analysis of a nonsmooth optimization approach to robust estimation in this paper we consider the problem of identifying a linear map from measurements which are subject to intermittent and arbitrarily large errors this is a fundamental problem in many estimation related applications such as fault detection state estimation in lossy networks hybrid system identification robust estimation etc the problem is hard because it exhibits some intrinsic combinatorial features therefore obtaining an effective solution necessitates relaxations that are both solvable at a reasonable cost and effective in the sense that they can return the true parameter vector the current paper discusses a nonsmooth convex optimization approach and provides a new analysis of its behavior in particular it is shown that under appropriate conditions on the data an exact estimate can be recovered from data corrupted by a large even infinite number of gross errors

adaptive identification of linear systems subject to gross errors in this note we investigate the convergence of a robust recursive identifier for linear models subject to impulsive disturbances under the assumption that the disturbance is unknown and can be of arbitrarily large magnitude the analyzed algorithm attempts to minimize online the sum of absolute errors so as to achieve a sparse prediction error sequence it is proved that the identifier converges exponentially fast into an euclidean ball whose size is determined by the richness properties of the estimation data the frequency of occurrence of impulsive errors and the parameters of the algorithm

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

on the representation of unity by binary cubic forms if f x y is a binary cubic form with integer coefficients such that f x 1 has at least two distinct complex roots then the equation f x y 1 possesses at most ten solutions in integers x and y nine if f has a nontrivial automorphism group if further f x y is reducible over z x y then this equation has at most 2 solutions unless f x y is equivalent under gl2 z action to either x x2 xy y2 or x x2 2y2 the proofs of these results rely upon the method of thue siegel as refined by evertse together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational diophantine approximation along the way we completely solve all thue equations f x y 1 for f cubic and irreducible of positive discriminant df 106 as corollaries we obtain bounds for the number of solutions to more general cubic thue equations of the form f x y m and to mordell s equation y2 x3 k where m and k are nonzero integers

recursive prediction error identification using the nonlinear wiener model the nonlinear wiener model consisting of a linear dynamic block in cascade with a static nonlinearity is considered a recursive prediction error identification algorithm based on the wiener model is derived the linear dynamic block is modelled as a siso transfer function operator and the static nonlinearity is approximated with a piecewise linear function a theoretical analysis of the method is carried out and conditions for local convergence to the true parameter vector are given in particular the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation a numerical example illustrates the performance of the algorithm further practical guidelines on how to apply the algorithm are also included in the paper

nonlinear diffusion effects on biological population spatial patterns motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations we investigate its implications in a paradigmatic model for the evolution of a single species density u x t the standard model includes growth and competition in a logistic expression and spreading is modeled through normal diffusion moreover the competition term is nonlocal which has been shown to give rise to spatial patterns we generalize the diffusion term through the nonlinear form tu x t d xxu x t with d 0 encompassing the cases where the state dependent diffusion coefficient either increases 1 or decreases 1 with the density yielding subdiffusion or superdiffusion respectively by means of numerical simulations and analytical considerations we display how that nonlinearity alters the phase diagram the type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape inducing fragmentation in the subdiffusive case the detection of the main persistent mode allows analytical prediction of the critical thresholds

a population model with nonlinear diffusion abstract a model is presented for a single species population moving in a limited one dimensional environment the birth death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age the diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation it is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant the interaction of the birth death and diffusion processes is studied and is shown to yield a modified birth death mechanism which is both time and space dependent

piecewise affine regression via recursive multiple least squares and multicategory discrimination in nonlinear regression choosing an adequate model structure is often a challenging problem while simple models such as linear functions may not be able to capture the underlying relationship among the variables over parametrized models described by a large set of nonlinear basis functions tend to overfit the training data leading to poor generalization on unseen data piecewise affine pwa models can describe nonlinear and possible discontinuous relationships while maintaining simple local affine regressor to output mappings with extreme flexibility when the polyhedral partitioning of the regressor space is learned from data rather than fixed a priori in this paper we propose a novel and numerically very efficient two stage approach for pwa regression based on a combined use of i recursive multi model least squares techniques for clustering and fitting linear functions to data and ii linear multi category discrimination either offline batch via a newton like algorithm for computing a solution of unconstrained optimization problems with objective functions having a piecewise smooth gradient or online recursive via averaged stochastic gradient descent

