Gaussian Mixture Distribution Model Research Articles

Statistical mirroring: A robust method for statistical dispersion estimation

This study introduces statistical mirroring as an innovative approach to statistical dispersion estimation, drawing inspiration from the Kabirian-based isomorphic optinalysis model, aimed at enhancing robustness and mitigating biases in estimation methods. Beyond scale-invariant characteristics, the proposed estimators emphasize scaloc-invariant robustness, thereby addressing a critical gap in dispersion estimation. By highlighting statistical meanic mirroring, alongside other forms of proposed statistical mirroring, the study underscores the adaptability and customization potential. Through extensive Monte Carlo simulations and real-life applications, in comparison with classical estimators, the results of the performance evaluation of the proposed estimators demonstrate robustness, efficiency, and transformations-invariance. The research offers a paradigm shift in addressing longstanding challenges in dispersion estimation, offering a new category of dispersion estimation and increased resistance to outliers. Notable limitations include selecting and evaluating the proposed statistical meanic mirroring under Gaussian and Gaussian mixture model distributions. This research paper significantly contributes to statistical methodologies, offering avenues for expanding knowledge in dispersion estimation. It recommends further exploration of proposed estimators across various statistical mirroring types and encourages comparative studies to establish their effectiveness, thereby advancing statistical knowledge and tools for precise data analysis.•The proposed methodology involves preprocessing transformations, statistical mirror design, and optimization to transform a univariate set into a bivariate one, facilitating the fitting of an isomorphic optinalysis model.•Estimators rely on a foundational bijective mapping of isoreflective pairs, deducing the probability of proximity or deviation from any defined center. This contrasts with classical estimators that utilize average or median deviations from a mean or median center.

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.

Optimization of Protection Level of GBAS with Gaussian Mixture Model

The Gaussian mixture model (GMM) is commonly used to model the heavy tail of the ground-based augmentation system (GBAS) range error distribution. In practice, Gaussian over-bounding based on a GMM is used to over-bound the heavy tail of the ranging errors, but the GBAS protection levels (PLs) based on the Gaussian over-bounding tend to be overestimated. Based on the idea of solution separation and overcoming the shortcoming of its direct reference to GBAS, this paper analyses the constraint conditions and objective functions of the optimal protection level based on solution separation under a GMM distribution, and proposes that multi-hypothesis solution set classification can effectively reduce the computational complexity. At the same time, least squares optimization and dynamic allocation of integrity risk are used to further reduce the protection level. This paper verifies the validity of the parameters of the GMM based on actual airport GBAS data, performs simulation verification of the typical scenarios of CAT I and CAT II/IIIa global GBAS under the Beidou 3 constellation, and analyses the performance improvement effect under different solution set traversal depths. The results show that when the traversal depths of CAT I and CAT II/IIIa are 4 and 6, the vertical protection level component of the ground ranging error is reduced by 14% and the total vertical protection level is reduced by 10%.

Acoustic emission analysis of crack type identification of corroded concrete columns under eccentric loading: A comparative analysis of RA-AF method and Gaussian mixture model

Crack location and type identification of reinforced concrete columns in service is of great interest to researchers, since it influences the decision-making of maintenance and retrofit. Due to the existing of load and corrosion, the complex stress state leads to complicated crack development which has been a challenge for researchers to determine the crack type. Acoustic emission (AE) method gives a promising solution to non-destructively locate and identify the crack position and type. In this work, corroded reinforced concrete columns were subjected to compression test with different eccentric degree. AE signals during the loading process were recorded. The traditional RA-AF method and a newly proposed Gaussian mixture model (GMM) method were applied to process the data for crack type identification. It was found that due to the large data set, RA-AF cannot identify the crack type in between the shear crack and the tensile crack types. The probability density distribution of GMM result gives an attractive offer to determine the crack type. A simple 4-step method based on GMM plot is proposed to identify the crack type which can calculate the ratio of shear crack and tensile crack in collected signals.

