an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

low eigenvalues of laplacian matrices of large random graphs for each n 2 let an ij be an n n symmetric matrix with diagonal entries equal to zero and the entries in the upper triangular part being independent with mean n and standard deviation n the laplacian matrix is defined by bf delta n rm diag sum j 1 n xi ij 1 leq i leq n bf a n in this paper we obtain the laws of large numbers for n k n the k 1 th smallest eigenvalue of n through the study of the order statistics of weakly dependent random variables under certain moment conditions on ij s we prove that as n rm i quad frac lambda n k bf delta n n mu n sigma n sqrt n log n to sqrt 2 quad a s for any k 1 further if n n 2 are independent with n 0 and n 1 then ii the sequence left frac lambda n k bf delta n sqrt n log n n geq 2 right is dense in left sqrt 2 2 k 1 1 sqrt 2 right a s for any k 0 in particular i holds for the erdos renyi random graphs similar results are also obtained for the largest eigenvalues of n

a state space formulation for optimal hankel norm approximations the optimal hankel norm approximation problem studied in 1 is reformulated in a state space setting the crucial extension theorem is reestablished in this framework the minimal degree optimal approximation is then derived in terms of state space parameters

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

statistical inference for point process models of rainfall in this paper we develop maximum likelihood procedures for parameter estimation and model selection that apply to a large class of point process models that have been used to model rainfall occurrences including cox processes neyman scott processes and renewal processes the statistical inference procedures are based on the stochastic intensity t lims 0 s 0 1 s e n t s n t n u u t the likelihood function of a point process is shown to have a simple expression in terms of the stochastic intensity the main result of this paper is a recursive procedure for computing stochastic intensities the procedure is applicable to a broad class of point process models including renewal cox process with markovian intensity processes and an important class of neyman scott processes the model selection procedure we propose which is based on likelihood ratios allows direct comparison of two classes of point processes to determine which provides a better model for a given data set the estimation and model selection procedures are applied to two data sets of simulated cox process arrivals and a data set of daily rainfall occurrences in the potomac river basin

conditional distributions of neyman scott models for storm arrivals and their use in irrigation scheduling count for the widely recognized but often neglected statistical dependence in storm arrivals gabriel and newmann 1957 1962 smith and schreiber 1973 kavvas and delleur 1975 recently authors like kavvas and delleur 1975 and gupta and waymire 1979 have argued that this dependence is due to clustering of rainfall in time and space they suggest modeling storm arrivals with cluster point processes like the neyman scott model this work uses the concept of cluster models for storm arrivals the conditional distribution of the occurrence of future events given a history of the timing of past events is derived for the neyman scott model this new result is then used with a reformulated model of the irrigation scheduling problem now the decision of when and how much to irrigate is based on the time since the last observed storm the effective soil moisture content and the amount of available irrigation water a case study is presented to illustrate the solution significantly there is no important difference between the theoretical net benefits derived from assuming a homogeneous independent poisson model or a dependent neyman scott n s model during the growing season a nonhomogeneous poisson model yields higher theoretical benefits as expected the conditional n s model leads to improved irrigation control as compared to the unconditional n s model from this case study however no definite conclusion can be drawn regarding whether the neyman scott model better represents the process of storm arrivals as opposed to the poisson model however the expected opportunity loss of choosing the wrong model can be significant interesting implications of the results are that data manipulation can have a significant effect on the ultimate model

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

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

convergence and convergence rate of the balanced realization truncations for infinite dimensional discrete time systems abstract in this paper we study approximation via balanced realizations for a large class of infinite dimensional discrete time linear systems we give properties of the truncated system and prove that the approximation is l 2 convergent in the case of nuclear systems we prove convergence in nuclear norm and give an estimation of the l convergence rate

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

density of states of sparse random matrices a supersymmetric formalism is used to derive a set of equations giving the density of states of any real symmetric sparse random matrix as a function of the distribution of non zero elements and the mean number of non zero elements per row p in the matrix where the non zero elements take the values or 1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method as e to 0 the density of states rho e behaves as 1 e ln e 2 the more general case where or 1 occur with unequal probabilities is also considered

generalized scale invariance in the atmosphere and fractal models of rain in recent years there has been considerable interest in stochastic rain models by developing new ideas about scale invariance and intermittency we argue that the scope of such models can be greatly extended the notions of scale invariance intermittency and the associated idea of fractal dimension have lately gained considerable ground particularly in the context of extremely variable phenomena such as those found in the mesoscale and in hydrodynamic turbulence we review some relevant work and argue that the atmosphere respects a symmetry principle that we call generalized scale invariance in which the statistical properties of the small and large scale are related to each other by a magnification coupled with a differential stratification due to gravity and differential rotation due to the coriolis force we further argue that the extremely erratic intermittent nature of the atmosphere is characterized by scale invariant fractal measures leading to hyperbolic fat tailed probability distributions of the fluctuations the standard statistical methods based on exponential fall offs in both correlations and probability distributions are inappropriate when the variability is of this type we show how both the scaling and intermittency may be exploited to develop extremely variable stochastic models of rain although the models examined here are the simplest of a family of extremely variable processes depending on only two radar determined parameters they lead to realizations possessing many realistic features of the rainfield including complexity of form clustering and bands at all scales as well as differential stratification and rotation finally we point to weaknesses in the models in particular their monodimensional nature and suggest possible improvements

on the selection of the eigenvalues to be retained in the reduced order models this note points out that the model order selection criteria proposed earlier for davison s and marshall s reduced order models are not applicable in many of the practical cases a new criterion is proposed to overcome the deficiencies of the earlier methods

a nonparametric wet dry spell model for resampling daily precipitation a nonparametric wet dry spell model is developed for resampling daily precipitation at a site the model considers alternating sequences of wet and dry days in a given season of the year all marginal joint and conditional probability densities of interest e g dry spell length wet spell length precipitation amount and wet spell length given prior to dry spell length are estimated nonparametrically using at site data and kernel probability density estimators procedures for the disaggregation of wet spell precipitation into daily precipitation and for the generation of synthetic sequences are proffered an application of the model for generating synthetic precipitation traces at a site in utah is presented

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

on binary time series obtained from continuous time point processes describing rainfall recent work on modeling rainfall occurrence has used continuous time point process models in many cases the available data is not sufficiently precise to properly fit such a model for example if only occurrence or nonoccurrence of precipitation is given daily a more appropriate model would be a binary time series in this work we relate some parameters of the time series thus obtained to parameters of the underlying but not observed continuous time process we carry out the details for several models proposed in the literature such as the neyman scott poisson cluster process and the smith karr renewal cox process an application of the theory is illustrated on some washington state rainfall data

a spectral theory of rainfall intensity at the meso β scale the available empirical descriptions of extratropical cyclonic storms are employed to formulate a physically realistic stochastic representation of the ground level rainfall intensity field in space and time the stochastic representation is based on three component stochastic point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscale areas within synoptic storms certain scale idealizations and assumptions on functional forms which qualitatively reflect the physical features lead to a closed form expression for the covariance function i e the real space time spectrum of the rainfall intensity field the theoretical spectrum explains the empirical spectral features observed by zawadzki almost a decade ago of particular interest and importance in this connection is an explanation of the empirical observation that the taylorian propogation of the fine scale structure via a transformation of time to space through the storm velocity holds only for a small time lag and not throughout the results here indicate the extent of this lag in terms of the characteristic scales associated with cell durations cellular birthrates and velocities etc

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

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

the effect of certain restrictions imposed on the interarrival times of flood events on the poisson distribution used for modeling flood counts in the application of partial duration series models of flood analysis it is occasionally observed that successive exceedances are correlated to reduce this correlation some investigators tend to impose certain restrictions on the interarrival times of flood events in order that these events will not occur close together in bunches we show analytically how such restrictions interfere with the underlying hypotheses of the poisson process commonly used to model flood counts and we caution against imposing restrictions that may render this simple and appealing model inapplicable

