modeling the searching behavior of social monkeys we discuss various features of the trajectories of spider monkeys lookingfor food in a tropical forest as observed recently in an extensive in situ study some of the features observed can be interpreted as the result of social interactions in addition a simple model of deterministic walk in a random environment reproduces the observed angular correlations between successive steps and in some cases the emergence of l evy distributions for the length of the steps c

gene stanley the n vector model and random walks with absorbing boundaries abstract gene stanley s early contribution to the n vector model is briefly reviewed also discussed is a problem solved by mccrea and whipple in 1940 namely random walks with absorbing boundaries we have found the exact solution of the analogous problem on the triangular lattice with a true finite boundary this problem was previously considered to be intractable

statistics of visits to sites in random walks abstract this paper analyses the statistics of visits to a site in random walks first general formulas will be derived and next they will be applied to simple symmetric random walks on finite and infinite lattices the dependence of the statistics on the lattices dimensionality will also be studied finally an application will be presented to explain the structure of the radio maps that trace the supernova remnants

principal components and orthogonal regression based on robust scales both principal components analysis pca and orthogonal regression deal with finding a p dimensional linear manifold minimizing a scale of the orthogonal distances of the m dimensional data points to the manifold the main conceptual difference is that in pca p is estimated from the data to attain a small proportion of unexplained variability whereas in orthogonal regression p equals m 1 the two main approaches to robust pca are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust univariate dispersion measure this article is more akin to second approach but rather than finding the components one by one we directly undertake the problem of finding for a given p a p dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold the scale may be either a smooth m scale or a trimmed scale an iterative algorithm is developed that is shown to converge to a local minimum a

the masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

levy brownian motion on finite intervals mean first passage time analysis we present the analysis of the first passage time problem on a finite interval for the generalized wiener process that is driven by levy stable noises the complexity of the first passage time statistics mean first passage time cumulative first passage time distribution is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non gaussian noises the validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2 recovering in this limit the standard results for the fokker planck dynamics driven by gaussian white noise

searchability of networks we investigate the searchability of complex systems in terms of their interconnectedness associating searchability with the number and size of branch points along the paths between the nodes we f

residence times and other functionals of reflected brownian motion we study the residence and local times for a brownian particle confined by reflecting boundaries and propose a general solution to the problem of finding the related probability distribution its fourier transform characteristic function and laplace transform survival probability are obtained in a compact matrix form involving the laplace operator eigenbasis explicit combinatorial relations are derived for the moments and the probability distribution is shown to be nearly gaussian when the exploration time is long enough when the eigenbasis or a part of it is known the numerical computation of the residence time distributions is straightforward and accurate the present approach can also be applied to investigate other functionals of reflected brownian motion describing in particular restricted diffusion in an external field or potential e g nuclei diffusing in an inhomogeneous magnetic field theoretical results for the local times are confronted with monte carlo simulations on the unit interval disk and sphere

escape driven by α stable white noises we explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the brownian particle is driven by white levy noise in a dynamical regime where inertial effects can safely be neglected the behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise induced discontinuity which is caused by the generalized brownian paths that jump beyond the barrier location without actually hitting it this fact implies that the boundary conditions for the mean first passage time mfpt are no longer determined by the well known local boundary conditions that characterize the case with normal diffusion by numerically implementing properly the set up boundary conditions we investigate the survival probability and the average escape time as a function of the corresponding levy white noise parameters depending on the value of the skewness beta of the levy noise the escape can either become enhanced or suppressed a negative asymmetry parameter beta typically yields a decrease for the escape rate while the rate itself depicts a non monotonic behavior as a function of the stability index alpha that characterizes the jump length distribution of levy noise exhibiting a marked discontinuity at alpha 1 we find that the typical factor of 2 that characterizes for normal diffusion the ratio between the mfpt for well bottom to well bottom and well bottom to barrier top no longer holds true for sufficiently high barriers the survival probabilities assume an exponential behavior versus time distinct non exponential deviations occur however for low barrier heights

data driven spatio temporal modeling using the integro difference equation a continuous in space discrete in time dynamic spatio temporal model known as the integro difference equation ide model is presented in the context of data driven modeling a novel decomposition of the ide is derived leading to state space representation that does not couple the number of states with the number of observation locations or the number of parameters based on this state space model an expectation maximization em algorithm is developed in order to jointly estimate the ide model s spatial field and spatial mixing kernel the resulting modeling framework is demonstrated on a set of examples

