moment equation and path integral techniques for wave propagation in random media differential equations for all moments of the field of a wave propagating through a random medium are derived under the parabolic approximation and the markov approximation but including anisotropy in the random medium and a deterministic background refractive index mathematical equivalence is demonstrated between these moment equations and path integral expressions for the moments obtained under the same approximations a discussion of approximations that are weaker than markov is given

a practical guide to heavy tails statistical techniques and applications part 1 applications heavy tailed probability distributions in the world wide web m e crovella et al self similarity and heavy tails structural modelling of network traffic w willinger et al heavy tails in high frequency financial data u a muller et al stable paretian modelling in finance some empirical and theoretical aspects s mittnik et al risk management and quantile estimation f bassi et al part 2 time series analyzing stable time series r j adler et al inference for linear processes with stable noise m calder r a davis on estimating the intensity of long range dependence in finite and infinite variance time series m s taqqu v teverovsky why non linearities can ruin the heavy tailed modeller s day s i resnick periodogram estimates from heavy tailed data t mikosch bayesian inference for time series with infinite variance stable innovations n ravishanker z qiou part 3 heavy tail estimation hill bootstrap and jackknife estimators for heavy tails o v pictet et al characteristic function based estimation of stable distribution parameters s m kogan d b williams part 4 regression bootstrapping signs and permutations for regression with heavy tailed errors a robust resampling r lepage et al linear regression with stable disturbances j h mcculloch part 5 signal processing deviation from normality in statistical signal processing parameter estimation with alpha stable distributions p tsakalides c l nikias statistical modelling and receiver design for multi user communication networks g a tsihrintzis part 6 model structures subexponential distributions c m goldie c kluppelberg structure of stationary stable processes j rosinski tail behaviour of some shot noise processes g samorodnitsky part 7 numerical procedures numerical approximation of the symmetric stable distribution and density j h mcculloch table of the maximally skewed stable distributions j h mcculloch d b panton multivariate stable distributions approximation estimation simulation and identification j p nolan univariate stable distributions parametrizations and software j p nolan

horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

a 1 488 approximation algorithm for the uncapacitated facility location problem we present a 1 488 approximation algorithm for the metric uncapacitated facility location ufl problem previously the best algorithm was due to byrka 2007 byrka proposed an algorithm parametrized by c and used it with c 1 6774 by either running his algorithm or the algorithm proposed by jain mahdian and saberi email protected 02 byrka obtained an algorithm that gives expected approximation ratio 1 5 we show that if c is randomly selected the approximation ratio can be improved to 1 488 our algorithm cuts the gap with the 1 463 approximability lower bound by almost 1 3

on using polynomial chaos for modeling uncertainty in acoustic propagation the use of polynomial chaos for incorporating environmental variability into propagation models is investigated in the context of a simplified one dimensional model which is relevant for acoustic propagation when the random sound speed is independent of depth environmental variability is described by a spectral representation of a stochastic process and the chaotic representation of the wave field then consists of an expansion in terms of orthogonal random polynomials issues concerning implementation of the relevant equations the accuracy of the approximation uniformity of the expansion over the propagation range and the computational burden necessary to evaluate different field statistics are addressed when the correlation length of the environmental fluctuations is small low order expansions work well while for large correlation lengths the convergence of the expansion is highly range dependent and requires high order approximants these conclusions also apply in higher dimensional propagation

power law statistics and universal scaling in the absence of criticality critical states are sometimes identified experimentally through power law statistics or universal scaling functions we show here that such features naturally emerge from networks in self sustained irregular regimes away from criticality in these regimes statistical physics theory of large interacting systems predict a regime where the nodes have independent and identically distributed dynamics we thus investigated the statistics of a system in which units are replaced by independent stochastic surrogates and found the same power law statistics indicating that these are not sufficient to establish criticality we rather suggest that these are universal features of large scale networks when considered macroscopically these results put caution on the interpretation of scaling laws found in nature

radial structure of the internet the structure of the internet at the autonomous system as level has been studied by the physics mathematics and computer science communities we extend this work to include features of the core and the periphery taking a radial perspective on as network structure new methods for plotting as data are described and they are used to analyse datasets that have been extended to contain edges missing from earlier collections the average distance from one vertex to the rest of the network is used as the baseline metric for investigating radial structure common vertex specific quantities are plotted against this metric to reveal distinctive characteristics of central and peripheral vertices two datasets are analysed

