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.

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.

A practical approach for outdoors distributed target localization in wireless sensor networks

Wireless sensor networks are posed as the new communication paradigm where the use of small, low-complexity, and low-power devices is preferred over costly centralized systems. The spectra of potential applications of sensor networks is very wide, ranging from monitoring, surveillance, and localization, among others. Localization is a key application in sensor networks and the use of simple, efficient, and distributed algorithms is of paramount practical importance. Combining convex optimization tools with consensus algorithms we propose a distributed localization algorithm for scenarios where received signal strength indicator readings are used. We approach the localization problem by formulating an alternative problem that uses distance estimates locally computed at each node. The formulated problem is solved by a relaxed version using semidefinite relaxation technique. Conditions under which the relaxed problem yields to the same solution as the original problem are given and a distributed consensusbased implementation of the algorithm is proposed based on an augmented Lagrangian approach and primaldual decomposition methods. Although suboptimal, the proposed approach is very suitable for its implementation in real sensor networks, i.e., it is scalable, robust against node failures and requires only local communication among neighboring nodes. Simulation results show that running an additional local search around the found solution can yield performance close to the maximum likelihood estimate.

Estimating the modal parameters from multiple measurement setups using a joint state space model

Computing the modal parameters of structural systems often requires processing data from multiple non-simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors, which are fixed for all measurements, while the other sensors change their position from one setup to the next. One possibility is to process the setups separately resulting 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 mode shapes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a new state space model that processes all setups at once. The result is that the global mode shapes are obtained automatically, and only a value for the natural frequency and damping ratio of each mode is estimated. We also investigate the estimation of this model using maximum likelihood and the Expectation Maximization algorithm, and apply this technique to simulated and measured data corresponding to different structures.

Bases metodológicas para un sistema de apoyo a la selección de especies en la restauración de la vegetación

RESUMEN El apoyo a la selección de especies a la restauración de la vegetación en España en los últimos 40 años se ha basado fundamentalmente en modelos de distribución de especies, también llamados modelos de nicho ecológico, que estiman la probabilidad de presencia de las especies en función de las condiciones del medio físico (clima, suelo, etc.). Con esta tesis se ha intentado contribuir a la mejora de la capacidad predictiva de los modelos introduciendo algunas propuestas metodológicas adaptadas a los datos disponibles actualmente en España y enfocadas al uso de los modelos en la selección de especies. No siempre se dispone de datos a una resolución espacial adecuada para la escala de los proyectos de restauración de la vegetación. Sin embrago es habitual contar con datos de baja resolución espacial para casi todas las especies vegetales presentes en España. Se propone un método de recalibración que actualiza un modelo de regresión logística de baja resolución espacial con una nueva muestra de alta resolución espacial. El método permite obtener predicciones de calidad aceptable con muestras relativamente pequeñas (25 presencias de la especie) frente a las muestras mucho mayores (más de 100 presencias) que requería una estrategia de modelización convencional que no usara el modelo previo. La selección del método estadístico puede influir decisivamente en la capacidad predictiva de los modelos y por esa razón la comparación de métodos ha recibido mucha atención en la última década. Los estudios previos consideraban a la regresión logística como un método inferior a técnicas más modernas como las de máxima entropía. Los resultados de la tesis demuestran que esa diferencia observada se debe a que los modelos de máxima entropía incluyen técnicas de regularización y la versión de la regresión logística usada en las comparaciones no. Una vez incorporada la regularización a la regresión logística usando penalización, las diferencias en cuanto a capacidad predictiva desaparecen. La regresión logística penalizada es, por tanto, una alternativa más para el ajuste de modelos de distribución de especies y está a la altura de los métodos modernos con mejor capacidad predictiva como los de máxima entropía. A menudo, los modelos de distribución de especies no incluyen variables relativas al suelo debido a que no es habitual que se disponga de mediciones directas de sus propiedades físicas o químicas. La incorporación de datos de baja resolución espacial proveniente de mapas de suelo nacionales o continentales podría ser una alternativa. Los resultados de esta tesis sugieren que los modelos de distribución de especies de alta resolución espacial mejoran de forma ligera pero estadísticamente significativa su capacidad predictiva cuando se incorporan variables relativas al suelo procedente de mapas de baja resolución espacial. La validación es una de las etapas fundamentales del desarrollo de cualquier modelo empírico como los modelos de distribución de especies. Lo habitual es validar los modelos evaluando su capacidad predictiva especie a especie, es decir, comparando en un conjunto de localidades la presencia o ausencia observada de la especie con las predicciones del modelo. Este tipo de evaluación no responde a una cuestión clave en la restauración de la vegetación ¿cuales son las n especies más idóneas para el lugar a restaurar? Se ha propuesto un método de evaluación de modelos adaptado a esta cuestión que consiste en estimar la capacidad de un conjunto de modelos para discriminar entre las especies presentes y ausentes de un lugar concreto. El método se ha aplicado con éxito a la validación de 188 modelos de distribución de especies leñosas orientados a la selección de especies para la restauración de la vegetación en España. Las mejoras metodológicas propuestas permiten mejorar la capacidad predictiva de los modelos de distribución de especies aplicados a la selección de especies en la restauración de la vegetación y también permiten ampliar el número de especies para las que se puede contar con un modelo que apoye la toma de decisiones. SUMMARY During the last 40 years, decision support tools for plant species selection in ecological restoration in Spain have been based on species distribution models (also called ecological niche models), that estimate the probability of occurrence of the species as a function of environmental predictors (e.g., climate, soil). In this Thesis some methodological improvements are proposed to contribute to a better predictive performance of such models, given the current data available in Spain and focusing in the application of the models to selection of species for ecological restoration. Fine grained species distribution data are required to train models to be used at the scale of the ecological restoration projects, but this kind of data are not always available for every species. On the other hand, coarse grained data are available for almost every species in Spain. A recalibration method is proposed that updates a coarse grained logistic regression model using a new fine grained updating sample. The method allows obtaining acceptable predictive performance with reasonably small updating sample (25 occurrences of the species), in contrast with the much larger samples (more than 100 occurrences) required for a conventional modeling approach that discards the coarse grained data. The choice of the statistical method may have a dramatic effect on model performance, therefore comparisons of methods have received much interest in the last decade. Previous studies have shown a poorer performance of the logistic regression compared to novel methods like maximum entropy models. The results of this Thesis show that the observed difference is caused by the fact that maximum entropy models include regularization techniques and the versions of logistic regression compared do not. Once regularization has been added to the logistic regression using a penalization procedure, the differences in model performance disappear. Therefore, penalized logistic regression may be considered one of the best performing methods to model species distributions. Usually, species distribution models do not consider soil related predictors because direct measurements of the chemical or physical properties are often lacking. The inclusion of coarse grained soil data from national or continental soil maps could be a reasonable alternative. The results of this Thesis suggest that the performance of the models slightly increase after including soil predictors form coarse grained soil maps. Model validation is a key stage of the development of empirical models, such as species distribution models. The usual way of validating is based on the evaluation of model performance for each species separately, i.e., comparing observed species presences or absence to predicted probabilities in a set of sites. This kind of evaluation is not informative for a common question in ecological restoration projects: which n species are the most suitable for the environment of the site to be restored? A method has been proposed to address this question that estimates the ability of a set of models to discriminate among present and absent species in a evaluation site. The method has been successfully applied to the validation of 188 species distribution models used to support decisions on species selection for ecological restoration in Spain. The proposed methodological approaches improve the predictive performance of the predictive models applied to species selection in ecological restoration and increase the number of species for which a model that supports decisions can be fitted.

