estimating pagerank on graph streams this article focuses on computations on large graphs e g the web graph where the edges of the graph are presented as a stream the objective in the streaming model is to use small amount of memory preferably sub linear in the number of nodes n and a smaller number of passes in the streaming model we show how to perform several graph computations including estimating the probability distribution after a random walk of length l the mixing time m and other related quantities such as the conductance of the graph by applying our algorithm for computing probability distribution on the web graph we can estimate the pagerank p of any node up to an additive error of se p e in o sm passes and o min n 1 esm 1 e m nsm 1 e sm space for any i 0 1 specifically for e e m n e m 1 2 we can compute the approximate pagerank values in o nm 1 4 space and o m3 4 passes in comparison a standard implementation of the pagerank algorithm will take o n space and o m passes we also give an approach to approximate the pagerank values in just o 1 passes although this requires o nm space

a generalized point process model for rainfall this paper further develops a stochastic model for rainfall at a single site in which storms arrive in a poisson process each storm generating a cluster of rain cells with each cell having a random duration and a random intensity the model is generalized by allowing each generated cell to be of n types the duration of each cell is an exponential random variable that has parameter dependent on the cell type the distribution of cell intensity is also dependent on the cell type so that the generalized model provides a correlation between the intensities and durations of the generated rain cells the case for two cell types is considered in some detail the cells are categorized as either heavy or light9 where the heavy cells have a shorter expected lifetime than the light cells which agrees with observational studies on precipitation fields properties are derived and used to fit the model to rainfall data at a single site the adequacy of fit is assessed by considering properties not used in the fitting procedure but which are of interest in applications e g extreme values further properties are derived which enable the model to be fitted to multi site rainfall data

complex analysis of a piece of toda lattice we study a small piece of two dimensional toda lattice as a complex dynamical system in particular it is shown analytically how the julia set which appears when the piece is deformed disappears as the system approaches the integrable limit

effects of high order correlations on personalized recommendations for bipartite networks in this paper we introduce a modified collaborative filtering mcf algorithm which has remarkably higher accuracy than the standard collaborative filtering in the mcf instead of the cosine similarity index the user user correlations are obtained by a diffusion process furthermore by considering the second order correlations we design an effective algorithm that depresses the influence of mainstream preferences simulation results show that the algorithmic accuracy measured by the average ranking score is further improved by 20 45 and 33 25 in the optimal cases of movielens and netflix data more importantly the optimal value depends approximately monotonously on the sparsity of the training set given a real system we could estimate the optimal parameter according to the data sparsity which makes this algorithm easy to be applied in addition two significant criteria of algorithmic performance diversity and popularity are also taken into account numerical results show that as the sparsity increases the algorithm considering the second order correlation can outperform the mcf simultaneously in all three criteria

on the correlation structure of continuous and discrete point rainfall 1 this paper proposes new analytical results based on the theory of stochastic processes that establish a theoretical dependence of point rainfall variance on the scale of observation under somewhat general assumptions it is shown that power law scaling commonly assumed by analyses and models of point rainfall cannot hold over all scales of observation and that an inner regime exists for small sampling intervals where the variance is a quadratic function of interval length a power law scaling regime is shown to exist for larger aggregation intervals whose characteristics depend on the memory of the continuous rainfall process the presence of a transition regime between the inner and the scaling regime is also shown and may explain the deviations from a power law scaling behavior in observed rainfall reported in the literature furthermore the application of the theory to rainfall data from a representative variety of climate types shows that fractal and multifractal analysis techniques and models may indeed fail to capture observed rainfall properties over wide ranges of aggregation scales finally correlation structures with short and long term memory are considered and different theoretical relationships expressing rainfall variance as a function of aggregation interval are derived and tested against rainfall observed at different locations with aggregation timescales ranging from 15 min to 2 days the theoretical and observational analyses show that both finite and infinite memory rainfall processes may be observed in nature

copula based daily rainfall disaggregation model 1 a daily rainfall disaggregation model which uses a copula to model the dependence structure between total depth total duration of wet periods and the maximum proportional depth of a wet period is presented the wet 1 dry 0 binary sequence is modeled by the nonrandomized bartlett lewis model with diurnal effect incorporated before superimposing the ar 1 depth process submodel unlike previous studies the model is structured such that all wet day data available are considered in the analysis without the need to discard any good quality daily data embedded in a month having some missing data this increased the data size thus improving the modeling process further the daily data are classified according to the total duration of wet periods duration within the day in this way a large proportion of the model parameters become seasonal invariant the overriding factor being the total duration of wet periods the potential of the developed model has been demonstrated by disaggregating both the data set used in developing the model parameters and also a 12 year continuous rainfall data set not used in the model parameterization gross rainfall statistics of several aggregation levels down to 6 min have been very well reproduced by the disaggregation model the copula dependence structure and the variation of the depth process submodel parameters with the total duration of wet periods are also very well captured by the presented model

the mathematical structure of rainfall representations 3 some applications of the point process theory to rainfall processes part 3 constitutes the final part of three parts devoted to the mathematical structure of rainfall the objective is to illustrate the scope of the tools developed in part 2 in the mathematical description of rainfall and rainfall driven processes a general overview of the three part series is given in part 1

a general method of community detection by identifying community centers with affinity propagation abstract detection of community structures is beneficial to analyzing the structures and properties of networks it is of theoretical interest and practical significance in modern science so far a large number of algorithms have been proposed to detect community structures in complex networks but most of them are suitable for a specific network structure in this paper a novel method called cdmic is proposed to detect the communities in un weighted weighted un directed directed and signed networks by constructing a dissimilarity distance matrix of network and identifying community centers with maximizing modularity for a given network we first estimate the distance between all pairs of nodes for constructing the dissimilarity distance matrix of the network then this distance matrix is input to the affinity propagation ap algorithm to extract a candidate center set of community thirdly we rank these centers in descending order according to the sum of their availability and responsibility finally we determine the community structure by selecting the center subset from the candidate center set in an incremental manner to make the modularity maximization on three real world networks and some synthetic networks experimental results show that our cdmic method has higher performance in terms of classification accuracy and normalized mutual information nmi and ability to tolerate the resolution limitation