a fast algorithm for the minimum covariance determinant estimator the minimum covariance determinant mcd method of rousseeuw is a highly robust estimator of multivariate location and scatter its objective is to find h observations out of n whose covariance matrix has the lowest determinant until now applications of the mcd were hampered by the computation time of existing algorithms which were limited to a few hundred objects in a few dimensions we discuss two important applications of larger size one about a production process at philips with n 677 objects and p 9 variables and a dataset from astronomy with n 137 256 objects and p 27 variables to deal with such problems we have developed a new algorithm for the mcd called fast mcd the basic ideas are an inequality involving order statistics and determinants and techniques which we call selective iteration and nested extensions for small datasets fast mcd typically finds the exact mcd whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders

random walks on carbon nanotubes and quasicrystals an analytic formula for the total number of k step walks between given sites on a carbon nanotube is obtained by using a new mathematical model based on a three axes description of the honeycomb lattice the new model represents an alternate mathematical description which may be useful in certain applications it is similar to the four axes description existing in the case of hexagonal crystals the use of one or more additional axes is a fundamental method in quasicrystal physics we show that the mathematical model we use for the honeycomb lattice can be defined in terms of a strip projection method and present a method to associate some finite graphs to a quasicrystal the random walks on these graphs are connected with random walks on a quasicrystal

first exit times and residence times for discrete random walks on finite lattices in this paper we derive explicit formulas for the surface averaged first exit time of a discrete random walk on a finite lattice we consider a wide class of random walks and lattices including random walks in a nontrivial potential landscape we also compute quantities of interest for modeling surface reactions and other dynamic processes such as the residence time in a subvolume the joint residence time of several particles and the number of hits on a reflecting surface

spatiotemporal multi resolution approximation of the amari type neural field model neural fields are spatially continuous state variables described by integro differential equations which are well suited to describe the spatiotemporal evolution of cortical activations on multiple scales here we develop a multi resolution approximation mra framework for the integro difference equation ide neural field model based on semi orthogonal cardinal b spline wavelets in this way a flexible framework is created whereby both macroscopic and microscopic behavior of the system can be represented simultaneously state and parameter estimation is performed using the expectation maximization em algorithm a synthetic example is provided to demonstrate the framework

the generalized lasso in the last few years the support vector machine svm method has motivated new interest in kernel regression techniques although the svm has been shown to exhibit excellent generalization properties in many experiments it suffers from several drawbacks both of a theoretical and a technical nature the absence of probabilistic outputs the restriction to mercer kernels and the steep growth of the number of support vectors with increasing size of the training set in this paper we present a different class of kernel regressors that effectively overcome the above problems we call this approach generalized lasso regression it has a clear probabilistic interpretation can handle learning sets that are corrupted by outliers produces extremely sparse solutions and is capable of dealing with large scale problems for regression functionals which can be modeled as iteratively reweighted least squares irls problems we present a highly efficient algorithm with guaranteed global convergence this defies a unique framework for sparse regression models in the very rich class of irls models including various types of robust regression models and logistic regression performance studies for many standard benchmark datasets effectively demonstrate the advantages of this model over related approaches

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

robpca a new approach to robust principal component analysis we introduce a new method for robust principal component analysis pca classical pca is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations two robust approaches have been developed to date the first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an s estimator and is limited to relatively low dimensional data the second approach is based on projection pursuit and can handle high dimensional data here we propose the robpca approach which combines projection pursuit ideas with robust scatter matrix estimation robpca yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data robpca can be computed rapidly and is able to detect exact fit situations as a by product robpca produces a diagnostic plot that displays and classifies the outliers we apply the algorithm to several datasets from chemometrics and engineering