integrated approach to the assessment of long range correlation in time series data to assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods the spectral method and the random walk analysis the first objective of this work is to show that each one of these methods used alone can be susceptible to producing false results we thus advocate an integrated approach which requires the use of both methods in a consistent fashion we provide the theoretical foundation of this approach and illustrate the main ideas using examples the second objective relates to the observation of long range anticorrelation hurst exponent hl1 2 in real world time series data the very peculiar nature of such processes is emphasized in light of the stringent condition under which such processes can occur using examples we discuss the possible factors that could contribute to the false claim of long range anticorrelations and demonstrate the particular importance of the integrated approach in this case

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

complex network based time series analysis abstract recent works show that complex network theory may be a powerful tool in time series analysis we propose in this paper a reliable procedure for constructing complex networks from the correlation matrix of a time series an original stock time series the corresponding return series and its amplitude series are considered the degree distribution of the original series can be well fitted with a power law while that of the return series can be well fitted with a gaussian function the degree distribution of the amplitude series contains two asymmetric gaussian branches reconstruction of networks from time series is a common problem in diverse research the proposed strategy may be a reasonable solution to this problem

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

validity of the markov approximation in ocean acoustics moment equations and path integrals for wave propagation in random media have been applied to many ocean acoustics problems both these techniques make use of the markov approximation the expansion parameter which must be less than one for the markov approximation to be valid is the subject of this paper there is a standard parameter the kubo number which various authors have shown to be sufficient fourth moment equations have been successfully used to predict the experimentally measured frequency spectrum of intensity in the mid ocean acoustic transmission experiment mate yet in spite of this success the kubo number is greater than 1 for the measured index of refraction variability for mate arriving at a contradiction here that contradiction is resolved by showing that the kubo parameter is far too pessimistic for the ocean case using the methodology of van kampen another parameter is found which appears to be both necessary and sufficient and is much smaller than the kubo number when phase fluctuations are dominated by large scales in the medium this parameter is shown to be small for the experimental regime of mate justifying the applications of the moment equations to that experiment

empirical and quadrature approximation of acoustic field and array response probability density functions numerical methods are presented for approximating the probability density functions pdf s of acoustic fields and receiver array responses induced by a given joint pdf of a set of acoustic environmental parameters an approximation to the characteristic function of the random acoustic field the inverse fourier transform of the field pdf is first obtained either by construction of the empirical characteristic function ecf from a random sample of the acoustic parameters or by application of generalized gaussian quadrature to approximate the integral defining the characteristic function the fourier transform is then applied to obtain an approximation of the pdf by a continuous function of the field variables application of both the ecf and generalized gaussian quadrature is demonstrated in an example of a shallow water ocean waveguide with two dimensional uncertainty of sound speed and attenuation coefficient in the ocean bottom both approximations lead to a smoother estimate of the field pdf than that provided by a histogram with generalized gaussian quadrature providing a smoother estimate at the tails of the pdf potential applications to acoustic system performance quantification and to nonparametric acoustic signal processing are discussed

a nearly linear time approximation scheme for the euclidean k median problem this paper provides a randomized approximation scheme for the k median problem when the input points lie in the d dimensional euclidean space the worst case running time is o 2 o log 1 epsilon varepsilon d 1 n log d 6 n which is nearly linear for any fixed varepsilon and d moreover our method provides the first polynomial time approximation scheme for and uncapacitated facility location instances in d dimensional euclidean space for any fixed d 2 our work extends techniques introduced originally by arora for the euclidean traveling salesman problem tsp to obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions the theorem is based on an adaptive decomposition scheme which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition we believe that our methodology is of independent interest and may find applications to further geometric problems