an efficient algorithm for link prediction in temporal uncertain social networks present a link prediction algorithm in temporal uncertain social networks we transform the link prediction in uncertain network to a random walk in a deterministic one we compute the similarity scores between a node and its neighbors within a subgraph instead of computations in the whole network present an o v 3 method to compute the similarity matrix instead of enumerating the 2 e possible worlds in the network we propose a method for integrating temporal and global topological information in temporal uncertain networks due to the inaccuracy incompleteness and noise in data from real applications uncertainty is a natural feature of real world networks in such networks each edge is associated with a probability value indicating its existence in the network predicting links in uncertain networks is computationally more challenging and differs semantically from predicting connections in deterministic networks this paper presents a method for link prediction in temporal uncertain networks in our method the predicting problem is formalized by designing a random walk in temporal uncertain networks the algorithm first transforms the link prediction problem in uncertain networks to a random walk in a deterministic network then the similarity scores between a node and its neighbors are computed within a sub graph around this node to reduce the computational time the proposed method integrates temporal and global topological information in temporal uncertain networks and can obtain more accurate results experimental results on real social networks show that our method can predict future links efficiently in temporal uncertain social networks and achieves higher quality results than other similar methods

accuracy test for link prediction in terms of similarity index the case of ws and ba models abstract link prediction is a technique that uses the topological information in a given network to infer the missing links in it since past research on link prediction has primarily focused on enhancing performance for given empirical systems negligible attention has been devoted to link prediction with regard to network models in this paper we thus apply link prediction to two network models the watts strogatz ws model and barabasi albert ba model we attempt to gain a better understanding of the relation between accuracy and each network parameter mean degree the number of nodes and the rewiring probability in the ws model through network models six similarity indices are used with precision and area under the roc curve auc value as the accuracy metrics we observe a positive correlation between mean degree and accuracy and size independence of the auc value

nonlinear quantum dynamical system based on the kadomtsev petviashvili ii equation the structure of soliton solutions of classical integrable nonlinear evolution equations which can be solved through the hirota transformation suggests a new way for the construction of nonlinear quantum dynamical systems that are based on the classical equations in the new approach the classical soliton solution is mapped into an operator u which is a nonlinear functional of the particle number operators over a fock space of quantum particles u obeys the evolution equation the classical soliton solutions are the eigenvalues of u in multi particle states in the fock space the construction easily allows for the incorporation of particle interactions which generate soliton effects that do not have a classical analog in this paper this new approach is applied to the case of the kadomtsev petviashvili ii equation the nonlinear quantum dynamical system describes a set of m 2s 1 particles with intrinsic spin s which interact in clusters of 1 n m 1 particles

hybrid nlie for the mirror ads5 s5 we revisit the derivation of hybrid nonlinear integral equations of the xxx model starting from the linearization of the t system related to spinon variables we obtain two sets of equations corresponding to two linearly independent solutions of the a1 tq relation recalling that the tq relations in the horizontal strips of the hook is of a1 type we replace the corresponding y functions by a finite number of auxiliary variables

a spatial temporal point process model with a continuous distribution of storm types 1 a point process rainfall model is further developed that has storm origins occurring in space time according to a poisson process where each storm origin has a random radius so that storms occur as circular regions in two dimensional space where the storm radii are taken to be independent exponential random variables each storm origin is of random type z where z follows a continuous probability distribution cell origins occur in a further spatial poisson process and have arrival times that follow a neyman scott point process each cell origin has a radius so that cells form discs in two dimensional space where the cell radii are independent exponential random variables each cell has a random lifetime and an intensity that remains constant over both the cell lifetime and cell disk area statistical properties up to third order are given for the model using these properties the model is fitted to 10 min series taken from 23 sites across the rome region italy distributional properties of the observed annual maxima are compared to equivalent values sampled from series that are simulated using the fitted model the results indicate that the model will be of use in urban drainage projects for the rome region

link prediction with node clustering coefficient predicting missing links in incomplete complex networks efficiently and accurately is still a challenging problem the recently proposed cannistrai alanis ravai car index shows the power of local link triangle information in improving link prediction accuracy inspired by the idea of employing local link triangle information we propose a new similarity index with more local structure information in our method local link triangle structure information can be conveyed by clustering coefficient of common neighbors directly the reason why clustering coefficient has good effectiveness in estimating the contribution of a common neighbor is that it employs links existing between neighbors of a common neighbor and these links have the same structural position with the candidate link to this common neighbor in our experiments three estimators precision aup and auc are used to evaluate the accuracy of link prediction algorithms experimental results on ten tested networks drawn from various fields show that our new index is more effective in predicting missing links than car index especially for networks with low correlation between number of common neighbors and number of links between common neighbors

link sign prediction by variational bayesian probabilistic matrix factorization with student t prior abstract in signed social networks link sign prediction refers to using the observed link signs to infer the signs of the remaining links which is important for mining and analyzing the evolution of social networks the widely used matrix factorization based approach bayesian probabilistic matrix factorization bmf assumes that the noise between the real and predicted entry is gaussian noise and the prior of latent features is multivariate gaussian distribution however gaussian noise model is sensitive to outliers and is not robust gaussian prior model neglects the differences between latent features that is it does not distinguish between important and non important features thus gaussian assumption based models perform poorly on real world sparse datasets to address these issues a novel variational bayesian probabilistic matrix factorization with student t prior model tbmf is proposed in this paper a univariate student t distribution is used to fit the prediction noise and a multivariate student t distribution is adopted for the prior of latent features due to the high kurtosis of student t distribution tbmf can select informative latent features automatically characterize long tail cases and obtain reasonable representations on many real world datasets experimental results show that tbmf improves the prediction performance significantly compared with the state of the art algorithms especially when the observed links are few

compatibility of continuous rainfall occurrence models with discrete rainfall observations inferences about the underlying rainfall generating mechanism made through a model fitted to the sampled rainfall series at a particular time scale are meaningful only if they are invariant under the time scale of measurement at which the fitting of the model is made following the work of i rodriguez iturbe et al 1984 this assertion is tested by examining the compatibility of the neyman scott n s model for the continuous underlying rainfall with sampled realizations taken over intervals ranging from one hour to one day the results suggest that the n s model is inconsistent at different time scales and therefore mathematical descriptions of rainfall using this model are inevitably limited to the time scale of measurement at which a description of rainfall is sought these findings are quite general in that they apply to any model with rainfall deposited instantaneously at times described by a n s process regardless of how the amounts associated with the events are distributed and of the dependence structure of the amounts