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.

Operational modal analisys using joint statistical analysis of multiple records

In Operational Modal Analysis (OMA) of a structure, the data acquisition process may be repeated many times. In these cases, the analyst has several similar records for the modal analysis of the structure that have been obtained at di�erent time instants (multiple records). The solution obtained varies from one record to another, sometimes considerably. The differences are due to several reasons: statistical errors of estimation, changes in the external forces (unmeasured forces) that modify the output spectra, appearance of spurious modes, etc. Combining the results of the di�erent individual analysis is not straightforward. To solve the problem, we propose to make the joint estimation of the parameters using all the records. This can be done in a very simple way using state space models and computing the estimates by maximum-likelihood. The method provides a single result for the modal parameters that combines optimally all the records.

Ice volume estimates from ground-penetrating radar surveys, western Nordenskiöld Land glaciers, Svalbard

As part of ongoing work within the SvalGlac project aimed to obtain a reliable estimate of the total ice volume of Svalbard glaciers and their potential contribution to sea level rise, in this contribution we present volume calculations, with detailed error estimates, for ten glaciers on western Nordenskiöld Land, central Spitsbergen, Svalbard. The volume estimates are based upon a dense net of GPR-retrieved ice thickness data collected over several field campaigns spanning the period 1999-2012, all of them except one within 2010-2012. The total area and volume of the ensemble are 113.38±0.09 km2 and 10.439±0.185 km3, respectively, while the individual areas, volumes and average ice thickness lie within 2.5-49.1 km2, 0.08-5.48 km3 and 29-108 m, respectively. The maximum recorded ice thickness, 265±15 m, corresponds to Fridtjovbreen, which has also the largest average thickness (108±1m). Available empirical formulae for Svalbard glaciers overestimate the total volume of these glaciers by 24% with respect to our calculation. On the basis of the pattern of scattering in the radargrams, we also analyse the hydrothermal structure of these glaciers. Nine out of ten are polythermal, while only one is entirely cold.