spatiotemporal system identification with continuous spatial maps and sparse estimation we present a framework for the identification of spatiotemporal linear dynamical systems we use a state space model representation that has the following attributes 1 the number of spatial observation locations are decoupled from the model order 2 the model allows for spatial heterogeneity 3 the model representation is continuous over space and 4 the model parameters can be identified in a simple and sparse estimation procedure the model identification procedure we propose has four steps 1 decomposition of the continuous spatial field using a finite set of basis functions where spatial frequency analysis is used to determine basis function width and spacing such that the main spatial frequency contents of the underlying field can be captured 2 initialization of states in closed form 3 initialization of state transition and input matrix model parameters using sparse regression the least absolute shrinkage and selection operator method and 4 joint state and parameter estimation using an iterative kalman filter sparse regression algorithm to investigate the performance of the proposed algorithm we use data generated by the kuramoto model of spatiotemporal cortical dynamics the identification algorithm performs successfully predicting the spatiotemporal field with high accuracy whilst the sparse regression leads to a compact model

principal component analysis based on robust estimators of the covariance or correlation matrix influence functions and efficiencies a robust principal component analysis can be easily performed by computing the eigenvalues and eigenvectors of a robust estimator of the covariance or correlation matrix in this paper we derive the influence functions and the corresponding asymptotic variances for these robust estimators of eigenvalues and eigenvectors the behaviour of several of these estimators is investigated by a simulation study it turns out that the theoretical results and simulations favour the use of s estimators since they combine a high efficiency with appealing robustness properties 2000 biometrika trust

exact solution for random walks on the triangular lattice with absorbing boundaries the problem of a random walk on a finite triangular lattice with a single interior source point and zig zag absorbing boundaries is solved exactly this problem has been previously considered intractable

levy flights search patterns of biological organisms we discuss recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l leads to an optimal random search strategy for organisms that can search efficiently for randomly located objects that can only be detected in the limited vicinity of the searcher and can be revisited any number of times we explore the extent to which these findings may be dependent on the dimensionality of the search space and the presence of short range correlations in the step lengths and directions

on the joint residence time of n independent two dimensional brownian motions we study the behaviour of several joint residence times of n independent brownian particles in a disc of radius r in two dimensions we consider i the time tn t spent by all n particles simultaneously in the disc within the time interval 0 t ii the time t m n t which at least m out of n particles spend together in the disc within the time interval 0 t and iii the time m n t which exactly m out of n particles spend together in the disc within the time interval 0 t we obtain very simple exact expressions for the expectations of these three residence times in the limit t

diffusion on complex networks a way to probe their large scale topological structures a diffusion process on complex networks is introduced in order to uncover their large scale topological structures this is achieved by focusing on the slowest decaying diffusive modes of the network the proposed procedure is applied to real world networks like a friendship network of known modular structure and an internet routing network for the friendship network its known structure is well reproduced in case of the internet where the structure is far less well known one indeed finds a modular structure and modules can roughly be associated with individual countries quantitatively the modular structure of the internet manifests itself in an approximately 10 times larger participation ratio of its slowest decaying modes as compared to the null model a random scale free network the extreme edges of the internet are found to correspond to russian and us military sites c 2004 elsevier b v all rights reserved less

territory covered by n random walkers the problem of evaluating the number of distinct sites s n t covered up to time t by n random walkers is revisited for the nontrivial time regime and for n 1 we show how to get the asymptotic behavior of s n t and we calculate the main and first two corrective terms the mth corrective term decays mildly as 1 ln m n for d dimensional d 1 2 3 simple cubic lattices the main term is the volume of the hypersphere of radius ln n 2 2dt d 1 2 d being the diffusion constant and the corrective terms account for the roughening of the surface of the set of visited sites

number of times a site is visited in two dimensional random walks in this paper formulas are derived to compute the mean number of times a site has been visited in a random walk on a two dimensional lattice asymmetric random walks are considered with or without drift for different boundary conditions it is shown that in case of absorbing boundaries the mean number of visits reaches stationary values over the lattice comparisons with a monte carlo simulation are also presented

asymptotic distributions of principal components based on robust dispersions algebraically principal components can be defined as the eigenvalues and eigenvectors of a covariance or correlation matrix but they are statistically meaningful as successive projections of the multivariate data in the direction of maximal variability an attractive alternative in robust principal component analysis is to replace the classical variability measure i e variance by a robust dispersion measure this projection pursuit approach was first proposed in li chen 1985 as a method of constructing a robust scatter matrix recent unpublished work of c croux and a ruiz gazen provided the influence functions of the resulting principal components the present paper focuses on the asymptotic distributions of robust principal components in particular we obtain the asymptotic normality of the principal components that maximise a robust dispersion measure we also explain the need to use a dispersion functional with a continuous influence function

robust functional estimation using the median and spherical principal components summary we present robust estimators for the mean and the principal components of a stochastic process in l2 r robustness and asymptotic properties of the estimators are studied theoretically by simulation and by example it is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components although their properties depend on the spacings of the eigenvalues of the covariance function

