robust kalman filter based on a generalized maximum likelihood type estimator a new robust kalman filter is proposed that detects and bounds the influence of outliers in a discrete linear system including those generated by thick tailed noise distributions such as impulsive noise besides outliers induced in the process and observation noises we consider in this paper a new type called structural outliers for a filter to be able to counter the effect of these outliers observation redundancy in the system is necessary we have therefore developed a robust filter in a batch mode regression form to process the observations and predictions together making it very effective in suppressing multiple outliers a key step in this filter is a new prewhitening method that incorporates a robust multivariate estimator of location and covariance the other main step is the use of a generalized maximum likelihood type gm estimator based on schweppe s proposal and the huber function which has a high statistical efficiency at the gaussian distribution and a positive breakdown point in regression the latter is defined as the largest fraction of contamination for which the estimator yields a finite maximum bias under contamination this gm estimator enables our filter to bound the influence of residual and position where the former measures the effects of observation and innovation outliers and the latter assesses that of structural outliers the estimator is solved via the iteratively reweighted least squares irls algorithm in which the residuals are standardized utilizing robust weights and scale estimates finally the state estimation error covariance matrix of the proposed gm kalman filter is derived from its influence function simulation results revealed that our filter compares favorably with the h filter in the presence of outliers

high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

dichotomies and asymptotic solutions of nonlinear differential systems the problem of asymptotic equivalence between two systems of ordinary differential equations has been studied by many authors l 5 it was first used by levinson in his theorem on asymptotic integration 6 71 in l 51 this problem is studied when system 1 2 is stable a t is constant and g t y b t is integrable in 3 theorems 5 111 it is assumed that the linear system 1 2 has an ordinary dichotomy and the nonlinear term g t y is lipschitz continuous in 2 the previous results are extended to nonlinear perturbations g t y which are not lipschitz continuous such that g t y l i r t y where r t y is monotone in y we emphasize that in all these works only the asymptotic equivalence between bounded solutions is established nothing is said about a possible correspondence between unbounded solutions of 1 1 and 1 2 in this paper we suppose that the nonlinear term g t y satisfies some integrability conditions that we summarize in hypothesis h2 these conditions permit one to consider systems with large nonlinearities for example equations where g t y is an oscillatory function of the variable y no lipschitz conditions or monotonicity properties are required of g t y however we will use dichotomic properties of system 1 2

robust estimation in structured linear regression a structured linear regression model is one in which there are permanent dependencies among some p row vectors of the n x p design matrix to study structured linear regression we introduce a new class of robust estimators called d estimators which can be regarded as a generalization of the least median of squares and least trimmed squares estimators they minimize a dispersion function of the ordered absolute residuals up to the rank h we investigate their breakdown point and exact fit point as a function of h in structured linear regression it is found that the d and s estimators can achieve the highest possible breakdown point for h appropriately chosen it is shown that both the maximum breakdown point and the corresponding optimal value of h h op are sample dependent they hinge on the design but not on the response the relationship between the breakdown point and the design vanishes when h is strictly larger than h op however when h is smaller than h op the breakdown point depends in a complicated way on the design as well as on the response

a central limit theorem for multivariate generalized trimmed k means a central limit theorem for generalized trimmed k means is obtained in a very general framework that covers the multivariate setting general penalty functions and general k 1 several applications including the location estimator case k 1 for elliptical distributions and the construction of multivariate not necessarily connected tolerance zones are also given

lp boundedness properties for volterra difference equations this work deals with l p boundedness of solutions of a class of volterra functional difference equation the effect of different kinds of the nonhomogeneous term lipschitz local lipschitz completely continuous are taken into consideration

soliton solutions of two bidirectional sixth order partial differential equations belonging to the kp hierarchy in this letter we analyse two bidirectional sixth order partial differential equations which are reductions in 1 1 dimensions of equations belonging to the kp hierarchy they have fourth order and fifth order lax pairs respectively we derive their backlund transformations and from the nonlinear superposition formula we can build their soliton solutions like a grammian the interesting dynamics of these solitons is that they may describe not only the overtaking collision but also the head on collision of solitary waves of different type and shape