theory of group representations and integration of nonlinear systems xa zz exp kx a abstract a constructive proof of complete integrability of the system xa zf z exp kx a and its explicit solution are given for k coinciding with the cartan matrix of simple lie algebras the technique for dealing with this system of essentially nonlinear two dimensional partial differential equations involves lax type representation and representation theory of the corresponding group

quantized representation of some nonlinear integrable evolution equations on the soliton sector the hirota algorithm for solving several integrable nonlinear evolution equations is suggestive of a simple construction of a quantized representation of these equations and their soliton solutions over a fock space of bosons or of fermions the classical nonlinear wave equation becomes a nonlinear equation for an operator the solution of this equation is constructed through an operator analog of the hirota transformation the classical n soliton solution is the expectation value of the solution operator in an n particle state in the fock space the effect of perturbations that modify soliton identity is demonstrated

a point process model for rainfall further developments a stochastic model for rainfall at a single site is studied in which storms arrive in a poisson process each storm consisting of a cluster of a random number of rain cells each cell having random duration and depth a model studied in an earlier paper is extended to provide a better fit to empirical experience the extension being based on the attachment of a single random variable to each storm to achieve in particular some correlation between the durations of the cells within a single storm the properties of the new model are developed its fitting to two sets of empirical data is described and the examination of adequacy of fit is studied in some detail via properties not used in the fitting procedure finally a theoretical study is made of short term prediction from the model

link prediction in multi relational networks based on relational similarity presented a belief propagation method to calculate the belief of each node in a network proposed a measurement of influence between different types of relations using belief vectors presented a nonnegative matrix factorization based algorithm for link prediction in multi relational networks theoretically proved the convergence and correctness of the proposed algorithm empirically demonstrated the proposed algorithm can achieve higher quality prediction results many real world networks contain multiple types of interactions and relations link prediction in such multi relational networks has become an important area in network analysis for link prediction in multi relational networks we should consider the similarity and influence between different types of relations in this paper we propose a link prediction algorithm in multi relational networks based on relational similarity in the algorithm a belief propagation method is presented to calculate the belief of each node and to construct the belief vector for each type of link we use the similarity between belief vectors to measure the influence between different types of relations based on the influence between different relations we present a nonnegative matrix factorization based method for link prediction in multi relational networks the convergence and correctness of the presented method are proved our experimental results show that our method can achieve higher quality prediction results than other similar algorithms

a hybrid model for point rainfall modeling a hybrid point rainfall model a product of two random processes is presented the model uses a jitter model a general technique for improving the fit of any point process based model we use the nonrandomized bartlett lewis rectangular pulse and an autoregressive model as a jitter first the five parameters of the bartlett lewis model are estimated by the method of moments using the mean of one aggregation level and the dry probabilities of all aggregation levels considered second the parameters of the autoregressive model are derived from the moments of the historical data and those given by the bartlett lewis model without additional cost in terms of parameter calibration using 15 min point rainfall data of capella central queensland australia as a test case the results of the hybrid model were better than the best results given by the randomized bartlett lewis model on its own the nonrandomized bartlett lewis model produced very poor results

bounded cascade models as nonstationary multifractals we investigate a class of bounded random cascade models which are multiplicative by construction yet additive with respect to some but not all of their properties we assume the multiplicative weights go to unity as the cascade proceeds then the resulting field has upper and lower bounds two largely complementary multifractal statistical methods of analysis are used singular measures and structure functions yielding respectively the exponent hierarchies ital d sub ital q and ital h sub ital q we study in more detail a specific subclass of one dimensional models with weights 1 plus minus 1 minus 2 ital p ital r sub ital n minus 1 sup ital h at relative scale ital r sub ital n 2 sup minus ital n after ital n cascade steps the parameter ital h gt 0 regulates the degree of nonstationarity at ital h 0 stationarity prevails and singular ital p model cascades meneveau and sreenivasan phys rev lett 59 1424 1987 are retrieved our model has at once large scale stationarity and small scale nonstationarity with stationary increments due to the boundedness the ital d sub ital q all converge to unity with increasing ital n the rate of convergence is estimated and the results are discussed in terms of residual multifractality a spurious singularity spectrum due to finite size effects the structure function exponents are moremore interesting ital h sub ital q min l brace ital h 1 ital q r brace in the limit ital n r arrow infinity we further focus on the cases ital q 1 related to the fractal structure of the graph ital q 2 related to the energy spectrum and ital q 1 ital h the critical order beyond which our multiplicative and multiscaling bounded cascade model can be statistically distinguished from fractional brownian motion the corresponding additive and monoscaling model less

search in power law networks many communication and social networks have power law link distributions containing a few nodes that have a very high degree and many with low degree the high connectivity nodes play the important role of hubs in communication and networking a fact that can be exploited when designing efficient search algorithms we introduce a number of local search strategies that utilize high degree nodes in power law graphs and that have costs scaling sublinearly with the size of the graph we also demonstrate the utility of these strategies on the gnutella peer to peer network

tight bounds for the cover time of multiple random walks we study the cover time of multiple random walks on undirected graphs g v e we consider k parallel independent random walks that start from the same vertex the speed up is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks recently alon et al 2008 5 derived several upper bounds on the cover time which imply a speed up of w k for several graphs however for many of them k has to be bounded by o logn they also conjectured that for any 1 2 our bounds are tight up to logarithmic factors our findings also reveal a surprisingly sharp threshold behaviour for certain graphs e g the d dimensional torus with d 2 and hypercubes there is a value t such that the speed up is approximately min t k for any 1

