variance analysis of identified linear miso models having spatially correlated inputs with application to parallel hammerstein models this contribution concerns variance analysis of linear multi input single output models when the inputs are temporally white but where different inputs may be correlated an expression is provided for the variance of a linearly parametrized estimate of the frequency response function from one block i e from one input to the output in particular this expression reveals that the variance increases in one block when the number of estimated parameters in another block is increased but levels off when the number of parameters in the other block reaches the number of parameters in the block in question it also quantifies exactly how correlation between inputs affects the resulting accuracy and a graphical representation is provided for this purpose the results are applicable to parallel miso hammerstein models when the nonlinearities are known and generalize an existing variance expression for this type of model

the effect of regularization on variance error this note addresses the problem of quantifying the effect of noise induced error so called variance error in system estimates found via a regularised cost criterion it builds on recent work by the authors in which expressions for nonregularised criterions are derived which are exact for finite model order those new expressions were established to be very different to previous quantifications that are widely used but based on asymptotic in model order arguments a key purpose of this note is to expose a rapprochement between these new finite model order and the pre existing asymptotic model order quantifications in so doing a further new result is established namely that variance error in the frequency domain is dependent on the choice of the point about which regularization is affected

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

unconditionally stable difference methods for delay partial differential equations this paper is concerned with the numerical solution of parabolic partial differential equations with time delay we focus in particular on the delay dependent stability analysis of difference methods that use a non constrained mesh i e the time step size is not required to be a submultiple of the delay we prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration when the following discretization is used a variant of the classical second order central differences to approximate the diffusion operator a linear interpolation to approximate the delay argument and finally the trapezoidal rule or the second order backward differentiation formula to discretize the time derivative we end the paper with some numerical experiments that confirm the theoretical results

black box identification of multivariable transfer functions asymptotic properties and optimal input design abstract the problem of estimating the transfer function of a multivariable linear stochastic system is considered the transfer function is parametrized as a black box and no order is chosen a priori this means that the model order may increase to infinity as the number of observed data tends to infinity the asymptotic covariance of the transfer function estimate is calculated and is found to be independent of the noise model when the model order increases to infinity the derived expressions are also used to determine optimal unprejudiced input signals for the identification experiment

a structure preserving method for a class of nonlinear dissipative wave equations with riesz space fractional derivatives abstract in this manuscript we consider an initial boundary value problem governed by a 1 1 dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics the model considers the presence of a spatial laplacian of fractional order which is defined in terms of riesz fractional derivatives as well as the inclusion of a generic continuously differentiable potential it is known that the undamped regime has an associated positive energy functional and we show here that it is preserved throughout time under suitable boundary conditions to approximate the solutions of this model we propose a finite difference discretization based on fractional centered differences some discrete quantities are proposed in this work to estimate the energy functional and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative moreover we establish suitable computational constraints under which the discrete energy of the system is positive the method is consistent of second order and is both stable and convergent the numerical simulations shown here illustrate the most important features of our numerical methodology

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

wavelet based iterative methods for a class of 2d partial integro differential equations abstract in this paper an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models two different approaches based on haar and legendre wavelets are studied to develop numerical methods in the first approach time domain is approximated with the help of forward finite difference approach in the second approach both time as well as space domains are approximated by wavelets appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature

two novel linear implicit momentum conserving schemes for the fractional korteweg de vries equation abstract we propose two conservative linear implicit schemes for the space fractional korteweg de vries fkdv equation one is the linear implicit crank nicolson scheme and the other is the linear implicit leap frog scheme in order to obtain a high order discretization in the space direction we adopt the fourier pseudospectral method the crank nicolson scheme and leap frog scheme are used for temporal discretization and those two schemes are efficient in practical computations because of their linear property furthermore we analyse the uniqueness boundness convergence of the two schemes numerical experiments are presented to validate the theoretical analysis

data driven local coordinates for multivariable linear systems and their application to system identification in this paper we introduce a new parametrization for state space systems data driven local coordinates ddlc the parametrization is obtained by restricting the full state space parametrization where all matrix entries are considered to be free to an affine plane containing a given nominal state space realization this affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state space systems at the nominal realization the application of the parametrization to prediction error identification is exemplified simulations indicate that the proposed parametrization has numerical advantages as compared to e g the more commonly used observable canonical form

