Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra

Approximating the nearest stable discrete-time system

In this paper, we consider the problem of stabilizing discrete-time linear systems by computing a nearby stable matrix to an unstable one. To do so, we provide a new characterization for the set of stable matrices. We show that a matrix $A$ is stable if and only if it can be written as $A=S^{-1}UBS$, where $S$ is positive definite, $U$ is orthogonal, and $B$ is a positive semidefinite contraction (that is, the singular values of $B$ are less or equal to 1). This characterization results in an equivalent non-convex optimization problem with a feasible set on which it is easy to project. We propose a very efficient fast projected gradient method to tackle the problem in variables $(S,U,B)$ and generate locally optimal solutions. We show the effectiveness of the proposed method compared to other approaches.Comment: 15 pages, new title, accepted in LA

Structured pseudospectra for small perturbations

In this paper we study the shape and growth of structured pseudospectra for small matrix perturbations of the form $A \leadsto A_\Delta=A+B\Delta C$, $\Delta \in \boldsymbol{\Delta}$, $\|\Delta\|\leq \delta$. It is shown that the properly scaled pseudospectra components converge to nontrivial limit sets as $\delta$ tends to 0. We discuss the relationship of these limit sets with $\mu$-values and structured eigenvalue condition numbers for multiple eigenvalues

μ-values and spectral value sets for linear perturbation classes defined by a scalar product

We study the variation of the spectrum of matrices under perturbations which are self- or skew-adjoint with respect to a scalar product. Computable formulas are given for the associated μ-values. The results can be used to calculate spectral value sets for the perturbation classes under consideration. We discuss the special case of complex Hamiltonian perturbations of a Hamiltonian matrix in detail

A comparison of treatment outcomes following whole-task and part-task methods for training scripts

functional purposes for individuals with nonfluent aphasia. The goal of script training is to generate fluent speech in a limited context by restoring portions of automatic, natural language production into the speech of individuals with aphasia. A whole task approach (e.g. facilitating production of a set of related sentences as a unit) has resulted in fluent production of scripts that generalizes to functional contexts (Youmans, Holland, Munoz, & Bourgeois, 2005). Further research is necessary to determine how script training compares to traditional linguistic-based therapy. Though script training is intended to improve functional communication, some participants have demonstrated generalized improvement in language, as evidence by increase in verbal output from pre-test to post-test discourse samples (Powers and Munoz, 2004). The purpose of this study is to explore how treatment outcomes from a whole-task script training method compare with outcomes from part-task (linguistic-based) method. Specifically, we examined response to treatment, generalization to functional contexts, and general increase in language output. Participant Description. A multiple baseline across behaviors single-subject design was used to examine the effects of the treatments. DP was a 46 year-old man 2 years post onset of a left CVA. He passed a hearing screening, a simple screening for visual acuity, an oral mechanism screening, and the Mini-Mental State Exam. Pre Treatment Assessment. DP exhibited Broca’s aphasia as demonstrated by performance on the Western Aphasia Battery (AQ=54). Additionally, narrative and conversational samples were obtained using the Cookie Theft picture from the Boston Diagnostic Aphasia Examination (BDAE) and the Picnic picture from the Western Aphasia Battery. Conversation samples were obtained by asking DP to provide the following information: “Tell me about your most frightening experience,” “Tell me about your stroke,” or “Tell me what you usually do on Saturdays.” Script Training. Three scripts were developed as a collaborative effort between the clinicians and DP. We identified three scripts that were unique, relevant, and functional. Scripts consisted of 4-5 syntactically complete sentences. One script was used in whole-task training (Script 1), one in part-task training (Script 2), and one served as an untrained probe (Script 3). DP attended 50 minute sessions two times per week. Additionally, 15 minutes a day he complete at home practice consistent with each treatment method. Scripts were probed at the start of each treatment session and audiotaped and transcribed for analysis using the Systematic Assessment of Language Transcription (SALT). Script production was analyzed for the following dependant variables: total number of main body words, number of maze words, and percentage of mazes and frequency counts for designated word codes (for example, Correct Script Word [CSW]). Whole-Task Training. Script 1 was trained using the whole-task approach. Repeated drill and a self-cuing hierarchy was used promote fluent production of the target script. Cueing sheets (see Figure 1 for an example) for each sentence were placed in separate pages of a talking photo album. Additionally, the use of a talking photo album allowed for the use of an audio cue (the clinician producing each sentence). The first script sentence was practiced; additional sentences were added in a forward chaining procedure when 100% accuracy had been reached in a single session following three consecutive attempts. Auditory and visual cues to support repetitive drill were controlled by the participants (i.e. the patient was taught to initially use the most facilitative cue [audio] then work his way up and down the hierarchy depending on effort) The criterion for attainment of script was 100% script accuracy during an independent probe. DP’s baseline performance on Script 1 was established over three sessions (Figure 3). He reached the 100% accuracy criterion following 8 treatment sessions. Accuracy of script production, as measured by percent correct script words (script words produced/total script words) was generally maintained at or above 80% after completion of training. The WAB was re-administered following Script 1 Training and DP demonstrated an increase in AQ (65, + 11 points). Additional data will be reported regarding pre- and post- test performance on discourse tasks, as well as additional measures obtained on script performance (such as error type). Interview with the participant indicated that he was using the script in a variety of functional contexts (for example, at church and on the phone). Part-Task Training. The syntactic structures of Script 2 were trained using a modified version of the Sentence Production Program for Aphasia (Helm-Estabrooks & Nicholas, 2000). The SPPA is a treatment program with hierarchically organized sentence types (beginning with easier sentences), 15 sentences per level. The syntactic structure is first trained with a model, then elicited with a cue (see Figure 2 as an example). The script sentences were integrated to SPPA sentences with a comparable syntactic structure. The criterion for attainment of script was 100% script accuracy during an independent probe. DP’s baseline performance on Script 2 was established over eleven sessions (Figure 4). Treatment was initiated at SPPA sentence type 4 (Wh- questions) and included the script sentences (also Wh- questions, see example Figure 2). DP exhibited frustration at Level 4 and could not score above a 2 (out of 15); therefore, treatment at SPAA Sentence Type 1 was initiated. DP progressed through the SPPA levels (using criteria provided in the treatment manual) until level 4 was reached, at which point the script sentences were once again integrated with the comparable SPPA sentences. DP met criteria for completion of SPAA level 4 which included correct production of Script sentences. However, correct production within the context of the SPPA cuing did not transfer to independent probes (Figure 4). DP did not reach the 100% criterion. Accuracy of script production did not exceed 11%. The WAB was re-administered following Script 2 training and DP demonstrated a minimal change in AQ (63, -2 points). Additional data will be reported regarding pre- and post- test performance on discourse tasks, as well as additional measures obtained on script performance (such as error type). Currently, we are in the process of completing the protocol with a second participant and expect to have the data to present. References Helm-Estabrooks, N., & Nicholas, M. (2000). Sentence Production Program for Aphasia. Austin, TX: Pro-ed. Youmans, G., Holland, A., Munoz, M. L., & Bourgeois, M. S. (2005). Script training and automaticity in two individuals with aphasia. Aphasiology, 19(3-5), 435-450. Powers, H. and Muñoz, M.L. (November, 2004) Treatment outcomes following script training for Broca’s aphasia. Poster session presented at the annual convention of the American Speech Language and Hearing Association