leverage and breakdown in l 1 regression abstract in this article the notion of leverage of a design point when fitting a linear regression model is interpreted geometrically in the case of least squares fitting the leverage indicators based on the diagonal of the hat matrix are widely applied by interpreting these hat matrix indicators geometrically leverage can be generalized to groups of design points as well as to other methods of fitting the article introduces a leverage indicator that is appropriate for l 1 regression and discusses some aspects of this new diagnostic it is shown that in the case of l 1 regression the leverage indicators have a precise interpretation they tell us about the breakdown and or the exactness of fit as an application the article considers the maximal possible breakdown value of l 1 regression and the choice of designs that achieve this maximum

computing the exact least median of squares estimate and stability diagnostics in multiple linear regression the difficulty in computing the least median of squares lms estimate in multiple linear regression is due to the nondifferentiability and many local minima of the objective function several approximate but not exact algorithms have been suggested this paper presents a method for computing the exact value of the lms estimate in multiple linear regression the lms estimate is a special case of the least quantile of squares lqs estimate which minimizes the qth smallest squared residual for a given data set for lms q n 2 p 1 2 where is the greatest integer function n is the sample size and p is the number of columns in the x matrix the algorithm can compute a range of exact lqs estimates in multiple linear regression by considering left begin array 20 c n p 1 end array right possible theta values it is based on the fact that each lqs estimate is the chebyshev or minimax fit to some q element subset of the data this yields a surprisingly ea

structured stability radii and exponential stability tests for volterra difference systems uniform exponential ue stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space the state space x is an abstract banach space we work with non fading phase spaces c 0 z x and z x and with exponentially fading phase spaces of the p and c 0 types for equations of the convolution type several criteria of ue stability are obtained in terms of the z transform email protected z of the convolution kernel k in terms of the input state operator and of the resolvent fundamental matrix these criteria do not impose additional positivity or compactness assumptions on coefficients k j time varying non convolution difference equations are studied via structured ue stability radii r t of convolution equations these radii correspond to a feedback scheme with delayed output and time varying disturbances we also consider stability radii r c associated with a time invariant disturbance operator unstructured stability radii and stability radii corresponding to delayed feedback for all these types of stability radii two sided estimates are obtained the estimates from above are given in terms of the z transform email protected z the estimate from below via the norm of the input output operator these estimates turn into explicit formulas if the state space x is hilbert or if disturbances are time invariant the results on stability radii are applied to obtain various exponential stability tests for non convolution equations several examples are provided

grammian n soliton solutions of a coupled kdv system from the lax pair and the binary darboux transformation of a coupled korteweg de vries system we show that its nonlinear superposition formula is identical to that obtained for the kaup kupershmidt equation therefore the n soliton solution can be associated with a determinant of the gram type

generalized kaup kupershmidt solitons and other solitary wave solutions of the higher order kdv equations new types of exact explicit solitary wave solutions of the higher order kdv equations are identified by applying a direct method designed specifically for constructing solitary wave solutions of evolution equations the first type is the generalized kaup kupershmidt gkk solitary waves which unify the structures of the sech2 kdv like soliton and the kaup kupershmidt soliton and also provide solutions of some other equations one of those equations is found to possess the multi soliton solutions which makes it a good candidate for being integrable in terms of the gkk solitons another type of solutions of the higher order kdv equations identified by applying the method represents the steady state localized structures the variety of equations possessing such solutions includes the integrable in terms of the sech2 solitons sawada kotera equation which thus appears to provide the localized solutions of two types

convergent solutions of linear functional difference equations in phase space by using weighted summable dichotomies and schauder s fixed point theorem we prove the existence of convergent solutions of linear functional difference equations we apply our result to volterra difference equations with infinite delay