E-Commerce Precision Marketing and Consumer Behavior Models Based on IoT Clustering Algorithm

This article aims to study e-commerce precision models and consumer behavior models based on clustering algorithms, and at the same time conduct detailed research on the Gaussian mixture distribution algorithm, consumer behavior and model construction, and precision marketing strategies in the clustering algorithm. First, a lot of analysis and demonstration of precision marketing strategies and the construction of consumer behavior models are carried out, and then the clustering algorithm-based electronic some experiments were carried out on the application of commercial precision marketing methods and consumer behavior models. The experimental results show that the precision marketing method using the clustering algorithm is more in line with the development of modern e-commerce. The application of the algorithm in the precision marketing methods of enterprises and consumer behavior models has promoted the vigorous development of enterprises, making the sales volume of enterprises reach 9.8 %.

Strong Representation Learning for Weakly Supervised Object Detection

To solve the problem that the feature maps generated by feature extraction network of traditional weakly supervised learning object detection algorithm is not strong in feature, and the mapping relationship between feature space and classification results is not strong, which restricts the performance of object detection, a weakly supervised object detection algorithm based on strong representation learning is proposed in this paper. Due to enhance the representation ability of feature maps, the algorithm weighted the channels of feature maps according to the importance of each channel, to strengthen the weight of crucial feature maps and ignore the significance of secondary feature maps. Meanwhile, a Gaussian Mixture distribution model with better classification performance was used to design the object instance classifier to enhance further the representation of the mapping between feature space and classification results, and a large-margin Gaussian Mixture (L-GM) loss was designed to increase the distance between sample categories and improve the generalization of the classifier. For verifying the effectiveness and advancement of the proposed algorithm, the performance of the proposed algorithm is compared with six classical weakly supervised target detection algorithms on VOC datasets. Experiments show that the weakly supervised target detection algorithm based on strong representation learning has outperformed other classical algorithms in average accuracy (AP) and correct location (CorLoc), with increases of 1.1%~14.6% and 2.8%~19.4%, respectively.

Developing an innovative bimodal model to characterize the dynamic radar cross section of aircrafts

The distribution of aircraft dynamic radar cross section (RCS) varies at different course short-cuts and heights and can behave as a bimodal shape. Classical models, such as the Weibull distribution, the chi-square distribution, and lognormal distribution, fit poorly with bimodal RCS distributions and are unable to accurately describe the statistical characteristics of aircraft dynamic RCS. In order to overcome the shortcoming of the classical models, we propose an exponential polynomial distribution model as the probability density function for studying the RCS statistical characteristics. The model parameters were acquired through the linear least-square method. Based on the RCS statistical data, the proposed distribution model was validated through simulation experiments. We calculated the fitting error between the new model and the RCS statistical distribution and determined the optimal exponential polynomial distribution model. We then computed the coefficients of the determination to evaluate the proposed model. The Kolmogorov–Smirnov test showed that the proposed approach yielded smaller values compared to conventional distribution models. For all the analyzed cases, the exponential polynomial model can fit very well against the RCS distribution data with bimodal shape, and the fitting effect is significantly better than the results of the three classical models. The advantages of the proposed model are further demonstrated by comparing it with the Gaussian mixture distribution model from fitting error, determination coefficient, goodness-of-fit, mean error and variance error. The results suggest that the exponential polynomial distribution model provides a more effective alternative in describing the fluctuation characteristics of aircraft dynamic RCS at the different course short-cuts and heights. Given the advantages exhibited by the exponential polynomial distribution model in this study, the methodology and findings in this study can be used to improve in early warning detection of radar targets and as a theoretical basis to enhance the overall performance of radars. 2009 Elsevier Ltd. All rights reserved.

