First-order Markov Chain Research Articles

Estimating stop over duration in the presence of trap-effects

Detection probability of individuals is increasingly taken into account during field monitoring schemes and in demographic models. Conversely, it is often taken for granted that trappability of animals will remain fairly constant and broadly similar between individuals present in a given area. However, animals may change their behaviour after being trapped. In this paper, we introduce a new hidden Markovian model to estimate stop over duration in the presence of trap-effects. This model combines nonhomogeneous Markovian states with semi-Markovian states in the non-observable state process, and simple distributions with first-order Markov chains as observation models. This model generalizes previously proposed models and enables the joint modeling of the time of residence and the trap effect. Two cases are considered, depending on whether or not emigration is time-dependent since arrival. We illustrate the latter with teal Anas crecca wintering in Camargue, Southern France and we demonstrate the importance of handling trap-effects.

The probability of potentially useful work days during the year in Londrina

The probabilities associated with the number of days suitable for working outdoors and for working with machinery in the field were estimated for the municipality of Londrina, state of Parana, Brazil. Days with less than 10 mm of precipitation were considered suitable for outdoor operations. The number of days per month and the number of ten-day periods satisfying this condition were modeled with a negative binomial distribution, and the probabilities were estimated at different levels. The days suitable for working with machinery in the field were specified as those days for which the values of water storage in the soil were at or below 90% of the available water capacity in the upper 30 cm of the soil and for which the precipitation was less than 5 mm. A first-order Markov chain was used to estimate the conditional probabilities of favorable days for working with machinery in the field. The results indicated that the best time to work outdoors in Londrina extends from April through September, with the most favorable period occurring between late July and early September. The highest probabilities of favorable days for working in the field occur during April and from June through August. The probabilities of two consecutive days favorable for working with machinery in the field are highest between June and August. These probabilities allow the determination of the needs for the performance of tasks involving mechanized equipment.

COMPLEX BEHAVIOR IN COUPLED CHAOTIC CIRCUITS RELATED BY INTERMITTENCY AND ITS MODELING METHODS

We discover a complex behavior when switching three different synchronization states in two coupled chaotic circuits related to intermittency. Two methods to model this interesting complex behavior are proposed. First, we model this complex behavior by a first-order Markov chain with four states. Second, the modeling method by using 1-D Poincaré map derived from one chaotic circuit is proposed. We confirm that both modeling methods agree very well with computer simulations. Finally, we discuss the characteristics of two modeling methods for calculation times, modeling accuracy and application of large-scale systems.

In silico prediction of mRNA poly(A) sites in Chlamydomonas reinhardtii

Accurately predicting polyadenylation [poly(A)] sites is important for defining the end of genes and understanding gene regulation mechanisms. Alternative polyadenylation (APA) has been demonstrated to play an important role in transcriptome diversity and regulating gene expression. To accurately predict poly(A) and APA sites in Chlamydomonas reinhardtii, a green alga that can be used to produce renewable energy, we proposed a novel model that integrated five methods for representing the features of these sites with a combined classifier. We presented a new grouping method based on pattern assembly to classify the poly(A) sites into four groups. We used five methods, involving the predicted RNA secondary structure, the term frequency-inverse document frequency weight, first-order Markov chain, pentamer ratio and a position weight matrix, to generate the feature space. We then developed a heuristic method to form the combined classifier by weighting multiple classifiers to predict poly(A) sites in each group. The high specificity and sensitivity of this model were demonstrated by testing the four groups of poly(A) sites and the intronic APA sites. The average prediction performance was approximately 8% higher than the performance of a previous prediction model. For the group without any conserved patterns, the prediction accuracy was 9% higher than for the accuracy with the previous technique. However, the prediction efficiency of this group was still significantly lower than that of the other groups, indicating the importance of identifying additional signal patterns for poly(A) site prediction. We also predicted the alternative poly(A) sites in introns with good accuracy. This prediction model was designed to be easily expanded with new classifiers or new features. Therefore, this model is applicable to new data or other species. Our model will be useful both in genome annotation because it predicts the end of a mature transcript and in genetic engineering because it enables researchers to eliminate undesirable poly(A) sites.

