of the encoder, which enhances the model’s ability in capturing discriminative features. To project the encoded features onto pixel space and obtain the segmentation maps, a cross-scale feature refinement module is built in the decoder with theoretical justification, which integrates the domain knowledge that radiologists segment NPC in a gradual refinement process. Experimental results verify the effectiveness of the proposed method for NPC segmentation. Remarkably, despite using only 10% of pixel-level annotation data, the domain knowledge-driven model outperforms recent deep neural networks that use the entire training data.

Bounded Flexible Scale Mixture of Normal Distributions with Application to Image Segmentation

Bounded Flexible Scale Mixture of Normal Distributions with Application to Image Segmentation

Mixture Gaussian process model with Gaussian mixture distribution for big data

In the era of chemical big data, the high complexity and strong interdependencies present in the datasets pose considerable challenges when constructing accurate parametric models. The Gaussian process model, owing to its non-parametric nature, demonstrates better adaptability when confronted with complex and interdependent data. However, the standard Gaussian process has two significant limitations. Firstly, the time complexity of inverting its kernel matrix during the inference process is O(n)3. Secondly, all data share a common kernel function parameter, which mixes different data types and reduces the model accuracy in mixing-category data identification problems. In light of this, this paper proposes a mixture Gaussian process model that addresses these limitations. This model reduces time complexity and distinguishes data based on different data features. It incorporates a Gaussian mixture distribution for the inducing variables to approximate the original data distribution. Stochastic Variational Inference is utilized to reduce the computational time required for parameter inference. The inducing variables have distinct parameters for the kernel function based on the data category, leading to improved analytical accuracy and reduced time complexity of the Gaussian process model. Numerical experiments are conducted to analyze and compare the performance of the proposed model on different-sized datasets and various data category cases.

Spatial Autoregressive Model with Mixture of Gaussian Distribution for the Random Effect: Formulation, Estimation and Application

Spatial Autoregressive Model with Mixture of Gaussian Distribution for the Random Effect: Formulation, Estimation and Application

Semiparametric regression models under skew scale mixtures of normal distributions

Semiparametric models (SM) are an important tool in modeling environmental data where generally a covariate presents an unknown nonlinear behavior. Usually, the error component is assumed to follow a normal distribution. However, in some situations, the response variable is skewed and heavy-tailed. This paper aims to extend the SMs allowing the errors to follow a skew scale mixture of normal distributions, increasing the model’s flexibility. In particular, we develop the EM algorithm for the proposed model, diagnostic analysis via global, local influence, and generalized leverage. A simulation study is also conducted to evaluate the efficiency of the EM algorithm. Finally, a suitable transformation is applied in a data set on ragweed pollen concentration to illustrate the utility of the proposed model.

A probabilistic proof of some integral formulas involving incomplete gamma functions

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ∫ 0 ∞ x − 2 ν cos ⁡ ( b x ) γ ( ν , α x 2 ) d x (for ν > 1 / 2 , b > 0, α > 0) and ∫ 0 ∞ x 2 ν − 1 cos ⁡ ( b x ) Γ ( − ν , α x 2 ) d x (for ν > 0, b > 0, α > 0), where γ(a, x) and Γ(a, x) are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas.