Mixture Distribution Model Research Articles

Libby-Novick Beta-Liouville Distribution for Enhanced Anomaly Detection in Proportional Data

We consider the problem of anomaly detection in proportional data by investigating the Libby-Novick Beta-Liouville distribution, a novel distribution merging the salient characteristics of Liouville and Libby-Novick Beta distributions. Its main benefit, compared to the typical distributions dedicated to proportional data such as Dirichlet and Beta-Liouville, is its adaptability and explanatory power when dealing with this kind of data. Our goal is to exploit this appropriateness for modeling proportional data to achieve great performance in the anomaly detection task. First, we develop generative models, namely finite mixture models of Libby-Novick Beta-Liouville distributions. Then, we propose two discriminative techniques: Normality scores based on selecting the given distribution to approximate the softmax output vector of a deep classifier and an improved version of Support Vector Machine (SVM) by suggesting a feature mapping approach. We demonstrate the benefits of the presented approaches through a variety of experiments on both image and non-image datasets. The results demonstrate that the proposed anomaly detectors based on the Libby-Novick Beta-Liouville distribution outperform the classical distributions as well as the baseline techniques.

A nonlinear filter based on GSK for relative navigation using relative orbital elements

Due to the sensitivity of the relative orbital elements model to the measurement noise, the non-stationary heavy-tailed noise(NSHT) induced by the time-varying environment during the relative navigation usually leads to filter divergence. To address this problem, a new nonlinear filter based on Gaussian-Student's-Multivariate K(GSK) mixture distribution is proposed in this paper. A Dirichlet stochastic mixture vector fusing Gaussian, Student's t, and Multivariate K distributions is introduced, thus proposing a GSK mixture distribution modeling measurement likelihood; then the Kullback-Leibler Divergence (KLD) of the true posteriori probability density function(PDF) and the approximate posteriori PDF are minimized by a variational Bayesian(VB) technique to solve for the state and parameter approximate a posteriori estimations, and finally a new nonlinear filter based on the GSK mixture distribution is derived for angles-only relative navigation in time-varying environments. Simulation outcomes indicate that the filter can realize state estimation in non-stationary states effectively with 45.16% higher estimation accuracy than the existing advanced filters.

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.

Intrinsic functional networks for distinct sources of error in visual working memory.

Visual working memory (VWM) is a core cognitive function wherein visual information is stored and manipulated over short periods. Response errors in VWM tasks arise from the imprecise memory of target items, swaps between targets and nontargets, and random guesses. However, it remains unclear whether these types of errors are underpinned by distinct neural networks. To answer this question, we recruited 80 healthy adults to perform delayed estimation tasks and acquired their resting-state functional magnetic resonance imaging scans. The tasks required participants to reproduce the memorized visual feature along continuous scales, which, combined with mixture distribution modeling, allowed us to estimate the measures of memory precision, swap errors, and random guesses. Intrinsic functional connectivity within and between different networks, identified using a hierarchical clustering approach, was estimated for each participant. Our analyses revealed that higher memory precision was associated with increased connectivity within a frontal-opercular network, as well as between the dorsal attention network and an angular-gyrus-cerebellar network. We also found that coupling between the frontoparietal control network and the cingulo-opercular network contributes to both memory precision and random guesses. Our findings demonstrate that distinct sources of variability in VWM performance are underpinned by different yet partially overlapping intrinsic functional networks.

Local neighborhood affects stem rehydration under drought: evidence from mixtures of European beech with two different conifers.