Particle Filters for Markov Switching Stochastic Volatility Models

This paper proposes an auxiliary particle filter algorithm for inference in regime switching stochastic volatility models in which the regime state is governed by a first-order Markov chain. We proposes an ongoing updated Dirichlet distribution to estimate the transition probabilities of the Markov chain in the auxiliary particle filter. A simulation-based algorithm is presented for the method which demonstrated that we are able to estimate a class of models in which the probability that the system state transits from one regime to a different regime is relatively high. The methodology is implemented to analyze a real time series: the foreign exchange rate of Australian dollars vs South Korean won.

Modelling the Arrival of Invasive Organisms via the International Marine Shipping Network: A Khapra Beetle Study

Species can sometimes spread significant distances beyond their natural dispersal ability by anthropogenic means. International shipping routes and the transport of shipping containers, in particular are a commonly recognised pathway for the introduction of invasive species. Species can gain access to a shipping container and remain inside, hidden and undetected for long periods. Currently, government biosecurity agencies charged with intercepting and removing these invasive species when they arrive to a county’s border only assess the most immediate point of loading in evaluating a shipping container’s risk profile. However, an invasive species could have infested a container previous to this point and travelled undetected before arriving at the border. To assess arrival risk for an invasive species requires analysing the international shipping network in order to identify the most likely source countries and the domestic ports of entry where the species is likely to arrive. We analysed an international shipping network and generated pathway simulations using a first-order Markov chain model to identify possible source ports and countries for the arrival of Khapra beetle (Trogoderma granarium) to Australia. We found Kaohsiung (Taiwan) and Busan (Republic of Korea) to be the most likely sources for Khapra beetle arrival, while the port of Melbourne was the most likely point of entry to Australia. Sensitivity analysis revealed significant stability in the rankings of foreign and Australian ports. This methodology provides a reliable modelling tool to identify and rank possible sources for an invasive species that could arrive at some time in the future. Such model outputs can be used by biosecurity agencies concerned with inspecting incoming shipping containers and wishing to optimise their inspection protocols.

Using Graphics Processors for High Performance SimRank Computation

Recently there has been a lot of interest in graph-based analysis. One of the most important aspects of graph-based analysis is to measure similarity between nodes in a graph. SimRank is a simple and influential measure of this kind, based on a solid graph theoretical model. However, existing methods on SimRank computation suffer from two limitations: 1) the computing cost can be very high in practice; and 2) they can only be applied on static graphs. In this paper, we exploit the inherent parallelism and high memory bandwidth of graphics processing units (GPU) to accelerate the computation of SimRank on large graphs. Furthermore, based on the observation that SimRank is essentially a first-order Markov Chain, we propose to utilize the iterative aggregation techniques for uncoupling Markov chains to compute SimRank scores in parallel for large graphs. The iterative aggregation method can be applied on dynamic graphs. Moreover, it can handle not only the link-updating problem but also the node-updating problem. We give the corresponding theoretical justification and analysis, propose three optimization strategies to further improve the computation efficiency, and extend the proposed algorithm to dynamic graphs. Extensive experiments on synthetic and real data sets verify that the proposed methods are efficient and effective.

Effectiveness of different footbathing frequencies using copper sulfate in the control of digital dermatitis in dairy cows

Two experiments were conducted to test the effectiveness of different footbathing frequencies using CuSO4 in the control of digital dermatitis (DD) in groups of dairy cows with low and high DD prevalence in an endemically infected herd. During the study, groups of cows walked through allocated footbath solutions after milking on 4 consecutive milkings each week, 2 wk, or monthly, depending on treatment. The footbath solutions were changed either after 200 cows had passed through the footbath or within 24h. All cows were scored weekly during milking for DD lesion stage on the hind feet using a 5-point nominal scale. A transition grade was assigned based on whether the DD lesions improved (1) or deteriorated or did not improve (0) from week to week, and was averaged for all cows in the group. Furthermore, from the longitudinal study data, all transitions between different DD lesion stages between the different time points were used in a discrete, first-order Markov chain (state transition matrix) model. In experiment 1, 70 Holstein-Friesian cows with DD lesions present on at least one of their hind feet were allocated to 1 of 2 footbath treatments for 14 wk: (1) 5% CuSO4 each week or (2) 5% CuSO4 every 2 wk. At the end of the study, no active DD lesions were observed in cows in either treatment group, but significantly more cows had no DD lesions (0.48±0.097 vs. 0.24±0.094) and fewer cows had lesions in the healing stage (0.52±0.104 vs. 0.77±0.090) for the weekly compared with the 2-wk footbathing regimen. The number of active DD lesions that the transition matrix model predicted over time was 3 and 8% for weekly and every 2 wk footbathing treatments, respectively. In experiment 2, 64 cows with no DD lesions on either of their hind feet were allocated to 1 of 2 footbath treatment regimens for 14 wk: (1) 5% CuSO4 every 2 wk or (2) 5% CuSO4 each month. At the end of the trial, significantly more cows had no DD (0.80±0.088 vs. 0.65±0.102) and fewer cows had DD lesions in the healing stage (0.20±0.088 vs. 0.35±0.102) in the every 2 wk regimen than in the monthly footbathing regimen. The number of active DD lesions that the transition matrix model predicted over time was low for both footbathing interventions (1 and 4%, respectively). Increasing the interval between footbaths does not appear the most appropriate mechanism for reducing CuSO4 usage.