a numerically efficient hamiltonian method for fractional wave equations abstract in this work we consider a partial differential equation that extends the well known wave equation the model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional laplacian of the riesz type homogeneous dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work the mathematical model has a fractional hamiltonian which is conserved when the damping coefficient is equal to zero and dissipated otherwise motivated by these facts we propose a finite difference method to approximate the solutions of the continuous model the method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model a discretized form of the hamiltoninan is also proposed in this work and we prove analytically that the method is capable of preserving dissipating the discrete energy when the continuous model preserves dissipates the energy we establish rigorously the properties of consistency stability and convergence of the method and provide some a priori bounds for the numerical solutions moreover we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime some computer simulations that assess the capability of the method to preserve dissipate the energy are carried out for illustration purposes

on traveling wave solutions of fisher s equation in two spatial dimensions it is shown for the quadratic fisher equation in two spatial dimensions that along with a plane wave there exist several other traveling waves with nontrivial front geometry some of the solutions are found in explicit form others are constructed approximately the dispersion relationship and velocity curvature dependence generated by these solutions are studied

an explicit positivity preserving finite difference scheme for the classical fisher kolmogorov petrovsky piscounov equation abstract in this work we introduce a family of explicit finite difference schemes to approximate positive solutions of the classical fisher kolmogorov petrovsky piscounov equation from population dynamics a neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters and numerical comparisons with analytical solutions are carried out in some detail

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

on a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model in this work we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion reaction model which appears in the context of population dynamics the existence of traveling wave solutions for the equation under consideration is a well known fact some of such solutions are positive bounded from above and monotone in both space and time motivated by these facts we propose an explicit nonlinear finite difference methodology to approximate consistently the solutions of the model under investigation in the linear regime the method is consistent of first order in time and of second order in space under certain flexible parameter conditions the method is capable of preserving the positivity the boundedness and the spatial and the temporal monotonicity of the traveling wave solutions moreover we establish analytically and numerically that the nonlinear method is conditionally stable a computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered

a front fixing numerical method for a free boundary nonlinear diffusion logistic population model the spatial temporal spreading of a new invasive species in a habitat has interest in ecology and is modeled by a moving boundary diffusion logistic partial differential problem where the moving boundary represents the unknown expanding front of the species in this paper a front fixing approach is applied in order to transform the original moving boundary problem into a fixed boundary one a finite difference method preserving qualitative properties of the theoretical solution is proposed results are illustrated with numerical experiments

on the frequency domain accuracy of closed loop estimates it has been argued that the frequency domain accuracy of high model order estimates obtained on the basis of closed loop data is largely invariant to whether direct or indirect approaches are used the analysis underlying this conclusion has employed variance expressions that are asymptotic both in the data length and the model order and hence are approximations when either of these are finite however recent work has provided variance expressions that are exact for finite possibly low model order and hence can potentially deliver more accurate quantification of estimation accuracy this paper and a companion one revisits the study of identification from closed loop data in light of these new quantifications and establishes that under certain assumptions there can be significant differences in the accuracy of frequency response estimates these discrepencies are established here and in the companion paper to be dependent on what type of direct indirect or joint input output identification strategy is pursued

the cholesky factorization of the inverse correlation or covariance matrix in multiple regression the lower triangular cholesky inverse root cir of the correlation matrix of the dependent and independent variables in a multiple regression problem is shown to be an excellent summary statistic yielding information about the full multiple regression and subsets in particular the cir may be used to select variables and identify alternative subsets it is also useful for interrelationship analysis

on the solution of hyperbolic two dimensional fractional systems via discrete variational schemes of high order of accuracy abstract in this work we consider a general class of damped wave equations in two spatial dimensions the model considers the presence of weyl space fractional derivatives as well as a generic nonlinear potential the system has an associated positive energy functional when damping is not present in which case the model is capable of preserving the energy throughout time meanwhile the energy of the system is dissipated in the damped scenario in this work the weyl space fractional derivatives are approximated through second order accurate fractional centered differences a high order compact difference scheme with fourth order accuracy in space and second order in time is proposed some associated discrete quantities are introduced to estimate the energy functional we prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative the positivity of the discrete energy of the system is also discussed the properties of consistency solvability stability and convergence of the proposed method are rigorously proved we provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models