Mixed-species forests are, for multiple reasons, promising options for forest management in Central Europe. However, the extent to which interspecific competition affects tree hydrological processes is not clear. High-resolution dendrometers capture sub-daily variations in stem diameter; they can simultaneously monitor stem growth (irreversible changes in diameter) and water status (reversible changes) of individual trees. Using the information on water status, we aimed to assess potential effects of tree species mixture, expressed as local neighborhood identity, on night-time rehydration and water stress. We deployed 112 sensors in pure and mixed forest stands of European beech, Norway spruce, and Douglas fir on four sites in north-western Germany, measuring stem diameter in 10-minute intervals for a period of four years (2019-2022). In a mixture distribution model, we used environmental variables, namely soil matric potential, atmospheric vapor pressure deficit, temperature, precipitation, and neighborhood identity to explain night-time rehydration, measured as the daily minimum tree water deficit (TWDmin). TWDmin was used as a daily indicator of water stress and the daily occurrence of sufficient water supply, allowing for stem growth (potential growth). We found that species and neighborhood identity affected night-time rehydration, but the impacts varied depending on soil water availability. While there was no effect at high water availability, increasing drought revealed species-specific patterns. Beech improved night-time rehydration in mixture with Douglas fir, but not in mixture with spruce. Douglas fir however, only improved rehydration at a smaller share of beech in the neighborhood, while beech dominance tended to reverse this effect. Spruce was adversely affected when mixed with beech. At species level and under dry conditions, we found that night-time rehydration was reduced in all species, but beech had a greater capacity to rehydrate under high to moderate soil water availability than the conifers, even under high atmospheric water demand. Our study gives new insights into neighborhood effects on tree water status and highlights the importance of species-specific characteristics for tree-water relations in mixed-species forests. It shows that drought stress of European beech can be reduced by admixing Douglas fir, which may point towards a strategy to adapt beech stands to climate change.

Weibull mixture estimation based on censored data with applications to clustering in reliability engineering

AbstractIt is the purpose of this paper to propose a novel clustering technique tailored to randomly censored data in reliability/survival analysis. It is based on an underlying mixture model of Weibull distributions and consists in estimating its parameters by means of a variant of the Expectation–Maximization method in the presence of random censorship. Beyond the description of the algorithm, model selection issues are addressed and we investigate its performance from an empirical perspective by applying it to possibly strongly censored (synthetic and real) survival data. The experiments carried out provide strong empirical evidence that our algorithm performs better than alternative methods standing as natural competitors in this framework.

Correlation analysis and joint probability density function model of wind pressures: Focusing on multivariate wind loads field on low-rise building under typhoon climate

The characteristics of the multivariate wind loads field on the roof are crucial to the wind-resistant design of low-rise buildings, which contain the correlation characteristics in space and probability characteristics in the time domain. This paper proposes a framework for constructing a Joint Probability Density Function (Joint PDF) model for a multivariate wind loads field. It provides a detailed correlation analysis for the first time. This paper employs wind pressure data collected from the roof of a low-rise building during Typhoon Muifa. It was found that the correlation becomes more robust with increasing roof pitch and the wind pressures are strongly correlated with a correlation coefficient exceeding 0.50 when the roof pitch is above 15°. The mixture distribution model is applied to the probability density function fitting procedure of wind pressure time series under typhoon climate, and the fitting effect is significantly better than other classical probability density functions. The optimal copula function is determined according to the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for estimating the Joint PDF. The results reveal that Gumbel-copula and Student-copula have the highest proportion in optimal copula functions, accounting for over 90% of the total copula functions. Then, a bivariate Joint PDF for wind pressures are established with the optimal copula function. Additionally, the comparison between measured bivariate Joint PDF and that constructed using copula functions verifies the accuracy of the proposed framework for constructing Joint PDFs. The Joint PDF of wind pressures can enhance the understanding of the stochastic characteristics of local wind load fields on roofs, and the correlation characteristics in space provide crucial references for improving the accuracy of wind load random field simulation and saving the cost of wind resistance design.

On the detection of population heterogeneity in causation between two variables: Finite mixture modeling of data collected from twin pairs.

Causal inference is inherently complex, often dependent on key assumptions that are sometimes overlooked. One such assumption is the potential for unidirectional or bidirectional causality, while another is population homogeneity, which suggests that the causal direction between two variables remains consistent across the study sample. Discerning these processes requires meticulous data collection through an appropriate research design and the use of suitable software to define and fit alternative models. In psychiatry, the co-occurrence of different disorders is common and can stem from various origins. A patient diagnosed with two disorders might have one recognized as primary and the other as secondary, suggesting the existence of two types of comorbidity within the population. For example, in some individuals, depression might lead to substance use, while in others, substance use could lead to depression. Identifying the primary disorder is crucial for developing effective treatment plans. This article explores the use of finite mixture models to depict within-sample heterogeneity. We begin with the Direction of Causation (DoC) model for twin data and extend it to a mixture distribution model. This extension allows for the calculation of the likelihood of each individual's data for the two alternate causal directions. Given twin data, there are four possible pairwise combinations of causal direction. Through simulations, we investigate the Direction of Causation Twin Mixture (mixCLPM) model's potential to detect and model heterogeneity due to varying causal directions.

Exploring Celestial Object Characteristics: An In-depth Analysis of Quasars, Stars, and White Dwarfs Using the Sloan Digital Sky Survey (SDSS) Dataset

This research utilizes the Sloan Digital Sky Survey (SDSS) dataset, examining 12,884 observations to explore quasars, stars, and white dwarf objects. Magnitude data and coordinates across five filter bands are analyzed, revealing unique features through statistical methods. The identification of 77,429 quasars with 15 dimensions enhances the dataset. Thorough analyses of stellar and white dwarf classes, coupled with visualization techniques, unveil variable relationships. Residual validation and Gaussian kernel density plots confirm significant class differences. Non-linear regression and a normal distribution mixture model depict complex variable relationships. A parallel coordinates plot aids in interpreting data patterns, while predictive modeling via regression exposes meaningful coefficients. Logistic regression effectively classifies astronomical objects in the SDSS training data. This research contributes to understanding celestial object characteristics, offering valuable insights for astronomers and astrophysicists in analyzing large-scale astronomical datasets.

Very short-term probabilistic and scenario-based forecasting of solar irradiance using Markov-chain mixture distribution modeling

This study investigates probabilistic and scenario-based forecasting of solar irradiance with Markov-chain mixture (MCM) distribution modeling, Persistence Ensemble (PeEn) and Climatology. Forecasts from MCM models with uniform and empirical emission distribution settings, respectively, are compared with PeEn and Climatology in terms of probabilistic forecasting performance. The MCM model is also extended with scenario generation capabilities and compared to scenario generation of the Climatology by means of Monte Carlo sampling. Forecasts were made on minute resolution normalized solar irradiance, i.e. the clear-sky index, from National Renewable Energy Laboratory and Swedish Meteorological and Hydrological Institute for two climatic regions: Oahu, Hawaii, USA and Norrköping, Sweden, respectively. Results show that the MCM models are neither necessarily the most reliable, nor the sharpest in terms of Prediction Normalized Average Width (PINAW), but they are the most accurate in terms of Continuous Ranked Probability Score (CRPS). MCM models with uniform and empirical emission distribution settings perform similar in the tested probabilistic forecasting metrics. In terms of scenario forecasting, MCM models with N=30 perform similar in probability distribution goodness-of-fit and autocorrelation Mean Absolute Error (MAE) and superior to N=2 and N=10 number of states. Mathematically, forecasts from the MCM model with empirical distribution setting are shown to correspond to PeEn and Climatology forecasts given special settings of the MCM model. Based on the conclusions, the suggestion is to use the MCM scenario forecast generator with uniform emission distribution setting as benchmark for scenario forecasts of very short-term solar irradiance.

Recursive copula-augmented space-correlated temporal very-short term spatiotemporal probabilistic forecasting of the clear-sky index

This study introduces a recursive model framework for augmenting separate temporal probabilistic forecasts of multiple correlated time-series with a copula correlation model. The model is applied to the Markov-chain mixture distribution (MCM) model to spatiotemporally forecast minute resolution normalized solar irradiance, c.f. the clear-sky index, derived from radiometer array measurements of Global Horizontal Irradiance (GHI) for 18 geographically adjacent stations at Oahu, Hawaii, USA. The results are evaluated by univariate and multivariate probabilistic forecast metrics in comparison with forecasts from purely temporal MCM, Climatology and the spatiotemporal Multivariate Persistence Ensemble (MuPEn) benchmark. Results show that the Climatology and the MuPEn forecasts are most reliable, while superiority in sharpness, based on Prediction Interval Normalized Average Width (PINAW), depends on forecast horizon. In terms of accuracy, measured with the univariate measure Continuous Ranked Probability Score (CRPS), the MCM model (with and without copula) forecasts are most accurate for the first two steps ahead forecasts, while the Climatology and MuPEn both have superior score for longer horizons. In terms of multivariate scores, the accuracy estimate Energy Score results for the forecasts are similar to the CRPS, while the Variogram Score, which takes into consideration the correlational structure of the multivariate time-series, is significantly improved by the copula-augmented MCM model compared to the univariate MCM model. The MuPEn model generated the lowest Variogram Score among all models. Tests with fewer stations and swapped training and test data gave similar model-to-model relative dynamics with variations in magnitude.

Tensile behaviour of pitting corroded steel bars: Laboratory investigation and probabilistic-based analysis

Localised corrosion in steel bars has been a long-standing issue in the durability of reinforced concrete structures, but a comprehensive scheme for the analysis of pitting corroded steel bars, especially with respect to the deformation capacity, is not currently available. In this study, the morphological characteristics of 27 pitting steel bars were captured using a 3D scanner. The measured data were used to establish the probability distribution model of the cross-sectional areas of the corroded bars. Uniaxial tensile tests were conducted, and the evolving deformation field of the corroded bars was recorded through Digital Image Correlation (DIC). Based on the 3D reconstructed model and DIC results, an analytical method for evaluating the mechanical properties of pitting steel bars was developed and validated. The results show that the two-component Gaussian mixture distribution model outperforms conventional unimodal distribution models. Comparison of the analytical results with experimental data demonstrates that the proposed procedure is capable of predicting not only the ultimate strength but also the gauge length-dependent ultimate strain of corroded bars. Additionally, there exists a strengthening effect in the ultimate stress at the critical sections and this effect should not be ignored for accurate predictions.

Rainfall frequency analysis using event-maximum rainfalls – An event-based mixture distribution modeling approach

Rainfall frequency analysis, an essential work for water resources management, is often conducted by using the annual maximum rainfall series. For rainfall stations with short record lengths and outliers presence, the use of annual maximum series for rainfall frequency analysis may yield design rainfall estimates of higher uncertainties. Moreover, for regions with cyclostationary climate patterns, the annual maximum rainfalls may be caused by different prevalent storm types, which differ in terms of their occurrence frequency and storm rainfall characteristics. In this study, we propose a novel event-maximum-rainfall-based mixture distribution modeling approach for rainfall frequency analysis. By considering the event-maximum rainfalls of individual storm events, the sample size for parameter estimation increases, and the uncertainty of design rainfall estimates reduces. Mixture distribution modeling enables a thorough investigation of the contributing probabilities of different storm types to the annual maximum rainfall. Through rigorous stochastic simulation, we demonstrated the superiority of the proposed approach over the conventional annual maximum rainfall approach. The proposed approach was applied to four representative rainfall stations in Taiwan, and the results revealed that the proposed approach is more robust than the conventional annual maximum rainfall approach. The results provide insights into the contributions of individual storm types to the annual maximum rainfall.