sampling based algorithm for link prediction in temporal networks present a fast sampling based method to predict the links in temporal network the method computes the similarity scores of potential links with a given node in a sub graph instead of in the whole network present a sampling based approach to construct the sub graph for a given node so that the error of the similarity estimated is lower than a given threshold present a quasi local index based method to compute the similarities in o n2 time while other similar methods require o n3 time experimental results show that the methods presented can obtain higher precision results in less time than other methods the problem of link prediction in temporal networks has attracted considerable recent attention from various domains such as sociology anthropology information science and computer science in this paper we propose a fast similarity based method to predict the potential links in temporal networks in this method we first combine the snapshots of the temporal network into a weighted graph a proper damping factor is used to assign greater importance to more recent snapshots then we construct a sub graph centered at each node in the weighted graph by a random walk from the node the sub graph constructed consists of a set of paths starting from the given node because the similarity score is computed within such small sub graphs centered at each node the algorithm can greatly reduce the computation time by choosing a proper number of sampled paths we can restrict the error of the estimated similarities within a given threshold while other random walk based algorithms require o n3 time for a network with n nodes the computation time of our algorithm is o n2 which is the lowest time complexity of a similarity based link prediction algorithm moreover because the proposed method integrates temporal and global topological information in the network it can yield more accurate results the experimental results on real networks show that our algorithm demonstrates the best or comparable quality results in less time than other methods

collaborative filtering based on multi channel diffusion abstract in this paper by applying a diffusion process we propose a new index to quantify the similarity between two users in a user object bipartite graph to deal with the discrete ratings on objects we use a multi channel representation where each object is mapped to several channels with the number of channels being equal to the number of different ratings each channel represents a certain rating and a user having voted an object will be connected to the channel corresponding to the rating diffusion process taking place on such a user channel bipartite graph gives a new similarity measure of user pairs which is further demonstrated to be more accurate than the classical pearson correlation coefficient under the standard collaborative filtering framework

an entropic stochastic representation of rainfall intermittency the origin of clustering and persistence 1 the well established physical and mathematical principle of maximum entropy interpreted as maximum uncertainty is used to explain the observed dependence properties of the rainfall occurrence process including the clustering behavior and persistence the conditions used for the maximization of entropy are as simple as possible i e that the rainfall processes is intermittent with dependent occurrences intermittency is quantified by the probability that a time interval is dry and dependence is quantified by the probability that two consecutive intervals are dry these two probabilities are used as constraints in a multiple scale entropy maximization framework which determines any conditional or unconditional probability of any sequence of dry and wet intervals at any timescale thus the rainfall occurrence process including its dependence structure is described by only two parameters this dependence structure appears to be non markovian application of this theoretical framework to the rainfall data set of athens indicates good agreement of theoretical predictions and empirical data at the entire range of scales for which probabilities dry and wet can be estimated from one hour to several months

stochastic daily precipitation models 1 a comparison of occurrence processes a first order markov chain and an alternating renewal process arp with a truncated geometric distribution of wet day intervals and a truncated negative binomial distribution of dry day intervals are compared as models describing the occurrence of sequences of wet and dry days numerical optimization techniques are used to obtain approximate maximum likelihood estimates of the fourier coefficients which describe the seasonal variation of the two markov chain parameters and the three parameters in the alternating renewal process for the four u s stations studied the markov chain model was superior to the arp using the minimum akaike information criterion

factorization of r matrix and baxter q operators for generic sl n spin chains we develop an approach for constructing the baxter operators for generic sl n spin chains the key element of our approach is the possibility of representing a solution of the yang baxter equation in the factorized form we prove that such a representation holds for a generic sl n invariant operator and find the explicit expression for the factorizing operators taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting baxter operators on the quantum space of the model we show that a generic transfer matrix factorizes into the product of n baxter operators and discuss an application of this representation for a derivation of functional relations for transfer matrices

integrable discretizations of the spin ruijsenaars schneider models integrable discretizations are introduced for the rational and hyperbolic spin ruijsenaars schneider models these discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous time counterparts explicit solutions are obtained for arbitrary flows of the hierarchies including the discrete time ones

the degree related clustering coefficient and its application to link prediction abstract link prediction plays a significant role in explaining the evolution of networks however it is still a challenging problem that has been addressed only with topological information in recent years based on the belief that network nodes with a great number of common neighbors are more likely to be connected many similarity indices have achieved considerable accuracy and efficiency motivated by the natural assumption that the effect of missing links on the estimation of a node s clustering ability could be related to node degree in this paper we propose a degree related clustering coefficient index to quantify the clustering ability of nodes unlike the classical clustering coefficient our new coefficient is highly robust when the observed bias of links is considered furthermore we propose a degree related clustering ability path dcp index which applies the proposed coefficient to the link prediction problem experiments on 12 real world networks show that our proposed method is highly accurate and robust compared with four common neighbor based similarity indices common neighbors cn adamic adar aa resource allocation ra and preferential attachment pa and the recently introduced clustering ability ca index

liouville field theory ist and poisson bracket structure the general solution for the liouville equation with any given number n of singularities is considered with the help of the inverse scattering transform method ist the set of regular continuous and discrete canonical variables is derived the dynamical generators of poincare and dilatation groups and n soliton solutions are constructed in terms of these variables

distributed random walks performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science including distributed computing in this article we focus on the problem of sampling random walks efficiently in a distributed network and its applications given bandwidth constraints the goal is to minimize the number of rounds required to obtain random walk samples all previous algorithms that compute a random walk sample of length e as a subroutine always do so naively that is in o e rounds the main contribution of this article is a fast distributed algorithm for performing random walks we present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk our algorithm performs a random walk of length e in o ed rounds o hides polylog n factors where n is the number of nodes in the network with high probability on an undirected network where d is the diameter of the network for small diameter graphs this is a significant improvement over the naive o e bound furthermore our algorithm is optimal within a poly logarithmic factor as there exists a matching lower bound nanongkai et al 2011 we further extend our algorithms to efficiently perform k independent random walks in o ked k rounds we also show that our algorithm can be applied to speedup the more general metropolis hastings sampling our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine we present two main applications first we give a fast distributed algorithm for computing a random spanning tree rst in an arbitrary undirected unweighted network which runs in o md rounds with high probability m is the number of edges our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network our algorithm is fully decentralized and can serve as a building block in the design of topologically aware networks