A New 40Ar/39Ar Analysis Method of Volcanoclastic Strata to Determine Eruption Periods—Example of Xintaimen, China

The Jehol Biota is famous for its diversity of Early Cretaceous fossils preserved under large amounts of pyroclastic materials. However, the precise determination on the ages of volcanoclastic strata is challenging, because of their complex provenance. The solution to this problem may lie in the greatly improved geochronological methods, such as the 40Ar/39Ar geochronology. In this study, we use the laser fusion 40Ar/39Ar dating method to date volcanoclastic samples and apply statistical analyses of the acquired apparent ages to determine the volcanic eruption periods. From the 12 volcanoclastic samples analyzed in this study, we obtained 262 apparent ages that vary from 85.5 to 153.6 Ma. By fitting these apparent ages with a Gaussian mixture distribution model, we find three age clusters that are normally distributed, with each cluster statistically representing an eruption period. The mean values of the three normal distributions are 90.37±2.72 (1σ), 109.89±4.94, and 125.26±10.01 Ma. A comparison of our results with a statistical analysis of literature data indicates that volcanic activities occurred over a wide spatial scale across western Liaoning Province during the two older periods, whereas during the last time period, eruptions occurred more locally and belonged to the final stage of volcanic eruptions in the basin. The method and results reported in this study provide a good example of (1) dating volcanoclastic materials using the 40Ar/39Ar method to determine the time periods of volcanic activities and (2) investigating the stratigraphic correlation in complex volcanoclastic basins.

An Objective Method for Probabilistic Forecasting of Multimodal Kuroshio States using Ensemble Simulation and Machine Learning

AbstractThis paper presents a method for detecting the ensemble means, spreads, and occurrence probabilities for each of the multiple Kuroshio states. This is accomplished by classifying the forecasts of the ensemble members with a Gaussian mixture distribution model, a machine learning method. Ensemble simulations with 80 members are conducted to reproduce possible occurrences of the multiple Kuroshio states, targeting the large meander event in 2017. To test its performance, first, the method is applied for the southernmost latitude, a conventional index that represents meander intensity. The results show that the Kuroshio initially taking the nearshore nonlarge meander state bifurcates into the large meander and offshore nonlarge meander states, which occur with similar probabilities. Both developments are accompanied by positive potential energy extraction rates, consistent with baroclinic instability. As a more objective approach, the method is then applied for the dominant modes derived from empirical orthogonal function (EOF) analysis of the sea surface height field in the entire Kuroshio region. Importantly, almost identical results can be achieved. In particular, the bimodality between the large meander and nonlarge meander is shown to appear on the axis of the first EOF mode. From a mathematical perspective, this mode can be interpreted as the singular vector which grows most rapidly following the time-evolution operator. Finally, the multimodality of the Kuroshio is reinterpreted as a phase transition phenomenon where the nearshore nonlarge meander constitutes the basic state.

Temporal and Spatial Characteristics of Multidimensional Extreme Precipitation Indicators: A Case Study in the Loess Plateau, China

Extreme precipitation can seriously affect the ecological environment, agriculture, human safety, and property resilience. A full-scale and scientific assessment in extreme precipitation characteristics is necessary for water resources management and providing decision-making support to mitigate the potential losses brought by extreme precipitation. In the present study, a multidimensional risk assessment framework is developed to investigate the spatial–temporal changes in different extreme precipitation indicators. The Gaussian mixture model (GMM) is applied to fit the distribution for each indicator and carry out single index risk assessment. The joint probabilistic features of multiple extreme indicators can be explored through coupling the GMM distributions into copulas. In addition, the moving window approach and the Mann–Kendall test are integrated to examine non-stationary risks (evaluated by “AND”, “OR”, and Kendall return periods) of multidimensional indicators along with their changing trends and significance. The proposed assessment framework is applied to the Loess Plateau, China. Four extreme precipitation indicators are characterized: the amount (P95), the number of days (D95), the intensity (I95), and the proportion (R95) of extreme precipitation. The spatial–temporal changes of these indicators and their multidimensional combinations (including six two-dimensional and three three-dimensional combinations) are fully identified and quantitatively evaluated.

