abstract: Functional regression models are widely considered in practice. To precisely understand an underlying functional mechanism, a good sampling schedule for collecting informative functional data is necessary, especially when data collection is limited. However, scarce research has been conducted on the optimal sampling schedule design for the functional regression model so far. To address this design issue, efficient approaches are proposed for generating the best sampling plan in the functional regression setting. First, three optimal experimental designs are considered under a function-on-function linear model: the schedule that maximizes the relative efficiency for recovering the predictor function, the schedule that maximizes the relative efficiency for predicting the response function, and the schedule that maximizes the mixture of the relative efficiencies of both the predictor and response functions. The obtained sampling plan allows a precise recovery of the predictor function and a precise prediction of the response function. The proposed approach can also be reduced to identify the optimal sampling plan for the problem with a scalar-on-function linear regression model. In addition, the optimality criterion on predicting a scalar response using a functional predictor is derived when the quadratic relationship between these two variables is present, and proofs of important properties of the derived optimality criterion are also provided. To find such designs, an algorithm that is comparably fast, and can generate nearly optimal designs is proposed. As the optimality criterion includes quantities that must be estimated from prior knowledge (e.g., a pilot study), the effectiveness of the suggested optimal design highly depends on the quality of the estimates. However, in many situations, the estimates are unreliable; thus, a bootstrap aggregating (bagging) approach is employed for enhancing the quality of estimates and for finding sampling schedules stable to the misspecification of estimates. Through case studies, it is demonstrated that the proposed designs outperform other designs in terms of accurately predicting the response and recovering the predictor. It is also proposed that bagging-enhanced design generates a more robust sampling design under the misspecification of estimated quantities. / Dissertation/Thesis / Doctoral Dissertation Statistics 2020

abstract: I focus on algorithms that generate good sampling points for function approximation. In 1D, it is well known that polynomial interpolation using equispaced points is unstable. On the other hand, using Chebyshev nodes provides both stable and highly accurate points for polynomial interpolation. In higher dimensional complex regions, optimal sampling points are not known explicitly. This work presents robust algorithms that find good sampling points in complex regions for polynomial interpolation, least-squares, and radial basis function (RBF) methods. The quality of these nodes is measured using the Lebesgue constant. I will also consider optimal sampling for constrained optimization, used to solve PDEs, where boundary conditions must be imposed. Furthermore, I extend the scope of the problem to include finding near-optimal sampling points for high-order finite difference methods. These high-order finite difference methods can be implemented using either piecewise polynomials or RBFs. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Nesse trabalho, apresentaremos o método econômico desenvolvido por Taguchi para monitoramento on line da qualidade para atributos. O propósito deste método é obter o intervalo de inspeção que minimiza o custo esperado por item produzido em um processo industrial. Em seguida, mostraremos o modelo alternativo proposto por Nayebpour e Woodall (1993) e a derivação dos estimadores de máxima verossimilhança e de Bayes desenvolvida por Borges, Esteves e Wechsler (2005). Finalmente, apresentaremos uma nova solução para o problema de determinação do intervalo de inspeção ótimo sob a perspectiva da Teoria de Decisão Bayesiana. A última solução será ilustrada com alguns exemplos. / In this work, we present Taguchi\'s economic on line quality monitoring method for attributes. The aim of this method is to obtain the inspection interval that minimizes the expected cost per item produced in an industrial process. After that, we will consider the alternative method proposed by Nayebpour e Woodall (1993) and the derivation of the maximum likelihood and Bayes estimators developed by Borges, Esteves e Wechsler (2005). Finally, we will propose a new solution to the problem of inspection interval determination through Bayesian Decision Theory. This last solution will be implemented and its performance will be illustrated with a few examples.

