String-averaging expectation-maximization for maximum likelihood estimation in emission tomography

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Maximum Likelihood Estimation of Modal Parameters in Structures Using the Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on maximum likelihood estimation (MLE) with the expectation maximization (EM) algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. The benchmark structure is a four-story, two-bay by two-bay steel-frame scale model structure built in the Earthquake Engineering Research Laboratory at the University of British Columbia, Canada. This paper focuses on Phase I of the analytical benchmark studies. A MATLAB-based finite element analysis code obtained from the IASC-ASCE SHM Task Group web site is used to calculate the dynamic response of the prototype structure. A number of 100 simulations have been made using this MATLAB-based finite element analysis code in order to evaluate the proposed identification method. There are several techniques to realize system identification. In this work, stochastic subspace identification (SSI)method has been used for comparison. SSI identification method is a well known method and computes accurate estimates of the modal parameters. The principles of the SSI identification method has been introduced in the paper and next the proposed MLE with EM algorithm has been explained in detail. The advantages of the proposed structural identification method can be summarized as follows: (i) the method is based on maximum likelihood, that implies minimum variance estimates; (ii) EM is a computational simpler estimation procedure than other optimization algorithms; (iii) estimate more parameters than SSI, and these estimates are accurate. On the contrary, the main disadvantages of the method are: (i) EM algorithm is an iterative procedure and it consumes time until convergence is reached; and (ii) this method needs starting values for the parameters. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using both the SSI method and the proposed MLE + EM method. The numerical results show that the proposed method identifies eigenfrequencies, damping ratios and mode shapes reasonably well even in the presence of 10% measurement noises. These modal parameters are more accurate than the SSI estimated modal parameters.

Influence of reconstruction parameters during filtered backprojection and ordered-subset expectation maximization in the measurement of the left-ventricular volumes and function during gated SPECT

A crucial method for investigating patients with coronary artery disease (CAD) is the calculation of the left ventricular ejection fraction (LVEF). It is, consequently, imperative to precisely estimate the value of LVEF--a process that can be done with myocardial perfusion scintigraphy. Therefore, the present study aimed to establish and compare the estimation performance of the quantitative parameters of the reconstruction methods filtered backprojection (FBP) and ordered-subset expectation maximization (OSEM). Methods: A beating-heart phantom with known values of end-diastolic volume, end-systolic volume, and LVEF was used. Quantitative gated SPECT/quantitative perfusion SPECT software was used to obtain these quantitative parameters in a semiautomatic mode. The Butterworth filter was used in FBP, with the cutoff frequencies between 0.2 and 0.8 cycles per pixel combined with the orders of 5, 10, 15, and 20. Sixty-three reconstructions were performed using 2, 4, 6, 8, 10, 12, and 16 OSEM subsets, combined with several iterations: 2, 4, 6, 8, 10, 12, 16, 32, and 64. Results: With FBP, the values of end-diastolic, end-systolic, and the stroke volumes rise as the cutoff frequency increases, whereas the value of LVEF diminishes. This same pattern is verified with the OSEM reconstruction. However, with OSEM there is a more precise estimation of the quantitative parameters, especially with the combinations 2 iterations × 10 subsets and 2 iterations × 12 subsets. Conclusion: The OSEM reconstruction presents better estimations of the quantitative parameters than does FBP. This study recommends the use of 2 iterations with 10 or 12 subsets for OSEM and a cutoff frequency of 0.5 cycles per pixel with the orders 5, 10, or 15 for FBP as the best estimations for the left ventricular volumes and ejection fraction quantification in myocardial perfusion scintigraphy.

Influence of reconstruction parameters during filtered backprojection and ordered-subset expectation maximization in the measurement of the left-ventricular volumes and function during gated SPECT.

A crucial method for investigating patients with coronary artery disease (CAD) is the calculation of the left ventricular ejection fraction (LVEF). It is, consequently, imperative to precisely estimate the value of LVEF--a process that can be done with myocardial perfusion scintigraphy. Therefore, the present study aimed to establish and compare the estimation performance of the quantitative parameters of the reconstruction methods filtered backprojection (FBP) and ordered-subset expectation maximization (OSEM). METHODS: A beating-heart phantom with known values of end-diastolic volume, end-systolic volume, and LVEF was used. Quantitative gated SPECT/quantitative perfusion SPECT software was used to obtain these quantitative parameters in a semiautomatic mode. The Butterworth filter was used in FBP, with the cutoff frequencies between 0.2 and 0.8 cycles per pixel combined with the orders of 5, 10, 15, and 20. Sixty-three reconstructions were performed using 2, 4, 6, 8, 10, 12, and 16 OSEM subsets, combined with several iterations: 2, 4, 6, 8, 10, 12, 16, 32, and 64. RESULTS: With FBP, the values of end-diastolic, end-systolic, and the stroke volumes rise as the cutoff frequency increases, whereas the value of LVEF diminishes. This same pattern is verified with the OSEM reconstruction. However, with OSEM there is a more precise estimation of the quantitative parameters, especially with the combinations 2 iterations × 10 subsets and 2 iterations × 12 subsets. CONCLUSION: The OSEM reconstruction presents better estimations of the quantitative parameters than does FBP. This study recommends the use of 2 iterations with 10 or 12 subsets for OSEM and a cutoff frequency of 0.5 cycles per pixel with the orders 5, 10, or 15 for FBP as the best estimations for the left ventricular volumes and ejection fraction quantification in myocardial perfusion scintigraphy.

Three-dimensional ordered-subset expectation maximization iterative protocol for evaluation of left ventricular volumes and function by quantitative gated SPECT: a dynamic phantom study.

The purposes of this study were to characterize the performance of a 3-dimensional (3D) ordered-subset expectation maximization (OSEM) algorithm in the quantification of left ventricular (LV) function with (99m)Tc-labeled agent gated SPECT (G-SPECT), the QGS program, and a beating-heart phantom and to optimize the reconstruction parameters for clinical applications. METHODS: A G-SPECT image of a dynamic heart phantom simulating the beating left ventricle was acquired. The exact volumes of the phantom were known and were as follows: end-diastolic volume (EDV) of 112 mL, end-systolic volume (ESV) of 37 mL, and stroke volume (SV) of 75 mL; these volumes produced an LV ejection fraction (LVEF) of 67%. Tomographic reconstructions were obtained after 10-20 iterations (I) with 4, 8, and 16 subsets (S) at full width at half maximum (FWHM) gaussian postprocessing filter cutoff values of 8-15 mm. The QGS program was used for quantitative measurements. RESULTS: Measured values ranged from 72 to 92 mL for EDV, from 18 to 32 mL for ESV, and from 54 to 63 mL for SV, and the calculated LVEF ranged from 65% to 76%. Overall, the combination of 10 I, 8 S, and a cutoff filter value of 10 mm produced the most accurate results. The plot of the measures with respect to the expectation maximization-equivalent iterations (I x S product) revealed a bell-shaped curve for the LV volumes and a reverse distribution for the LVEF, with the best results in the intermediate range. In particular, FWHM cutoff values exceeding 10 mm affected the estimation of the LV volumes. CONCLUSION: The QGS program is able to correctly calculate the LVEF when used in association with an optimized 3D OSEM algorithm (8 S, 10 I, and FWHM of 10 mm) but underestimates the LV volumes. However, various combinations of technical parameters, including a limited range of I and S (80-160 expectation maximization-equivalent iterations) and low cutoff values (< or =10 mm) for the gaussian postprocessing filter, produced results with similar accuracies and without clinically relevant differences in the LV volumes and the estimated LVEF.

An approach to operational modal analysis using the expectation maximization algorithm

This paper presents the Expectation Maximization algorithm (EM) applied to operational modal analysis of structures. The EM algorithm is a general-purpose method for maximum likelihood estimation (MLE) that in this work is used to estimate state space models. As it is well known, the MLE enjoys some optimal properties from a statistical point of view, which make it very attractive in practice. However, the EM algorithm has two main drawbacks: its slow convergence and the dependence of the solution on the initial values used. This paper proposes two different strategies to choose initial values for the EM algorithm when used for operational modal analysis: to begin with the parameters estimated by Stochastic Subspace Identification method (SSI) and to start using random points. The effectiveness of the proposed identification method has been evaluated through numerical simulation and measured vibration data in the context of a benchmark problem. Modal parameters (natural frequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using SSI and the EM algorithm. On the whole, the results show that the application of the EM algorithm starting from the solution given by SSI is very useful to identify the vibration modes of a structure, discarding the spurious modes that appear in high order models and discovering other hidden modes. Similar results are obtained using random starting values, although this strategy allows us to analyze the solution of several starting points what overcome the dependence on the initial values used.

Operational Modal Analysis using Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on Maximum Likelihood Estimation and the Expectation Maximization algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated applying the proposed identification method to a set of 100 simulated cases. The numerical results show that the proposed method estimates all the modal parameters reasonably well in the presence of 30% measurement noise even. Finally, advantages and disadvantages of the method have been discussed.

Exact property estimation from diffusion Monte Carlo with minimal stochastic reconfiguration /

Our objective is to develop a diffusion Monte Carlo (DMC) algorithm to estimate the exact expectation values, ($o|^|^o), of multiplicative operators, such as polarizabilities and high-order hyperpolarizabilities, for isolated atoms and molecules. The existing forward-walking pure diffusion Monte Carlo (FW-PDMC) algorithm which attempts this has a serious bias. On the other hand, the DMC algorithm with minimal stochastic reconfiguration provides unbiased estimates of the energies, but the expectation values ($o|^|^) are contaminated by ^, an user specified, approximate wave function, when A does not commute with the Hamiltonian. We modified the latter algorithm to obtain the exact expectation values for these operators, while at the same time eliminating the bias. To compare the efficiency of FW-PDMC and the modified DMC algorithms we calculated simple properties of the H atom, such as various functions of coordinates and polarizabilities. Using three non-exact wave functions, one of moderate quality and the others very crude, in each case the results are within statistical error of the exact values.

Estimation du modèle GARCH à changement de régimes et son utilité pour quantifier le risque de modèle dans les applications financières en actuariat

Le modèle GARCH à changement de régimes est le fondement de cette thèse. Ce modèle offre de riches dynamiques pour modéliser les données financières en combinant une structure GARCH avec des paramètres qui varient dans le temps. Cette flexibilité donne malheureusement lieu à un problème de path dependence, qui a empêché l'estimation du modèle par le maximum de vraisemblance depuis son introduction, il y a déjà près de 20 ans. La première moitié de cette thèse procure une solution à ce problème en développant deux méthodologies permettant de calculer l'estimateur du maximum de vraisemblance du modèle GARCH à changement de régimes. La première technique d'estimation proposée est basée sur l'algorithme Monte Carlo EM et sur l'échantillonnage préférentiel, tandis que la deuxième consiste en la généralisation des approximations du modèle introduites dans les deux dernières décennies, connues sous le nom de collapsing procedures. Cette généralisation permet d'établir un lien méthodologique entre ces approximations et le filtre particulaire. La découverte de cette relation est importante, car elle permet de justifier la validité de l'approche dite par collapsing pour estimer le modèle GARCH à changement de régimes. La deuxième moitié de cette thèse tire sa motivation de la crise financière de la fin des années 2000 pendant laquelle une mauvaise évaluation des risques au sein de plusieurs compagnies financières a entraîné de nombreux échecs institutionnels. À l'aide d'un large éventail de 78 modèles économétriques, dont plusieurs généralisations du modèle GARCH à changement de régimes, il est démontré que le risque de modèle joue un rôle très important dans l'évaluation et la gestion du risque d'investissement à long terme dans le cadre des fonds distincts. Bien que la littérature financière a dévoué beaucoup de recherche pour faire progresser les modèles économétriques dans le but d'améliorer la tarification et la couverture des produits financiers, les approches permettant de mesurer l'efficacité d'une stratégie de couverture dynamique ont peu évolué. Cette thèse offre une contribution méthodologique dans ce domaine en proposant un cadre statistique, basé sur la régression, permettant de mieux mesurer cette efficacité.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Estimation of the degree of polarization through computational sensing

The degree of polarization of a refected field from active laser illumination can be used for object identifcation and classifcation. The goal of this study is to investigate methods for estimating the degree of polarization for refected fields with active laser illumination, which involves the measurement and processing of two orthogonal field components (complex amplitudes), two orthogonal intensity components, and the total field intensity. We propose to replace interferometric optical apparatuses with a computational approach for estimating the degree of polarization from two orthogonal intensity data and total intensity data. Cramer-Rao bounds for each of the three sensing modalities with various noise models are computed. Algebraic estimators and maximum-likelihood (ML) estimators are proposed. Active-set algorithm and expectation-maximization (EM) algorithm are used to compute ML estimates. The performances of the estimators are compared with each other and with their corresponding Cramer-Rao bounds. Estimators for four-channel polarimeter (intensity interferometer) sensing have a better performance than orthogonal intensities estimators and total intensity estimators. Processing the four intensities data from polarimeter, however, requires complicated optical devices, alignment, and four CCD detectors. It only requires one or two detectors and a computer to process orthogonal intensities data and total intensity data, and the bounds and estimator performances demonstrate that reasonable estimates may still be obtained from orthogonal intensities or total intensity data. Computational sensing is a promising way to estimate the degree of polarization.

Maximum likelihood estimation of new state space models for operational modal analysis

The modal analysis of a structural system consists on computing its vibrational modes. The experimental way to estimate these modes requires to excite the system with a measured or known input and then to measure the system output at different points using sensors. Finally, system inputs and outputs are used to compute the modes of vibration. When the system refers to large structures like buildings or bridges, the tests have to be performed in situ, so it is not possible to measure system inputs such as wind, traffic, . . .Even if a known input is applied, the procedure is usually difficult and expensive, and there are still uncontrolled disturbances acting at the time of the test. These facts led to the idea of computing the modes of vibration using only the measured vibrations and regardless of the inputs that originated them, whether they are ambient vibrations (wind, earthquakes, . . . ) or operational loads (traffic, human loading, . . . ). This procedure is usually called Operational Modal Analysis (OMA), and in general consists on to fit a mathematical model to the measured data assuming the unobserved excitations are realizations of a stationary stochastic process (usually white noise processes). Then, the modes of vibration are computed from the estimated model. The first issue investigated in this thesis is the performance of the Expectation- Maximization (EM) algorithm for the maximum likelihood estimation of the state space model in the field of OMA. The algorithm is described in detail and it is analysed how to apply it to vibration data. After that, it is compared to another well known method, the Stochastic Subspace Identification algorithm. The maximum likelihood estimate enjoys some optimal properties from a statistical point of view what makes it very attractive in practice, but the most remarkable property of the EM algorithm is that it can be used to address a wide range of situations in OMA. In this work, three additional state space models are proposed and estimated using the EM algorithm: • The first model is proposed to estimate the modes of vibration when several tests are performed in the same structural system. Instead of analyse record by record and then compute averages, the EM algorithm is extended for the joint estimation of the proposed state space model using all the available data. • The second state space model is used to estimate the modes of vibration when the number of available sensors is lower than the number of points to be tested. In these cases it is usual to perform several tests changing the position of the sensors from one test to the following (multiple setups of sensors). Here, the proposed state space model and the EM algorithm are used to estimate the modal parameters taking into account the data of all setups. • And last, a state space model is proposed to estimate the modes of vibration in the presence of unmeasured inputs that cannot be modelled as white noise processes. In these cases, the frequency components of the inputs cannot be separated from the eigenfrequencies of the system, and spurious modes are obtained in the identification process. The idea is to measure the response of the structure corresponding to different inputs; then, it is assumed that the parameters common to all the data correspond to the structure (modes of vibration), and the parameters found in a specific test correspond to the input in that test. The problem is solved using the proposed state space model and the EM algorithm. Resumen El análisis modal de un sistema estructural consiste en calcular sus modos de vibración. Para estimar estos modos experimentalmente es preciso excitar el sistema con entradas conocidas y registrar las salidas del sistema en diferentes puntos por medio de sensores. Finalmente, los modos de vibración se calculan utilizando las entradas y salidas registradas. Cuando el sistema es una gran estructura como un puente o un edificio, los experimentos tienen que realizarse in situ, por lo que no es posible registrar entradas al sistema tales como viento, tráfico, . . . Incluso si se aplica una entrada conocida, el procedimiento suele ser complicado y caro, y todavía están presentes perturbaciones no controladas que excitan el sistema durante el test. Estos hechos han llevado a la idea de calcular los modos de vibración utilizando sólo las vibraciones registradas en la estructura y sin tener en cuenta las cargas que las originan, ya sean cargas ambientales (viento, terremotos, . . . ) o cargas de explotación (tráfico, cargas humanas, . . . ). Este procedimiento se conoce en la literatura especializada como Análisis Modal Operacional, y en general consiste en ajustar un modelo matemático a los datos registrados adoptando la hipótesis de que las excitaciones no conocidas son realizaciones de un proceso estocástico estacionario (generalmente ruido blanco). Posteriormente, los modos de vibración se calculan a partir del modelo estimado. El primer problema que se ha investigado en esta tesis es la utilización de máxima verosimilitud y el algoritmo EM (Expectation-Maximization) para la estimación del modelo espacio de los estados en el ámbito del Análisis Modal Operacional. El algoritmo se describe en detalle y también se analiza como aplicarlo cuando se dispone de datos de vibraciones de una estructura. A continuación se compara con otro método muy conocido, el método de los Subespacios. Los estimadores máximo verosímiles presentan una serie de propiedades que los hacen óptimos desde un punto de vista estadístico, pero la propiedad más destacable del algoritmo EM es que puede utilizarse para resolver un amplio abanico de situaciones que se presentan en el Análisis Modal Operacional. En este trabajo se proponen y estiman tres modelos en el espacio de los estados: • El primer modelo se utiliza para estimar los modos de vibración cuando se dispone de datos correspondientes a varios experimentos realizados en la misma estructura. En lugar de analizar registro a registro y calcular promedios, se utiliza algoritmo EM para la estimación conjunta del modelo propuesto utilizando todos los datos disponibles. • El segundo modelo en el espacio de los estados propuesto se utiliza para estimar los modos de vibración cuando el número de sensores disponibles es menor que vi Resumen el número de puntos que se quieren analizar en la estructura. En estos casos es usual realizar varios ensayos cambiando la posición de los sensores de un ensayo a otro (múltiples configuraciones de sensores). En este trabajo se utiliza el algoritmo EM para estimar los parámetros modales teniendo en cuenta los datos de todas las configuraciones. • Por último, se propone otro modelo en el espacio de los estados para estimar los modos de vibración en la presencia de entradas al sistema que no pueden modelarse como procesos estocásticos de ruido blanco. En estos casos, las frecuencias de las entradas no se pueden separar de las frecuencias del sistema y se obtienen modos espurios en la fase de identificación. La idea es registrar la respuesta de la estructura correspondiente a diferentes entradas; entonces se adopta la hipótesis de que los parámetros comunes a todos los registros corresponden a la estructura (modos de vibración), y los parámetros encontrados en un registro específico corresponden a la entrada en dicho ensayo. El problema se resuelve utilizando el modelo propuesto y el algoritmo EM.