Crash Risk Assessment of Off-Ramps, Based on the Gaussian Mixture Model Using Video Trajectories

The focus of this paper is the crash risk assessment of off-ramps in Xi’an. The time-to-collision (TTC) is used for the measurement and cross-comparison of the crash risk of each location. Five sites from the urban expressway in Xi’an were selected to explore the TTC distribution. An unmanned aerial vehicle and a camera were used to collect traffic flow data for 20 min at each site. The parameters, including speed, deceleration rate, truck percentage, traffic volume, and vehicle trajectories, were extracted from video images. The TTCs were calculated for each vehicle. The Gaussian mixture model (GMM) was proposed to predict the TTC probability density functions (PDFs) and cumulative density functions (CDFs) for five sites. The Kolmogorov–Smirnov (K-S) test indicated that the samples followed the estimated GMM distribution. The relationship between the crash risk level and influencing factors was studied by an ordinal logistic regression model and a naive Bayesian model. The results showed that the naive Bayesian model had an accuracy of 86.71%, while the ordinal logistic regression model had an accuracy of 84.81%. The naive Bayesian model outperformed the ordinal logistic regression model, and it could be applied to the real-time collision warning system.

Probability Distribution of the Hull Motion and Mooring Line Tension of Two Floating Systems

The statistical and distribution characteristics of the responses of a floater and its mooring lines are essential in designing floating/mooring systems. In general, the dynamic responses of offshore structures obey a Gaussian distribution, assuming that the structural system, and sea loads are linear or weakly nonlinear. However, mooring systems and wave loads are considerably nonlinear, and the dynamic responses of hull/mooring systems are non-Gaussian. In this study, the dynamic responses of two types of floaters, semi-submersible and spar platforms, and their mooring lines are computed using coupled dynamic analysis in the time domain. Herein, the statistical characteristics and distributions of the hull motion and mooring line tension are discussed and compared. The statistical distributions of the dynamic responses have strong non-Gaussianity and are unreasonably fitted by a Gaussian distribution for the two floating and mooring systems. Then, the effects of water depth, wave parameters, and low-frequency and wave-frequency components on the non-Gaussianity of the hull motion, and mooring line tension are investigated and discussed. A comparison of the statistical distributions of the responses with various probability density functions, including the Gamma, Gaussian, General Extreme Value, Weibull, and Gaussian Mixture Model (GMM) distributions, shows that the GMM distribution is better than the others for characterizing the statistical distributions of the hull motion, and mooring line tension responses. Furthermore, the GMM distribution has the best accuracy of response prediction.

Hydrological Uncertainty Processor (HUP) with Estimation of the Marginal Distribution by a Gaussian Mixture Model

Uncertainty assessments of hydrological prediction results can reflect additional hydrological information and reveal important hydrological characteristics of river basins, which is of great significance to disaster prevention and reduction. The hydrological uncertainty processor (HUP), which is a key part of the Bayesian forecasting system (BFS), has derived a variety of methods for hydrological uncertainty forecasting. The HUP allows for any form of marginal distributions of hydrological data and does not require a unified estimation structure for the marginal distribution function. The Gaussian mixture model (GMM) is a probability distribution estimation model that can approximate any probability distribution with arbitrary precision. In this paper, the GMM was used to estimate the marginal distribution of observed and modelled data, and this method is called HUP-GMM. The uncertainty of river discharge at the Yichang hydrological station on the main stem of the Yangtze River in China is predicted by the HUP-GMM. The Weibull and Gamma distributions, which are commonly used hydrological probability distributions, are compared to analyse the performance of the GMM. In June, when the measured flow h3 is 13,850 m3/s and the GMM, Gamma and Weibull distributions are used, the prior probabilities are 1.63E-04, 1.05E-04 and 9.50E-05 and the posterior probabilities are 2.57E-04, 1.61E-04 and 1.38E-04, respectively. In September, when the measured flow h3 is 35,400 m3/s and the GMM, Gamma and Weibull distributions are used, the prior probabilities are 5.98E-05, 2.21E-05 and 2.18E-05 and the posterior probabilities are 1.64E-04, 9.15E-05 and 8.43E-05, respectively. The results show that the performance of the uncertainty estimation of the prior and posterior probability distributions in the HUP-GMM has been improved.

Natural Image Quality Assessment Based on Visual Biological Cognitive Mechanism

With the focus of the main problems in no-reference natural image quality assessment (NR-IQA), the researchers propose a more universal, efficient and integrated resolution based on visual biological cognitive mechanism. First, the authors bring up an inspiring visual cognitive computing model (IVCCM) on the basis of visual heuristic principles. Second, the authors put forward an asymmetric generalized gaussian mixture distribution model (AGGMD), and the model can describe the probability distribution density of the images more precisely. Third, the authors extract the quality-aware multiscale local invariant features (QAMLIF) statistic and perceptive from natural images and form quality-aware uniform features descriptors (QAUFD) based on clustering and encoding the visual quality features. Fourth, the authors build topic semantic model and realize the resolution with Bayesian inference with IVCCM, AGGDM and QAUFD to implement NR-IQA. Theoretical research and experimental results show that the proposed resolution perform better with biological cognitive mechanism.

Emotional speech analysis using harmonic plus noise model and Gaussian mixture model

Extracting the valuable information from the emotional speech is one of the major challenges in the areas of emotion recognition and human-machine interfaces. Most of the research in emotion recognition is based on the analysis of fundamental frequency, energy contour, duration of silence, formant, Mel-band energies, linear prediction cepstral coefficients, and Mel frequency cepstral coefficients. It was observed that emotion classification using sinusoidal features perform better as compared to the linear prediction and cepstral features. Harmonic models are considered as a variant of the sinusoidal model. In order to improve emotional speech classification rate and conversion of neutral speech to emotional speech, analysis using different harmonic features of emotional speech is a critical step. In this paper, investigations have been carried out using Berlin emotional speech database to analyze gender-based emotional speech using harmonic plus noise model (HNM) features and Gaussian mixture model (GMM). Analysis has been performed with the HNM features like pitch, harmonic amplitude, maximum voiced frequency and noise components. From the results, it can be observed that different emotional speech of male and female speakers can be represented with K components of GMM distribution. The optimal number of GMM components have been decided on the basis of Akaike information criterion (AIC) score.

Functional connectivity based parcellation of early visual cortices.

Human brain can be divided into multiple brain regions based on anatomical and functional properties. Recent studies showed that resting-state connectivity can be utilized for parcellating brain regions and identifying their distinctive roles. In this study, we aimed to parcellate the primary and secondary visual cortices (V1 and V2) into several subregions based on functional connectivity and to examine the functional characteristics of each subregion. We used resting-state data from a research database and also acquired resting-state data with retinotopy results from a local site. The long-range connectivity profile and three different algorithms (i.e., K-means, Gaussian mixture model distribution, and Ward's clustering algorithms) were adopted for the parcellation. We compared the parcellation results within V1 and V2 with the eccentric map in retinotopy. We found that the boundaries between subregions within V1 and V2 were located in the parafovea, indicating that the anterior and posterior subregions within V1 and V2 corresponded to peripheral and central visual field representations, respectively. Next, we computed correlations between each subregion within V1 and V2 and intermediate and high-order regions in ventral and dorsal visual pathways. We found that the anterior subregions of V1 and V2 were strongly associated with regions in the dorsal stream (V3A and inferior parietal gyrus), whereas the posterior subregions of V1 and V2 were highly related to regions in the ventral stream (V4v and inferior temporal gyrus). Our findings suggest that the anterior and posterior subregions of V1 and V2, parcellated based on functional connectivity, may have distinct functional properties.

Bivariate Gaussian mixture–based equivalent linearization method for stochastic seismic analysis of nonlinear structures

SummaryTo address challenges in stochastic seismic analysis of nonlinear structures, this paper further develops a recently proposed Gaussian mixture–based equivalent linearization method (GM‐ELM). The GM‐ELM uses a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. Using properties of the Gaussian mixture model, GM‐ELM can decompose the non‐Gaussian response of a nonlinear system into multiple Gaussian responses of linear single–degree of freedom oscillators. With the set of the equivalent linear systems identified by GM‐ELM, response statistics as crossing rate and first‐passage probability can be computed conveniently using theories of linear random vibration analysis. However, the original version of GM‐ELM may lead to an inaccurate estimate because of the heuristic parameters of the linear system introduced to supplement insufficient information. To overcome this limitation and define unique equivalent linear systems, this paper proposes a further developed version of GM‐ELM, which uses a mixture of bivariate Gaussian densities instead of univariate models. Moreover, to facilitate the use of elastic response spectra for estimating the mean peak responses of a nonlinear structure, a new response spectrum combination rule is proposed for GM‐ELM. Two numerical examples of hysteretic structural systems are presented in this paper to illustrate the application of the bivariate GM‐ELM to nonlinear stochastic seismic analysis. The analysis results obtained by the bivariate GM‐ELM are compared with those obtained by the univariate GM‐ELM, the conventional equivalent linearization method, the tail equivalent linearization method, and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/yisangri/GitHub‐bGM‐ELM‐code.git