Offshore Wind Energy Assessment with a Clustering Approach to Mixture Model Parameter Estimation

In wind resource assessment research, mixture models are gaining importance due to the complex characteristics of wind data. The precision of parameter estimations for these models is paramount, as it directly affects the reliability of wind energy forecasts. Traditionally, the expectation–maximization (EM) algorithm has served as a primary tool for such estimations. However, challenges are often encountered with this method when handling complex probability distributions. Given these limitations, the objective of this study is to propose a new clustering algorithm, designed to transform mixture distribution models into simpler probability clusters. To validate its efficacy, a numerical experiment was conducted, and its outcomes were compared with those derived from the established EM algorithm. The results demonstrated a significant alignment between the new method and the traditional EM approach, indicating that comparable accuracy can be achieved without the need for solving complex nonlinear equations. Moreover, the new algorithm was utilized to examine the joint probabilistic structure of wind speed and air density in China’s coastal regions. Notably, the clustering algorithm demonstrated its robustness, with the root mean square error value being notably minimal and the coefficient of determination exceeding 0.9. The proposed approach is suggested as a compelling alternative for parameter estimation in mixture models, particularly when dealing with complex probability models.

Extreme wind turbine response extrapolation with the Gaussian mixture model

Abstract. The wind turbine extreme response estimation based on statistical extrapolation necessitates using a minimal number of simulations to calculate a low exceedance probability. The target exceedance probability associated with a 50-year return period is 3.8×10-7, which is challenging to evaluate with a small prediction error. The situation is further complicated by the fact that the distribution of the wind turbine response might be multimodal, and the extremes belong to a different statistical population than the main body of the distribution. Traditional theoretical probability distributions, mostly unimodal, may not be suitable for this task. The problem could be alleviated by applying a fit specifically on the tail of the distribution. Yet, a single unimodal distribution may not be sufficient for modeling diverse wind turbine responses, and an inappropriate distribution model could lead to significant prediction errors, including bias and variance errors. The Gaussian mixture model, a probabilistic and flexible mixture distribution model used extensively for clustering and density estimation tasks, is infrequently applied in the wind energy sector. This paper proposes using the Gaussian mixture model to extrapolate extreme wind turbine responses. The performance of two approaches is evaluated: (1) parametric fitting first and aggregation afterward and (2) data aggregation first followed by fitting. Different distribution models are benchmarked against the Gaussian mixture model. The results show that the Gaussian mixture model is capable of estimating a low exceedance probability with minor bias error, even with limited simulation data, and demonstrates flexibility in modeling the distributions of varying response variables.

Accounting for COVID-19-type shocks in mortality modeling: a comparative study

AbstractMortality shocks such as the one induced by the COVID-19 pandemic have substantial impact on mortality models. We describe how to deal with them in the period effect of the Lee–Carter model. The main idea is to not rely on the usual normal distribution assumption as it is not always justified. We consider a mixture distribution model based on the peaks-over-threshold method, a jump model, and a regime switching model and introduce a modified calibration procedure to account for the fact that varying amounts of data are necessary for calibrating different parts of these models. We perform an extensive empirical study for nine European countries, comparing the models with respect to their parameters, quality of fit, and forecasting performance. Moreover, we define five exemplary scenarios regarding the future development of pandemic-related mortality. As a result of our evaluations, we recommend the peaks-over-threshold approach for applications with a possibility of extreme mortality events.

Mixture models of probability distributions applied to rainfall in the state of Pernambuco, Brazil

The Brazilian semi-arid region is recurrently affected by the scarcity of water that marks the landscape as it prints periods of severe drought. Therefore, rainfall in this region greatly influences plant growth in regional hydrological processes that affect droughts or floods. It is of practical interest to assess how changes in rainfall patterns occur to anticipate hydrological dynamics. However, this is not easy as climate change reshapes global hydrology. Thus, assertive forecasting has become rare and imputed estimates of a reasonable degree of uncertainty. The objective of this work was to verify from the mixture of exponential, gamma, beta, log-normal, Weibull, normal, log-logistic, and exponentiated log-logistic distributions, which best fits the monthly rainfall of the state of Pernambuco, Brazil. The data used came from 133 monthly rainfall series (1950 to 2012) distributed over the state of Pernambuco. The Maximum Likelihood Method estimated all parameters. The Kolmogorov-Smirnov adherence test was applied at 5% probability to assess the adjustments. The least rejected distributions in the adherence test were Weibull, gamma, and beta; October presented the smallest number of distributions considered adequate to model monthly rainfall. More than 99% of the rain gauge stations had some adequate probabilistic distribution to model monthly rainfall in March. For most months, except for March, the Weibull distribution was the most suitable for modeling the monthly rainfall in the vast majority of rain gauge stations of Pernambuco.

Modeling Heterogeneity in Temporal Dynamics: Extending Latent State-Trait Autoregressive and Cross-lagged Panel Models to Mixture Distribution Models

Longitudinal models suited for the analysis of panel data, such as cross-lagged panel or autoregressive latent-state trait models, assume population homogeneity with respect to the temporal dynamics of the variables under investigation. This assumption is likely to be too restrictive in a myriad of research areas. We propose an extension of autoregressive and cross-lagged latent state-trait models to mixture distribution models. The models allow researchers to model unobserved person heterogeneity and qualitative differences in longitudinal dynamics based on comparatively few observations per person, while taking into account temporal dependencies between observations as well as measurement error in the variables. The models are extended to include categorical covariates, to investigate the distribution of encountered latent classes across observed groups. The potential of the models is illustrated with an application to self-esteem and affect data in patients with borderline personality disorder, an anxiety disorder, and healthy control participants. Requirements for the models’ applicability are investigated in an extensive simulation study and recommendations for model applications are derived.

White matter microstructure is associated with the precision of visual working memory

Visual working memory is critical for goal-directed behavior as it maintains continuity between previous and current visual input. Functional neuroimaging studies have shown that visual working memory relies on communication between distributed brain regions, which implies an important role for long-range white matter connections in visual working memory performance. Here, we characterized the relationship between the microstructure of white matter association tracts and the precision of visual working memory representations. To that purpose, we devised a delayed estimation task which required participants to reproduce visual features along a continuous scale. A sample of 80 healthy adults performed the task and underwent diffusion-weighted MRI. We applied mixture distribution modelling to quantify the precision of working memory representations, swap errors, and guess rates, all of which contribute to observed responses. Latent components of microstructural properties in sets of anatomical tracts were identified by principal component analysis. We found an interdependency between fibre coherence in the bilateral superior longitudinal fasciculus (SLF) I, SLF II, and SLF III, on one hand, and the bilateral inferior fronto-occipital fasciculus (IFOF), on the other, in mediating the precision of visual working memory in a functionally specific manner. We also found that individual differences in axonal density in a network comprising the bilateral inferior longitudinal fasciculus (ILF) and SLF III and right SLF II, in combination with a supporting network located elsewhere in the brain, form a common system for visual working memory to modulate response precision, swap errors, and random guess rates.

Pollution threshold assessment and risk area delineation of heavy metals in soils through the finite mixture distribution model and Bayesian maximum entropy theory

Pollution threshold and high-risk area determination for heavy metals is important for effectively developing pollution control strategies. Based on heavy metal contents in 3627 dense samples, an integrated framework combining the finite mixture distribution model and Bayesian maximum entropy (BME) theory was proposed to assess pollution thresholds, contamination levels and risk areas in an uncertain environment for soil heavy metals. The results showed that the average heavy metal contents were in the order Zn > Cr > Pb > Cu > Ni > As > Cd > Hg, with strong/moderate variation, and the corresponding pollution thresholds were 158.39, 84.29, 47.84, 49.75, 28.95, 18.01, 0.49 and 0.16 mg/kg, respectively. The thresholds were consistently greater than the Zhejiang Province backgrounds but lower than the national risk screening values, except for Cd. Approximately 27.9% of the samples were classified as contaminated at various levels, and they were distributed in the northern, northwestern and eastern regions of the study area. Additionally, 3.73%, 5.34% and 8.22% of the total area were classified as at-risk areas under confidence levels of 95%, 90% and 75%, respectively, through BME theory. The findings provide a reasonable classification system and suggestions for heavy metal pollution management and control.