Structured pseudospectra and the condition of a nonderogatory eigenvalue

Let $\lambda$ be a nonderogatory eigenvalue of $A\in\mathbb{C}^{n\times n}$ of algebraic multiplicity m. The sensitivity of $\lambda$ with respect to matrix perturbations of the form $A\leadsto A+\Delta$, $\Delta\in\boldsymbol{\Delta}$, is measured by the structured condition number $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$. Here $\boldsymbol{\Delta}$ denotes the set of admissible perturbations. However, if $\boldsymbol{\Delta}$ is not a vector space over $\mathbb{C}$, then $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$ provides only incomplete information about the mobility of $\lambda$ under small perturbations from $\boldsymbol{\Delta}$. The full information is then given by the set $K_{\boldsymbol{\Delta}}(x,y)=\{y^*\Delta x;$ $\Delta\in\boldsymbol{\Delta},$ $\|\Delta\|\leq1\}\subset\mathbb{C}$ that depends on $\boldsymbol{\Delta}$, a pair of normalized right and left eigenvectors $x,y$, and the norm $\|\cdot\|$ that measures the size of the perturbations. We always have $\kappa_{\boldsymbol{\Delta}}(A,\lambda)=\max\{|z|^{1/m};$ $z\in K_{\boldsymbol{\Delta}}(x,y)\}$. Furthermore, $K_{\boldsymbol{\Delta}}(x,y)$ determines the shape and growth of the $\boldsymbol{\Delta}$-structured pseudospectrum in a neighborhood of $\lambda$. In this paper we study the sets $K_{\boldsymbol{\Delta}}(x,y)$ and obtain methods for computing them. In doing so we obtain explicit formulae for structured eigenvalue condition numbers with respect to many important perturbation classes

Rubric scoring of a clinical test of executive functioning

Executive functions (EF) are complex abilities that allow one to successfully complete independent, deliberate, and novel goal-directed activities (Lezak, Howieson, & Loring, 2004). EF tests require solving problems with minimal direction from the examiner (Baddeley, 1992; Shallice & Burgess, 1991). Because EF skills tend to show up globally, Lezak et al. (2004) suggested clinicians will learn more about one’s EF abilities by observing how he or she goes about solving a problem than from a test score. If this is the case, a “rubric” score that took into consideration “how” a problem was solved may provide the clinician with better information for treatment planning than a test score, as long as it did not greatly affect test sensitivity or specificity. Sensitivity and specificity are important factors in determining the usefulness of EF tests. Sensitivity refers to the probability of identifying abnormal functioning in an impaired individual or “hit rate” of a test, whereas specificity reflects the probability of correctly identifying healthy individual with the test (Cartoni & Lincoln, 2005; Kiel & Kaszniak, 2002). Rubrics are useful scoring tools that divide tasks into component elements and provide a description of levels of performance for each element (Goodrich, 2005; Hanna & Smith, 1998). Rubrics have been widely used to assess student performance (Andrade, 2000; Falchikov, 1986; Goodrich, 1997), but have not been used to score EF tests. The aim of this study was to examine sensitivity and specificity for a clinical test of EF, the Rapid Assessment of Problem Solving test (RAPS) when scored with a rubric that allowed the examiner to describe the quality of performance using a standard that was developed from a large body of normative research. This differs from using the three traditional test scores from the RAPS that require time intensive calculations

Nonlinear eigenvalue problems with specified eigenvalues

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T, we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. We consider backward errors that only allow constant perturbations, which do not depend on the eigenvalue parameter. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general setting involving several eigenvalues is ignificantly more difficult. Under mild assumptions, we derive a singular value optimization characterization for the minimal perturbation that addresses the general case

Structured eigenvalue backward errors of matrix pencils and polynomials with Hermitian and related structures

We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, *-even, and *-odd. Numerical experiments suggest that in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations