L'imagerie cérébrale connaît un intérêt grandissant, en tant que phénotype intermédiaire, dans la compréhension du chemin complexe qui relie les gènes à un phénotype comportemental ou clinique. Dans ce contexte, un premier objectif est de proposer des méthodes capables d'identifier la part de variabilité génétique qui explique une certaine part de la variabilité observée en neuroimagerie. Les approches univariées classiques ignorent les effets conjoints qui peuvent exister entre plusieurs gènes ou les covariations potentielles entre régions cérébrales.Notre première contribution a été de chercher à améliorer la sensibilité de l'approche univariée en tirant avantage de la nature multivariée des données génétiques, au niveau local. En effet, nous adaptons l'inférence au niveau du cluster en neuroimagerie à des données de polymorphismes d'un seul nucléotide (SNP), en cherchant des clusters 1D de SNPs adjacents associés à un même phénotype d'imagerie. Ensuite, nous prolongeons cette idée et combinons les clusters de voxels avec les clusters de SNPs, en utilisant un test simple au niveau du "cluster 4D", qui détecte conjointement des régions cérébrale et génomique fortement associées. Nous obtenons des résultats préliminaires prometteurs, tant sur données simulées que sur données réelles.Notre deuxième contribution a été d'utiliser des méthodes multivariées exploratoires pour améliorer la puissance de détection des études d'imagerie génétique, en modélisant la nature multivariée potentielle des associations, à plus longue échelle, tant du point de vue de l'imagerie que de la génétique. La régression Partial Least Squares et l'analyse canonique ont été récemment proposées pour l'analyse de données génétiques et transcriptomiques. Nous proposons ici de transposer cette idée à l'analyse de données de génétique et d'imagerie. De plus, nous étudions différentes stratégies de régularisation et de réduction de dimension, combinées avec la PLS ou l'analyse canonique, afin de faire face au phénomène de sur-apprentissage dû aux très grandes dimensions des données. Nous proposons une étude comparative de ces différentes stratégies, sur des données simulées et des données réelles d'IRM fonctionnelle et de SNPs. Le filtrage univarié semble nécessaire. Cependant, c'est la combinaison du filtrage univarié et de la PLS régularisée L1 qui permet de détecter une association généralisable et significative sur les données réelles, ce qui suggère que la découverte d'associations en imagerie génétique nécessite une approche multivariée. / Brain imaging is increasingly recognised as an interesting intermediate phenotype to understand the complex path between genetics and behavioural or clinical phenotypes. In this context, a first goal is to propose methods to identify the part of genetic variability that explains some neuroimaging variability. Classical univariate approaches often ignore the potential joint effects that may exist between genes or the potential covariations between brain regions. Our first contribution is to improve the sensitivity of the univariate approach by taking advantage of the multivariate nature of the genetic data in a local way. Indeed, we adapt cluster-inference techniques from neuroimaging to Single Nucleotide Polymorphism (SNP) data, by looking for 1D clusters of adjacent SNPs associated with the same imaging phenotype. Then, we push further the concept of clusters and we combined voxel clusters and SNP clusters, by using a simple 4D cluster test that detects conjointly brain and genome regions with high associations. We obtain promising preliminary results on both simulated and real datasets .Our second contribution is to investigate exploratory multivariate methods to increase the detection power of imaging genetics studies, by accounting for the potential multivariate nature of the associations, at a longer range, on both the imaging and the genetics sides. Recently, Partial Least Squares (PLS) regression or Canonical Correlation Analysis (CCA) have been proposed to analyse genetic and transcriptomic data. Here, we propose to transpose this idea to the genetics vs. imaging context. Moreover, we investigate the use of different strategies of regularisation and dimension reduction techniques combined with PLS or CCA, to face the overfitting issues due to the very high dimensionality of the data. We propose a comparison study of the different strategies on both a simulated dataset and a real fMRI and SNP dataset. Univariate selection appears to be necessary to reduce the dimensionality. However, the generalisable and significant association uncovered on the real dataset by the two-step approach combining univariate filtering and L1-regularised PLS suggests that discovering meaningful imaging genetics associations calls for a multivariate approach.