ALTERNATIVE APPROACHES FOR ECONOMETRIC ANALYSIS OF PANEL COUNT DATA USING DYNAMIC LATENT CLASS MODELS (WITH APPLICATION TO DOCTOR VISITS DATA)

Cross-sectional latent class regression models, also known as switching regressions or hidden Markov models, cannot identify transitions between classes that may occur over time. This limitation can potentially be overcome when panel data are available. For such data, we develop a sequence of models that combine features of the static cross-sectional latent class (finite mixture) models with those of hidden Markov models. We model the probability of movement between categories in terms of a Markovian structure, which links the current state with a previous state, where state may refer to the category of an individual. This article presents a suite of mixture models of varying degree of complexity and flexibility for use in a panel count data setting, beginning with a baseline model which is a two-component mixture of Poisson distribution in which latent classes are fixed and permanent. Sequentially, we extend this framework (i) to allow the mixing proportions to be smoothly varying continuous functions of time-varying covariates, (ii) to add time dependence to the benchmark model by modeling the class-indicator variable as a first-order Markov chain and (iii) to extend item (i) by making it dynamic and introducing covariate dependence in the transition probabilities. We develop and implement estimation algorithms for these models and provide an empirical illustration using 1995-1999 panel data on the number of doctor visits derived from the German Socio-Economic Panel.

Prediction of hydrological drought durations based on Markov chains: case of the Canadian prairies

Hydrological drought durations (lengths) in the Canadian prairies were modelled using the standardized hydrological index (SHI) sequences derived from the streamflow series at annual, monthly and weekly time scales. The rivers chosen for the study present high levels of persistence (as indicated by values exceeding 0.95 for lag-1 autocorrelation in weekly SHI sequences), because they encompass large catchment areas (2210–119 000 km2) and traverse, or originate in, lakes. For such rivers, Markov chain models were found to be simple and efficient tools for predicting the drought duration (year, month, or week) based on annual, monthly and weekly SHI sequences. The prediction of drought durations was accomplished at threshold levels corresponding to median flow (Q50) (drought probability, q = 0.5) to Q95 (drought probability, q = 0.05) exceedence levels in the SHI sequences. The first-order Markov chain or the random model was found to be acceptable for the prediction of annual drought lengths, based on the Hazen plotting position formula for exceedence probability, because of the small sample size of annual streamflows. On monthly and weekly time scales, the second-order Markov chain model was found to be satisfactory using the Weibull plotting position formula for exceedence probability. The crucial element in modelling drought lengths is the reliable estimation of parameters (conditional probabilities) of the first- and second-order persistence, which were estimated using the notions implicit in the discrete autoregressive moving average class of models. The variance of drought durations is of particular significance, because it plays a crucial role in the accurate estimation of persistence parameters. Although, the counting method of the estimation of persistence parameters was found to be unsatisfactory, it proved useful in setting the initial values and also in subsequent adjustment of the variance-based estimates of persistence parameters. At low threshold levels corresponding to q < 0.20, even the first-order Markov chain can be construed as a satisfactory model for predicting drought durations based on monthly and weekly SHI sequences. Editor D. Koutsoyiannis; Associate editor C. Onof Citation Sharma, T.C. and Panu, U.S., 2012. Prediction of hydrological drought durations based on Markov chains in the Canadian prairies. Hydrological Sciences Journal, 57 (4), 705–722.

