Mixture Of Gamma Distributions Research Articles

Statistical structural inference from edge weights using a mixture of gamma distributions

Abstract The inference of reliable and meaningful connectivity information from weights representing the affinity between nodes in a graph is an outstanding problem in network science. Usually, this is achieved by simply thresholding the edge weights to distinguish true links from false ones and to obtain a sparse set of connections. Tools developed in statistical mechanics have provided particularly effective ways to locate the optimal threshold so as to preserve the statistical properties of the network structure. Thermodynamic analogies together with statistical mechanical ensembles have been proven to be useful in analysing edge-weighted networks. To extend this work, in this article, we use a statistical mechanical model to describe the probability distribution for edge weights. This models the distribution of edge weights using a mixture of Gamma distributions. Using a two-component Gamma mixture model with components describing the edge and non-edge weight distributions, we use the Expectation–Maximization algorithm to estimate the corresponding Gamma distribution parameters and mixing proportions. This gives the optimal threshold to convert weighted networks to sets of binary-valued connections. Numerical analysis shows that it provides a new way to describe the edge weight probability. Furthermore, using a physical analogy in which the weights are the energies of molecules in a solid, the probability density function for nodes is identical to the degree distribution resulting from a uniform weight on edges. This provides an alternative way to study the degree distribution with the nodal probability function in unweighted networks. We observe a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted and unweighted networks reveal an improved performance for inferring binary edge connections from edge weights.

Gamma Mixture Density Networks and their application to modelling insurance claim amounts

We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.

Inferring phenomenological models of first passage processes.

Biochemical processes in cells are governed by complex networks of many chemical species interacting stochastically in diverse ways and on different time scales. Constructing microscopically accurate models of such networks is often infeasible. Instead, here we propose a systematic framework for building phenomenological models of such networks from experimental data, focusing on accurately approximating the time it takes to complete the process, the First Passage (FP) time. Our phenomenological models are mixtures of Gamma distributions, which have a natural biophysical interpretation. The complexity of the models is adapted automatically to account for the amount of available data and its temporal resolution. The framework can be used for predicting behavior of FP systems under varying external conditions. To demonstrate the utility of the approach, we build models for the distribution of inter-spike intervals of a morphologically complex neuron, a Purkinje cell, from experimental and simulated data. We demonstrate that the developed models can not only fit the data, but also make nontrivial predictions. We demonstrate that our coarse-grained models provide constraints on more mechanistically accurate models of the involved phenomena.

Daily Precipitation Generation using a Hidden Markov Model with Correlated Emissions for the Potomac River Basin

AbstractA daily precipitation generator based on a hidden Markov model with Gaussian copulas (HMM‐GC) is constructed using remote sensing data from GPM‐IMERG for the Potomac river basin on the East Coast of the USA. Daily precipitation over the basin from 2001–2018 for the wet season months of July to September is modeled using a 4‐state HMM, and correlated precipitation amounts are generated from a mixture of Gamma distributions using Gaussian copulas for each state. Synthetic data from a model using a mixture of two Gamma distributions for the non‐zero precipitation is shown to replicate the historical data better than a model using a single Gamma distribution.

A general statistical model for waiting times until collapse of a system

The distribution of waiting times until the occurrence of a critical event is a crucial statistical problem across several disciplines in Science. In this work we present a statistical model in which a relevant quantity X accumulates until overcoming a threshold X∗, which defines the collapse. The obtained waiting time distribution is a mixture of gamma distributions, which in turn can be approximated as an effective gamma distribution.

On Mixtures of Gamma distributions, distributions with hyperbolically monotone densities and Generalized Gamma Convolutions (GGC)

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y ·X and Y/X are ...

Fitting Gamma Mixture Density Networks with Expectation-Maximization Algorithm

We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.

Performance analysis of one- and two-way relays for underwater optical wireless communications

In this paper, the amplify and forward (AF) cooperative technique for one- and two-way relays has been implemented for underwater optical wireless communication (UOWC). UOWC suffers from scattering, absorption, and turbulence effects. The distance of communication between UOWC devices is typically within the range of 100 m. So relay-based UWOC has been proposed to improve the performance of device-to-device (D2D) based UWOC by increasing the effective link range. Performance analysis of unidirectional and bidirectional relay-based systems has been carried out in terms of outage probability and average symbol error probability (ASEP) for log-normal underwater fading channels. The analytical results have been validated by means of Monte Carlo simulations. Closed form expressions for ASEP have been obtained by using a mixture of gamma distributions, which was not possible using log-normal distributions. It has been observed that bidirectional relays, even though they have a better data rate than unidirectional relays, suffer in terms of outage probability.

