The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors. In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor. These methods are also implemented in MATLAB. We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-09-02T13:38:23Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_Rafaella de Figueiredo Sales.pdf: 2788155 bytes, checksum: 95e4d0c2ca7494069abdc945b186152e (MD5) / Made available in DSpace on 2016-09-02T13:38:23Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_Rafaella de Figueiredo Sales.pdf: 2788155 bytes, checksum: 95e4d0c2ca7494069abdc945b186152e (MD5) Previous issue date: 2016-02-19 / PETROBRÁS / PRH - 28 / O biodiesel é considerado um potencial substituto para os combustíveis de origem fóssil. Geralmente, esse biocombustível é produzido através da transesterificação de óleos vegetais ou gordura animal utilizando um álcool de cadeia curta. A vasta aplicação industrial da transesterificação para produção de biodiesel requer um cuidadoso monitoramento do processo e modelagem dessa reação, orientados para um melhor entendimento e para a otimização do processo. Além disso, o controle do processo é de fundamental importância para garantir a qualidade do biodiesel de forma constante, uniforme e atendendo às especificações necessárias para a sua comercialização. Dentre as técnicas existentes para o monitoramento de reações, a espectroscopia no infravermelho próximo permite uma análise rápida e não destrutiva, podendo ser utilizada para o monitoramento in-line de processos. Nesse contexto, o presente trabalho descreve o monitoramento in-line da produção de biodiesel por transesterificação de óleo de soja com metanol utilizando a espectroscopia na região do infravermelho próximo. As reações foram conduzidas em batelada com controle de temperatura e velocidade de agitação. Dois estudos diferentes foram então realizados a partir dos dados espectroscópicos coletados ao longo das reações. A primeira abordagem consistiu na implementação de estratégias de controle estatístico multivariado e de qualidade, baseadas em métodos de projeção multivariada. Nesse estudo, modelos de referência baseados em dois métodos diferentes (tempo real e após o término da batelada) foram desenvolvidos usando bateladas sob condições normais de operação (concentração de catalisador NaOH de 0,75% em massa em relação à massa de óleo de soja, temperatura de 55°C e velocidade de agitação de 500 rpm). Foram avaliadas diferentes técnicas de pré-processamento espectral, bem como estratégias para desdobramento da matriz de dados espectrais. Posteriormente, bateladas submetidas a diferentes perturbações (relacionadas à adição de água e a mudanças na temperatura, na velocidade de agitação, na concentração de catalisador e na matéria prima utilizada) foram produzidas para avaliar o desempenho dos diferentes modelos em relação às suas capacidades de detectar falhas de operação. De um modo geral, as cartas de controle desenvolvidas foram capazes de detectar a maioria das falhas intencionalmente produzidas ao longo do processo. Apenas pequenas perturbações na velocidade de agitação não foram identificadas. A segunda abordagem baseou-se no estudo do comportamento cinético da metanólise do óleo de soja em diferentes condições de temperatura (20, 45 e 55°C) e de concentração de catalisador (0,75 e 1,0% m/m de NaOH). Nesse estudo, a modelagem cinética foi realizada a partir dos perfis de concentração do éster metílico estimados utilizando o método de Resolução multivariada de curvas com mínimos quadrados alternantes com restrição de correlação. Os perfis de concentração obtidos permitiram o desenvolvimento de um estudo cinético simplificado da reação, a qual foi considerada como seguindo uma cinética de pseudo-primeira ordem baseada na concentração do éster metílico para a reação global de transesterificação. O modelo cinético incluiu apenas o regime pseudo-homogêneo, em que é observada uma rápida taxa de reação e a cinética está controlada pela reação química. Para esse regime, obteve-se uma energia de ativação de 32,29 kJ.mol-1, com base na equação de Arrhenius. / Biodiesel is considered a potential substitute for fossil fuels. In general, this biofuel is produced by transesterification of vegetable oils or animal fats using a short-chained alcohol. The wide industrial application of transesterification for biodiesel production requires a thorough process monitoring and modeling of this reaction, oriented to a better understanding and optimization of the process. Moreover, process control is of utmost importance to ensure in a constant and uniform way the biodiesel quality in order to meet commercialization requirements. Among the existing techniques applied