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

robust low rank approximation of data matrices with elementwise contamination we propose a robust method to approximate an n p data matrix with one of rank q the method is based on yohai s regression mm estimates it is intended to be resistant against the existence of both atypical rows and of scattered atypical cells and also to be able to cope with missing values we propose an algorithm based on alternating m regressions and a starting estimate based on successive rank one fits which involves o npq operations simulations show that our estimate outperforms competing estimates in terms of both efficiency and resistance three high dimensional real data sets are analyzed the running time of our estimate for large data sets is shown to be less than that of its competitors

robust pca via outlier pursuit singular value decomposition svd and principal component analysis pca is one of the most widely used techniques for dimensionality reduction successful and efficiently computable it is nevertheless plagued by a well known well documented sensitivity to outliers recent work has considered the setting where each point has a few arbitrarily corrupted components yet in applications of svd or pca such as robust collaborative filtering or bioinformatics malicious agents defective genes or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted we present an efficient convex optimization based algorithm that we call outlier pursuit which under some mild assumptions on the uncorrupted points satisfied e g by the standard generative assumption in pca problems recovers the exact optimal low dimensional subspace and identifies the corrupted points such identification of corrupted points that do not conform to the low dimensional approximation is of paramount interest in bioinformatics financial applications and beyond our techniques involve matrix decomposition using nuclear norm minimization however our results setup and approach necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition since we develop an approach to recover the correct column space of the uncorrupted matrix rather than the exact matrix itself in any problem where one seeks to recover a structure rather than the exact initial matrices techniques developed thus far relying on certificates of optimality will fail we present an important extension of these methods which allows the treatment of such problems

an expectation maximization method for spatio temporal blind source separation using an ar mog source model in this paper we develop a maximum likelihood ml spatio temporal blind source separation bss algorithm where the temporal dependencies are explained by assuming that each source is an autoregressive ar process and the distribution of the associated independent identically distributed i i d innovations process is described using a mixture of gaussians unlike most ml methods the proposed algorithm takes into account both spatial and temporal information optimization is performed using the expectation maximization em method the source model is adapted to maximize the likelihood and the update equations have a simple analytical form the proposed method which we refer to as autoregressive mixture of gaussians ar mog outperforms nine other methods for artificial mixtures of real audio we also show results for using ar mog to extract the fetal cardiac signal from real magnetocardiographic mcg data

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

optimizing the encounter rate in biological interactions levy versus brownian strategies an important application involving two species reaction diffusion systems relates to the problem of finding the best statistical strategy for optimizing the encounter rate between organisms we investigate the general problem of how the encounter rate depends on whether organisms move in levy or brownian random walks by simulating a limiting generalized searcher target model e g predator prey mating partner pollinator flower we find that levy walks confer a significant advantage for increasing encounter rates when the searcher is larger or moves rapidly relative to the target and when the target density is low

random strategies of contact tracking abstract one of several critical issues in the development of optimal disease containment and eradication strategies is the knowledge of underlying contacts between individuals here we employ random search strategies to identify all possible links representing direct or indirect interactions between individuals building up the system in order to recognize all contacts the searcher performs symmetric levy flights onto the accessible area we investigate the influence of local and non local information the exponent characterizing asymptotic behavior of levy flights boundary conditions density of links and type of a search strategy on the efficiency of the search process monte carlo examination of the suggested model reveals that the efficiency of the search process is sensitive to the type of boundary conditions depending on the assumed type of boundary conditions efficiency of the search process can be a monotonic or non monotonic function of the exponents characterizing asymptotic behavior of levy flights consequently among the whole spectrum of exponents characterizing the power law behavior of jumps length there exist distinguished values of stability index representing the most efficient search processes these exponents correspond to extreme minimal or maximal or intermediate values of stability index associated with gaussian maximally heavy tailed or cauchy like strategies respectively

langevin picture of levy walks and their extensions in this paper we derive langevin picture of levy walks applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled levy walk next we introduce extensions of levy walks in which jump sizes are some functions of waiting times we prove that under proper scaling conditions such generalized levy walks converge in distribution to the appropriate limiting processes we also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes showing that different coupling functions lead to various types of anomalous diffusion

dynamic models for spatiotemporal data we propose a model for non stationary spatiotemporal data to account for spatial variability we model the mean function at each time period as a locally weighted mixture of linear regressions to incorporate temporal variation we allow the regression coefficients to change through time the model is cast in a gaussian state space framework which allows us to include temporal components such as trends seasonal effects and autoregressions and permits a fast implementation and full probabilistic inference for the parameters interpolations and forecasts to illustrate the model we apply it to two large environmental data sets tropical rainfall levels and atlantic ocean temperatures

first passage quantities of brownian motion in a bounded domain with multiple targets a unified approach in this paper we introduce a general computation method to systematically determine the mean first passage time mfpt the global mean first passage time gmfpt and splitting probabilities for a continuous brownian motion in a confined 2d or 3d domain with multiple absorbing targets in the bulk or on the boundary this method is applied to spherically symmetric domains in the limit of small sized targets and asymptotic expansions of the mpft gmfpt and splitting probabilities are obtained in four distinct cases 3d domains with targets in the bulk 3d domains with targets on the boundary 2d domains with targets on the bulk and 2d domains with targets on the boundary this approach gives a unified description of existing exact results which were obtained using specific technics and also yields new results in particular for n targets splitting probabilities

multidimensional levy walk and its scaling limits in this paper we obtain the scaling limit of a multidimensional levy walk and describe the detailed structure of the limiting process the scaling limit is a subordinated stable levy motion with the parent process and subordinator being strongly dependent processes the corresponding langevin picture is derived we also introduce a useful method of simulating levy walks with a predefined spectral measure which controls the direction of each jump our approach can be applied in the analysis of real life data we are able to recover the spectral measure from the data and obtain the full characterization of a levy walk we also give examples of some useful spectral measures which cover a large class of possible scenarios in the modeling of real life phenomena

dynamics of populations in a changing environment we present an individual based model of a population that lives in a changing environment the individuals forming the population are subject to mutations and selection pressure using monte carlo simulations we have shown that depending on the values of the mutation rate and selection the population may reach either an active phase it will survive or an absorbing phase it will become extinct we have determined that the transition between the two states phases is continuous we have shown that when the selection is weaker the population lives in all available space while if the selection is stronger it will move to the regions where the living conditions are better avoiding those with more difficult conditions the dependence of the mean time to extinction on the rate of mutations has been determined and discussed

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

random walks on finite lattice tubes exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends explicit formulas are derived for the absorption probabilities at the ends and for the expectations that a random walk will visit a particular lattice site before being absorbed results are obtained for lattice tubes of arbitrary size and each of the regular lattice types square triangular and honeycomb the results include an adjustable parameter to model the effects of strain such as surface curvature on the surface diffusion results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig zag carbon nanotubes with open ends

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation

robust functional principal components a projection pursuit approach in many situations data are recorded over a period of time and may be regarded as realizations of a stochastic process in this paper robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting our approach combines robust projection pursuit with different smoothing methods consistency of the estimators are shown under mild assumptions the performance of the classical and robust procedures are compared in a simulation study under different contamination schemes 1 introduction analogous to classical principal components analysis pca the projection pursuit approach to robust pca is based on finding projections of the data which have maximal dispersion instead of using the variance as a measure of dispersion a robust scale estimator sn is used for the maximization problem this approach was introduced by li and chen 1985 who proposed estimators based on maximizing or minimizing a robust scale in this way given a sample xi r d 1 i n the first robust principal component vector is defined as

estimation and model selection for an ide based spatio temporal model a state space model of the stochastic spatio temporal integro difference equation ide is derived based on multidimensional sampling theory the dimensions of the state space and parameter space of the model are identified from the spatial bandwidth of the system and the support of the redistribution kernel of the ide when both the bandwidth and the kernel support are unknown a method to propose a number of state space and parameter space dimensions is presented these chosen dimensions result in a number of candidate model structures bayesian model selection making use of bayes factor the data augmentation algorithm and importance sampling is then used to identify the model best suited to represent the data in a maximum a posteriori sense