A Chance-Constrained Economic Dispatch Model in Wind-Thermal-Energy Storage System

As a type of renewable energy, wind energy is integrated into the power system with more and more penetration levels. It is challenging for the power system operators (PSOs) to cope with the uncertainty and variation of the wind power and its forecasts. A chance-constrained economic dispatch (ED) model for the wind-thermal-energy storage system (WTESS) is developed in this paper. An optimization model with the wind power and the energy storage system (ESS) is first established with the consideration of both the economic benefits of the system and less wind curtailments. The original wind power generation is processed by the ESS to obtain the final wind power output generation (FWPG). A Gaussian mixture model (GMM) distribution is adopted to characterize the probabilistic and cumulative distribution functions with an analytical expression. Then, a chance-constrained ED model integrated by the wind-energy storage system (W-ESS) is developed by considering both the overestimation costs and the underestimation costs of the system and solved by the sequential linear programming method. Numerical simulation results using the wind power data in four wind farms are performed on the developed ED model with the IEEE 30-bus system. It is verified that the developed ED model is effective to integrate the uncertain and variable wind power. The GMM distribution could accurately fit the actual distribution of the final wind power output, and the ESS could help effectively decrease the operation costs.

Equivalent linearization method using Gaussian mixture (GM-ELM) for nonlinear random vibration analysis

A new equivalent linearization method is developed for nonlinear random vibration analysis. The method employs a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. The parameters of the Gaussian mixture model are estimated by an optimization algorithm which requires a few rounds of dynamic analysis of the nonlinear system. Due to properties of the Gaussian mixture distribution model, the proposed Gaussian mixture based equivalent linearization method (GM-ELM) can decompose the non-Gaussian response of a nonlinear system into multiple Gaussian responses of linear single-degree-of-freedom oscillators. Using a probabilistic combination technique, the linear system of GM-ELM can provide the response probability distribution equal to the Gaussian mixture estimation of the nonlinear response distribution. Using the linear system of GM-ELM in conjunction with linear random vibration theories, response statistics such as the mean up-crossing rate and first-passage probability of the nonlinear system can be conveniently computed. In order to facilitate applications of GM-ELM in earthquake engineering practice, a response spectrum formula is also proposed to compute the mean peak response of the nonlinear system by using the elastic response spectra representing the peak responses of the linear single-degree-of-freedom oscillators. Finally, two numerical examples are presented to illustrate and test GM-ELM. The analysis results obtained from GM-ELM are compared with those obtained from the conventional ELM and Monte-Carlo simulation. The supporting source code and data are available for download at https://github.com/ziqidwang/GitHub-GM-ELM-code.git.

Inside-Out Propagation: Developing a Unified Model for the Interference in 5G Networks

This article presents a generalized indoor-to-outdoor signal propagation model based on ray-tracing and exhaustive measurements campaign with the aim of developing a unified model for the interference in fifth-generation (5G) networks with femtocells underlying macrocells. It considers large buildings and typical residential houses. The objective is to capture the complex effects of the architectural layout, fading, shadowing, and building materials as well as to develop a practical nonsite-specific composite model to be used in the interference analysis and planning of next-generation networks, where not all information about the position of the transmitters or the building layouts is available. To that end, many different floor plans are considered, and the positions of the indoor transmitter are changed randomly. The obtained results reveal that the signal attenuation in the outdoor environment can be best characterized by a Gaussian-mixture (GM) distribution model.