SMC^2: an efficient algorithm for sequential analysis of state-space models

We consider the generic problem of performing sequential Bayesian inference in a state-space model with observation process y, state process x and fixed parameter theta. An idealized approach would be to apply the iterated batch importance sampling (IBIS) algorithm of Chopin (2002). This is a sequential Monte Carlo algorithm in the theta-dimension, that samples values of theta, reweights iteratively these values using the likelihood increments p(y_t|y_1:t-1, theta), and rejuvenates the theta-particles through a resampling step and a MCMC update step. In state-space models these likelihood increments are intractable in most cases, but they may be unbiasedly estimated by a particle filter in the x-dimension, for any fixed theta. This motivates the SMC^2 algorithm proposed in this article: a sequential Monte Carlo algorithm, defined in the theta-dimension, which propagates and resamples many particle filters in the x-dimension. The filters in the x-dimension are an example of the random weight particle filter as in Fearnhead et al. (2010). On the other hand, the particle Markov chain Monte Carlo (PMCMC) framework developed in Andrieu et al. (2010) allows us to design appropriate MCMC rejuvenation steps. Thus, the theta-particles target the correct posterior distribution at each iteration t, despite the intractability of the likelihood increments. We explore the applicability of our algorithm in both sequential and non-sequential applications and consider various degrees of freedom, as for example increasing dynamically the number of x-particles. We contrast our approach to various competing methods, both conceptually and empirically through a detailed simulation study, included here and in a supplement, and based on particularly challenging examples.Comment: 27 pages, 4 figures; supplementary material available on the second author's web pag

Multi-tier framework for the inferential measurement and data-driven modeling

A framework for the inferential measurement and data-driven modeling has been proposed and assessed in several real-world application domains. The architecture of the framework has been structured in multiple tiers to facilitate extensibility and the integration of new components. Each of the proposed four tiers has been assessed in an uncoupled way to verify their suitability. The first tier, dealing with exploratory data analysis, has been assessed with the characterization of the chemical space related to the biodegradation of organic chemicals. This analysis has established relationships between physicochemical variables and biodegradation rates that have been used for model development. At the preprocessing level, a novel method for feature selection based on dissimilarity measures between Self-Organizing maps (SOM) has been developed and assessed. The proposed method selected more features than others published in literature but leads to models with improved predictive power. Single and multiple data imputation techniques based on the SOM have also been used to recover missing data in a Waste Water Treatment Plant benchmark. A new dynamic method to adjust the centers and widths of in Radial basis Function networks has been proposed to predict water quality. The proposed method outperformed other neural networks. The proposed modeling components have also been assessed in the development of prediction and classification models for biodegradation rates in different media. The results obtained proved the suitability of this approach to develop data-driven models when the complex dynamics of the process prevents the formulation of mechanistic models. The use of rule generation algorithms and Bayesian dependency models has been preliminary screened to provide the framework with interpretation capabilities. Preliminary results obtained from the classification of Modes of Toxic Action (MOA) indicate that this could be a promising approach to use MOAs as proxy indicators of human health effects of chemicals.Finally, the complete framework has been applied to three different modeling scenarios. A virtual sensor system, capable of inferring product quality indices from primary process variables has been developed and assessed. The system was integrated with the control system in a real chemical plant outperforming multi-linear correlation models usually adopted by chemical manufacturers. A model to predict carcinogenicity from molecular structure for a set of aromatic compounds has been developed and tested. Results obtained after the application of the SOM-dissimilarity feature selection method yielded better results than models published in the literature. Finally, the framework has been used to facilitate a new approach for environmental modeling and risk management within geographical information systems (GIS). The SOM has been successfully used to characterize exposure scenarios and to provide estimations of missing data through geographic interpolation. The combination of SOM and Gaussian Mixture models facilitated the formulation of a new probabilistic risk assessment approach.Aquesta tesi proposa i avalua en diverses aplicacions reals, un marc general de treball per al desenvolupament de sistemes de mesurament inferencial i de modelat basats en dades. L'arquitectura d'aquest marc de treball s'organitza en diverses capes que faciliten la seva extensibilitat així com la integració de nous components. Cadascun dels quatre nivells en que s'estructura la proposta de marc de treball ha estat avaluat de forma independent per a verificar la seva funcionalitat. El primer que nivell s'ocupa de l'anàlisi exploratòria de dades ha esta avaluat a partir de la caracterització de l'espai químic corresponent a la biodegradació de certs compostos orgànics. Fruit d'aquest anàlisi s'han establert relacions entre diverses variables físico-químiques que han estat emprades posteriorment per al desenvolupament de models de biodegradació. A nivell del preprocés de les dades s'ha desenvolupat i avaluat una nova metodologia per a la selecció de variables basada en l'ús del Mapes Autoorganitzats (SOM). Tot i que el mètode proposat selecciona, en general, un major nombre de variables que altres mètodes proposats a la literatura, els models resultants mostren una millor capacitat predictiva. S'han avaluat també tot un conjunt de tècniques d'imputació de dades basades en el SOM amb un conjunt de dades estàndard corresponent als paràmetres d'operació d'una planta de tractament d'aigües residuals. Es proposa i avalua en un problema de predicció de qualitat en aigua un nou model dinàmic per a ajustar el centre i la dispersió en xarxes de funcions de base radial. El mètode proposat millora els resultats obtinguts amb altres arquitectures neuronals. Els components de modelat proposat s'han aplicat també al desenvolupament de models predictius i de classificació de les velocitats de biodegradació de compostos orgànics en diferents medis. Els resultats obtinguts demostren la viabilitat d'aquesta aproximació per a desenvolupar models basats en dades en aquells casos en els que la complexitat de dinàmica del procés impedeix formular models mecanicistes. S'ha dut a terme un estudi preliminar de l'ús de algorismes de generació de regles i de grafs de dependència bayesiana per a introduir una nova capa que faciliti la interpretació dels models. Els resultats preliminars obtinguts a partir de la classificació dels Modes d'acció Tòxica (MOA) apunten a que l'ús dels MOA com a indicadors intermediaris dels efectes dels compostos químics en la salut és una aproximació factible.Finalment, el marc de treball proposat s'ha aplicat en tres escenaris de modelat diferents. En primer lloc, s'ha desenvolupat i avaluat un sensor virtual capaç d'inferir índexs de qualitat a partir de variables primàries de procés. El sensor resultant ha estat implementat en una planta química real millorant els resultats de les correlacions multilineals emprades habitualment. S'ha desenvolupat i avaluat un model per a predir els efectes carcinògens d'un grup de compostos aromàtics a partir de la seva estructura molecular. Els resultats obtinguts desprès d'aplicar el mètode de selecció de variables basat en el SOM milloren els resultats prèviament publicats. Aquest marc de treball s'ha usat també per a proporcionar una nova aproximació al modelat ambiental i l'anàlisi de risc amb sistemes d'informació geogràfica (GIS). S'ha usat el SOM per a caracteritzar escenaris d'exposició i per a desenvolupar un nou mètode d'interpolació geogràfica. La combinació del SOM amb els models de mescla de gaussianes dona una nova formulació al problema de l'anàlisi de risc des d'un punt de vista probabilístic

Statistical Methods for Semiconductor Manufacturing

In this thesis techniques for non-parametric modeling, machine learning, filtering and prediction and run-to-run control for semiconductor manufacturing are described. In particular, algorithms have been developed for two major applications area: - Virtual Metrology (VM) systems; - Predictive Maintenance (PdM) systems. Both technologies have proliferated in the past recent years in the semiconductor industries, called fabs, in order to increment productivity and decrease costs. VM systems aim of predicting quantities on the wafer, the main and basic product of the semiconductor industry, that may be physically measurable or not. These quantities are usually ’costly’ to be measured in economic or temporal terms: the prediction is based on process variables and/or logistic information on the production that, instead, are always available and that can be used for modeling without further costs. PdM systems, on the other hand, aim at predicting when a maintenance action has to be performed. This approach to maintenance management, based like VM on statistical methods and on the availability of process/logistic data, is in contrast with other classical approaches: - Run-to-Failure (R2F), where there are no interventions performed on the machine/process until a new breaking or specification violation happens in the production; - Preventive Maintenance (PvM), where the maintenances are scheduled in advance based on temporal intervals or on production iterations. Both aforementioned approaches are not optimal, because they do not assure that breakings and wasting of wafers will not happen and, in the case of PvM, they may lead to unnecessary maintenances without completely exploiting the lifetime of the machine or of the process. The main goal of this thesis is to prove through several applications and feasibility studies that the use of statistical modeling algorithms and control systems can improve the efficiency, yield and profits of a manufacturing environment like the semiconductor one, where lots of data are recorded and can be employed to build mathematical models. We present several original contributions, both in the form of applications and methods. The introduction of this thesis will be an overview on the semiconductor fabrication process: the most common practices on Advanced Process Control (APC) systems and the major issues for engineers and statisticians working in this area will be presented. Furthermore we will illustrate the methods and mathematical models used in the applications. We will then discuss in details the following applications: - A VM system for the estimation of the thickness deposited on the wafer by the Chemical Vapor Deposition (CVD) process, that exploits Fault Detection and Classification (FDC) data is presented. In this tool a new clustering algorithm based on Information Theory (IT) elements have been proposed. In addition, the Least Angle Regression (LARS) algorithm has been applied for the first time to VM problems. - A new VM module for multi-step (CVD, Etching and Litography) line is proposed, where Multi-Task Learning techniques have been employed. - A new Machine Learning algorithm based on Kernel Methods for the estimation of scalar outputs from time series inputs is illustrated. - Run-to-Run control algorithms that employ both the presence of physical measures and statistical ones (coming from a VM system) is shown; this tool is based on IT elements. - A PdM module based on filtering and prediction techniques (Kalman Filter, Monte Carlo methods) is developed for the prediction of maintenance interventions in the Epitaxy process. - A PdM system based on Elastic Nets for the maintenance predictions in Ion Implantation tool is described. Several of the aforementioned works have been developed in collaborations with major European semiconductor companies in the framework of the European project UE FP7 IMPROVE (Implementing Manufacturing science solutions to increase equiPment pROductiVity and fab pErformance); such collaborations will be specified during the thesis, underlying the practical aspects of the implementation of the proposed technologies in a real industrial environment