separating multipaths by global optimization of a multidimensional matched filter a transmitted signal can arrive at a receiver via several refracted fermat paths if the paths are independent in the fresnel sense then the received signal can be modeled as the sum of amplitude scaled and time shifted copies of a predetermined replica plus white noise we present an algorithm that uses the replica to determine the time shifts and amplitudes for each path it is referred to as an n dimensional matched filter algorithm by analogy with the well known matched filter algorithm the cross correlation between the received signal and the replica oscillates near the center frequency of the transmitted signal this causes the n dimensional matched filter output to have many local maxima that are not globally optimal the time shifts and amplitude scalings for the fermat paths are determined by maximizing the output of the n dimensional matched filter the algorithm is more robust and efficient than others currently available simulated realizations of received signals were generated with multipath and noise characteristics similar to an ocean acoustic transmission case these realizations were then separated into arrival times and corresponding amplitudes by the algorithm the results of these tests and the general limitations of the algorithm are discussed

approximation algorithms for metric facility location and k median problems using the primal dual schema and lagrangian relaxation we present approximation algorithms for the metric uncapacitated facility location problem and the metric k median problem achieving guarantees of 3 and 6 respectively the distinguishing feature of our algorithms is their low running time o m log m and o m log m l log n respectively where n and m are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities the main algorithmic ideas are a new extension of the primal dual schema and the use of lagrangian relaxation to derive approximation algorithms

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

a method for approximating acoustic field amplitude uncertainty caused by environmental uncertainties in underwater acoustics the accuracy of computational field predictions is commonly limited by uncertainty in environmental parameters an approximate technique for determining the probability density function pdf of computed field amplitude a from known environmental uncertainties is presented here the technique can be applied to several n uncertain parameters simultaneously requires n 1 field calculations and can be used with any acoustic field model the technique implicitly assumes independent input parameters and is based on finding the optimum spatial shift between field calculations completed at two different values of each uncertain parameter this shift information is used to convert uncertain environmental parameter distributions into pdf a the technique s accuracy is good when the shifted fields match well its accuracy is evaluated in range independent underwater sound channels via an l1 error norm defined between approximate and numerically converged results for pdf a in 50 m a

hybrid uncertainty propagation in structural acoustic systems based on the polynomial chaos expansion and dimension wise analysis abstract hybrid uncertainties are ubiquitous in the structural acoustic analysis and greatly affect the behaviors of the coupled system however requirements of small uncertainties and rewriting simulation codes for the numerical analysis are always necessary for current methods in this context a non intrusive dual layer analysis procedure i e a hybrid method of the polynomial chaos expansion and dimension wise analysis pce dw is proposed in this paper the pce procedure based on the sparse grid collocation strategy is utilized to handle the random uncertainty while the dw procedure is employed to evaluate the interval bounds of statistical characteristics of the system response the pce dw also applies to the structural acoustic analysis with only random or interval parameters the investigation of two structural acoustic systems with hybrid uncertainties finally demonstrates the accuracy efficiency and capability of the proposed method

on the fractal characterization of paretian poisson processes abstract paretian poisson processes are poisson processes which are defined on the positive half line have maximal points and are quantified by power law intensities paretian poisson processes are elemental in statistical physics and are the bedrock of a host of power law statistics ranging from pareto s law to anomalous diffusion in this paper we establish evenness based fractal characterizations of paretian poisson processes considering an array of socioeconomic evenness based measures of statistical heterogeneity we show that amongst the realm of poisson processes which are defined on the positive half line and have maximal points paretian poisson processes are the unique class of fractal processes exhibiting scale invariance the results established in this paper are diametric to previous results asserting that the scale invariance of poisson processes with respect to physical randomness based measures of statistical heterogeneity is characterized by exponential poissonian intensities

understanding the determinants of volatility clustering in terms of stationary markovian processes abstract volatility is a key variable in the modeling of financial markets the most striking feature of volatility is that it is a long range correlated stochastic variable i e its autocorrelation function decays like a power law for large time lags in the present work we investigate the determinants of such feature starting from the empirical observation that the exponent of a certain stock s volatility is a linear function of the average correlation of such stock s volatility with all other volatilities we propose a simple approach consisting in diagonalizing the cross correlation matrix of volatilities and investigating whether or not the diagonalized volatilities still keep some of the original volatility stylized facts as a result the diagonalized volatilities result to share with the original volatilities either the power law decay of the probability density function and the power law decay of the autocorrelation function this would indicate that volatility clustering is already present in the diagonalized un correlated volatilities we therefore present a parsimonious univariate model based on a non linear langevin equation that well reproduces these two stylized facts of volatility the model helps us in understanding that the main source of volatility clustering once volatilities have been diagonalized is that the economic forces driving volatility can be modeled in terms of a smoluchowski potential with logarithmic tails

approximation algorithms for the set covering and vertex cover problems we propose a heuristic that delivers in o n 3 steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

embedding uncertainty into ocean acoustic propagation models l a probabilistic formalism is proposed for the direct inclusion of environmental uncertainty into an acoustic model that describes propagation in an ocean waveguide incomplete environmental knowledge is characterized by a spectral representation of uncertainty using expansions of random processes in terms of orthogonal random polynomials a brief summary of the method is presented and a set of coupled differential equations describing the propagation of both the acoustic field and its associated uncertainty is derived for the case where the uncertain environment is attributed to a lack of complete information concerning the waveguide s sound speed distribution

hurst exponent estimation of self affine time series using quantile graphs abstract in the context of dynamical systems time series analysis is frequently used to identify the underlying nature of a phenomenon of interest from a sequence of observations for signals with a self affine structure like fractional brownian motions fbm the hurst exponent h is one of the key parameters here the use of quantile graphs qgs for the estimation of h is proposed a qg is generated by mapping the quantiles of a time series into nodes of a graph h is then computed directly as the power law scaling exponent of the mean jump length performed by a random walker on the qg for different time differences between the time series data points the qg method for estimating the hurst exponent was applied to fbm with different h values comparison with the exact h values used to generate the motions showed an excellent agreement for a given time series length estimation error depends basically on the statistical framework used for determining the exponent of the power law model the qg method is numerically simple and has only one free parameter q the number of quantiles nodes with a simple modification it can be extended to the analysis of fractional gaussian noises

growing networks of overlapping communities with internal structure we introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks the model comprises two complementary mechanisms one mechanism accounts for the evolution of the internal link structure of a single community and the second mechanism coordinates the growth of multiple overlapping communities the first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates we demonstrate that this simple mechanism gives rise to an effective maximal degree within communities this observation is related to the anthropological theory known as dunbar s number i e the empirical observation of a maximal number of ties which an average individual can sustain within its social groups the second mechanism is based on a recently proposed generalization of preferential attachment to community structure appropriately called structural preferential attachment spa the combination of these two mechanisms into a single model spa allows us to reproduce a number of the global statistics of real networks the distribution of community sizes of node memberships and of degrees the spa model also predicts a three qualitative regimes for the degree distribution within overlapping communities and b strong correlations between the number of communities to which a node belongs and its number of connections within each community we present empirical evidence that support our findings in real complex networks

on the bias of traceroute sampling or power law degree distributions in regular graphs understanding the graph structure of the internet is a crucial step for building accurate network models and designing efficient algorithms for internet applications yet obtaining this graph structure can be a surprisingly difficult task as edges cannot be explicitly queried for instance empirical studies of the network of internet protocol ip addresses typically rely on indirect methods like traceroute to build what are approximately single source all destinations shortest path trees these trees only sample a fraction of the network s edges and a paper by lakhina et al 2003 found empirically that the resulting sample is intrinsically biased further in simulations they observed that the degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is poisson in this article we study the bias of traceroute sampling mathematically and for a very general class of underlying degree distributions explicitly calculate the distribution that will be observed as example applications of our machinery we prove that traceroute sampling finds power law degree distributions in both regular and poisson distributed random graphs thus our work puts the observations of lakhina et al on a rigorous footing and extends them to nearly arbitrary degree distributions

a systematic approach to bound factor revealing lps and its application to the metric and squared metric facility location problems a systematic technique to bound factor revealing linear programs is presented we show how to derive a family of upper bound factor revealing programs upfrp and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm obtaining an upfrp is straightforward and can be used as an alternative to analytical proofs that are usually very long and tedious we apply this technique to the metric facility location problem mflp and to a generalization where the distance function is a squared metric we call this generalization the squared metric facility location problem smflp and prove that there is no approximation factor better than 2 04 2 04 assuming mathrm p ne mathrm np p np then we analyze the best known algorithms for the mflp based on primal dual and lp rounding techniques when they are applied to the smflp we prove very tight bounds for these algorithms and show that the lp rounding algorithm achieves a ratio of 2 04 2 04 and therefore has the best possible factor for the smflp we use upfrps in the dual fitting analysis of the primal dual algorithms for both the smflp and the mflp improving some of the previous analysis for the mflp