on elliptic lax systems on the lattice and a compound theorem for hyperdeterminants a general elliptic n n matrix lax scheme is presented leading to two classes of elliptic lattice systems one which we interpret as the higher rank analogue of the landau lifschitz equations while the other class we characterize as the higher rank analogue of the lattice krichever novikov equation or adler s lattice we present the general scheme but focus mainly on the latter type of models in the case n 2 we obtain a novel lax representation of adler s elliptic lattice equation in its so called 3 leg form the case of rank n 3 is analyzed using cayley s hyperdeterminant of format yielding a multi component system of coupled 3 leg quad equations

on space time scaling of cumulated rainfall fields in the study of space time rainfall it is particularly important to establish characteristic properties to guide both theoretical and modeling research efforts in the present paper new observational analyses on the scaling properties of time evolving cumulated rainfall fields are presented and a theoretical framework for their interpretation is introduced it is found that the time evolution of the spatial organization of a cumulated rainfall field produces scaling relationships of spatial variance versus time and characteristic values for the scaling exponent the reproduction of these values constitutes a basic requirement for spatial temporal field generators in order to model important properties of real rainfall fields it is then shown on theoretical grounds what properties the instantaneous rainfall intensity fields must obey in order to reproduce the experimental observations and how the size of the observation domain affects the scaling relationships some current stochastic models of space time precipitation are finally discussed and analyzed in the light of the tools introduced to show under what circumstances the models considered give acceptable results furthermore it is shown that the assumption of an exponential time correlation function used in many current rainfall models is not compatible with observations

continuous time versus discrete time point process models for rainfall occurrence series several authors have had apparent success in applying continuous time point process models to rainfall occurrence sequences in this paper it is shown that if rainfall occurrences are interpreted as the events of a point process and not as a censored sample the continuous time point process methodology and estimation procedures are not directly applicable since they fail to account for the time discreteness of the sample process this is demonstrated analytically by studying the effects of discretization on selected statistical properties of a poisson process a neyman scott process and a renewal cox process with markovian intensity in general the study of rainfall occurrences under the continuous time point process framework may result in misleading inferences regarding clustering dispersion and consequently incorrect interpretations of the underlying rainfall generating mechanisms for example daily rainfall occurrence structures underdispersed relative to the poisson process are usually overdispersed relative to the bernoulli process the discrete time analogue of the poisson these findings are confirmed by the statistical analysis of six daily rainfall records representative of a range of u s climates two of which are described in detail

multiscaling properties of rainfall and bounded random cascades the connection among different types of exponents characterizing multiscaling properties of rainfall and a criterion for stationarity of random fields are discussed and a new phenomenological model for rainfall time series simulation is proposed the bounded random cascade model presented here is a generalization of the well known model with the multiplicative weights of the generator converging to unity as the cascade proceeds to smaller scales this allows one to directly simulate a multiscaling random field with an energy spectrum exponent 1 which is typical for rainfall time series data but which cannot be produced by a standard model a procedure is proposed for estimating the cascade parameters of this new bounded model from observed data parameters are estimated from two data sets with different degrees of intermittency and different spectral exponents the bounded model simulations using these parameters produced realistic rainfall time series with spectral exponents similar to their observed counterparts

the mathematical structure of rainfall representations 2 a review of the theory of point processes part 2 constitutes the second of three parts devoted to the mathematical structure of rainfall an expository yet current treatment of those features of point process theory which are essential to descriptions of the hydrologic processes is developed a general overview of this three part series is given in part 1

a nonparametric model for stochastic generation of daily rainfall occurrence 1 this paper presents a nonparametric model for generating single site daily rainfall occurrence the model is formulated to reproduce longer term variability and low frequency features such as drought and sustained wet periods while still reproducing characteristics at daily time scales parsimony is achieved within a markovian framework by using aggregate predictor variables that describe how wet it has been over a period of time the use of a moving window to give a seasonally representative sample at any given time of year ensures an accurate representation of the seasonal variations present in the rainfall time series actual simulation proceeds by resampling from the historical record of rainfall occurrence conditional to the current values of the associated predictors the model is applied using historical daily rainfall occurrence from sydney australia we find that the use of multiple predictors specified at a daily level and at seasonal annual and multiyear aggregation levels leads to sequences that more closely reproduce the longer term variability present in the historic records compared to sequences produced by models incorporating only one or two predictors

a fast algorithm for predicting links to nodes of interest present a fast method to predict the links to nodes of interest the method computes the similarity scores of potential links connecting to a given node in a sub graph instead of in the whole network present an approach to construct the sub graph for the given node so that the errors of the similarities estimated are lower than a given threshold present a quasi global index based method to compute the similarities in the network in o n time while other similar methods require o n3 time experimental results show that the methods presented can obtain high precision results in less time than other methods the problem of link prediction has recently attracted considerable attention in various domains such as sociology anthropology information science and computer science in many real world applications we must predict similarity scores only between pairs of vertices in which users are interested rather than predicting the scores of all pairs of vertices in the network in this paper we propose a fast similarity based method to predict links related to nodes of interest in the method we first construct a sub graph centered at the node of interest by choosing the proper size for such a sub graph we can restrict the error of the estimated similarities within a given threshold because the similarity score is computed within a small sub graph the algorithm can greatly reduce computation time the method is also extended to predict potential links in the whole network to achieve high process speed and accuracy experimental results on real networks demonstrate that our algorithm can obtain high accuracy results in less time than other methods can

modelling extremes of the areal rainfall process risk assessment for many hydrological structures requires an estimate of the extremal behaviour of the rainfall regime within a specified catchment region in most cases it is the spatially aggregated rainfall which is the key process though in practice only pointwise rainfall measurements from a network of sites over the region are available in this paper we address the problem of making inferences about the extremal properties of the aggregated process from the pointwise data working within the usual extreme value paradigm a model is derived in which the resulting distribution is determined by the marginal tail behaviour and spatial dependence at extreme levels of the process data collected from a region in the south west of england are used to illustrate the procedure

classical a n w geometry by analyzing theextrinsic geometry of two dimensional surfaces chirally embedded incp n thecp n w surface 1 we give exact treatments in various aspects of the classical w geometry in the conformal gauge first the basis of tangent and normal vectors are defined at regular points of the surface such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the conformal a n toda lax pair since the latter is known to be intrinsically related with the w symmetries this gives the geometrical meaning of thea n w algebra second w surfaces are put in one to one correspondence with solutions of the conformally reduced wznw model which is such that the toda fields give the cartan part in the gauss decomposition of its solutions third the additional variables of the toda hierarchy are used as coordinates ofcp n this allows us to show that w transformations may be extended as particular diffeomorphisms of this target space higher dimensional generalizations of the wznw equations are derived and related with the zakharov shabat equations of the toda hierarchy fourth singular points are studied from a global viewpoint using our earlier observation 1 that w surfaces may be regarded as instantons the global indices of the w geometry which are written in terms of the toda fields are shown to be the instanton numbers for associated mappings of w surfaces into the grassmannians the relation with the singularities of w surface is derived by combining the toda equations with the gauss bonnet theorem

the mathematical structure of rainfall representations 1 a review of the stochastic rainfall models this is the first of a three part series on the mathematical structure of rainfall models several important attempts at modeling rainfall are reviewed special attention is given to the mathematical structures that arise in the rainfall descriptions a general overview of the three part series is given as preface to this part

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

daily spatiotemporal precipitation simulation using latent and transformed gaussian processes 1 a daily stochastic spatiotemporal precipitation generator that yields spatially consistent gridded quantitative precipitation realizations is described the methodology relies on a latent gaussian process to drive precipitation occurrence and a probability integral transformed gaussian process for intensity at individual locations the model reduces to a markov chain for precipitation occurrence and a gamma distribution for precipitation intensity allowing statistical parameters to be included in a generalized linear model framework statistical parameters are modeled as spatial gaussian processes which allows for interpolation to locations where there are no direct observations via kriging one advantage of such a model for the statistical parameters is that stochastic generator parameters are immediately available at any location with the ability to adapt to spatially varying precipitation characteristics a second advantage is that parameter uncertainty generally unavailable with deterministic interpolators can be immediately quantified at all locations the methodology is illustrated on two data sets the first in iowa and the second over the pampas region of argentina in both examples the method is able to capture the local and domain aggregated precipitation behavior fairly well at a wide range of time scales including daily monthly and annually

some models for discretized series of events a discretized series of events is a binary time series that indicates whether or not events of a point process in the line occur in successive intervals such data are common in environmental applications we describe a class of models for them based on an unobserved continuous time discrete state markov process which determines the rate of a doubly stochastic poisson process from which the binary time series is constructed by discretization we discuss likelihood inference for these processes and their second order properties and extend them to multiple series an application involves modeling the times of exposures to air pollution at a number of receptors in western europe

classical scattering in liouville field theory a detailed description of classical scattering in liouville field theory lft is presented contrary to widespread belief lft scattering is shown to be non trivial although it is finite dimensional in some sense in particular for certain phase spaces the lft s matrix is represented as a transformation of the poisson group sl 2 r the completeness and conformal invariance of the scattering are also indicated singular fields are treated on an equal footing with regular ones except that only the latter are given a consistent hamiltonian interpretation a number of unexpected peculiarities of lft scattering are revealed first for some exceptional field configurations the asymptotic fields are not solutions of d alembert s equation rather they are a sum of the d alembert and liouville components second the scattering occurs in two or three spaces and last depending on the choice of the algebra of observables the conventional splitting of the d alembert field into leftward and rightward components is either in general impossible or essentially non unique

a sublinear time distributed algorithm for minimum weight spanning trees this paper considers the question of identifying the parameters governing the behavior of fundamental global network problems many papers on distributed network algorithms consider the task of optimizing the running time successful when an o n bound is achieved on an n vertex network we propose that a more sensitive parameter is the network s diameter diam this is demonstrated in the paper by providing a distributed minimum weight spanning tree algorithm whose time complexity is sublinear in n but linear in diam specifically o diam n varepsilon cdot log n for varepsilon frac ln 3 ln 6 0 6131 our result is achieved through the application of graph decomposition and edge elimination by pipelining techniques that may be of independent interest

a stochastic cluster model of daily rainfall sequences a two level point stochastic model for the rainfall occurrences at a given rainfall station is constructed in the time dimension the model is a cluster process of the neyman scott type the model has the rainfall generating mechanisms as its primary level and the rainfalls that are generated by these mechanisms as the secondary level it uses infinite superposition of rainfalls and has a very flexible dependence structure the model is fitted to daily rainfall sequences in indiana after these are stationarized by a transformation the fit of the model is then tested in terms of its correlation and marginal probability characteristics the present form of the neyman scott cluster model is time homogeneous therefore the neyman scott process as presented in this paper may be of practical use only for modeling the stationary rainfall occurrences

a nonparametric approach for representing interannual dependence in monthly streamflow sequences 1 the estimation of risks associated with water management plans requires generation of synthetic streamflow sequences the mathematical algorithms used to generate these sequences at monthly timescales are found lacking in two main respects inability in preserving dependence attributes particularly at large seasonal to interannual time lags and a poor representation of observed distributional characteristics in particular representation of strong asymmetry or multimodality in the probability density function proposed here is an alternative that naturally incorporates both observed dependence and distributional attributes in the generated sequences use of a nonparametric framework provides an effective means for representing the observed probability distribution while the use of a variable kernel ensures accurate modeling of streamflow data sets that contain a substantial number of zero flow values a careful selection of prior flows imparts the appropriate short term memory while use of an aggregate flow variable allows representation of interannual dependence the nonparametric simulation model is applied to monthly flows from the beaver river near beaver utah and the burrendong dam inflows new south wales australia

exact quantum s matrix in the liouville field theory classical scattering in the liouville field theory lft is essentially finite dimensional and in some cases the classical s matrix can be represented as a transformation of the poisson group sl 2 r motivated by this and using conjectural quantum analogues of some ingredients of the classical model we find an exact quantum s matrix without constructing the quantum lft in full the quantum s matrix is explicitly represented as a transformation of the quantum group slq 2 r and for a particular implementation the s matrix is shown to be unitary unitarily generated

gauge and dual symmetries and linearization of hirota s bilinear equations in hirota s hiroshima university technical report nos a 6 a 9 1981 j phys soc jpn 50 3785 1981 bilinear difference equation which is satisfied by solutions to the kadomtsev petviashvili kp hierarchy gauge and dual symmetries are found which enable one to reduce the problem of solving the nonlinear equation to solving a single linear equation