a nonstandard finite difference scheme for convection diffusion equations having constant coefficients in this note we derive using the subequation method a new nonstandard finite difference scheme nsfd for a class of convection diffusion equations having constant coefficients despite the fact that this scheme has nonlinear denominator functions of the step sizes even for linear pdes it has a couple of favourable properties it is explicit and due to its construction it reproduces important properties of the solution of the parabolic pde this proposed method conserves by construction the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the black scholes equation for the valuation of standard options since it avoids negative values for the calculated prices finally we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

least squares estimation of a class of frequency functions a finite sample variance expression a new expression for the variance of scalar frequency functions estimated using the least squares method is presented the expression is valid for finite sample size and for a class of model structures which includes finite impulse response laguerre and kautz models when the number of estimated parameters coincides with the number of excitation frequencies of the input the expression gives direct insight into how excitation frequencies and amplitudes affect the accuracy of frequency function estimates with the help of this expression a severe sensitivity of the accuracy with respect to the excitation frequencies is exposed the relevance of the expression when more excitation frequencies are used is also discussed

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

input design via lmis admitting frequency wise model specifications in confidence regions a framework for reformulating input design problems in prediction error identification as convex optimization problems is presented for linear time invariant single input single output systems this framework unifies and extends existing results on open loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided a wide range of model quality constraints can be handled in particular different frequency by frequency constraints can be used this opens up new applications of input design in areas such as robust control furthermore quality specifications can be imposed on all models in a confidence region thus allowing for statements such as with at least 99 probability the model quality specifications will be satisfied

conditional full stability of positivity preserving finite difference scheme for diffusion advection reaction models this work has been partially supported by the ministerio de economia y competitividad spanish grant mtm2017 89664 p

difference methods for parabolic equations with robin condition abstract classical solutions of nonlinear second order partial differential functional equations of parabolic type with the robin condition are approximated in the paper by solutions of associated boundedness preserving implicit difference functional equations it is proved that the discrete solutions uniquely exist they are uniformly bounded with respect to meshes and the numerical method is convergent and stable we also find the error estimate and its asymptotic behavior the properties of some auxiliary nonlinear discrete recurrent equations are showed the proofs are based on the comparison technique and the banach fixed point theorem

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

numerical treatment of the spherically symmetric solutions of a generalized fisher kolmogorov petrovsky piscounov equation in the present work the connection of the generalized fisher kpp equation to physical and biological fields is noted radially symmetric solutions to the generalized fisher kpp equation are considered and analytical results for the positivity and asymptotic stability of solutions to the corresponding time independent elliptic differential equation are quoted an energy analysis of the generalized theory is carried out with further physical applications in mind and a numerical method that consistently approximates the energy of the system and its rate of change is presented the method is thoroughly tested against analytical and numerical results on the classical fisher kpp equation the heaviside equation and the generalized fisher kpp equation with logistic nonlinearity and heaviside initial profile obtaining as a result that our method is highly stable and accurate even in the presence of discontinuities as an application we establish numerically that under the presence of suitable initial conditions there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value an analytical prediction is provided for the heaviside equation against which we verify the validity of our computational code and numerical approximations are provided for several generalized fisher kpp problems

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart

the information matrix of multiple input single output time series models expressions are given for the information matrix of the parameters of the multiple input single output time series model for correlated and uncorrelated inputs allowing lags between inputs the model under consideration is a generalization of the multiple regression model with autocorrelated errors the transfer function model and the autoregressive moving average exogenous armax model the elements of the fisher matrix are evaluated using algorithms developed for the univariate arma model

an analysis of the parametrization by data driven local coordinates for multivariable linear systems in this paper we study a novel parametrization for state space systems namely data driven local coordinates ddlc which have recently been introduced and applied even though ddlc has meanwhile become the default parametrization used in the system identification toolbox of the software package matlab an analysis of properties of ddlc which are relevant to identification has not been performed up to now in this paper we provide insights into the geometry and topology of the ddlc construction and show a number of results which are important for actual identification such as maximum likelihood type estimation