Data-driven Approximation of Cumulative Distribution Function Using Particle Swarm Optimization based Finite Mixtures of Logistic Distribution

Gaussian mixtures (i.e. linear combinations of multivariate Gaussian probability densities) appear in numerous applications due to their universal ability to approximate multimodal probability distributions. Finding the modes (maxima) of a Gaussian mixture is a fundamental problem arising in many practical applications such as machine learning and digital image processing. In this paper, we propose a computationally efficient method for finding a significant mode of the Gaussian mixture. Such a mode represents an area of large probability, and it often coincides with the global mode of the mixture. The proposed method uses a Gaussian convolution in order to remove undesired local maxima of the Gaussian mixture and preserve its underlying structure. The transformation between the maximizers of the smoothed Gaussian mixture and the original one is formulated as a differential equation. A robust trust region method for tracing the solution curve of this differential equation is described. Our formulation also allows mixtures with negative weights or even negative values, which occur in some applications related to machine learning or quantum mechanics. The applicability of the method to mode-finding of Gaussian kernel density estimates obtained from experimental data is illustrated. Finally, some numerical results are given to demonstrate the ability of the method to find significant modes of Gaussian mixtures and kernel density estimates.

In order to have a reliable estimate of wind energy potential of a site, high frequency wind speed and direction data recorded for an extended period of time is required. Weibull distribution function is commonly used to approximate the recorded data distribution for estimation of wind energy. In the present study a comparison of Weibull function and Gaussian mixture model (GMM) as theoretical functions are used. The data set used for the study consists of hourly wind speeds and wind directions of 54 years duration recorded at Ijmuiden wind site located in north of Holland. The entire hourly data set of 54 years is reduced to 12 sets of hourly averaged data corresponding to 12 months. Authenticity of data is assessed by computing descriptive statistics on the entire data set without average and on monthly 12 data sets. Additionally, descriptive statistics show that wind speeds are positively skewed and most of the wind data points are observed to be blowing in south-west direction. Cumulative distribution and probability density function for all data sets are determined for both Weibull function and GMM. Wind power densities on monthly as well as for the entire set are determined from both models using probability density functions of Weibull function and GMM. In order to assess the goodness-of-fit of the fitted Weibull function and GMM, coefficient of determination (R2) and Kolmogorov-Smirnov (K-S) tests are also determined. Although R2 test values for Weibull function are much closer to ‘1’ compared to its values for GMM. Nevertheless, overall performance of GMM is superior to Weibull function in terms of estimated wind power densities using GMM which are in good agreement with the power densities estimated using wind data for the same duration. It is reported that wind power densities for the entire wind data set are 307 W/m2 and 403.96 W/m2 estimated using GMM and Weibull function, respectively.

Time-series anomaly detection is critical for maintaining the operational integrity of industrial systems, yet it is often hampered by the scarcity of labeled anomalous data and the complexity of non-stationary temporal patterns. This paper proposes GPTAD (Generative Pre-trained Anomaly Detector), a novel framework that integrates an end-to-end synthetic data generation pipeline with a specialized Dual-Path Transformer (DPT) architecture. Unlike conventional methods that process raw signals uniformly, the proposed DPT backbone utilizes a series decomposition mechanism to independently handle seasonal and trend components. The seasonal path employs a sparse attention-based Transformer to capture high-frequency dependencies, while the trend path utilizes a linear projection to maintain global consistency. To address the data scarcity problem, we introduce an improved synthesis method that leverages Gaussian Mixture Models (GMM) and Kernel Density Estimation (KDE) to generate high-fidelity pre-training datasets. Experimental results demonstrate that the proposed two-phase "pre-train and fine-tune" strategy significantly enhances reconstruction accuracy and anomaly detection sensitivity, outperforming existing state-of-the-art models on multiple industrial benchmarks.

Efficient similarity search for uncertain data is a challenging task in many modern data mining applications like image retrieval, speaker recognition and stock market analysis. A common way to model the uncertainty of data objects is using probability density functions in the form of Gaussian Mixture Models (GMMs), which have an ability to approximate arbitrary distribution. However, due to the possible unequal length of mixture models, the use of existing index techniques has serious problems for the objects modeled by GMMs. Either the techniques cannot handle GMMs or they have too many limitations. Hence, we propose a dynamic index structure, Gaussian Component based Index (GCI), for GMMs. GCI decomposes GMMs into the single, pairs, or n-lets of Gaussian components, stores these components into well studied index trees such as U-tree and Gauss-Tree, and refines the corresponding GMMs in a conservative but tight way. GCI supports both k-most-likely queries and probability threshold queries by means of Matching Probability. Extensive experimental evaluations of GCI demonstrate a considerable speed-up of similarity search on both synthetic and real-world data sets.

This paper proposes a probabilistic approach for statistical modeling of the wind power. The cumulative distribution functions (CDFs) of wind power show a number of variations and cannot be represented by any specific distribution. This paper represents the wind power CDFs based on Gaussian mixture model. A novel method of parameters obtainment of Gaussian mixture model is proposed to fit the actual CDF of wind power with more accuracy. An actual wind farm is studied to prove the accuracy of the proposed statistical modeling and method of parameters obtainment.

In many modern applications such as biometric identification systems, sensor networks, medical imaging, geology, and multimedia databases, the data objects are not described exactly. Therefore, recent solutions propose to model data objects by probability density functions(pdf). Since a pdf describing an uncertain object is often not explicitly known, approximation techniques like Gaussian mixture models(GMM) need to be employed. In this paper, we introduce a method for efficiently indexing and querying GMMs allowing fast object retrieval for arbitrary shaped pdf. We consider probability ranking queries which are very important for probabilistic similarity search. Our method stores the components and weighting functions of each GMM in an index structure. During query processing the mixture models are dynamically reconstructed whenever necessary. In an extensive experimental evaluation, we demonstrate that GMMs yield a compact and descriptive representation of video clips. Additionally, we show that our new query algorithm outperforms competitive approaches when answering the given probabilistic queries on a database of GMMs comprising about 100.000 single Gaussians.

To improve the performance of a conventional rescue equipment, we designed a novel multi-functional rescue end-effector comprising a separation mechanism and a grasping mechanism. The failure causes of the separation mechanism are analyzed, and three main failure modes are determined. Meanwhile, their performance functions are established as well. To solve the complex correlations among multiple failure modes, a novel mixed copula function that can capture the upper and lower tail dependencies simultaneously is proposed, which comprises the Gumbel copula and the Clayton copula. Further, a novel procedure that adopts the non-parametric kernel density estimation (NKDE) technique to estimate the probability density function (PDF) and the cumulative distribution function (CDF) for each performance function of the separation mechanism is proposed. The parameters of the mixed copula function are estimated using the method of maximum likelihood estimation. Subsequently, to further study the separation mechanism’s reliability during working, a time-variant parameter estimation method is introduced to build the time-variant mixed copula function. Finally, both the reliability and time-variant reliability of the separation mechanism are analyzed via the proposed integral methods. Comparisons with other analysis schemes show that the proposed integral method is not only effective but also efficient in analyzing the reliability of our target mechanism.

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.

Quantifying uncertainties in data and model: a prediction model for extreme rainfall events with application to Beijing subway.

This paper is devoted to study a new mixture model. This newly suggested model is a mixture of flexible Weibull extension and Burr XII distributions. We study the characteristics of the proposed distribution that includes probability density function, cumulative distribution function, reliability function and ageing behaviour through the failure rate function. Some statistical properties of the proposed distribution such as quantile function, the mode, median, r-th moments, the moment generating function and the order statistics are discussed. Moreover, the method of maximum likelihood is proposed for estimating the model parameters and also the Fisher's information matrix is derived. Finally, the performance of the proposed distribution is compared with existing distributions using applications of two real data sets to show the flexibility and efficiency of the new model.

We extend the State-Based Anomaly Detection approach by introducing precise and imprecise anomaly detectors using the Bayesian and credal combination operators, where evidences over time are combined into a joint evidence. We use imprecision in order to represent the sensitivity of the classification regarding an object being normal or anomalous. We evaluate the detectors on a real-world maritime dataset containing recorded AIS data and show that the anomaly detectors outperform previously proposed detectors based on Gaussian mixture models and kernel density estimators. We also show that our introduced anomaly detectors perform slightly better than the State-Based Anomaly Detection approach with a sliding window.

With the development of LiDAR technology and the availability of topo‐bathymetric data of high quality, new methods are emerging to describe and understand more accurately fluvial geomorphology. We explore the capacities of probability density functions (PDFs) of detrended dimensionless elevations extracted from digital elevation models (DEMs) to document channels morphology and configuration. These DEMs were obtained from a topo‐bathymetric LiDAR survey of 450 km performed on the middle and lower reaches of the largest river of France: the Loire River. The objective is to propose a thorough and complete method for PDF analysis in order to assess the vertical structuration of river reaches presenting various fluvial patterns. The analysis was conducted on the PDFs of active channel belt of four sites selected for the diversity of their morphological configurations (anabranching, braided, channelized and meandering). Results showed that PDFs appear to be specific of a reach and that they are useful for describing its vertical structure. The simplification with Gaussian mixture model (GMM) is effective in slicing PDF and the resulting Gaussians components are related to morphological units (MUs). The distance between Gaussian curves obtained using GMMs can be considered as an indicator of topo‐bathymetric connectivity between MUs (e.g., main and side channels) on sites presenting multichannel or channelized channel configurations. The study showed the high potential of PDF and GMM analysis in the field of fluvial geomorphology and suggests interesting developments for river management and restoration.

Scattering kernels are of paramount importance in modeling gas–surface interactions for rarefied gas flows. However, most existing empirical models need one or several accommodation coefficients (ACs) to be determined before applications. In this paper, an unsupervised machine learning technique, known as the Gaussian mixture (GM) model, is applied to establish a new scattering kernel based on the simulated data collected by molecular dynamics (MD) simulations. The main work is devoted to the scattering of diatomic molecules under thermal non-equilibrium conditions. Correspondingly, different MD simulations on the scattering process of nitrogen molecules from a platinum surface have been performed involving rotational and translational excitation. Here, we evaluate the performance of the GM and Cercignani–Lampis–Lord models against the MD approach by comparing the velocity correlation distributions and the relevant outgoing velocity probability density function as well as the computed ACs. The presented comparisons have demonstrated the superiority of the GM model in matching with MD results. Therefore, in the case of diatomic gases, the GM model can be employed as a promising strategy to derive the generalized boundary conditions.

In this paper we seek a Gaussian mixture model (GMM) of an n-variate probability density function. Usually the parameters of GMMs are determined in the original n-dimensional space by optimizing a maximum likelihood (ML) criterion. A practical deficiency of this method of fitting GMMs is its poor performance when dealing with high-dimensional data since a large sample size is needed to match the accuracy that is possible in low dimensions. We propose a method for fitting the GMM based on the projection pursuit strategy. This GMM is highly constrained and hence its ability to model structure in subspaces is enhanced, compared to a direct ML fitting of a GMM in high dimensions. Our method is closely related to recently developed independent factor analysis (IFA) mixture models. The comparisons with ML fitting of GMM in n-dimensions and IFA mixtures show that the proposed method is an attractive choice for fitting GMMs using small sizes of training sets.

The rise of machine learning (ML) has made ML models lucrative targets for adversarial attacks. One of these attacks is Beta Poisoning, which is a recently proposed training-time attack based on heuristic poisoning of the training dataset. While Beta Poisoning was shown to be effective against linear ML models, it was originally developed with a fixed Gaussian Kernel Density Estimator (KDE) for likelihood estimation, and its effectiveness against more advanced, non-linear ML models has not been explored. In this paper, we advance the state of the art in Beta Poisoning attacks by making three novel contributions. First, we extend the attack so that it can be executed with arbitrary KDEs and norm functions. We integrate Gaussian, Laplacian, Epanechnikov and Logistic KDEs with three norm functions, and show that the choice of KDE can significantly impact attack effectiveness, especially when attacking linear models. Second, we empirically show that Beta Poisoning attacks are ineffective against non-linear ML models (such as neural networks and multi-layer perceptrons), even with our extensions. Results imply that the effectiveness of the attack decreases as model non-linearity and complexity increase. Finally, our third contribution is the development of a discriminator-based defense against Beta Poisoning attacks. Results show that our defense strategy achieves 99% and 93% accuracy in identifying poisoning samples on MNIST and CIFAR-10 datasets, respectively.