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

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

multiscaling pulse representation of temporal rainfall 1 we develop a pulse based representation of temporal rainfall with multifractal properties in the small scale limit the representation combines a traditional model for the exterior process at the synoptic scale with a novel hierarchical pulse model for the event interiors for validation we apply the model to a temporal rainfall record from florence italy although the model has only six parameters four for the exterior process and two for the event interiors it accurately reproduces a wide range of empirical statistics including the distribution of dry and wet periods the distribution of rainfall intensity up to extreme fractiles the spectral density the moment scaling function k q and the distribution of the partition coefficients for rainfall disaggregation the model also reproduces observed deviations of physical rainfall from perfect scaling multiscaling behavior

further developments of the neyman scott clustered point process for modeling rainfall this paper provides some useful results for modeling rainfall it extends work on the neyman scott cluster model for simulating rainfall time series several important properties have previously been found for the model for example the expectation and variance of the amount of rain captured in an arbitrary time interval rodriguez iturbe et al 1987a in this paper additional properties are derived such as the probability of an arbitrary interval of any chosen length being dry in applications this is a desirable property to have and is often used for fitting stochastic rainfall models to field data the model is currently being used in rainfall time series research directed toward improving sewage systems in the united kingdom to illustrate the model s performance an example is given where the model is fitted to 10 years of hourly data taken from blackpool england

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

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

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

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

ltv model reduction with upper error bounds a new approach for computing upper error bounds for reduced order models of linear time varying systems is presented it is based on a transformation technique of the hankel singular values using positive real odd incremented functions by applying such time varying functions the singular values to be removed can be forced to become equal and constant so that they can be reduced two variations of this method are proposed one for finite time horizons and the other for infinite time problems including periodic systems

eigenvalue distribution of large weighted random graphs we study eigenvalue distribution of the adjacency matrix a n p of weighted random graphs n p we assume that the graphs have n vertices and the average number of edges attached to one vertex is p to each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite in the first part of the paper we consider the moments of normalized eigenvalue counting function n p of a n p assuming all moments of a finite we obtain recurrent relations that determine the moments of the limiting measure p limn n p the method developed is applied to the laplace operator closely related with a n p using the recurrent relations we analyze the form of p for the both of random matrix families in the second part of the paper we consider the resolvents g a z of a n p and of n p and study the functions fn a z u 1 n k 1n exp ugkk a z in the limit n we derive closed equations that uniquely determine the limiting functio

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

scaling limits and self similarity in precipitation fields it has been recognized in the literature on rainfall models in hydrology that the exact calculations of probabilities for models which incorporate cluster structure is typically not possible consequently one is led to considerations of limiting approximations however because of the presence of dependencies and or fat tails in several of the current models the nature of the approximations involves some important issues for both theoretical as well as experimental consideration in precipitation research the objective of the present paper is to systematically describe these issues as they pertain to rainfall models some specific large scale approximations are given for models of the type currently in the literature

relative error h sub infinity identification from autocorrelation data a stochastic realization method a variant of the balanced stochastic truncation bst method for approximated realization of power spectrum matrices is shown to form the basis for an identification procedure that is well suited to the task of determining relative error bounded approximate plant models for use in control design from input output cross correlation data central to the theory is a novel l sub infinity norm bound on the relative error between an exact realization of the data and bst approximate realization

a markov renewal model for rainfall occurrences a probabilistic model for the temporal description of daily rainfall occurrences at a single location is presented by defining an event as a day with measurable precipitation the model is cast into the discrete time point process framework in the proposed model the sequence of times between events is formed by sampling from two geometric distributions according to transition probabilities specified by a markov chain the model belongs to the class of markov renewal processes and exhibits clustering relative to the independent bernoulli process as a special case it reduces to a renewal model with a mixture distribution for the interarrival times the rainfall occurrence model coupled with a mixed exponential distribution for the nonzero daily rainfall amounts was applied to the daily rainfall series for snoqualmie falls washington and was successful in preserving the short term structure of the occurrence process as well as the distributional properties of the seasonal rainfall amounts

model reduction for discrete unstable systems based on generalized normal representations the input and output normal representations of stable systems are generalized for unstable systems these are then used to reduce the order of unstable discrete time linear multivariable systems resulting in reduced order models with the same number of unstable poles as the full order model and with an a priori upper bound on the reduction error the generalized normal representations are then modified to include a frequency weighting resulting in frequency weighted generalized normal representations these are used to derive frequency weighted reduced order models for unstable discrete systems with the same number of unstable poles as the full order model and with an a priori upper bound on the frequency weighted reduction error

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

state space modeling of aerodynamic forces on plate using singular value decomposition this technique employs the singular value decomposition of a block hankel matrix constructed from the aerodynamic matrix impulse response resulting in a system that has common global poles