Submitted by Valquíria Barbieri (kikibarbi@hotmail.com) on 2018-05-11T20:59:15Z No. of bitstreams: 1 DISS_2015_Karen Janones da Rocha.pdf: 4327569 bytes, checksum: 1207959001ffdfc221d56edf601d488a (MD5) / Approved for entry into archive by Jordan (jordanbiblio@gmail.com) on 2018-05-24T14:50:15Z (GMT) No. of bitstreams: 1 DISS_2015_Karen Janones da Rocha.pdf: 4327569 bytes, checksum: 1207959001ffdfc221d56edf601d488a (MD5) / Made available in DSpace on 2018-05-24T14:50:15Z (GMT). No. of bitstreams: 1 DISS_2015_Karen Janones da Rocha.pdf: 4327569 bytes, checksum: 1207959001ffdfc221d56edf601d488a (MD5) Previous issue date: 2015-02-06 / CAPES / O objetivo geral do presente estudo foi caracterizar um fragmento secundário de Floresta Estacional Semidecidual localizado em Tapurah-MT, quanto a sua estrutura e composição florestal, verificar a formação de grupos florísticos e, ainda, explorar possíveis relações com o ambiente. Aplicou-se o método de área fixa com conglomerados retangulares de dimensões de 10 x 250 m, foram alocados e medidos cinco conglomerados com cinco subunidades cada de 10 X 50 m. Em cada subunidade amostral foi considerada todas as espécies arbóreas e arbustivas com diâmetro à altura do peito (DAP) superior ou igual a 10 cm. A composição florística foi analisada quanto ao número de famílias, gêneros e espécies botânicas encontradas no levantamento do componente arbóreo e a suficiência amostral do levantamento florístico foi verificada pelo procedimento Bootstrap. A similaridade florística entre as subunidades amostrais foi obtida através do Índice de Jaccard e a diversidade de espécies nas subunidades amostrais foi medida pelo Índice de Shannon (H’) e pelo Índice de Equabilidade de Pielou (J’). A caracterização da estrutura horizontal da vegetação foi feita a partir dos parâmetros fitossociológicos e a estrutura diamétrica pelo procedimento de Spiegel. A presença de grupos florísticos foi verificada pelo método de associação das espécies e o número de grupos foi estabelecido pelo coeficiente de concordância de Kendall, onde para cada grupo florístico foi analisada a estrutura horizontal, o padrão de distribuição espacial das espécies pelo índice de Payandeh e a estrutura diamétrica dos indivíduos pelo procedimento de Spiegel. A construção da matriz das variáveis edáficas foi realizada através de uma análise preliminar para identificar variáveis semelhantes entre as subunidades amostrais, as quais não apresentaram influência foram retiradas. A correlação entre os dados de vegetação e dados ambientais foi realizada por meio da Análise de Correlação Canônica, que permitiu confirmar se os nutrientes do solo influenciam na presença das espécies e pela Análise de Redundância Canônica para avaliar quais as variáveis ambientais apresentaram maior influência sobre os indivíduos. No fragmento foi verificada uma alta variabilidade florística e estrutural, que pode ser explicada pelos históricos de perturbação local a que este fragmento foi submetido no passado. De uma forma geral, a vegetação corresponde a de florestas secundárias jovens e apresenta uma comunidade estável e autorregenerativa, além de preservar características da estrutura original. Através da análise de agrupamento foi verificado que as características autoecológicas das espécies assim como os DAP’s médios de cada espécie foram os principais responsáveis pela associação e similaridade entre os grupos. Também foi verificada que apesar das perturbações no ambiente que salientam a saturação do sítio florestal, o fragmento está se recuperando. A heterogeneidade das variáveis edáficas relacionadas influencia no comportamento florístico-estrutural do fragmento secundário de Floresta Estacional Semidecidual. Sendo, as espécies das famílias Fabaceae, Lauraceae, Moraceae e Vochysiaceae as mais influentes para o presente estudo. Destacando a Qualea paraensis Ducke quanto à importância ecológica e a sua adaptabilidade ao ambiente. / The general objective of this study was to characterize a secondary fragment of semideciduous forest located in Tapurah-MT, as its structure and forest composition, verify the formation of floristic groups and also explore possible relationships with the environment. Was applied the fixed area method with five rectangular clusters of 10 x 250 m, they were measured and allocated to five subunits of 10 x 50 m each. In each sample subunit was considered all tree and shrub species with diameter at breast height (DBH) greater than or equal to 10 cm. The floristic composition was analyzed for the number of families, genera and plant species found in the survey of the tree component and sampling sufficiency of floristic survey was tested by bootstrap procedure. The floristic similarity between plots was obtained through the Jaccard index and the diversity of species in the sample subunits was measured by the Shannon Index (H') and equability index of Pielou (J'). The characterization of the horizontal structure of vegetation was made from the phytosociological parameters and the structure diameter by Spiegel procedure. The presence of floristic groups was verified by the association method of species and the number of groups was established by Kendall concordance coefficient, where for each floristic group was analyzed horizontal structure, the pattern of spatial distribution of species by Payandeh index and the diameter distribution of individuals by Spiegel procedure. The construction of the matrix of the soil variables was performed in a preliminary analysis to identify variables similar to the sample subunits, which showed no influence were dropped. The correlation between the data of vegetation and environmental data was performed by Canonical Correlation Analysis, which allowed confirm that soil nutrients influence the presence of the species and the Canonical Redundancy Analysis to evaluate which environmental variables had the greatest influence on the individuals. In the studied fragment was observed high variability floristic and structural, which can be explained by historical local disturbance that this fragment was in the past. In general, the vegetation corresponds to young secondary forests and presents a stable and self-regenerative community, besides preserving the original structure characteristics. Through cluster analysis it was found that the ecological self characteristics of the species as well as the average DBH of each species were mainly responsible for the association and similarity between the groups. We also observed that despite the disturbances in the environment that emphasize the saturation of forest site, the fragment is recovering. The soil variables heterogeneity related influence the floristic-structural behavior of the secondary fragment of semideciduous forest. Being, the species of the families: Fabaceae, Lauraceae, Moraceae and Vochysiaceae the most influential for the present study. Highlighting the Qualea paraensis Ducke about its ecological significance and adaptability to the environment.