Serum IgG Responses and Seroconversion Patterns to Cryptosporidium gp15 among Children in a Birth Cohort in South India

The correlates of protective immunity to cryptosporidiosis are not well understood. This study was conducted to assess the effect of maternal serum IgG against Cryptosporidium gp15 on responses to this antigen in children with (cases) and without (controls) PCR-confirmed cryptosporidial diarrhea. Maternal sera (n = 129) and sera from cases (n = 39) and controls (n = 90) collected at 3.5, 9, and 24 months of age were tested for serum IgG against Cryptosporidium gp15 by enzyme-linked immunosorbent assay (ELISA). Seroconversion patterns were evaluated by estimating probabilities of seroconversion along three time points based on the transition pathways by using a first-order Markov chain process and empirical Bayesian estimates. There was no difference in serum IgG levels or seropositivity rates to gp15 between cases and controls across all time points in children or in IgG levels to this antigen between mothers of cases and controls. The most common transition pathway can be described as a seronegative child at 3.5 months who seroconverts at 9 months and remains seropositive at 24 months. This pattern remained stable irrespective of the serological status of the mother or the case or control status of the child. Children were most likely to be exposed to Cryptosporidium for the first time between the ages of 3 and 9 months, and most of the children seroconverted by 24 months. The high degree of seroconversion among control children is suggestive of high rates of asymptomatic transmission in this region.

Modeling reproductive trajectories of roe deer females: Fixed or dynamic heterogeneity?

The relative role of dynamic and fixed heterogeneity in shaping the individual heterogeneity observed in most life-history traits remains difficult to quantify. In a recent work, Tuljapurkar et al. (2009) suggested modeling individual heterogeneity in lifetime reproductive success by a null model building reproductive trajectories from a first-order Markov chain. According to this model, among-individual differences in reproductive trajectories would be generated by the stochastic transitions among reproductive states (such as breeder and non-breeder) due to dynamic heterogeneity. In this work, we analyze the individual variation in three reproductive metrics (reproductive status, fecundity, and reproductive success) in two populations of roe deer intensively monitored using Tuljapurkar et al. (2009)’s dynamic model. Moreover, we challenge the Tuljapurkar model previously used as a biological null model to test whether the observed distribution of reproductive success over the lifetime was generated by a stochastic process by modifying two steps of the previous model to build a full stochastic model. We show that a distribution generated by the full dynamic model proposed by Tuljapurkar et al. (2009) can be consistently interpreted as only generated from a stochastic biological process provided that the probabilities of transition among reproductive states used are independent of the current reproductive state and that the positive co-variation that usually occurs between survival and reproduction among individuals is removed. Only the reproductive status of roe deer females could be restricted to a stochastic process described by the full stochastic model, probably because most females (>90%) were breeders in a given year. The fecundity of roe deer females could not be adequately described by the full dynamic and full stochastic model, and the observed distribution of female reproductive success differed from the one generated by a full dynamic model in which each individual reproductive trajectory was independent of the individual lifespan (second step of the full dynamic model changed). While there was clear evidence that dynamic heterogeneity occurred and accounted for a large part of the observed among-individual variation in reproductive trajectories of roe deer females, a stochastic process did not provide a suitable model for all reproductive metrics. Consequently, models including additional fixed and dynamic traits need to be built in order to separate the relative role of fixed and dynamic heterogeneities in generating reproductive trajectories.

Multisite statistical downscaling model for daily precipitation combined by multivariate multiple linear regression and stochastic weather generator

This study provides a multi-site hybrid statistical downscaling procedure combining regression-based and stochastic weather generation approaches for multisite simulation of daily precipitation. In the hybrid model, the multivariate multiple linear regression (MMLR) is employed for simultaneous downscaling of deterministic series of daily precipitation occurrence and amount using large-scale reanalysis predictors over nine different observed stations in southern Quebec (Canada). The multivariate normal distribution, the first-order Markov chain model, and the probability distribution mapping technique are employed for reproducing temporal variability and spatial dependency on the multisite observations of precipitation series. The regression-based MMLR model explained 16 % ~ 22 % of total variance in daily precipitation occurrence series and 13 % ~ 25 % of total variance in daily precipitation amount series of the nine observation sites. Moreover, it constantly over-represented the spatial dependency of daily precipitation occurrence and amount. In generating daily precipitation, the hybrid model showed good temporal reproduction ability for number of wet days, cross-site correlation, and probabilities of consecutive wet days, and maximum 3-days precipitation total amount for all observation sites. However, the reproducing ability of the hybrid model for spatio-temporal variations can be improved, i.e. to further increase the explained variance of the observed precipitation series, as for example by using regional-scale predictors in the MMLR model. However, in all downscaling precipitation results, the hybrid model benefits from the stochastic weather generator procedure with respect to the single use of deterministic component in the MMLR model.