resonant triads for two bidirectional equations in 1 1 dimensions the two soliton solution of the kdv equation is known not to degenerate into a solution describing resonant triads of solitons in 1 1 dimensions however we show that two integrable coupled kdv systems of the drinfeld sokolov class possess solutions which may degenerate into resonant triads these solutions are associated with a resonance relation which generalizes the usual one previously considered by hirota and ito

on bidirectional fifth order nonlinear evolution equations lax pairs and directionally dependent solitary waves in this paper lax pairs are constructed for two fifth order nonlinear evolution equations of boussinesq type which govern wave propagation in two opposite directions one of the equations is related to the well known sawada kotera sk equation and through its bilinear form is identified with the ramani equation the second equation about which very little seems to be known may be considered a bidirectional version of the kaup kupershmidt kk equation and is the main focus of this study the anomalous solitary wave of this latter equation is derived and is found to possess the remarkable property that its profile depends on the direction of propagation this type of directional dependence would appear to be quite unusual and to our knowledge has not been reported in the literature before now by taking an appropriate undirectional long wave limit it is shown that neither the ramani nor the bidirectional kaup kupershmidt bkk equation can be classified as truly boussinesq in character a distinction that is made precise in the study recursion formulas are given for generating an infinity of conserved densities for both equations these are used to obtain the first few conservation laws of the bkk and ramani equations explicitly not surprisingly they exhibit the same lacunary behavior as their unidirectional counterparts in conclusion a canonical interpretation of the n soliton solution of the bkk equation is proposed which provides a basis for constructing these anomalous solitons in a future work

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

calculating the breakdown point of sparse linear models in robust statistics the concept of breakdown point was introduced to quantify the robustness of an estimator in a linear regression model computing the breakdown point is useful in tuning some robust regression estimators e g the least trimmed squares estimator computing the breakdown point for a structured linear model i e one with dependencies among some p rows of the n p design matrix x can be very demanding this article presents an algorithm for calculating the maximum breakdown point for sparse linear models which are a special type of structured linear model whose design matrix has many zero entries the algorithm decomposes a sparse design matrix into smaller submatrixes on which the computation is performed thereby leading to substantial savings in computation an assembly process along with a few numerical examples illustrate the application of the algorithm and demonstrate its computational benefits

nonlinear superposition formula for the kaup kupershmidt partial differential equation abstract the fifth order kaup kupershmidt kk equation is one of the solitonic equations related to the integrable cases of the henon heiles system as opposed to the sawada kotera equation which is its dual equation the construction of the n soliton solutions of kk is not an easy task in using a perturbation scheme or the hirota bilinear formalism from the backlund transformation obtained by singularity analysis considerations we here establish the permutability theorem for the kk equation this allows us to explain the anomalous structure of the n soliton solution empirically obtained by parker physica d 137 2000 25 33 34 48

the integrable kdv6 equations multiple soliton solutions and multiple singular soliton solutions abstract in this work the completely integrable sixth order nonlinear equations kdv6 are studied three distinct cases of these equations are selected for a reliable treatment multiple soliton solutions and multiple singular soliton solutions are formally derived for these equations the cole hopf transformation method combined with the hirota s bilinear sense are used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

on one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

the cole hopf transformation and improved tanh coth method applied to new integrable system kdv6 abstract in this paper we study a new coupled system kdv6 from the point of view of constructing exact solutions for it by using the cole hopf transformation and one improved tanh coth method traveling wave solution which includes periodic and soliton solutions are formally derived

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

roughness of h k dichotomies abstract in this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations the so called h k dichotomies these include a great variety of dichotomies among them the ordinary and exponential dichotomies exponential ordinary dichotomies levinson dichotomies for diagonal systems etc we prove that under general uniform conditions upon the functions h and k these dichotomies are not destroyed by integrable perturbations this result generalizes the property of roughness of ordinary dichotomies under integrable perturbations

asymptotic properties of solutions to nonautonomous volterra difference systems with infinite delay abstract we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear nonautonomous volterra difference system with infinite delay and its perturbed system under additional assumptions this correspondence is elevated to asymptotic equivalence on the other side we prove the existence of weighted convergent solutions

generalized s estimators abstract in this article we introduce a new type of positive breakdown regression method called a generalized s estimator or gs estimator based on the minimization of a generalized m estimator of residual scale we compare the class of gs estimators with the usual s estimators including least median of squares it turns out that gs estimators attain a much higher efficiency than s estimators at the cost of a slightly increased worst case bias we investigate the breakdown point the maxbias curve and the influence function of gs estimators we also give an algorithm for computing gs estimators and apply it to real and simulated data

