an analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems in this paper we study a novel parametrization for state space systems namely separable least squares data driven local coordinates slsddlc the parametrization by slsddlc has recently been successfully applied to maximum likelihood estimation of linear dynamic systems in a simulation study the use of slsddlc has led to numerical advantages in comparison to the use of more conventional parametrizations including data driven local coordinates ddlc however an analysis of properties of slsddlc 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 slsddlc construction and show a number of results which are important for actual identification in particular for maximum likelihood estimation we also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

matrix rational h 2 approximation a gradient algorithm based on schur analysis this paper deals with the rational approximation of specified order n to transfer functions which are assumed to be matrix valued functions in the hardy space for the complement of the closed unit disk endowed with the l2 norm an approach is developed leading to a new algorithm the first one to our knowledge which concerns matrix transfer functions in l2 norm this approach generalizes the ideas presented in l baratchart m cardelli and m olivi automatica 27 1991 pp 413 418 in the scalar case but involves substantial new difficulties using the douglas shapiro shields factorization of transfer functions the criterion for the rational approximation problem above is expressed in terms of inner matrix functions of mcmillan degree n these functions which possess a manifold structure are represented by means of local coordinate maps obtained in d alpay l baratchart and a gombani oper theory adv appl 73 1994 pp 30 66 from a tangential schur algorithm and for which the coordinates range over n copies of the unit ball a gradient algorithm is then employed to solve the approximation problem using the coordinate maps to describe the manifold locally and changing from one coordinate map to another when required however while processing the gradient algorithm a boundary point can be reached it is proved that such a point can be considered as an initial point for searching for a local minimum of lower degree while a local minimum of mcmillan degree k n provides a starting point for searching for a local minimum at degree k 1 the minimization process then pursues through different degrees the convergence of this algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example

parametric imaging with bayesian priors a validation study with 11c altropane pet abstract it has been suggested that bayesian estimation methods may be used to improve the signal to noise ratio of parametric images however there is little experience with the method and some of the underlying assumptions and performance properties of bayesian estimation remain to be investigated we used a sample population of 54 subjects studied previously with 11 c altropane to empirically evaluate the assumptions performance and some practical issues in forming parametric images by using normality tests we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80 of voxels the standard deviation of the binding potential can be reduced 30 50 using bayesian estimation without introducing substantial bias the sample size required to form the a priori information was found to be modest as little as ten subjects may be sufficient and the choice of specific subjects has little effect on bayesian estimation a realistic simulation study showed that detection of localized differences in parametric images e g by statistical parametric mapping spm could be made more reliable and or conducted with smaller sample size using bayesian estimation in conclusion bayesian estimation can improve the snr of parametric images and better detect localized changes in cohorts of subjects

linear fractional transformations in rings and modules abstract this report examines the basic properties of linear fractional transformations of a matrix argument with matrix coefficients and still more general entities the familiar properties of scalar transformations generalize surprisingly well for instance cross ratios of matrices are preserved up to similarity the questions of multiple transitivity and uniqueness of coefficients are examined and two engineering applications are outlined

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

comparing methods of analyzing fmri statistical parametric maps approaches for the analysis of statistical parametric maps spms can be crudely grouped into three main categories in which different philosophies are applied to delineate activated regions these being type i error control thresholding false discovery rate fdr control thresholding and posterior probability thresholding to better understand the properties of these main approaches we carried out a simulation study to compare the approaches as they would be used on real data sets using default settings we find that posterior probability thresholding is the most powerful approach and type i error control thresholding provides the lowest levels of type i error false discovery rate control thresholding performs in between the other approaches for both these criteria although for some parameter settings this approach can approximate the performance of posterior probability thresholding based on these results we discuss the relative merits of the three approaches in an attempt to decide upon an optimal approach we conclude that viewing the problem of delineating areas of activation as a classification problem provides a highly interpretable framework for comparing the methods within this framework we highlight the role of the loss function which explicitly penalizes the types of errors that may occur in a given analysis

analytical study of the effects of soft tissue artefacts on functional techniques to define axes of rotation abstract the accurate location of the main axes of rotation aor is a crucial step in many applications of human movement analysis there are different formal methods to determine the direction and position of the aor whose performance varies across studies depending on the pose and the source of errors most methods are based on minimizing squared differences between observed and modelled marker positions or rigid motion parameters implicitly assuming independent and uncorrelated errors but the largest error usually results from soft tissue artefacts sta which do not have such statistical properties and are not effectively cancelled out by such methods however with adequate methods it is possible to assume that sta only account for a small fraction of the observed motion and to obtain explicit formulas through differential analysis that relate sta components to the resulting errors in aor parameters in this paper such formulas are derived for three different functional calibration techniques geometric fitting mean finite helical axis and sara to explain why each technique behaves differently from the others and to propose strategies to compensate for those errors these techniques were tested with published data from a sit to stand activity where the true axis was defined using bi planar fluoroscopy all the methods were able to estimate the direction of the aor with an error of less than 5 whereas there were errors in the location of the axis of 30 40 mm such location errors could be reduced to less than 17 mm by the methods based on equations that use rigid motion parameters mean finite helical axis sara when the translation component was calculated using the three markers nearest to the axis

an unbiased parametric imaging algorithm for nonuniformly sampled biomedical system parameter estimation an unbiased algorithm of generalized linear least squares glls for parameter estimation of nonuniformly sampled biomedical systems is proposed the basic theory and detailed derivation of the algorithm are given this algorithm removes the initial values required and computational burden of nonlinear least regression and achieves a comparable estimation quality in terms of the estimates bias and standard deviation therefore this algorithm is particular useful in image wide pixel by pixel based parameter estimation e g to generate parametric images from tracer dynamic studies with positron emission tomography an example is presented to demonstrate the performance of this new technique this algorithm is also generally applicable to other continuous system parameter estimation

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

a general method of bayesian estimation for parametric imaging of the brain abstract we report a general method of bayesian estimation that uses prior measurements to improve the signal to noise ratio of parametric images computed from dynamic pet scanning in our method the ordinary weighted least squares cost function is augmented by a penalty term to yield k s min k c f k t c 1 c f k s k s 0 k k t k 1 k k where c is a pet concentration history and c is its variance f is the model of the concentration history k k 1 k 2 k m t is the parameter vector k is the vector of population means for the model parameters k is its covariance k k s 0 is the conventional weighted sum of squares s 0 is chosen to control the balance between the prior and new data data from a prior population of subjects are analyzed with standard methods to provide maps of the mean parameter values and their variances as an example of this approach we used the dynamic image data of 10 normal subjects who had previously been studied with 11 c raclopride to estimate the prior distribution the dynamic data were transformed to stereotactic coordinates and analyzed by standard methods the resulting parametric maps were used to compute the voxel wise sample statistics then the cohort of priors was analyzed as a function of s using nonlinear least squares estimation and the cost function shown above as s is increased the standard error in estimating bp in single subjects was substantially reduced allowing measurement in bp in thalamus cortex brain stem etc additional studies demonstrate that a range of s values exist for which the bias is not excessive even when parameter values differ markedly from the sample mean this method can be used with any kinetic model so long as it is possible to compute a map of a priori mean parameters and their variances

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

on diffeomorphisms between classes of linear systems abstract in this paper we show that the following classes of linear dynamical systems are diffeomorphic to each other fixed order asymptotically stable strictly bounded real strictly positive real stable and minimum phase and invertible these results are then used to prove the diffeomorphisms between the state bundles associated with these classes of systems and between their associated principal fibre bundles we shall conclude this paper by discussing how these results can be used to deduce geometric properties of other classes of linear systems and to construct overlapping parametrizations

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

a state space calculus for rational probability density functions and applications to non gaussian filtering we propose what we believe to be a novel approach to performing calculations for rational density functions using state space representations of the densities by standard results from realization theory a rational probability density function is considered to be the transfer function of a linear system with generally complex entries the stable part of this system is positive real which we call the density summand the existence of moments is investigated using the markov parameters of the density summand moreover explicit formulae are given for the existing moments in terms of these markov parameters some of the main contributions of the paper are explicit state space descriptions for products and convolutions of rational densities as an application which is of interest in its own right the filtering problem is investigated for a linear time varying system whose noise inputs have rational probability density functions in particular state space formulations are derived for the calculation of the prediction and update equations the case of cauchy noise is treated as an illustrative example

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a balancing approach to the realization of systems with internal passivity and reciprocity this paper addresses the realization of positive real transfer functions which are symmetric with respect to some signature matrix we show that a realization that is jointly internally reciprocal and internally passive can be achieved by positive real balancing

new least squares solutions for estimating the average centre of rotation and the axis of rotation a new method is proposed for estimating the parameters of ball joints also known as spherical or revolute joints and hinge joints with a fixed axis of rotation the method does not require manual adjustment of any optimisation parameters and produces closed form solutions it is a least squares solution using the whole 3d motion data set we do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation this method is compared with other methods that use similar assumptions in the cases of random measurement errors systematic skin movements and skin movements with random measurement noise simulation results indicate that the new method is superior in terms of the algorithm used the closure of the solution consistency and minimal manual parameter adjustment the method can also be adapted to joints with translational movements

nudelman interpolation parametrizations of lossless functions and balanced realizations we investigate the parametrization issue for discrete time stable all pass multivariable systems by means of a schur algorithm involving a nudelman interpolation condition a recursive construction of balanced realizations is associated with it that possesses a very good numerical behavior several atlases of charts or families of local parametrizations are presented and for each atlas a chart selection strategy is proposed the last one can be viewed as a nice mutual encoding property of lossless functions and turns out to be very efficient these parametrizations allow for solving optimization problems within the fields of system identification and optimal control

a new balanced canonical form for stable multivariable systems a new balanced canonical form is presented for stable multivariable linear systems overlapping continuous block balanced canonical forms were introduced by hanson ober for the stable single input single output siso case as a generalisation of the balanced canonical form introduced by ober 1987 for the siso case in the search for a generalization of these results to the multivariable case a new multivariable balanced canonical form was discovered which is of interest in its own right and is presented in this paper the new canonical form has a number of nice properties the integer invariants that appear in the canonical form are the multiplicities of the hankel singular values and a number of new invariants which are in one to one objective correspondence with the kronecker indexes of subsystems truncation of the state vector leads to stable minimal models in canonical form in the siso case the canonical form coincides with ober s balanced canonical form the reachability matrix of a system in canonical form with identical singular values is positive upper triangular

overlapping block balanced canonical forms and parametrizations the stable siso case the balanced canonical form and parametrization of ober for the case of siso stable systems are extended to block balanced canonical forms and related input normal forms and parametrizations they form an overlapping atlas of parametrizations of the manifold of stable siso systems of given order this extends the usefulness of these parametrizations e g in gradient algorithms for system identification as an implication of our construction it follows that each of the subsets of the parametrization of r ober internat j control 46 1987 pp 643 670 corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable siso systems of fixed order

polynomial optimization and a jacobi davidson type method for commuting matrices in this paper we introduce a new jacobi davidson type eigenvalue solver for a set of commuting matrices called jdcomm used for the global optimization of so called minkowski norm dominated polynomials in several variables the stetter moller matrix method yields such a set of real non symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem a drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse however the other matrices are generally much sparser and have the same eigenvectors because of the commutativity this fact is used to design the jdcomm method for this problem the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix standard jacobi davidson and implicitly restarted arnoldi as well as conventional solvers for computing the global optimum i e sostools gloptipoly and phcpack

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

weighted h 2 rational approximation and consistency we investigate consistency properties of rational approximation of prescribed type in the weighted hardy space h 2 mu for the exterior of the unit disk where mu is a positive symmetric measure on the unit circle mathbb t the question of consistency which is especially significant for gradient algorithms that compute local minima concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational in addition to describing some basic properties of the approximation problem we prove for measures mu having a rational function distribution weight with respect to arclength on mathbb t that consistency holds only under rather restricted conditions

kalman smoothing improves the estimation of joint kinematics and kinetics in marker based human gait analysis we developed a kalman smoothing algorithm to improve estimates of joint kinematics from measured marker trajectories during motion analysis kalman smoothing estimates are based on complete marker trajectories this is an improvement over other techniques such as the global optimisation method gom kalman filtering and local marker estimation lme where the estimate at each time instant is only based on part of the marker trajectories we applied gom kalman filtering lme and kalman smoothing to marker trajectories from both simulated and experimental gait motion to estimate the joint kinematics of a ten segment biomechanical model with 21 degrees of freedom three simulated marker trajectories were studied without errors with instrumental errors and with soft tissue artefacts sta two modelling errors were studied increased thigh length and hip centre dislocation we calculated estimation errors from the known joint kinematics in the simulation study compared with other techniques kalman smoothing reduced the estimation errors for the joint positions by more than 50 for the simulated marker trajectories without errors and with instrumental errors compared with gom kalman smoothing reduced the estimation errors for the joint moments by more than 35 compared with kalman filtering and lme kalman smoothing reduced the estimation errors for the joint accelerations by at least 50 our simulation results show that the use of kalman smoothing substantially improves the estimates of joint kinematics and kinetics compared with previously proposed techniques gom kalman filtering and lme for both simulated with and without modelling errors and experimentally measured gait motion

a new method for estimating the axis of rotation and the center of rotation a new method is presented for estimating the parameters of two different models of a joint the two models are 1 a rotational joint with a fixed axis of rotation also referred to as a hinge joint and 2 a ball and socket model corresponding to a spherical joint given the motion of a set of markers it is shown how the parameters can be estimated utilizing the whole data set the parameters are estimated from motion data by minimizing two objective functions the method does not assume a rigid body motion but only that each marker rotates around the same fixed axis of rotation or center of rotation simulation results indicate that in situations where the rigid body assumption is valid and when measurement noise is present the proposed method is inferior to methods that utilize the rigid body assumption however when there are large skin movement artefacts simulation results show the proposed method to be more robust

the finite helical axis of the knee joint a non invasive in vivo study using fast pc mri an understanding of the in vivo knee joint kinematics is critical for the further improvement and validation of knee joint models and for the development of better surgical and rehabilitative protocols unfortunately most studies exploring the finite helical axis fha tend to produce excellent qualitative results but quantitative results are often lacking thus the purpose of this study was to non invasively and in vivo quantify the tibiofemoral fha in a relatively large normal population during volitional knee extension using fast pc mri to report the data relative to consistent coordinate systems making it available for modeling input experimental comparison and for device design to determine the variability of the fha to investigate the screw home mechanism and to test the hypothesis that knee joint kinematics are independent of gender intra and inter subject repeatability was excellent the intra inter subject repeatability of the fha orientation in the frontal and axial planes was 1 8 3 3 and 3 7 6 0 of the average value respectively at the beginning of extension the fha was directed laterally and slightly superiorly and at the end of extension it was directed in the lateral inferior direction indicative of the screw home mechanism the fha location was not fixed during extension there was small but significant differences in all fha parameters between genders and normalizing positional data relative to epicondylar width helped to reduce this difference the data obtained in the current study forms an excellent base for future knee joint modeling and clinical studies

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

a new method for estimating joint parameters from motion data joint centers and axes of rotation joint parameters are central to all branches of movement analysis in gait analysis the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation anthropometric regression equations and misplaced alignment devices several alternative methods have been proposed but to date none have been shown to be accurate and reliable enough for use in the clinical setting this article describes a new method for joint parameter estimation the new method can be summarized as follows i the motions of two adjacent segments spanning a single joint are tracked ii the axis of rotation between every pair of observed segment configurations is computed iii the most likely intersection of all axes effective joint center and most likely orientation of the axes effective joint axis is found initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject for the analog the center was found to be within 3 8 mm of the geometric center and 2 0 degrees of the geometric axis standard deviation for the adult subject hip centers varied on the order of 1 3 mm knee centers by 3 9 mm and knee axes by 2 0 degrees the results suggest that the new method is an objective precise and practical alternative to the standard clinical approach

map based kinetic analysis for voxel by voxel compartment model estimation detailed imaging of the cerebral glucose metabolism using fdg we propose a novel algorithm for voxel by voxel compartment model analysis based on a maximum a posteriori map algorithm voxel by voxel compartment model analysis can derive functional images of living tissues but it suffers from high noise statistics in voxel based pet data and extended calculation times we initially set up a feature space of the target radiopharmaceutical composed of a measured plasma time activity curve and a set of compartment model parameters and measured the noise distribution of the pet data the dynamic pet data were projected onto the feature space and then clustered using the mahalanobis distance our method was validated using simulation studies and compared with roi based ordinary kinetic analysis for fdg the parametric images exhibited an acceptable linear relation with the simulations and the roi based results and the calculation time took about 10 min we therefore concluded that our proposed map based algorithm is practical

system identification of microwave filters from multiplexers by rational interpolation microwave multiplexers are multi port structures composed of several two port filters connected to a common junction this paper addresses the de embedding problem in which the goal is to determine the filtering components given the measured scattering parameters of the overall multiplexer at several frequencies due to structural properties the transmission zeros of the filters play a crucial role in this problem and consequently in our approach we propose a system identification algorithm for deriving a rational model of the filters scattering matrix the approach is based on rational interpolation with derivative constraints with the interpolation conditions being located precisely at the filters transmission zeros