A number of studies have investigated the relationship between surface electromyogram (EMG) and torque exerted about a joint. The standard deviation of the recorded EMG signal is defined as the EMG amplitude. The EMG amplitude estimation technique varies with the study from conventional type of processing (i.e. rectification followed by low pass filtering) to further addition of different noise rejection and signal-to-noise ratio improvement stages. Advanced EMG amplitude processors developed recently that incorporate signal whitening and multiple-channel combination have been shown to significantly improve amplitude estimation. The main contribution of this research is a comparison of the performance of EMG-torque estimators with and without these advanced EMG amplitude processors. The experimental data are taken from fifteen subjects that produced constant-posture, non-fatiguing, force-varying contractions about the elbow while torque and biceps/triceps EMG were recorded. Utilizing system identification techniques, EMG amplitude was related to torque through a zeros-only (finite impulse response, FIR) model. The incorporation of whitening and multiple-channel combination separately reduced EMG-torque errors and their combination provided a cumulative improvement. A 15th-order linear FIR model provided an average estimation error of 6% of maximum voluntary contraction (or 90% of variance accounted for) when EMG amplitudes were obtained using a four-channel, whitened processor. The equivalent single-channel, unwhitened (conventional) processor produced an average error of 8% of maximum voluntary contraction (variance accounted for of 68%). This study also describes the occurrence of spurious peaks in estimated torque when the torque model is created from data with a sampling rate well above the bandwidth of the torque. This problem is anticipated when the torque data are sampled at the same rate as the EMG data. The problem is resolved by decimating the EMG amplitude prior to relating it to joint torque, in this case to an effective sampling rate of 40.96 Hz.

Nesse trabalho, apresentaremos o método econômico desenvolvido por Taguchi para monitoramento on line da qualidade para atributos. O propósito deste método é obter o intervalo de inspeção que minimiza o custo esperado por item produzido em um processo industrial. Em seguida, mostraremos o modelo alternativo proposto por Nayebpour e Woodall (1993) e a derivação dos estimadores de máxima verossimilhança e de Bayes desenvolvida por Borges, Esteves e Wechsler (2005). Finalmente, apresentaremos uma nova solução para o problema de determinação do intervalo de inspeção ótimo sob a perspectiva da Teoria de Decisão Bayesiana. A última solução será ilustrada com alguns exemplos. / In this work, we present Taguchi\'s economic on line quality monitoring method for attributes. The aim of this method is to obtain the inspection interval that minimizes the expected cost per item produced in an industrial process. After that, we will consider the alternative method proposed by Nayebpour e Woodall (1993) and the derivation of the maximum likelihood and Bayes estimators developed by Borges, Esteves e Wechsler (2005). Finally, we will propose a new solution to the problem of inspection interval determination through Bayesian Decision Theory. This last solution will be implemented and its performance will be illustrated with a few examples.

Le problème d'intérêt est d'estimer la fonction de concentration et l'aire sous la courbe (AUC) à travers l'estimation des paramètres d'un modèle de régression linéaire avec un processus d'erreur autocorrélé. On construit un estimateur linéaire sans biais simple de la courbe de concentration et de l'AUC. On montre que cet estimateur construit à partir d'un plan d'échantillonnage régulier approprié est asymptotiquement optimal dans le sens où il a exactement la même performance asymptotique que le meilleur estimateur linéaire sans biais (BLUE). De plus, on montre que le plan d'échantillonnage optimal est robuste par rapport à la misspecification de la fonction d'autocovariance suivant le critère du minimax. Lorsque des observations répétées sont disponibles, cet estimateur est consistant et a une distribution asymptotique normale. Les résultats obtenus sont généralisés au processus d'erreur de Hölder d'indice compris entre 0 et 2. Enfin, pour des tailles d'échantillonnage petites, un algorithme de recuit simulé est appliqué à un modèle pharmacocinétique avec des erreurs corrélées. / The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with autocorrelated error process. We construct a simple linear unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from a sampling design generated by an appropriate density is asymptotically optimal in the sense that it has exactly the same asymptotic performance as the best linear unbiased estimator (BLUE). Moreover, we prove that the optimal design is robust with respect to a misspecification of the autocovariance function according to a minimax criterion. When repeated observations are available, this estimator is consistent and has an asymptotic normal distribution. All those results are extended to the error process of Hölder with index including between 0 and 2. Finally, for small sample sizes, a simulated annealing algorithm is applied to a pharmacokinetic model with correlated errors.