clustering through continuous facility location problems we consider the continuous facility location problem conflp given a finite set of clients c r d and a number f r conflp consists in opening a set f r d of facilities each at cost f and connecting each client to an open facility the objective is to minimize the costs of opening facilities and connecting clients we reduce conflp to the standard facility location problem flp by using the so called approximate center sets this reduction preserves the approximation except for an error e and implies 1 488 e and 2 04 e approximations when the connection cost is given by the euclidean distance and the squared euclidean distance respectively moreover we obtain approximate center sets for the case that the connection cost is the th power of the euclidean distance achieving approximations for the corresponding problems for any 1 as a byproduct we also obtain a polynomial time approximation scheme for the k median problem with this cost function for any fixed k

power law distribution of degree degree distance a better representation of the scale free property of complex networks whether real world complex networks are scale free or not has long been controversial recently in broido and clauset a d broido a clauset nat commun 10 1017 2019 it was claimed that the degree distributions of real world networks are rarely power law under statistical tests here we attempt to address this issue by defining a fundamental property possessed by each link the degree degree distance the distribution of which also shows signs of being power law by our empirical study surprisingly although full range statistical tests show that degree distributions are not often power law in real world networks we find that in more than half of the cases the degree degree distance distributions can still be described by power laws to explain these findings we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted two way selection process the model does not always produce solid power law distributions but predicts that the degree degree distance distribution exhibits stronger power law behavior than the degree distribution of a finite size network especially when the network is dense we test the strength of our model and its predictive power by examining how real world networks evolve into an overly dense stage and how the corresponding distributions change we propose that being scale free is a property of a complex network that should be determined by its underlying mechanism e g preferential attachment rather than by apparent distribution statistics of finite size we thus conclude that the degree degree distance distribution better represents the scale free property of a complex network

efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules we estimated posterior model covariances by randomly perturbing data and start model vectors with samples drawn from prior probability distributions solving the corresponding deterministic inversion for each sample and integrating the resulting model vectors over all realizations the computational burden of this method is the number of samples required for the integration because each sample involves a solution to a nonlinear inverse problem for certain classes of smooth prior probability functions we present a new approach to the model covariance estimation problem using grids based on maximally sparse cubature integration rules the n 1 degree two and 2n degree three cubature rules exist to exactly integrate symmetric gaussian distributions in n stochastic dimensions these represent the two sparsest integration rules possible and thus provide the most efficient estimation method for model covariances using integration in contrast to monte carlo mc integration which requires hundreds of random samples to estimate covariances and has ill defined convergence sparse cubature quadrature rules have determined numbers of samples and definite convergence the trade off is that they also have limited degrees of accuracy using a simple 1d electromagnetic example with 67 stochastic variables we demonstrated that this sparse integration is three to five times more efficient than mc methods in terms of numbers of required samples

linear time approximation schemes for clustering problems in any dimensions we present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions more specifically our approach leads to simple randomized algorithms for the k means k median and discrete k means problems that yield 1 e approximations with probability 1 2 and running times of o 2 k e o 1 dn these are the first algorithms for these problems whose running times are linear in the size of the input nd for n points in d dimensions assuming k and e are fixed our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling

estimating heavy tail exponents through max self similarity in this paper a novel approach to the problem of estimating the heavy tail exponent 0 of a distribution is proposed it is based on the fact that block maxima of size m scale at a rate m1 for independent as well as for a number of dependent data this scaling rate can be captured well by the max spectrum plot of the data that leads to regression based estimators for consistency and asymptotic normality of these estimators is established for independent data under mild conditions on the behavior of the tail of the distribution the proposed estimators have an important computational advantage over existing methods namely they can be calculated and updated sequentially in an on line fashion without having to store the entire data set practical issues on the automatic selection of tuning parameters for the estimators and corresponding confidence intervals are also addressed extensive numerical simulations show that the proposed method is competitive for both small and large sample sizes robust to contaminants and continues to work under the presence of substantial amount of dependence the proposed estimators are used to illustrate the close connection between long range dependence and heavy tails over an internet traffic trace