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

asymptotic behavior in volterra difference systems with unbounded delay we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear volterra difference equation with unbounded delay and its perturbation this correspondence is elevated to asymptotic equivalence under suitable conditions

a bounded influence high breakdown efficient regression estimator abstract we consider the multiple linear regression model y i x i e i i 1 2 n with random carriers and focus on the estimation of this article s main contribution is to present an estimator that is affine regression and scale equivariant has both a high breakdown point and a bounded influence function and has an asymptotic efficiency greater than 95 versus least squares under gaussian errors we give conditions under which the estimator a one step general m estimator that uses schweppe weights and is based on a high breakdown initial estimator satisfies these properties the major conditions necessary for the estimator are a it must be based on a n consistent initial estimator with a 50 breakdown point and b it must be based on a psi function that is odd bounded and strictly increasing the advantage of this estimator over previous approaches is that it does not downweight high leverage points without first considering how they fit the bulk of the data methods of computi

aspects of robust linear regression section 1 of the paper contains a general discussion of robustness in section 2 the influence function of the hampel rousseeuw least median of squares estimator is derived linearly invariant weak metrics are constructed in section 3 it is shown in section 4 that s estimators satisfy an exact holder condition of order 1 2 at models with normal errors in section 5 the breakdown points of the hampel krasker dispersion and regression functionals are shown to be 0 the exact breakdown point of the krasker welsch dispersion functional is obtained as well as bounds for the corresponding regression functional section 6 contains the construction of a linearly equivariant high breakdown and locally lipschitz dispersion functional for any design distribution in section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point section 8 contains a linearly equivariant high breakdown regression functional which is lipschitz continuous at models with normal errors

dichotomies and asymptotic formulas for the solutions of differential equations abstract the new concept of h k dichotomy is studied extensively and is used in asymptotic equivalence problems

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

asymptotic integration of nonautonomous delay differential systems we prove existence theorems and asymptotic formulas for the solutions of a class of delay differential equations with time state dependent lag

generalized l m and r statistics abstract a class of statisticss generalizing u statistics and l statistics and containing other varieties of statistics as well such as trimmed u statistics is studied using the differentiable statistical function approach differential approximations are obtained and the influence curves of these generalized l statistics are derived these results are employed to establish asymptotic normality for such statistics parallel generalizations of m and r statistics are noted strong convergence berry esseen rates and computational aspects are discussed

asymptotic behavior of solutions of poincare recurrence systems abstract sufficient conditions are given for the poincare recurrence system y m 1 a p m y m to have a solution such that y m m 1 o 1 v as m where is an eigenvalue of the constant matrix a and v is an associated eigenvector the summability conditions on p permit conditional convergence and the o 1 terms are specified precisely

an asymptotic theory for retarded functional difference equations using discrete dichotomies and krasnoselsky s theorem we obtain existence and asymptotic behavior of convergent solutions for retarded functional difference equations we also will get some global properties for the set of convergent solutions applications on volterra difference equations with infinite delay are shown

about the behavior of solutions for volterra difference equations with infinite delay we study the existence of discrete almost automorphic solutions and asymptotic behavior for non linear volterra difference equations of convolution type with infinite delay where the nonlinear perturbation is considered not necessarily globally lipschitz the results are a consequence of application of different fixed point theorems namely the contraction principle the leray schauder the krasnoselkii and matkowski s fixed point technique

asymptotic property of linear volterra difference systems by using the resolvent matrix and the comparison principle we investigate the asymptotic behavior of linear volterra difference systems

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates