sample and ergodic properties of some min stable processes a random vector is min stable or jointly negative exponential if any weighted minimum of its components has a negative exponential distribution the vectors can be subordinated to a two dimensional homogeneous poisson point process through positive mathscr l 1 functions called spectral functions a critical feature of this representation is the point of the poisson process called the location that defines a min stable random variable a measure of association between min stable random variables is used to define mixing conditions for min stable processes the association between two min stable random variables x 1 and x 2 is defined as the probability that they share the same location and is denoted by q x 1 x 2 mixing criteria for a min stable process x t are defined by how fast the association between x t and x t s goes to zero as s rightarrow infty for some stationary processes including the moving minimum process conditions on the spectral functions are derived in order that the processes satisfy mixing conditions

probabilities of concurrent extremes the statistical modelling of spatial extremes has recently made major advances much of its focus so far has been on the modelling of the magnitudes of extreme events but little attention has been paid on the timing of extremes to address this gap this paper introduces the notion of extremal concurrence suppose that one measures precipitation at several synoptic stations over multiple days we say that extremes are concurrent if the maximum precipitation over time at each station is achieved simultaneously e g on a single day under general conditions we show that the finite sample concurrence probability converges to an asymptotic quantity deemed extremal concurrence probability using palm calculus we establish general expressions for the extremal concurrence probability through the max stable process emerging in the limit of the componentwise maxima of the sample explicit forms of the extremal concurrence probabilities are obtained for various max stable models and several estimators are introduced in particular we prove that the pairwise extremal concurrence probability for max stable vectors is precisely equal to the kendall s tau the estimators are evaluated by using simulations and applied to study the concurrence patterns of temperature extremes in the united states the results demonstrate that concurrence probability can provide a powerful new perspective and tools for the analysis of the spatial structure and impact of extremes

a numerical integration perspective on gaussian filters this paper proposes a numerical integration perspective on the gaussian filters a gaussian filter is approximation of the bayesian inference with the gaussian posterior probability density assumption being valid there exists a variation of gaussian filters in the literature that derived themselves from very different backgrounds from the numerical integration viewpoint various versions of gaussian filters are only distinctive from each other in their specific treatments of approximating the multiple statistical integrations a common base is provided for the first time to analyze and compare gaussian filters with respect to accuracy efficiency and stability factor this study is expected to facilitate the selection of appropriate gaussian filters in practice and to help design more efficient filters by employing better numerical integration methods

conditional simulation of max stable processes summary since many environmental processes such as heat waves or precipitation are spatial in extent it is likely that a single extreme event affects several loca tions and the areal modelling of extremes is therefore essential if the spatial dependence of e xtremes has to be appropriately taken into account this paper proposes a framework for conditional simulations of max stable processes and give closed forms for brown resnick and schlather processes we test the method on simulated data and give an application to extreme rainfall around zurich and extreme temperature in switzerland results show that the proposed framework provides accurate conditional simulations and can handle real sized problems

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

exact simulation of max stable processes max stable processes play an important role as models for spatial extreme events their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult algorithms based on finite approximations are often inexact and computationally inefficient we present a new algorithm for exact simulation of a max stable process at a finite number of locations it relies on the idea of simulating only the extremal functions that is those functions in the construction of a max stable process that effectively contribute to the pointwise maximum we further generalize the algorithm by dieker mikosch 2015 for brown resnick processes and use it for exact simulation via the spectral measure we study the complexity of both algorithms prove that our new approach via extremal functions is always more efficient and provide closed form expressions for their implementation that cover most popular models for max stable processes and multivariate extreme value distributions for simulation on dense grids an adaptive design of the extremal function algorithm is proposed

state estimation for invariant systems on lie groups with delayed output measurements this paper proposes a state estimation methodology for invariant systems on lie groups where outputs of the system are measured with delay the proposed method is based on cascading an observer and a predictor the observer uses delayed measurements and provides estimates of delayed states the predictor uses those estimates together with the current inputs of the system to compensate for the delay and to provide a prediction of the current state of the system we consider three classes of left invariant right invariant and mixed invariant systems and propose predictors tailored to each class the key contribution of the paper is to exploit the underlying symmetries of systems to design novel predictors that are computationally simple and generic in the sense that they can be combined with any stable observer or filter we provide a rigorous stability analysis demonstrating that the prediction of the current state converges to the current system trajectory if the observer state converges to the delayed system trajectory the good performance of the proposed approach is demonstrated using a sophisticated software in the loop simulator indicating the robustness of the observer predictor methodology even when large measurement delays are present

variational bayesian adaptive cubature information filter based on wishart distribution this paper presents a noise adaptive variational bayesian cubature information filter based on wishart distribution in the frame of recursive bayesian estimation the noise adaptive information filter propagating the information matrix and information state is derived and the integration of recursive bayesian estimation is approximated by cubature integration rule then the inverse of measurement noise matrix is modeled as a wishart distribution so the joint distribution of posterior state and measurement noise can be approximated by the product of independent gaussian and wishart furthermore the corresponding square root version is also derived to improve numerical characteristics simulation results with unknown and correlated measurement noise demonstrate the effectiveness of the proposed algorithms

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

some relations between extended and unscented kalman filters the unscented kalman filter ukf has become a popular alternative to the extended kalman filter ekf during the last decade ukf propagates the so called sigma points by function evaluations using the unscented transformation ut and this is at first glance very different from the standard ekf algorithm which is based on a linearized model the claimed advantages with ukf are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model we point out several less known links between ekf and ukf in terms of two conceptually different implementations of the kalman filter the standard one based on the discrete riccati equation and one based on a formula on conditional expectations that does not involve an explicit riccati equation first it is shown that the sigma point function evaluations can be used in the classical ekf rather than an explicitly linearized model second a less cited version of the ekf based on a second order taylor expansion is shown to be quite closely related to ukf the different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

extended information filter on matrix lie groups in this paper we propose a new state estimation algorithm called the extended information filter on lie groups the proposed filter is inspired by the extended kalman filter on lie groups and exhibits the advantages of the information filter with regard to multisensor update and decentralization while keeping the accuracy of stochastic inference on lie groups we present the theoretical development and demonstrate its performance on multisensor rigid body attitude tracking by forming the state space on the so 3 r3 group where the first and second component represent the orientation and angular rates respectively the performance of the filter is compared with respect to the accuracy of attitude tracking with parametrization based on euler angles and with respect to execution time of the extended kalman filter formulation on lie groups the results show that the filter achieves higher performance consistency and smaller error by tracking the state directly on the lie group and that it keeps smaller computational complexity of the information form with respect to high number of measurements

square root quadrature kalman filtering the quadrature kalman filter qkf is a recursive nonlinear filtering algorithm developed in the kalman filtering framework it computes the mean and covariance of all conditional densities using the gauss hermite quadrature rule in this correspondence we develop a square root extension of the quadrature kalman filter using matrix triangularizations the square root quadrature kalman filter sqkf propagates the mean and the square root of the covariance although equivalent to the qkf algebraically the sqkf exhibits excellent numerical characteristics but at the expense of increased computational complexity we also present possible refinements of the generic sqkf

square root sigma point information filtering the sigma point information filters employ a number of deterministic sigma points to calculate the mean and covariance of a random variable which undergoes a nonlinear transformation these sigma points can be generated by the unscented transform or stirling s interpolation which corresponds to the unscented information filter uif and the central difference information filter cdif respectively in this technical note we develop the square root extensions of uif and cdif which have better numerical properties than the original versions e g improved numerical accuracy double order precision and preservation of symmetry we also show that the square root unscented information filter sruif might lose the positive definiteness due to the negative cholesky update whereas the square root central difference information filter srcdif has only positive cholesky update therefore the srcdif is preferable to the sruif concerning the numerical stability

exploiting occurrence times in likelihood inference for componentwise maxima multivariate extreme value distributions arise as the limiting distributions of normalised componentwise maxima they are often used to model multivariate data that can be regarded as the componentwise maxima of some unobserved underlying multivariate process in many applications we have extra information we often know the locations of the maxima within the underlying process if the process is temporal this knowledge is frequently available through the dates on which the maxima are recorded we show how to incorporate this extra information into maximum likelihood procedures asymptotic and small sample efficiency results are presented for the dependence parameter in the logistic parametric sub class of bivariate extreme value distributions we conclude with an application to sea levels