for reaction monitoring, near infrared spectroscopy allows a fast and non-destructive analysis, being able to be used for in-line process monitoring. In this context, the present work describes the in-line monitoring of biodiesel production by transesterification of soybean oil with methanol using near infrared spectroscopy. Reactions were carried out in batch reactors with temperature and agitation speed control. Two different studies were then carried out from spectroscopic data collected throughout the reactions. The first approach consisted of the implementation of multivariate statistic and quality control strategies, based on multivariate projection methods. In this study, reference models based on two different methods (real time and end of batch) were developed using batches under normal operating condition (0.75 w/w% of catalyst NaOH with respect to the amount of oil, temperature of 55ºC and stirring speed of 500 rpm). Different techniques for pre-processing and unfolding the spectral data were evaluated. Afterwards, batches subjected to different disturbances (related with water addition and changes in the temperature, agitation speed, catalyst content and raw material used) were manufactured to assess the performance of the different models in terms of their capability for fault detection. In general, the control charts developed were able to detect most of the failures intentionally produced during the batch runs. Only small variations in the stirring spend were not detected. The second approach was based on the study of the kinetic behavior of the soybean oil methanolysiscarried out with different temperatures (20, 44 and 55°C) and catalyst (NaOH) concentrations (0.75 and 1.0 w/w% based onoil weight). In this study, the kinetic modelling was developed from methyl ester concentration profiles estimated by Multivariate curve resolution alternating least squares with correlation constraint. The obtained concentration profiles allowed the development of a simplified kinetic model of the reaction, which was considered to follow a pseudo-first order overall kinetics based on methyl ester concentration for global transesterification reaction. The kinetic model included only the pseudo-homogeneous regime where the overall process kinetics is under chemical reaction control and a fast methanolysis rate is observed. For this regime, the activation energy was 32.29 kJ.mol-1, based on Arrhenius equation.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-09-02T14:13:49Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_Rafaella de Figueiredo Sales.pdf: 2788155 bytes, checksum: 95e4d0c2ca7494069abdc945b186152e (MD5) / Made available in DSpace on 2016-09-02T14:13:49Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_Rafaella de Figueiredo Sales.pdf: 2788155 bytes, checksum: 95e4d0c2ca7494069abdc945b186152e (MD5) Previous issue date: 2016-02-19 / PETROBRAS/ PRH 28 / O biodiesel é considerado um potencial substituto para os combustíveis de origem fóssil. Geralmente, esse biocombustível é produzido através da transesterificação de óleos vegetais ou gordura animal utilizando um álcool de cadeia curta. A vasta aplicação industrial da transesterificação para produção de biodiesel requer um cuidadoso monitoramento do processo e modelagem dessa reação, orientados para um melhor entendimento e para a otimização do processo. Além disso, o controle do processo é de fundamental importância para garantir a qualidade do biodiesel de forma constante, uniforme e atendendo às especificações necessárias para a sua comercialização. Dentre as técnicas existentes para o monitoramento de reações, a espectroscopia no infravermelho próximo permite uma análise rápida e não destrutiva, podendo ser utilizada para o monitoramento in-line de processos. Nesse contexto, o presente trabalho descreve o monitoramento in-line da produção de biodiesel por transesterificação de óleo de soja com metanol utilizando a espectroscopia na região do infravermelho próximo. As reações foram conduzidas em batelada com controle de temperatura e velocidade de agitação. Dois estudos diferentes foram então realizados a partir dos dados espectroscópicos coletados ao longo das reações. A primeira abordagem consistiu na implementação de estratégias de controle estatístico multivariado e de qualidade, baseadas em métodos de projeção multivariada. Nesse estudo, modelos de referência baseados em dois métodos diferentes (tempo real e após o término da batelada) foram desenvolvidos usando bateladas sob condições normais de operação (concentração de catalisador NaOH de 0,75% em massa em relação à massa de óleo de soja, temperatura de 55°C e velocidade de agitação de 500 rpm). Foram avaliadas diferentes técnicas de pré-processamento espectral, bem como estratégias para desdobramento da matriz de dados espectrais. Posteriormente, bateladas submetidas a diferentes perturbações (relacionadas à adição de água e a mudanças na temperatura, na velocidade de agitação, na concentração de catalisador e na matéria prima utilizada) foram produzidas para avaliar o desempenho dos diferentes modelos em relação às suas capacidades de detectar falhas de operação. De um modo geral, as cartas de controle desenvolvidas foram capazes de detectar a maioria das falhas intencionalmente produzidas ao longo do processo. Apenas pequenas perturbações na velocidade de agitação não foram identificadas. A segunda abordagem baseou-se no estudo do comportamento cinético da metanólise do óleo de soja em diferentes condições de temperatura (20, 45 e 55°C) e de concentração de catalisador (0,75 e 1,0% m/m de NaOH). Nesse estudo, a modelagem cinética foi realizada a partir dos perfis de concentração do éster metílico estimados utilizando o método de Resolução multivariada de curvas com mínimos quadrados alternantes com restrição de correlação. Os perfis de concentração obtidos permitiram o desenvolvimento de um estudo cinético simplificado da reação, a qual foi considerada como seguindo uma cinética de pseudo-primeira ordem baseada na concentração do éster metílico para a reação global de transesterificação. O modelo cinético incluiu apenas o regime pseudo-homogêneo, em que é observada uma rápida taxa de reação e a cinética está controlada pela reação química. Para esse regime, obteve-se uma energia de ativação de 32,29 kJ.mol-1, com base na equação de Arrhenius. / Biodiesel is considered a potential substitute for fossil fuels. In general, this biofuel is produced by transesterification of vegetable oils or animal fats using a short-chained alcohol. The wide industrial application of transesterification for biodiesel production requires a thorough process monitoring and modeling of this reaction, oriented to a better understanding and optimization of the process. Moreover, process control is of utmost importance to ensure in a constant and uniform way the biodiesel quality in order to meet commercialization requirements. Among the existing techniques applied for reaction monitoring, near infrared spectroscopy allows a fast and non-destructive analysis, being able to be used for in-line process monitoring. In this context, the present work describes the in-line monitoring of biodiesel production by transesterification of soybean oil with methanol using near infrared spectroscopy. Reactions were carried out in batch reactors with temperature and agitation speed control. Two different studies were then carried out from spectroscopic data collected throughout the reactions. The first approach consisted of the implementation of multivariate statistic and quality control strategies, based on multivariate projection methods. In this study, reference models based on two different methods (real time and end of batch) were developed using batches under normal operating condition (0.75 w/w% of catalyst NaOH with respect to the amount of oil, temperature of 55ºC and stirring speed of 500 rpm). Different techniques for pre-processing and unfolding the spectral data were evaluated. Afterwards, batches subjected to different disturbances (related with water addition and changes in the temperature, agitation speed, catalyst content and raw material used) were manufactured to assess the performance of the different models in terms of their capability for fault detection. In general, the control charts developed were able to detect most of the failures intentionally produced during the batch runs. Only small variations in the stirring spend were not detected. The second approach was based on the study of the kinetic behavior of the soybean oil methanolysiscarried out with different temperatures (20, 44 and 55°C) and catalyst (NaOH) concentrations (0.75 and 1.0 w/w% based onoil weight). In this study, the kinetic modelling was developed from methyl ester concentration profiles estimated by Multivariate curve resolution alternating least squares with correlation constraint. The obtained concentration profiles allowed the development of a simplified kinetic model of the reaction, which was considered to follow a pseudo-first order overall kinetics based on methyl ester concentration for global transesterification reaction. The kinetic model included only the pseudo-homogeneous regime where the overall process kinetics is under chemical reaction control and a fast methanolysis rate is observed. For this regime, the activation energy was 32.29 kJ.mol-1, based on Arrhenius equation.

現今許多網路商店或電子商務將產品銷售給消費者的過程中，皆使用推薦系統的幫助來提高銷售量。如亞馬遜公司(Amazon)、Netflix，深入了解顧客的使用習慣，建構專屬的推薦系統並進行個性化的推薦商品給每一位顧客。 推薦系統應用的技術分為協同過濾和內容過濾兩大類，本研究旨在探討協同過濾推薦系統中潛在因子模型方法，利用矩陣分解法找出評分矩陣。在Koren等人(2009)中，將矩陣分解法的演算法大致分為兩種，隨機梯度下降法(Stochastic gradient descent)與交替最小平方法(Alternating least squares)。本研究主要研究目的有三項，一為比較交替最小平方法與隨機梯度下降法的預測能力，二為兩種矩陣分解演算法在加入偏誤項後的表現，三為先完成交替最小平方法與隨機梯度下降法，以其預測值對原始資料之遺失值進行資料插補，再利用奇異值分解法對完整資料做矩陣分解，觀察其前後方法的差異。 研究結果顯示，隨機梯度下降法所需的運算時間比交替最小平方法所需的運算時間少。另外，完成兩種矩陣分解演算法後，將預測值插補遺失值，進行奇異值分解的結果也顯示預測能力有提升。 / Recommender system has been largely used by Internet companies such Amazon and Netflix to make recommendations for Internet users. Techniques for recommender systems can be divided into content filtering approach and collaborative filtering approach. Matrix factorization is a popular method for collaborative filtering approach. It minimizes the object function through stochastic gradient descent and alternating least squares. This thesis has three goals. First, we compare the alternating least squares method and stochastic gradient descent method. Secondly, we compare the performance of matrix factorization method with and without the bias term. Thirdly, we combine singular value decomposition and matrix factorization. As expected, we found the stochastic gradient descent takes less time than the alternating least squares method, and the the matrix factorization method with bias term gives more accurate prediction. We also found that combining singular value decomposition with matrix factorization can improve the predictive accuracy.

Dans la première partie de cette thèse, on formule deux méthodes pour le calcul d'une décomposition polyadique canonique avec facteurs matriciels linéairement structurés (tels que des facteurs de Toeplitz ou en bande): un algorithme de moindres carrés alternés contraint (CALS) et une solution algébrique dans le cas où tous les facteurs sont circulants. Des versions exacte et approchée de la première méthode sont étudiées. La deuxième méthode fait appel à la transformée de Fourier multidimensionnelle du tenseur considéré, ce qui conduit à la résolution d'un système d'équations monomiales homogènes. Nos simulations montrent que la combinaison de ces approches fournit un estimateur statistiquement efficace, ce qui reste vrai pour d'autres combinaisons de CALS dans des scénarios impliquant des facteurs non-circulants. La seconde partie de la thèse porte sur la récupération de tenseurs de rang faible et, en particulier, sur le problème de reconstruction tensorielle (TC). On propose un algorithme efficace, noté SeMPIHT, qui emploie des projections séquentiellement optimales par mode comme opérateur de seuillage dur. Une borne de performance est dérivée sous des conditions d'isométrie restreinte habituelles, ce qui fournit des bornes d'échantillonnage sous-optimales. Cependant, nos simulations suggèrent que SeMPIHT obéit à des bornes optimales pour des mesures Gaussiennes. Des heuristiques de sélection du pas et d'augmentation graduelle du rang sont aussi élaborées dans le but d'améliorer sa performance. On propose aussi un schéma d'imputation pour TC basé sur un seuillage doux du coeur du modèle de Tucker et son utilité est illustrée avec des données réelles de trafic routier / In the first part of this thesis, we formulate two methods for computing a canonical polyadic decomposition having linearly structured matrix factors (such as, e.g., Toeplitz or banded factors): a general constrained alternating least squares (CALS) algorithm and an algebraic solution for the case where all factors are circulant. Exact and approximate versions of the former method are studied. The latter method relies on a multidimensional discrete-time Fourier transform of the target tensor, which leads to a system of homogeneous monomial equations whose resolution provides the desired circulant factors. Our simulations show that combining these approaches yields a statistically efficient estimator, which is also true for other combinations of CALS in scenarios involving non-circulant factors. The second part of the thesis concerns low-rank tensor recovery (LRTR) and, in particular, the tensor completion (TC) problem. We propose an efficient algorithm, called SeMPIHT, employing sequentially optimal modal projections as its hard thresholding operator. Then, a performance bound is derived under usual restricted isometry conditions, which however yield suboptimal sampling bounds. Yet, our simulations suggest SeMPIHT obeys optimal sampling bounds for Gaussian measurements. Step size selection and gradual rank increase heuristics are also elaborated in order to improve performance. We also devise an imputation scheme for TC based on soft thresholding of a Tucker model core and illustrate its utility in completing real-world road traffic data acquired by an intelligent transportation

Anthropogenic emissions into the troposphere can impact air quality, leading to poorer health outcomes in the affected areas. Volatile organic compounds (VOCs) are a group of chemical compounds, including some which are toxic, that are precursors in the formation of ground-level ozone and secondary organic aerosols. VOCs have a variety of sources, and the distribution of atmospheric VOCs differs significantly over time and space. Historically, the large number of chemical species present at low concentrations (parts-per-trillion to parts-per-billion by volume) have made VOCs difficult to measure in ambient air. However, with improvements in analytical instrumentation, these measurements are becoming more common place. Direct mass spectrometry (MS), such as membrane introduction mass spectrometry (MIMS) and proton-transfer reaction time-of-flight mass spectrometry (PTR-ToF-MS) facilitate real-time, continuous measurements of VOCs in air, with full scan mass spectral data capturing changes in chemical composition with high temporal resolution. Operated on-road, mobilized direct MS has been used for quantitative mapping of VOCs at the neighborhood scale, but identifying VOC sources based on the observed mixture of molecules in the full scan MS dataset has yet to be explored. This dissertation describes the use of chemometric techniques to interrogate full scan MS data, and the progression from discriminating VOC samples of known chemical composition based on full scan MIMS data through to the apportionment of VOC sources measured continuously with a PTR-ToF-MS system operating in a moving vehicle. Lab‐constructed VOC samples of known chemical composition and concentration demonstrated the use of principal component analysis (PCA) to discriminate, and k-nearest neighbours to classify, samples based on normalized full scan MIMS data. Furthermore, multivariate curve resolution-alternating least squares (MCR-ALS) was used to resolve mixtures into molecular component contributions. PCA was also used to discriminate ‘real-world’ VOC mixtures (e.g., woodsmoke VOCs, headspace above aqueous hydrocarbon samples) of unknown chemical composition measured by MIMS. Using vehicle mounted MIMS and PTR-ToF-MS systems, full scan MS data of ambient atmospheric VOCs were collected and PCA was applied to the normalized full scan MS data. A supervised analysis performed PCA on samples collected near known VOC sources, while an unsupervised analysis using PCA followed by cluster analysis was used to identify groups in a continuous, time series PTR-ToF-MS dataset measured between Nanaimo and Crofton, British Columbia (BC). In both the supervised and unsupervised analysis, samples impacted by emissions from different sources (e.g., internal combustion engines, sawmills, composting facilities, pulp mills) were discriminated. With PCA, samples were discriminated based on differences in the observed full scan MS data, however real-world samples are often impacted by multiple VOC sources. MCR-weighted ALS (MCR-WALS) was applied to the continuous, time series PTR-ToF-MS data from three field campaigns on Vancouver Island, BC for source apportionment. Variable selection based on signal-to-noise ratios was used to reduce the mass list while retaining the observed m/z that capture changes in the mixture of VOCs measured, improving model results, and reducing computation time. Both point (e.g., anthropogenic hydrocarbon emissions, pulp mill emissions) and diffuse (e.g., VOCs from forest fire smoke) VOC sources were identified in the data, and were apportioned to determine their contributions to the measured samples. The data analyzed captured fine scale changes in the ambient VOCs present in the air, and geospatial maps of each individual source, and of the source apportionment were used to visualize the distribution of VOC sources across the sampling area. This work represents the first use of MCR-WALS to identify and apportion ambient VOC sources based on continuous PTR-ToF-MS data measured from a moving vehicle. The methods described can be applied to larger scale field campaigns for the source apportionment of VOCs across multiple days to capture diurnal and seasonal variations. Identifying spatial and temporal trends in the sources of VOCs at the regional scale can help to identify pollution ‘hot spots’ and inform evidence-based public policy for improving air quality. / Graduate / 2022-08-17