Estimation of modal parameters in structures using multiple time history records

In Operational Modal Analysis of structures we often have multiple time history records of vibrations measured at different time instants. This work presents a procedure for estimating the modal parameters of the structure processing all the records, that is, using all available information to obtain a single estimate of the modal parameters. The method uses Maximum Likelihood Estimation and the Expectation Maximization algorithm. Finally, it has been applied to various problems for both simulated and real structures and the results show the advantage of the joint analysis proposed.

Joint statistical analysis of multiple measurement setups for the estimation of vibrational modes

Computing the modal parameters of large structures in Operational Modal Analysis often requires to process data from multiple non simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors that are fixed for all the measurements, while the other sensors are moved from one setup to the next. One possibility is to process the setups separately what result in different modal parameter estimates for each setup. Then the reference sensors are used to merge or glue the different parts of the mode shapes to obtain global modes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a state space model that can be used to process all setups at once so the global mode shapes are obtained automatically and subsequently only a value for the natural frequency and damping ratio of each mode is computed. We also present how this model can be estimated using maximum likelihood and the Expectation Maximization algorithm. We apply this technique to real data measured at a footbridge.

Using Inside-Outside Algorithm for Estimation of the Offspring Distribution in Multitype Branching Processes

Multitype branching processes (MTBP) model branching structures, where the nodes of the resulting tree are particles of different types. Usually such a process is not observable in the sense of the whole tree, but only as the “generation” at a given moment in time, which consists of the number of particles of every type. This requires an EM-type algorithm to obtain a maximum likelihood (ML) estimate of the parameters of the branching process. Using a version of the inside-outside algorithm for stochastic context-free grammars (SCFG), such an estimate could be obtained for the offspring distribution of the process.