ФОРМУЛА ОБРАЩЕНИЯ ДЛЯ РАЦИОНАЛЬНЫХ ХАРАКТЕРИСТИЧЕСКИХ ФУНКЦИЙ ВЕРОЯТНОСТНЫХ РАСПРЕДЕЛЕНИЙ

The paper deals with the problem of clarifying the well-known inversion formulas for distribution functions, usually describing the increment of these functions. The validity of the corresponding inversion formulas for the distribution function π and their densities has been proved for the particular case of distributions with rational characteristic functions. The obtained formulas for distribution functions, which include additionally constant terms equal to 0.5, were not previously known. Functions of positively distributed random variables and quantities distributed over the entire axis have been considered separately. To test the hypothesis of fairness of the obtained treatment formula, including a previously unknown term equal to 0.5, in the general case there have been given examples of calculating distribution functions, whose characteristic functions are not considered as rational functions: for constant and uniform laws. The verification confirmed the objectiveness of the formulated hypothesis about the obtained validity of the inversion form for arbitrary distribution functions. It has also been shown that any distribution function and any density can be represented as a limit of a mixture of gamma distributions (distribution functions and densities), having shifts along the abscissa axis and, possibly, with altered signs of the arguments. The obtained result proves that the set of gamma distributions with shifted arguments is uniformly dense in the set of all distributions.

A new [formula omitted]-deformed parametric model for the size distribution of wealth

It has been pointed out by Patriarca et al. (2005) that the power-law tailed equilibrium distribution in heterogeneous kinetic exchange models with a distributed saving parameter can be resolved as a mixture of Gamma distributions corresponding to particular subsets of agents. Here, we propose a new four-parameter statistical distribution which is a κ-deformation of the Generalized Gamma distribution with a power-law tail, based on the deformed exponential and logarithm functions introduced by Kaniadakis (2001). We found that this new distribution is also an extension to the κ-Generalized distribution proposed by Clementi et al. (2010), with an additional shape parameter ν, and properly reproduces the whole range of the distribution of wealth in such heterogeneous kinetic exchange models. We also provide various associated statistical measures and inequality measures.

МОДЕЛИРОВАНИЕ МНОГОКАНАЛЬНЫХ СИСТЕМ МАССОВОГО ОБСЛУЖИВАНИЯ КАК КОМПОНЕНТОВ СОЦИОТЕХНИЧЕСКИХ СИСТЕМ

For a multichannel queuing system, in which all calls have individual characteristics of arrival and maintenance in accordance with their characteristics in the required socio-technical system and switching from one call to another are performed in accordance with a specified switching function determined by the resolution policy adopted in the socio-technical system of conflict situations, recurrence relations have been obtained for the queue lengths, for the list of free instrument changes, for the list of call numbers waiting for service, and for a number of other characteristics at successive call termination services. The procedure of sequential calculation of all specified characteristics of the system is described taking into account their internal interrelation. In accordance with this procedure, in a strictly defined sequence, eleven characteristics of the system are calculated on the basis of recurrence relations. To increase the efficiency of the process of practical implementation of the modeling procedure, it is proposed to replace the initial distribution functions of random variables by their approximate values, which are mixtures of gamma distributions whose values can be calculated significantly faster than the values of the initial distributions. The problem of finding a set of exponential distributions for a given simulated distribution function is formalized, the mixture of which approximates the distribution function with a given accuracy.

AGGREGATION OF DEPENDENT RISKS IN MIXTURES OF EXPONENTIAL DISTRIBUTIONS AND EXTENSIONS

AbstractThe distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modelled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al., 2016), gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

New Results on the Fluctuating Two-Ray Model With Arbitrary Fading Parameters and Its Applications

The fluctuating two-ray (FTR) fading model provides a much better fit than other fading models for small-scale fading measurements in millimeter wave communications. In this paper, using a mixture of gamma distributions, new exact analytical expressions for the probability density and cumulative distribution functions of the FTR distribution with arbitrary fading parameters are presented. Moreover, the performance of digital communication systems over the FTR fading channel is evaluated in terms of the channel capacity and the bit error rate. The interaction between channel fading parameters and system performance is further investigated. Our newly derived results extend and complement previous knowledge of the FTR fading model.

Quantification of Muscle Tissue Properties by Modeling the Statistics of Ultrasound Image Intensities Using a Mixture of Gamma Distributions in Children With and Without Cerebral Palsy.

To investigate whether quantitative ultrasound (US) imaging, based on the envelope statistics of the backscattered US signal, can describe muscle properties in typically developing children and those with cerebral palsy (CP). Radiofrequency US data were acquired from the rectus femoris muscle of children with CP (n = 22) and an age-matched cohort without CP (n = 14) at rest and during maximal voluntary isometric contraction. A mixture of gamma distributions was used to model the histogram of the echo intensities within a region of interest in the muscle. Muscle in CP had a heterogeneous echo texture that was significantly different from that in healthy controls (P < .001), with larger deviations from Rayleigh scattering. A mixture of 2 gamma distributions showed an excellent fit to the US intensity, and the shape and rate parameters were significantly different between CP and control groups (P < .05). The rate parameters for both the single gamma distribution and mixture of gamma distributions were significantly higher for contracted muscles compared to resting muscles, but there was no significant interaction between these factors (CP and muscle contraction) for a mixed-model analysis of variance. Ultrasound tissue characterization indicates a more disorganized architecture and increased echogenicity in muscles in CP, consistent with previously documented increases in fibrous infiltration and connective tissue changes in this population. Our results indicate that quantitative US can be used to objectively differentiate muscle architecture and tissue properties.

Products of ratios of gamma functions — An application to the distribution of the test statistic for testing the equality of covariance matrices

An expansion of the ratio of two gamma functions is used to obtain an adequate representation for the product of ratios of gamma functions which enables the derivation of approximating expansions for the distribution of the test statistic used for testing the equality of several covariance matrices. These asymptotic approximations are based on mixtures of gamma distributions and are obtained using a new expression of the Fourier transform of the density function of the logarithm of the test statistic and also on a matching moments technique. Numerical studies show that these approximations are simple, precise and easy to implement. A computational module, developed for Mathematica software, is presented together with an example of application.

The gamma shape mixture model and influence of sample-unit size on estimation of tree diameter distributions: Forest modelling

The gamma shape mixture (GSM) model is based on a mixture of gamma distributions. A general Bayesian approach for estimating the unknown parameters of the GSM model was employed. Tree diameter at breast height (DBH) sets of complex and stratified forests are ideal data sources for testing the usefulness of theoretical distributions applied to modelling and simulating strongly differentiated data sets. The GSM model was useful in fitting these DBH structures. During the identification of homogenous forest patches of the similar DBH structures one should select sample plots of at least 0.15 ha in area, with a minimum of 30 trees. The use of GSM model has the potential to facilitate the presentation of not only DBH structures, but also other empirical distributions, describing e.g. various biological stages and processes. The GSM model can be employed especially for modelling multimodal, asymmetrical and heavy-tailed survey data.

Capacity Analysis of Two-Wave With Diffuse Power Fading Channels Using a Mixture of Gamma Distributions

In this letter, we analyze the capacity of a class of generalized fading channels represented by two-wave with diffuse power (TWDP) fading models. We obtain a new representation of TWDP fading model as a mixture of gamma distributions and apply the new representation to the approximate evaluation of TWDP channel capacity under different transmission policies. We also present upper bounds on approximation errors.

Skellam process with resetting: a neural spike train model.

This paper introduces the Skellam process with resetting. Resetting is a modification that accommodates the modeling of neural spike trains. We show this as a biologically plausible model, which codes the information content of neural spike trains with three, potentially, time-varying functions. We show that the interspike interval distribution under this model follows a mixture of gamma distributions, a flexible class covering a wide range of commonly used models. Through simulation studies and the analyses of connected retinal ganglion and lateral geniculate nucleus cells, we evaluate the performance of this model. Copyright © 2016 John Wiley & Sons, Ltd.

Ticking Away the Moments: Timing Regularity Helps to Better Predict Customer Activity

Accurate predictions of a customer’s activity status and future purchase propensities are crucial for managing customer relationships. This article extends the recency–frequency paradigm of customer-base analysis by integrating regularity in interpurchase timing in a modeling framework. By definition, regularity implies less variation in timing patterns and thus better predictability. Whereas most stochastic customer behavior models assume a Poisson process of “random” purchase occurrence, allowing for regularity in the purchase timings is beneficial in noncontractual settings because it improves inferences about customers’ latent activity status. This especially applies to those valuable customers who were previously very frequently active but have recently exhibited a longer purchase hiatus. A newly developed generalization of the well-known Pareto/NBD model accounts for varying degrees of regularity across customers by replacing the NBD component with a mixture of gamma distributions (labeled Pareto/GGG). The authors demonstrate the impact of incorporating regularity on forecasting accuracy using an extensive simulation study and a range of empirical applications. Even for mildly regular timing patterns, it is possible to improve customer-level predictions; the stronger the regularity, the greater the gain. Furthermore, the cost in terms of data requirements is marginal because only one additional summary statistic, in addition to recency and frequency, is needed that captures historical transaction timing. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mksc.2015.0963 .

Analysis of UWB communication over IEEE 802.15.3a channel by superseding lognormal shadowing by Mixture of Gamma distributions

In this paper, we calculate the bit error rate (BER) of the ultra-wide band (UWB) wireless communication system over IEEE 802.15.3a channel model by superseding the lognormal (LN) shadow fading distribution to the Mixture of Gamma (MG) distributions. In general, shadow fading is the effect of random fluctuation of received signal power around the mean path loss and it is modeled as LN distribution. In this work, we approximate the LN shadow fading distribution to the MG distributions, because of intractable, indefinite mathematical expression of the characteristic function (CF) and moment generating function (MGF) of the LN distribution, which are required to evaluate BER of the UWB communication system. To estimate the approximate MG distributions parameters, we use rth moment algorithm with least square non-linear curve fitting criteria. This work employs the L-fingers RAKE receiver which is needed to collect the total energy at the receiver in the UWB system. The results show that 5-MG distributions is the good fit for LN shadow fading distribution of the UWB communication system. The proposed analytical BER is closely matched to the simulated BER for all the four IEEE 802.15.3a channel models. It verifies the accuracy of the approximation considered in this BER analysis.