Ice volume estimates from ground-penetrating radar surveys, Wedel Jarlsberg Land glaciers, Svalbard

One of the aims of the SvalGlac project is to obtain an improved estimate, with reliable error estimates, of the volume of Svalbard glaciers and their potential contribution to sea level rise. As part of this work, we present volume calculations, with detailed error estimates, for eight glaciers on Wedel Jarlsberg Land, southern Spitsbergen, Svalbard. The volume estimates are based upon a dense net of GPR-retrieved ice thickness data collected over several field campaigns spanning the period 2004-2011. The total area and volume of the ensemble are 502.9±18.6 km2 and 80.72±2.85 km3, respectively. Excluding Ariebreen (a tiny glacier, menor que 0.4 km2 in area), the individual areas, volumes and average ice thickness lie within 4.7-141.0 km2, 0.30-25.85 km3 and 64-183 m, respectively. The maximum recorded ice thickness, ca. 619±13 m, is found in Austre Torellbreen. To estimate the ice volume of small non-echo-sounded tributary glaciers, we used a function providing the best fit to the ice thickness along the centre line of a collection of such tributaries where echo-soundings were available, and assuming parabolic cross-sections. We did some tests on the effect on the measured ice volumes of the distinct radio-wave velocity (RWV) of firn as compared to ice, and cold versus temperate ice, concluding that the changes in volume implied by such corrections were within the error bounds of our volume estimate using a constant RWV for the entire glacier inferred from common mid-point measurements on the upper ablation area.

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 the EM algorithm to estimate the state space model for OMAX

This paper presents a time-domain stochastic system identification method based on Maximum Likelihood Estimation and the Expectation Maximization algorithm that is applied to the estimation of modal parameters from system input and output data. The effectiveness of this structural identification method is evaluated through numerical simulation. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the simulated structure are estimated applying the proposed identification method to a set of 100 simulated cases. The numerical results show that the proposed method estimates the modal parameters with precision in the presence of 20% measurement noise even. Finally, advantages and disadvantages of the method have been discussed.

Analysis of advanced European nuclear fuel cycle scenarios including transmutation and economic estimates

Four European fuel cycle scenarios involving transmutation options (in coherence with PATEROS and CPESFR EU projects) have been addressed from a point of view of resources utilization and economic estimates. Scenarios include: (i) the current fleet using Light Water Reactor (LWR) technology and open fuel cycle, (ii) full replacement of the initial fleet with Fast Reactors (FR) burning U?Pu MOX fuel, (iii) closed fuel cycle with Minor Actinide (MA) transmutation in a fraction of the FR fleet, and (iv) closed fuel cycle with MA transmutation in dedicated Accelerator Driven Systems (ADS). All scenarios consider an intermediate period of GEN-III+ LWR deployment and they extend for 200 years, looking for long term equilibrium mass flow achievement. The simulations were made using the TR_EVOL code, capable to assess the management of the nuclear mass streams in the scenario as well as economics for the estimation of the levelized cost of electricity (LCOE) and other costs. Results reveal that all scenarios are feasible according to nuclear resources demand (natural and depleted U, and Pu). Additionally, we have found as expected that the FR scenario reduces considerably the Pu inventory in repositories compared to the reference scenario. The elimination of the LWR MA legacy requires a maximum of 55% fraction (i.e., a peak value of 44 FR units) of the FR fleet dedicated to transmutation (MA in MOX fuel, homogeneous transmutation) or an average of 28 units of ADS plants (i.e., a peak value of 51 ADS units). Regarding the economic analysis, the main usefulness of the provided economic results is for relative comparison of scenarios and breakdown of LCOE contributors rather than provision of absolute values, as technological readiness levels are low for most of the advanced fuel cycle stages. The obtained estimations show an increase of LCOE ? averaged over the whole period ? with respect to the reference open cycle scenario of 20% for Pu management scenario and around 35% for both transmutation scenarios. The main contribution to LCOE is the capital costs of new facilities, quantified between 60% and 69% depending on the scenario. An uncertainty analysis is provided around assumed low and high values of processes and technologies.

Ice volume estimates from ground-penetrating radar surveys, Wedel Jarlsberg Land glaciers, Svalbard

We present ground-penetrating radar (GPR)—based volume calculations, with associated error estimates, for eight glaciers on Wedel Jarlsberg Land, southwestern Spitsbergen, Svalbard, and compare them with those obtained from volume-area scaling relationships. The volume estimates are based upon GPR ice-thickness data collected during the period 2004–2013. The total area and volume of the ensemble are 502.91 ± 18.60 km2 and 91.91 ± 3.12 km3, respectively. The individual areas, volumes, and average ice thickness lie within 0.37–140.99 km2, 0.01–31.98 km3, and 28–227 m, respectively, with a maximum recorded ice thickness of 619 ± 13 m on Austre Torellbreen. To estimate the ice volume of unsurveyed tributary glaciers, we combine polynomial cross-sections with a function providing the best fit to the measured ice thickness along the center line of a collection of 22 surveyed tributaries. For the time-to-depth conversion of GPR data, we test the use of a glacierwide constant radio-wave velocity chosen on the basis of local or regional common midpoint measurements, versus the use of distinct velocities for the firn, cold ice, and temperate ice layers, concluding that the corresponding volume calculations agree with each other within their error bounds.