computing all power integral bases of cubic fields applying baker s effective method and the reduction procedure of baker and davenport we present several lists of solutions of index form equations in totally real and complex cubic algebraic number fields these solutions yield all power integral bases of these fields 1 introduction let k be a cubic algebraic number field and denote by zk the ring of integers of k a power integral basis of k is an integral basis of the form 1 2 with some a e zk if there exists such an a then we say that k is monogenic since zk z e obviously if a has this property then it holds also for a k with any k e z from a practical point of view it is important to know whether there exists a power integral basis of k and if so what are the numerical values of a if the galois group of k is cyclic then the discriminant of k is a full square the problem of monogeneity in cyclic cubic fields was considered by m n gras 12 13 archinard 1 and dummit and kisilevsky 4 m n gras and archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic moreover dummit and kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases for arbitrary algebraic number fields l it was proved by gyory 14 that up to obvious translations by elements of z there are only finitely many a e zl with zl z and he gave effective but rather large bounds for the sizes of these a in this paper we present a method which allows us to determine all possible values of a up to translation with rational integers such that 1 2 is a power integral

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

piecewise linear solution paths with application to direct weight optimization recently pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression this paper presents a generalization of these algorithms to a wider class of problems it is shown that the approach can be applied to the nonparametric system identification method direct weight optimization dwo and be used to enhance the computational efficiency of this method the most important design parameter in the dwo method is a parameter l controlling the bias variance trade off and the use of parametric optimization with piecewise linear solution paths means that the dwo estimates can be efficiently computed for all values of l simultaneously this allows for designing computationally attractive adaptive bandwidth selection algorithms one such algorithm for dwo is proposed and demonstrated in two examples

complete solutions to families of quartic thue equations using a method due to e thomas we prove that if lai 9 9 1027 then the diophantine equations 4 3 2 2 3 4 x ax y x y axy y and 4 3 2 2 3 4 x ax y 3x y axy y i1 have exactly twelve solutions namely x y 0 1 1 0 1 1 f 1 1 a 1 l fa and eight solutions x y o 1 i 0 i 1 i f1 respectively

spectral bayesian estimation for general stochastic hybrid systems abstract general stochastic hybrid systems gshs have been formulated to represent various types of uncertainties in hybrid dynamical systems in this paper we propose computational techniques for bayesian estimation of gshs in particular the fokker planck equation that describes the evolution of uncertainty distributions along gshs is solved by spectral techniques where an arbitrary form of probability density of the hybrid state is represented by a mixture of fourier series the method is based on splitting the fokker planck equation represented by an integro partial differential equation into the partial differentiation part for continuous diffusion and the integral part for discrete transition and integrating the solution of each part the propagated density function is used with a likelihood function in the bayes formula to estimate the hybrid state for given sensor measurements the unique feature of the proposed technique is that the probability density describing complete stochastic properties of the hybrid state is constructed without relying on the common gaussian assumption in contrast to other methods that compute certain expected values such as mean or variance we apply the proposed method to the estimation of two examples the bouncing ball model and the dubins vehicle model we show that the proposed technique yields propagated densities consistent with monte carlo simulations more accurate estimates over the gaussian based approach and some computational benefits over a particle filter

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

forward and backward least angle regression for nonlinear system identification a forward and backward least angle regression lar algorithm is proposed to construct the nonlinear autoregressive model with exogenous inputs narx that is widely used to describe a large class of nonlinear dynamic systems the main objective of this paper is to improve model sparsity and generalization performance of the original forward lar algorithm this is achieved by introducing a replacement scheme using an additional backward lar stage the backward stage replaces insignificant model terms selected by forward lar with more significant ones leading to an improved model in terms of the model compactness and performance a numerical example to construct four types of narx models namely polynomials radial basis function rbf networks neuro fuzzy and wavelet networks is presented to illustrate the effectiveness of the proposed technique in comparison with some popular methods

estimation of jump box jenkins models abstract jump box jenkins bj models are a collection of a finite set of linear dynamical submodels in bj form that switch over time according to a markov chain this paper addresses the problem of maximum a posteriori estimation of jump bj models from a given training input output dataset the proposed solution method estimates the coefficients of the bj submodels the state transition probabilities of the markov chain regulating the switching of operating modes and the corresponding mode sequence hidden in the dataset in particular the posterior distribution of all the unknown variables characterizing the jump bj model is derived and then maximized using a coordinate ascent algorithm the resulting estimation algorithm alternates between gauss newton optimization of the coefficients of the bj submodels a method derived based on an instance of prediction error methods tailored to bj models with switching coefficients and approximated dynamic programming for optimization of the sequence of active modes the quality of the proposed estimation approach is evaluated on a numerical example based on synthetic data and in a case study related to segmentation of honeybee dances