Markov Chain CFAR Detection for Polarimetric Data Using Data Fusion

This paper proposes a new Markov-chain-based constant false alarm rate (CFAR) detector for polarimetric data using low-level data fusion and high-level decision fusion. The Markov-chain-based CFAR detector extends traditional probability density function (pdf) based CFAR detection to first-order Markov chain model by considering both correlation between neighboring pixels and pdf information in CFAR detection. With the additional correlation information, the proposed approach results in advancing the performance of conventional CFAR detectors. Moreover, to take advantage of full polarizations of polarimetric data, various data fusion methods are considered to improve detection performance, including polarimetric transformation, principal component analysis, and decision fusion. Our experimental results confirm the superiority of the new Markov chain polarimetric CFAR detector over conventional pdf-based CFAR detectors.

Markov Chain Models for Hydrological Drought Characteristics

Abstract Estimation of drought characteristics such as probabilities and return periods of droughts of various lengths is of major importance in drought forecast and management and in solving water resources problems related to water quality and navigation. This study aims at applying first- and second-order Markov chain models to dry and wet periods of annual streamflow series to reproduce the stochastic structure of hydrological droughts. Statistical evaluation of drought duration and intensity is usually carried out using runs analysis. First-order Markov chain model (MC1) for dry and wet periods is not adequate when autocorrelation of the original hydrological series is high. A second-order Markov chain model (MC2) is proposed to estimate the probabilities and return periods of droughts. Results of these models are compared with those of a simulation study assuming a lag-1 autoregressive [AR(1)] process widely used to model annual streamflows. Probability distribution and return periods of droughts of various lengths are estimated and compared with the results of MC1 and MC2 models using efficacy evaluation statistics. It is found that the MC2 model in general gives results that are in better agreement with simulation results as compared with the MC1 model. Skewness is found to have little effect on return periods except when autocorrelation is very high. MC1 and MC2 models are applied to droughts observed in some annual streamflow series, with the result that the MC2 model has a relatively good agreement considering the limited duration of the records.

Optimal structural inference of signaling pathways from unordered and overlapping gene sets.

A plethora of bioinformatics analysis has led to the discovery of numerous gene sets, which can be interpreted as discrete measurements emitted from latent signaling pathways. Their potential to infer signaling pathway structures, however, has not been sufficiently exploited. Existing methods accommodating discrete data do not explicitly consider signal cascading mechanisms that characterize a signaling pathway. Novel computational methods are thus needed to fully utilize gene sets and broaden the scope from focusing only on pairwise interactions to the more general cascading events in the inference of signaling pathway structures. We propose a gene set based simulated annealing (SA) algorithm for the reconstruction of signaling pathway structures. A signaling pathway structure is a directed graph containing up to a few hundred nodes and many overlapping signal cascades, where each cascade represents a chain of molecular interactions from the cell surface to the nucleus. Gene sets in our context refer to discrete sets of genes participating in signal cascades, the basic building blocks of a signaling pathway, with no prior information about gene orderings in the cascades. From a compendium of gene sets related to a pathway, SA aims to search for signal cascades that characterize the optimal signaling pathway structure. In the search process, the extent of overlap among signal cascades is used to measure the optimality of a structure. Throughout, we treat gene sets as random samples from a first-order Markov chain model. We evaluated the performance of SA in three case studies. In the first study conducted on 83 KEGG pathways, SA demonstrated a significantly better performance than Bayesian network methods. Since both SA and Bayesian network methods accommodate discrete data, use a 'search and score' network learning strategy and output a directed network, they can be compared in terms of performance and computational time. In the second study, we compared SA and Bayesian network methods using four benchmark datasets from DREAM. In our final study, we showcased two context-specific signaling pathways activated in breast cancer. Source codes are available from http://dl.dropbox.com/u/16000775/sa_sc.zip.

Modeling the role of rainfall patterns in seasonal malaria transmission

Seasonal total precipitation is well known to affect malaria transmission because Anopheles mosquitoes depend on standing water for breeding habitat. However, the within-season temporal pattern of the rainfall influences persistence of standing water and thus rainfall patterns can also affect mosquito population dynamics in water-limited environments. Here, using a numerical simulation, I show that intraseasonal rainfall pattern accounts for 39% of the variance in simulated mosquito abundance in a Niger Sahel village where malaria is endemic but highly seasonal. I apply a field validated coupled hydrology and entomology model. Using synthetic rainfall time series generated using a stationary first-order Markov Chain model, I hold all variables except hourly rainfall constant, thus isolating the contribution of rainfall pattern to variance in mosquito abundance. I further show the utility of hydrology modeling using topography to assess precipitation effects by analyzing collected water. Time-integrated surface area of pools explains 70% of the variance in simulated mosquito abundance from a mechanistic model, and time-integrated surface area of pools persisting longer than 7 days explains 82% of the variance. Correlations using the hydrology model output explain more variance in mosquito abundance than the 60% from rainfall totals. I extend this analysis to investigate the impacts of this effect on malaria vector mosquito populations under climate shift scenarios, holding all climate variables except precipitation constant. In these scenarios, rainfall mean and variance change with climatic change, and the modeling approach evaluates the impact of non-stationarity in rainfall and the associated rainfall patterns on expected mosquito activity.

Robustness of keystroke-dynamics based biometrics against synthetic forgeries

Biometric systems including keystroke-dynamics based authentication have been well studied in the literature. The attack model in biometrics typically considers impersonation attempts launched by human imposters. However, this attack model is not adequate, as advanced attackers may utilize programs to forge data. In this paper, we consider the effects of synthetic forgery attacks in the context of biometric authentication systems. Our study is performed in a concrete keystroke-dynamic authentication system. The main focus of our work is evaluating the security of keystroke-dynamics authentication against synthetic forgery attacks. Our analysis is performed in a remote authentication framework called TUBA that we design and implement for monitoring a user’s typing patterns. We evaluate the robustness of TUBA through experimental evaluation including two series of simulated bots. The keystroke sequences forged by the two bots are modeled using first-order Markov chains. Support vector machine is used for classification. Our results, based on 20 users’ keystroke data, are reported. Our work shows that keystroke dynamics is robust against the two specific types of synthetic forgery attacks studied, where attacker draws statistical samples from a pool of available keystroke dataset other than the target. We also describe TUBA’s use for detecting anomalous activities on remote hosts, and present its use in a specific cognition-based anomaly detection system. The use of TUBA provides high assurance on the information collected from the hosts and enables remote security diagnosis and monitoring.

Improved synthetic wind speed generation using modified Mycielski approach

SUMMARY In this paper, novel approaches for wind speed data generation using Mycielski algorithm are developed and presented. To show the accuracy of developed approaches, we used three-year collected wind speed data belonging to deliberately selected two different regions of Turkey (Izmir and Kayseri) to generate artificial wind speed data. The data belonging to the first two years are used for training, whereas the remaining one-year data are used for testing and accuracy comparison purposes. The concept of distinct synthetic data production with correlation-wise and distribution-wise similar statistical properties constitutes the main idea of the proposed methods for a successful artificial wind speed generation. Generated data are compared with test data for both regions in the sense of basic statistics, Weibull distribution parameters, transition probabilities, spectral densities, and autocorrelation functions; and are also compared with the data generated by the classical first-order Markov chains method. Results indicate that the accuracy and realistic behavior of the proposed method is superior to the classical method in the literature. Comparisons and results are discussed in detail. Copyright © 2011 John Wiley & Sons, Ltd.

Finite-state Markov channel modelling under jointly varying amplitude and phase in time-varying flat fading channels

In this study, the authors investigate the first-order finite-state Markov chain modelling of time-varying flat fading channels. The scope of this work is to consider the joint effects of unknown channel phase and amplitude on the received signal. Performance measures such as level crossing rate, steady-state probability, state transition probability and state time duration of the fading models are derived, plotted and analysed. Comparison of the performance measures are made for Rician, Lognormal and Weibull fading distributions.