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

performance evaluation of ukf based nonlinear filtering the performance of the modified unscented kalman filter ukf for nonlinear stochastic discrete time system with linear measurement equation is investigated it is proved that under certain conditions the estimation error of the ukf remains bounded furthermore it is shown that the design of noise covariance matrix plays an important role in improving the stability of the algorithm error behavior of the ukf is then derived in terms of mean square error mse and the cramer rao lower bound crlb is introduced as a performance measure the modified ukf is found to approach the crlb if the difference between the real noise covariance matrix and the selected one is small enough these results are verified by using monte carlo simulations on two example systems

a class of stable square root nonlinear information filters information filters can process nonlinear systems with uncertain prior knowledge and the particular square root form of adaptive filters can improve numerical stability based on a square root decomposition of information matrix and an extra positive definite matrix the unscented transform and the cubature rule are applied to the information filtering architecture for nonlinear estimation a class of stable square root nonlinear information filters is then proposed in this technical note in addition the boundedness of their estimation errors is also proven results from simulations of filtering a chaotic map demonstrate that the proposed square root nonlinear filters can improve numerical stability and has better filtering performance than other information filters

comment on a new method for the nonlinear transformation of means and covariances in filters and estimators with authors reply the above paper julier et al ieee trans automat contr vol 45 pp 477 82 2000 generalizes the kalman filter to nonlinear systems by transforming approximations of the probability distributions through the nonlinear process and measurement functions this comment derives exactly the same estimator by linearizing the process and measurement functions by a statistical linear regression through some regression points in contrast with the extended kalman filter which uses an analytic linearization in one point this insight allows one 1 to understand predict the performance of the estimator for specific applications and 2 to make adaptations to the estimator i e the choice of the regression points and their weights in those cases where the original formulation does not assure good results in reply the authors state that the commenters conclusion is unnecessarily narrow interpretation of results

likelihood based inference for max stable processes the last decade has seen max stable processes emerge as a common tool for the statistical modeling of spatial extremes however their application is complicated due to the unavailability of the multivariate density function and so likelihood based methods remain far from providing a complete and flexible framework for inference in this article we develop inferentially practical likelihood based methods for fitting max stable processes derived from a composite likelihood approach the procedure is sufficiently reliable and versatile to permit the simultaneous modeling of marginal and dependence parameters in the spatial context at a moderate computational cost the utility of this methodology is examined via simulation and illustrated by the analysis of united states precipitation extremes

on the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max stable distributions full likelihood based inference for high dimensional multivariate extreme value distributions or max stable processes is feasible when incorporating occurrence times of the maxima without this information ddd dimensional likelihood inference is usually precluded due to the large number of terms in the likelihood however some studies have noted bias when performing high dimensional inference that incorporates such event information particularly when dependence is weak we elucidate this phenomenon showing that for unbiased inference in moderate dimensions dimension ddd should be of a magnitude smaller than the square root of the number of vectors over which one takes the componentwise maximum a bias reduction technique is suggested and illustrated on the extreme value logistic model

geostatistics of extremes we describe a prototype approach to flexible modelling for maxima observed at sites in a spatial domain based on fitting of max stable processes derived from underlying gaussian random fields the models we propose have generalized extreme value marginal distributions throughout the spatial domain consistent with statistical theory for maxima in simpler cases and can incorporate both geostatistical correlation functions and random set components parameter estimation and fitting are performed through composite likelihood inference applied to observations from pairs of sites with occurrence times of maxima taken into account if desired and competing models are compared using appropriate information criteria diagnostics for lack of model fit are based on maxima from groups of sites the approach is illustrated using annual maximum temperatures in switzerland with risk analysis proposed using simulations from the fitted max stable model drawbacks and possible developments of the approach are discussed