recursive identification method for piecewise arx models a sparse estimation approach this paper deals with the identification of nonlinear systems using piecewise linear models by means of a sparse overparameterization this challenging problem is turned into a convex optimization problem the proposed method uses a likelihood based methodology which adaptively penalizes model complexity and directly leads to a recursive implementation in this sparse estimation approach the tuning of user parameters is avoided and the computational complexity is kept linear in the number of data samples numerical examples with both simulated and experimental data are presented and the results are compared with previously published methods

error bounds for piecewise smooth and switching regression this paper deals with regression problems in which the nonsmooth target is assumed to switch between different operating modes specifically piecewise smooth pws regression considers target functions switching deterministically via a partition of the input space while switching regression considers arbitrary switching laws this paper derives generalization error bounds in these two settings by following the approach based on rademacher complexities for pws regression our derivation involves a chaining argument and a decomposition of the covering numbers of pws classes in terms of the ones of their component function classes and the capacity of the classifier partitioning the input space this yields error bounds with a radical dependence on the number of modes for switching regression the decomposition can be performed directly at the level of the rademacher complexities which yields bounds with a linear dependence on the number of modes by using once more chaining and decomposition at the level of covering numbers we show how to recover a radical dependence examples of applications are given in particular for pws and swichting regression with linear and kernel based component functions

a bounded error approach to piecewise affine system identification this paper proposes a three stage procedure for parametric identification of piecewise affine autoregressive exogenous pwarx models the first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems min pfs problem for a suitable set of linear complementary inequalities derived from data second a refinement procedure reduces misclassifications and improves parameter estimates the third stage determines a polyhedral partition of the regressor set via two class or multiclass linear separation techniques as a main feature the algorithm imposes that the identification error is bounded by a quantity spl delta such a bound is a useful tuning parameter to trade off between quality of fit and model complexity the performance of the proposed pwa system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick and place machine

variational depth from focus reconstruction this paper deals with the problem of reconstructing a depth map from a sequence of differently focused images also known as depth from focus dff or shape from focus we propose to state the dff problem as a variational problem including a smooth but nonconvex data fidelity term and a convex nonsmooth regularization which makes the method robust to noise and leads to more realistic depth maps in addition we propose to solve the nonconvex minimization problem with a linearized alternating directions method of multipliers allowing to minimize the energy very efficiently a numerical comparison to classical methods on simulated as well as on real data is presented

large integral points on elliptic curves to ml friend dan shanks abstract we describe several methods which permit one to search for big integral points on certain elliptic curves i e for integral solutions x of certain diophantine equations of the form v2 x3 ax b a b e z in a large range ixi iy i b in time polynomial in loglogb we also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point

on the resolution of inhomogeneous norm form equations in two dominating variables applying baker s well known method and the reduction procedure described by baker and davenport we give a numerical algorithm for finding all solutions of inhomogeneous thue equations of type nk q x ay x 1 in the variables x y e z and a e zic with a max x y ll2 where k q a is a totally real cubic field

a difference of convex functions algorithm for switched linear regression this paper deals with switched linear system identification and more particularly aims at solving switched linear regression problems in a large scale setting with both numerous data and many parameters to learn we consider the recent minimum of error framework with a quadratic loss function in which an objective function based on a sum of minimum errors with respect to multiple submodels is to be minimized the paper proposes a new approach to the optimization of this nonsmooth and nonconvex objective function which relies on difference of convex dc functions programming in particular we formulate a proper dc decomposition of the objective function which allows us to derive a computationally efficient dc algorithm numerical experiments show that the method can efficiently and accurately learn switching models in large dimensions and from many data points

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

identification of switched linear regression models using sum of norms regularization this paper proposes a general convex framework for the identification of switched linear systems the proposed framework uses over parameterization to avoid solving the otherwise combinatorially forbidding identification problem and takes the form of a least squares problem with a sum of norms regularization a generalization of the 1 regularization the regularization constant regulates the complexity and is used to trade off the fit and the number of submodels

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

on the complexity of piecewise affine system identification the paper provides results regarding the computational complexity of hybrid system identification more precisely we focus on the estimation of piecewise affine pwa maps from input output data and analyze the complexity of computing a global minimizer of the error previous work showed that a global solution could be obtained for continuous pwa maps with a worst case complexity exponential in the number of data in this paper we show how global optimality can be reached for a slightly more general class of possibly discontinuous pwa maps with a complexity only polynomial in the number of data however with an exponential complexity with respect to the data dimension this result is obtained via an analysis of the intrinsic classification subproblem of associating the data points to the different modes in addition we prove that the problem is np hard and thus that the exponential complexity in the dimension is a natural expectation for any exact algorithm

model based modes detection and discernibility for switched affine discrete time systems this paper addresses the problem of modes detection for switching discrete time linear and affine systems a unified presentation of the model based methods with the corresponding modes detectors is carried out and a comparative study is conducted next the different notions of discernibility proposed in the literature are presented such notions are central since they guarantee the uniqueness of the solution delivered by the modes detectors a complete step by step procedure of modes detection is proposed the paper includes whenever necessary some extensions of the existing results

identification of piecewise affine systems based on statistical clustering technique this paper is concerned with the identification of a class of piecewise affine systems called a piecewise affine autoregressive exogenous pwarx model the pwarx model is composed of arx sub models each of which corresponds to a polyhedral region of the regression space under the temporary assumption that the number of sub models is known a priori the input output data are collected into several clusters by using a statistical clustering algorithm we utilize support vector classifiers to estimate the boundary hyperplane between two adjacent regions in the regression space in each cluster the parameter vector of the sub model is obtained by the least squares method it turns out that the present statistical clustering approach enables us to estimate the number of sub models based on the information criteria such as caic and mdl the estimate of the number of sub models is performed by applying the identification procedure several times to the same data set after having fixed the number of sub models to different values finally we verify the applicability of the present identification method through a numerical example of a hammerstein model

brief paper a continuous optimization framework for hybrid system identification we propose a new framework for hybrid system identification which relies on continuous optimization this framework is based on the minimization of a cost function that can be chosen as either the minimum or the product of loss functions the former is inspired by traditional estimation methods while the latter is inspired by recent algebraic and support vector regression approaches to hybrid system identification in both cases the identification problem is recast as a continuous optimization program involving only the real parameters of the model as variables thus avoiding the use of discrete optimization this program can be solved efficiently by using standard optimization methods even for very large data sets in addition the proposed framework easily incorporates robustness to different kinds of outliers through the choice of the loss function

synchronization of life cycles by collective transport and emergence of supercycles a model of physiological age accompanied by nonlinear diffusion in space is studied analytically and numerically and is shown to develop nonstationary traveling population waves a window of intermediate growth rates is found where collective supercycles are formed from individual stochastic life cycles supercycle periods can be considerably different larger or smaller than the average longevities of contributing individuals while the time averaged spatial expansion rate has a local maximum in the supercycling mode a method of adiabatic similarity solutions is used to derive dependencies of the solution parameters on source and sink inhomogeneities and obtain closed coupled dynamic equations for the age structure and leading and trailing fronts analytical results are compared with numerically computed similarity and full solutions for several types of population waves we discuss possible model applications to development of lichen thallus multiyear patterns of agricultural crop yields and autocorrelation of locust swarming

a new kernel based approach to hybrid system identification all the approaches for hybrid system identification appeared in the literature assume that model complexity is known popular models are e g piecewise arx with a priori fixed orders furthermore the developed numerical procedures have been tested only on simple systems e g composed of arx subsystems of order 1 or at most 2 this represents a major drawback for real applications this paper proposes a new regularized technique for identification of piecewise affine systems namely the hybrid stable spline algorithm hss hss exploits the recently introduced stable spline kernel to model the submodels impulse responses as zero mean gaussian processes including information on submodels predictor stability the algorithm consists of a two step procedure first exploiting the bayesian interpretation of regularization the problem of classifying and distributing the data to the subsystems is cast as marginal likelihood optimization we show how an approximated optimization can be efficiently performed by a markov chain monte carlo scheme then the stable spline algorithm is used to reconstruct each subsystem numerical experiments on real and simulated data are included to test the new procedure they show that hss not only solves all the most popular benchmark problems proposed in the literature without having exact information on arx subsystems order but can also identify more complex high order piecewise affine systems matlab code implementing the approach called hybrid stable spline toolbox is also made available

reduced size kernel models for nonlinear hybrid system identification this brief paper focuses on the identification of nonlinear hybrid dynamical systems i e systems switching between multiple nonlinear dynamical behaviors thus the aim is to learn an ensemble of submodels from a single set of input output data in a regression setting with no prior knowledge on the grouping of the data points into similar behaviors to be able to approximate arbitrary nonlinearities kernel submodels are considered however in order to maintain efficiency when applying the method to large data sets a preprocessing step is required in order to fix the submodel sizes and limit the number of optimization variables this brief paper proposes four approaches respectively inspired by the fixed size least squares support vector machines the feature vector selection method the kernel principal component regression and a modification of the latter in order to deal with this issue and build sparse kernel submodels these are compared in numerical experiments which show that the proposed approach achieves the simultaneous classification of data points and approximation of the nonlinear behaviors in an efficient and accurate manner