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

identification of microwave filters by analytic and rational h2 approximation in this paper an original approach to frequency identification is explained and demonstrated through an application in the domain of microwave filters this approach splits into two stages a stable and causal model of high degree is first computed from the data completion stage then model reduction is performed to get a rational low order model in the first stage the most is made of the data taking into account the expected behavior of the filter a reduced order model is then computed by rational h 2 approximation a new and efficient method has been developed improved over the years and implemented to solve this problem it heavily relies on the underlying hilbert space structure and on a nice parameterization of the optimization set this approach guarantees the stability of the mimo approximant of prescribed mcmillan degree

fast and reliable estimation of multiple parametric images using an integrated method for dynamic spect dynamic single photon emission computed tomography spect has demonstrated the potential to quantitatively estimate physiological parameters in the brain and the heart the generalized linear least square glls method is a well established method for solving linear compartment models with fast computational speed however the high level of noise intrinsic in the spect data leads to reliability and instability problems of glls for generating parametric images an integrated method is proposed to restrict the noise in both the temporal and spatial domains to estimate multiple parametric images for dynamic spect this method comprises three steps which are optimum image sampling schedule in the projection space cluster analysis applied postreconstruction and parametric image generation with glls the simulation and experimental studies for the neuronal nicotine acetylcholine receptor tracer of 5 123i iodo a 85380 were employed to evaluate the performance of the proposed method the results of influx rate of k1 and volume of distribution of vd demonstrated that the integrated method was successful in generating low noise parametric images for high noise spect data without enhancing the partial volume effect furthermore the integrated method is computationally efficient for potential clinical applications

on the differentiable manifold of fixed order stable linear systems abstract it is shown that the real analytic differentiable manifold of time invariant linear dynamical systems of fixed mcmillan degree n with m dimensional inputs and m dimensional outputs is real analytically diffeomorphic to the submanifold of i o stable i e asymptotically stable systems this generalizes results of helmke ober and hanzon about the topology of such manifolds similar results hold for the corresponding state bundles and their associated principal bundles this implies in fact that there is a one to one correspondence between smooth local canonical forms for finite dimensional linear systems of fixed order and for finite dimensional i o stable linear systems of fixed order

a study on statistically reliable and computationally efficient algorithms for generating local cerebral blood flow parametric images with positron emission tomography with the advent of positron emission tomography pet a variety of techniques have been developed to measure local cerebral blood flow lcbf noninvasively in humans a potential class of techniques which includes linear least squares ls linear weighted least squares wls linear generalized least squares gls and linear generalized weighted least squares gwls is proposed the statistical characteristics of these methods are examined by computer simulation the authors present a comparison of these four methods with two other rapid estimation techniques developed by huang et al 1982 and alpert 1984 and two classical methods the unweighted and weighted nonlinear least squares regression the results show that these methods can take full advantage of the contribution from the fine temporal sampling data of modern tomographs and thus provide statistically reliable estimates that are comparable to those obtained from nonlinear ls regression these methods also have high computational efficiency and the parameters can be estimated directly from operational equations in one single step therefore they can potentially be used in image wide estimation of local cerebral blood flow and distribution volume with pet

constructive algebra methods for the l 2 problem for stable linear systems abstract we investigate the feasibility of using computer algebra for solving l 2 system approximation problems the first order optimality conditions yield a set of polynomial equations which can in principle be solved using grobner basis methods a general solution along these lines would be tremendously useful although it does not appear to be feasible at present except for rather low mcmillan degrees we demonstrate that it can be feasible for specific examples in such cases global optima can be found reliably we show that the use of a schwarz like canonical form simplifies the structure of the problem

canonical lossless state space systems staircase forms and the schur algorithm a new finite atlas of overlapping balanced canonical forms for multivariate discrete time lossless systems is presented the canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns this is a generalization of a similar balanced canonical form for continuous time lossless systems it is shown that this atlas is in fact a finite sub atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential schur algorithm such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices the connection between these balanced canonical forms for lossless systems and the tangential schur algorithm for lossless systems is a generalization of the same connection in the siso case that was noted before the results are directly applicable to obtain a finite sub atlas of multivariate input normal canonical forms for stable linear systems of given fixed order which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold

functional knee axis based on isokinetic dynamometry data comparison of two methods mri validation and effect on knee joint kinematics this paper compares geometry based knee axes of rotation transepicondylar axis and geometric center axis and motion based functional knee axes of rotation faor two algorithms are evaluated to calculate faors gamage and lasenby s sphere fitting algorithm gl and ehrig et al s axis transformation algorithm sara calculations are based on 3d motion data acquired during isokinetic dynamometry aors are validated with the equivalent axis based on static mr images we quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes and the difference in location as the distance between the intersection points of the axes with the sagittal plane maximum differences between faors resulting from gl and sara were 5 7 and 15 4mm respectively maximum differences between faors resulting from gl or sara and the equivalent axis were 5 4 11 5mm and 8 6 12 8mm respectively differences between geometry based axes and ea are larger than differences between faor and ea both in orientation maximum 10 6 and location maximum 20 8mm knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using kalman smoothing for the joint angles the maximum rms difference with the mri based equivalent axis which was used as a reference was 3 for the knee joint accelerations the maximum rms difference with the equivalent axis was 20 s 2 functional knee axes of rotation describe knee motion better than geometry based axes gl performs better than sara for calculations based on experimental dynamometry

balanced realizations of discrete time stable all pass systems and the tangential schur algorithm in this paper the connections are investigated between two different approaches towards the parametrization of multivariable stable all pass systems in discrete time the first approach involves the tangential schur algorithm which employs linear fractional transformations it stems from the theory of reproducing kernel hilbert spaces and enables the direct construction of overlapping local parametrizations using schur parameters and interpolation points the second approach proceeds in terms of state space realizations in the scalar case a balanced canonical form exists that can also be parametrized by schur parameters this canonical form can be constructed recursively using unitary matrix operations here this procedure is generalized to the multivariable case by establishing the connections with the first approach it gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential schur algorithm

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

improved signal to noise ratio in parametric images by cluster analysis abstract parametric images are formed by analyzing the concentration history of every voxel in pet data sets because pet concentration data at the voxel level are rather noisy noise propagation into the parametric image is often quite noticeable to address this problem a model based clustering method has been developed to generate parametric images the basic idea of the clustering method is to average over voxels whose concentration histories have the same shape we applied the method to a two parameter k 1 k 2 compartment model of local cerebral blood flow the statistic r tc t dt c t dt te k 2 t c a t dt e k 2 t c a t dt classifies curves in terms of k 2 where c t and c a t denote the tissue and blood concentration histories respectively and is the convolution operator simulation studies of noise propagation in the clustering statistic showed that 30 voxel noise yielded a 2 standard deviation in r parametric images of blood flow and partition coefficient were computed for an o 15 study with and without clustering cluster size affected bias statistical precision and computation time with clusters of 400 voxels the variance of the flow parameter was around 1 50 smaller with clustering with negligible bias and a computation time of 30 s on a 64 mhz workstation for 15 128 128 images with matlab 5 1

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

riemannian optimal identification method for linear systems with symmetric positive definite matrix this article develops identification methods for linear continuous time symmetric systems such as electrical network systems multiagent network systems and temperature dynamics in buildings to this end we formulate three system identification problems for the corresponding discrete time systems the first is a least squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive definite matrices and two euclidean spaces in the second problem to reduce the search dimensions the product manifold is replaced with the quotient set under a specified group action by the orthogonal group in the third problem the manifold of symmetric positive definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements in particular we examine the quotient geometry in the second problem we propose riemannian conjugate gradient methods for the three problems and select initial points using a popular subspace method the effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the gauss newton method which is one of the most popular approach for solving least squares problems

an evaluation of the algorithms for determining local cerebral metabolic rates of glucose using positron emission tomography dynamic data measurement of the local cerebral metabolic rate of glucose lcmrglc and the individual rate constant parameters of the sup 18 f 2 fluoro 2 deoxy d glucose fdg model can provide a clearer understanding and insight to the physiological processes in the human brain and a quicker and more accurate means of diagnosis in clinical applications a systematic study using simulated and clinical tissue time activity data is presented to evaluate several existing and newly developed major algorithms used for determining lcmrglc and the individual rate constants from positron emission tomography dynamic data the computational and statistical properties of the autoradiographic approach weighted and unweighted nonlinear least squares methods patlak graphic approach weighted integration method linear least squares and generalized linear least squares methods are investigated and discussed in this paper