stochastic stability of the discrete time extended kalman filter the authors analyze the error behavior for the discrete time extended kalman filter for general nonlinear systems in a stochastic framework in particular it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough this result is verified by numerical simulations for an example system

stationary max stable fields associated to negative definite functions let wi i be independent copies of a zero mean gaussian process w t t d with stationary increments and variance 2 t independently of wi let i 1 ui be a poisson point process on the real line with intensity e y dy we show that the law of the random family of functions vi i where vi t ui wi t 2 t 2 is translation invariant in particular the process t i 1 vi t is a stationary max stable process with standard gumbel margins the process arises as a limit of a suitably normalized and rescaled pointwise maximum of n i i d stationary gaussian processes as n if and only if w is a nonisotropic fractional brownian motion on d under suitable conditions on w the process has a mixed moving maxima representation

a dependence measure for multivariate and spatial extreme values properties and inference we present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems for multivariate problems the dependence measure characterises dependence at the bivariate level for all pairs and all higher orders up to and including the dimension of the variable necessary and sufficient conditions are given for subsets of dependence measures to be self consistent that is to guarantee the existence of a distribution with such a subset of values for the dependence measure for pairwise dependence these conditions are given in terms of positive semidefinite matrices and non differentiable positive definite functions we construct new nonparametric estimators for the dependence measure which unlike all naive nonparametric estimators impose these self consistency properties as the new estimators provide an improvement on the naive methods both in terms of the inferential and interpretability properties their use in exploratory extreme value analyses should aid the identification of appropriate dependence models the methods are illustrated through an analysis of simulated multivariate data which shows that a lack of self consistency is frequently a problem with the existing estimators and by a spatial analysis of daily rainfall extremes in south west england which finds a smooth decay in extremal dependence with distance

estimation of husler reiss distributions and brown resnick processes summary estimation of extreme value parameters from observations in the max domain of attraction of a multivariate max stable distribution commonly uses aggregated data such as block maxima multivariate peaks over threshold methods in contrast exploit additional information from the non aggregated large observations we introduce an approach based on peaks over thresholds that provides several new estimators for processes in the max domain of attraction of the frequently used husler reiss model and its spatial extension brown resnick processes the method relies on increments t0 conditional on t0 exceeding a high threshold where t0 is a fixed location when the marginals are standardized to the gumbel distribution these increments asymptotically form a gaussian process resulting in computationally simple estimates of the husler reiss parameter matrix and particularly enables parametric inference for brown resnick processes based on high dimensional multivariate densities this is a major advantage over composite likelihood methods that are commonly used in spatial extreme value statistics since they rely only on bivariate densities a simulation study compares the performance of the new estimators with other commonly used methods as an application we fit a non isotropic brown resnick process to the extremes of 12 year data of daily wind speed measurements

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

efficient inference for spatial extreme value processes associated to log gaussian random functions max stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes and thus form an important class of models for the extreme values of spatial processes until recently inference for max stable processes has been restricted to the use of pairwise composite likelihoods due to intractability of higher dimensional distributions in this work we consider random fields that are in the domain of attraction of a widely used class of max stable processes namely those constructed via manipulation of log gaussian random functions for this class we exploit limiting d dimensional multivariate poisson process intensities of the underlying process for inference on all d vectors exceeding a high marginal threshold in at least one component employing a censoring scheme to incorporate information below the marginal threshold we also consider the d dimensional distributions for the equivalent max stable process and perform full likelihood inference by exploiting the methods of stephenson tawn 2005 where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood the stephenson tawn likelihood is in fact simply a special case of the censored poisson process likelihood we assess the improvements in inference from both methods over pairwise likelihood methodology by simulation