evaluation of cluster based rectangular pulses point process models for rainfall this technical note presents a comparison of cluster based point rainfall models using the historical hourly rainfall data observed between 1949 and 1976 at denver colorado the denver data are used to analyze the performance of three classes of models namely the bartlett lewis model the geometric neyman scott model and the poisson neyman scott model the original formulation of the structure of each model as well as the modified description developed in order to improve the zero depth probability is considered in this study rodriguez iturbe et al 1987a concluded that it is unlikely that empirical analysis of rainfall data can be used to choose between the bartlett lewis model and the neyman scott model in a subsequent paper rodriguez iturbe et al 1987b argued that the choice of the distribution of the number of cells per storm for the neyman scott model either geometric or poisson has no general bias effect on the stochastic structure some investigators e g burlando and rosso 1991 however reported results contradictory to those of the previous authors in light of these observations this note investigates the performance of the cluster based models for the denver data the geometric neyman scott model yields better results compared to the poisson neyman scott model moreover the bartlett lewis model is shown to be very sensitive to the sets of moment equations used in the parameter estimation this sensitivity is not observed in the neyman scott scheme and is believed to be a drawback for applying the bartlett lewis model in hydrologic simulation studies

an inhomogeneous lax representation for the hirota equation motivated by recent work on quantum integrable models without u 1 symmetry we show that the sl 2 hirota equation admits a lax representation with inhomogeneous terms the compatibility of the auxiliary linear problem leads to a new consistent family of hirota like equations

information filtering based on transferring similarity in this brief report we propose an index of user similarity namely the transferring similarity which involves all high order similarities between users accordingly we design a modified collaborative filtering algorithm which provides remarkably higher accurate predictions than the standard collaborative filtering more interestingly we find that the algorithmic performance will approach its optimal value when the parameter contained in the definition of transferring similarity gets close to its critical value before which the series expansion of transferring similarity is convergent and after which it is divergent our study is complementary to the one reported in e a leicht p holme and m e j newman phys rev e 73 026120 2006 and is relevant to the missing link prediction problem

information filtering via weighted heat conduction algorithm abstract in this paper by taking into account effects of the user and object correlations on a heat conduction hc algorithm a weighted heat conduction whc algorithm is presented we argue that the edge weight of the user object bipartite network should be embedded into the hc algorithm to measure the object similarity the numerical results indicate that both the accuracy and diversity could be improved greatly compared with the standard hc algorithm and the optimal values reached simultaneously on the movielens and netflix datasets the algorithmic accuracy measured by the average ranking score can be improved by 39 7 and 56 1 in the optimal case respectively and the diversity could reach 0 9587 and 0 9317 when the recommendation list equals to 5 further statistical analysis indicates that in the optimal case the distributions of the edge weight are changed to the poisson form which may be the reason why hc algorithm performance could be improved this work highlights the effect of edge weight on a personalized recommendation study which maybe an important factor affecting personalized recommendation performance

predicting link directions using local directed path abstract link prediction in directed network is attracting growing interest among many network scientists compared with predicting the existence of a link determining its direction is more complicated in this paper we propose an efficient solution named local directed path to predict link direction by adding an extra ground node to the network we solve the information loss problem in sparse network which makes our method effective and robust as a quasi local method our method can deal with large scale networks in a reasonable time empirical analysis on real networks shows that our method can correctly predict link directions which outperforms some local and global methods

spatial gossip and resource location protocols the dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm exchanging information in each communication step in a variety of contexts the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches in many settings such as a cluster of distributed computing hosts new information is generated at individual nodes and is most interesting to nodes that are nearby thus we propose distance based propagation bounds as a performance measure for gossip mechanisms a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d and is independent of the size of the network for nodes arranged with uniform density in euclidean space we present natural gossip mechanisms called spatial gossip that satisfy such a guarantee new information is spread to nodes at distance d with high probability in o log1 e d time steps such a bound combines the desirable qualitative features of uniform gossip in which information is spread with a delay that is logarithmic in the full network size and deterministic flooding in which information is spread with a delay that is linear in the distance and independent of the network size our mechanisms and their analysis resolve a conjecture of demers et al 1987 we further show an application of our gossip mechanisms to a basic resource location problem in which nodes seek to rapidly learn the location of the nearest copy of a resource in a network this problem which is of considerable practical importance can be solved by a very simple protocol using spatial gossip whereas we can show that no protocol built on top of uniform gossip can inform nodes of their approximately nearest resource within poly logarithmic time the analysis relies on an additional useful property of spatial gossip namely that information travels from its source to sinks along short paths not visiting points of the network far from the two nodes

a comparative copula based bivariate frequency analysis of observed and simulated storm events a case study on bartlett lewis modeled rainfall 1 because of a lack of historical rainfall time series of considerable length one often has to rely on simulated rainfall time series e g in the design of hydraulic structures one way to simulate such time series is by means of stochastic point process rainfall models such as the bartlett lewis type of model for the evaluation of model performance with a focus on the reproduction of extreme rainfall events often a univariate extreme value analysis is performed recently developed concepts in statistical hydrology now offer other means of evaluating the overall performance of such models in this study a copula based frequency analysis of storms is proposed as a tool to evaluate differences between the return periods of several types of observed and modeled storms first this study performs an analysis of several storm variables which indicates a problem with the modeling of the temporal structure of rainfall by the models thereafter the bivariate frequency analysis of storms defined by their duration and volume illustrates the underestimation and overestimation of the return period of the storms simulated by the models which is partially explained by a large difference in the marginal distribution functions of the storm duration and storm volume the difference in the degree of association between the latter and a different mean storm interarrival time the proposed methodology allows for the identification of some problems with the rainfall simulations from which recommendations for possible improvements to rainfall models can be made

enhanced speciation in particle swarm optimization for multi modal problems abstract in this paper we present a novel multi modal optimization algorithm for finding multiple local optima in objective function surfaces we build from species based particle swarm optimization spso by using deterministic sampling to generate new particles during the optimization process by implementing proximity based speciation coupled with speciation of isolated particles and by including turbulence regions around already found solutions to prevent unnecessary function evaluations instead of using error threshold values the new algorithm uses the particle s experience geometric mean and exclusion factor to detect local optima and stop the algorithm the performance of each extension is assessed with leave it out tests and the results are discussed we use the new algorithm called isolated speciation based particle swarm optimization ispso and a benchmark algorithm called niche particle swarm optimization nichepso to solve a six dimensional rainfall characterization problem for 192 rain gages across the united states we show why it is important to find multiple local optima for solving this real world complex problem by discussing its high multi modality solutions found by both algorithms are compared and we conclude that ispso is more reliable than nichepso at finding optima with a significantly lower objective function value

non power law scale properties of rainfall in space and time 1 the identification of general relationships linking statistical properties of rainfall aggregated at different temporal and spatial scales possesses clear theoretical and practical relevance among other properties it is important to characterize the scale dependence of rainfall variability which may for many purposes be summarized by its variance this paper presents theoretical results and observational analyses connecting the variance of temporal and spatial rainfall to the scale of observation under an assumption of second order stationarity previous results regarding the theoretical form of the temporal variance as a function of aggregation are refined and extended to the spatial case it is shown that the variance of aggregated rainfall exhibits an overall nonscaling form within the range of scales of usual hydrological interest it is further shown that the theoretically derived form of the variance as a function of scale is incompatible with power law relations usually assumed for the second order statistical moment within multiscaling approaches to rainfall modeling predictions of the theoretical derivations are validated by use of observations representing a wide variety of resolutions from 2 s to 1 hour in time and 4 km in space climates and measurement instruments the validation shows a good agreement between observations and the theoretically predicted forms of the variance as a function of aggregation the proposed theoretical framework thus provides a useful context for rainfall analysis modeling and downscaling in space and time and suggests careful reexaminations of usual multiscaling assumptions

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

poisson structure of the liouville field theory a new analysis of the poisson structure of the liouville field theory lft in an infinite volume is presented the second poisson structure of the korteweg de vries equation is thoroughly investigated as an essential part of the approach and a variety of correct poisson brackets is found this along with other causes leads to a variety of correct lft poisson structures special attention is paid to the most important lft property of conformal invariance in particular a maximal conformal group suitable for the adopted lft phase spaces is found and various hamiltonian representations of the conformal algebra are described the properties of local commutativity and canonicity are proved

quantum integrable systems and elliptic solutions of classical discrete nonlinear equations in spite of the diversity of solvable models of quantum field theory and the vast variety of methods the final results display dramatic unification the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies e sum limits i varepsilon left u i right 1 1 where u i obey a system of algebraic or transcendental equations known as bethe equations 1 2 the major ingredients of bethe equations are determined by the algebraic structure of the problem a typical example of a system of bethe equations related to a i type models with elliptic r matrix is e 4 eta nu frac phi left u j right phi left u j 2 right mathop pi limits k frac sigma left eta left u j u k 2 right right sigma left eta left u j u k 2 right right 1 2 where x is the weierstrass function and phi left u right mathop pi limits k 1 n sigma left eta left u y k right right 1 3

probabilistic representation of the temporal rainfall process by a modified neyman scott rectangular pulses model parameter estimation and validation the capability of the neyman scott clustered stochastic point process model of rainfall to preserve various observed statistics is considered randomization of the cell duration parameter from storm to storm is shown to considerably improve the wet dry period joint distribution and extreme value statistics a simple procedure for parameter estimation is introduced and applied

coupling stochastic models of different timescales a methodology is proposed for coupling stochastic models of hydrologic processes applying to different timescales so that time series generated by the different models be consistent given two multivariate time series generated by two separate unrelated stochastic models of the same hydrologic process each applying to a different timescale a transformation is developed referred to as a coupling transformation that appropriately modifies the time series of the lower level finer timescale so that this series becomes consistent with the time series of the higher level coarser timescale without affecting the second order stochastic structure of the former and also establishes appropriate correlations between the two time series the coupling transformation is based on a developed generalized mathematical proposition which ensures preservation of marginal and joint second order statistics and of linear relationships between lower and higher level processes several specific forms of the coupling transformation are studied from the simplest single variate to the full multivariate in addition techniques for evaluating parameters of the coupling transformation based on second order moments of the lower level process are studied furthermore two methods are proposed to enable preservation of the skewness of the processes in addition to that of second order statistics the overall methodology can be applied to problems involving disaggregation of annual to seasonal and seasonal to subseasonal timescales as well as problems involving finer timescales e g daily to hourly with the only requirement that a specific stochastic model is available for each involved timescale the performance of the methodology is demonstrated by means of a detailed numerical example

the relationship between a continuous time renewal model and a discrete markov chain model of precipitation occurrence a relationship between a continuous wet dry renewal model with gamma distributed dry intervals and a markov chain model for daily rainfall is derived the relationship requires numerical evaluation for the general case but closed form solutions are available for the markov process case in which dry intervals are exponentially distributed the markov process equations are equivalent to those derived by j r green 1964 for first order transition probabilities but are presented in a simplified nonrecursive form the markov process relationships are examined for convergence and accuracy with simulation studies and comparison to observed rainfall records the markov process model is shown to provide a good representation in certain parts of the united states in other areas where the markov property is inappropriate due to event clustering or other phenomena the gamma model provides an improved characterization of the relationship between continuous and discrete rainfall occurrence

seasonal generalized exponential probability models with application to interstorm and storm durations a simple generalization of the exponential probability model is presented which provides a flexible method for identifying and fitting probability distributions when coupled with likelihood based estimation this approach enables a parsimonious description of seasonal dependence using harmonic functions a case study involving modeling of interstorm and storm durations for several australian state capitals illustrates practical issues in calibration and identification because the rainfall data were stored in a binned format consisting of rainfall depths accumulated over fixed time intervals the start and end times of storms were known only to the resolution of the fixed time interval a likelihood function is developed which properly makes use of such information likelihood ratio statistics along with monthly distribution plots are used to select the number of harmonics necessary to model seasonal dependence it is shown for melbourne that a single harmonic adequately describes seasonal dependence thereby reducing the number of parameters from 48 to 6