線性迴歸模型中共線性的問題是導致模型不適當的重大原因之一。共線性 的存在不止會影響到參數的估計，使參數的變異變大，還會妨礙我們評估 自變數對模型重要性的能力，甚至會使我們忽略或去除掉重要的自變數。 而變異膨脹因子是診斷線性迴歸模型共線性問題時常用而有效的方法之一 ，但它只是考慮單一自變數的情況。本文則對於模型同時加入一組自變數 時影響原模型共線性的問題，先推導出廣義的判定係數，再利用它推導出 變異膨脹矩陣。再應用這個變異膨脹矩陣發展出六個準則，使得變異膨脹 矩陣有一個單一的指標來對模型的共線性做診斷。最後並以一個例子以實 際的數據，用六個準則對不同的模型做診斷，並嘗試找出各準則的指標。

<p>A Real-Time Brain-Computer Interface is a technical system classifying increased or decreased brain activity in Real-Time between different body movements, actions performed by a person. Focus in this thesis will be on testing algorithms and settings, finding the initial time interval and how increased activity in the brain can be distinguished and satisfyingly classified. The objective is letting the system give an output somewhere within 250ms of a thought of an action, which will be faster than a persons reaction time. </p><p>Algorithms in the preprocessing were Blind Signal Separation and the Fast Fourier Transform. With different frequency and time interval settings the algorithms were tested on an offline Electroencephalographic data file based on the "Ten Twenty" Electrode Application System, classified using an Artificial Neural Network. </p><p>A satisfying time interval could be found between 125-250ms, but more research is needed to investigate that specific interval. A reduction in frequency resulted in a lack of samples in the sample window preventing the algorithms from working properly. A high frequency is therefore proposed to help keeping the sample window small in the time domain. Blind Signal Separation together with the Fast Fourier Transform had problems finding appropriate correlation using the Ten-Twenty Electrode Application System. Electrodes should be placed more selectively at the parietal lobe, in case of requiring motor responses.</p>

A Real-Time Brain-Computer Interface is a technical system classifying increased or decreased brain activity in Real-Time between different body movements, actions performed by a person. Focus in this thesis will be on testing algorithms and settings, finding the initial time interval and how increased activity in the brain can be distinguished and satisfyingly classified. The objective is letting the system give an output somewhere within 250ms of a thought of an action, which will be faster than a persons reaction time. Algorithms in the preprocessing were Blind Signal Separation and the Fast Fourier Transform. With different frequency and time interval settings the algorithms were tested on an offline Electroencephalographic data file based on the "Ten Twenty" Electrode Application System, classified using an Artificial Neural Network. A satisfying time interval could be found between 125-250ms, but more research is needed to investigate that specific interval. A reduction in frequency resulted in a lack of samples in the sample window preventing the algorithms from working properly. A high frequency is therefore proposed to help keeping the sample window small in the time domain. Blind Signal Separation together with the Fast Fourier Transform had problems finding appropriate correlation using the Ten-Twenty Electrode Application System. Electrodes should be placed more selectively at the parietal lobe, in case of requiring motor responses.

The development of modern information technology has enabled collecting data of unprecedented size and complexity. Examples include web text data, microarray & proteomics, and data from scientific domains (e.g., meteorology). To learn from these high dimensional and complex data, traditional machine learning techniques often suffer from the curse of dimensionality and unaffordable computational cost. However, learning from large-scale high-dimensional data promises big payoffs in text mining, gene analysis, and numerous other consequential tasks. Recently developed sparse learning techniques provide us a suite of tools for understanding and exploring high dimensional data from many areas in science and engineering. By exploring sparsity, we can always learn a parsimonious and compact model which is more interpretable and computationally tractable at application time. When it is known that the underlying model is indeed sparse, sparse learning methods can provide us a more consistent model and much improved prediction performance. However, the existing methods are still insufficient for modeling complex or dynamic structures of the data, such as those evidenced in pathways of genomic data, gene regulatory network, and synonyms in text data. This thesis develops structured sparse learning methods along with scalable optimization algorithms to explore and predict high dimensional data with complex structures. In particular, we address three aspects of structured sparse learning: 1. Efficient and scalable optimization methods with fast convergence guarantees for a wide spectrum of high-dimensional learning tasks, including single or multi-task structured regression, canonical correlation analysis as well as online sparse learning. 2. Learning dynamic structures of different types of undirected graphical models, e.g., conditional Gaussian or conditional forest graphical models. 3. Demonstrating the usefulness of the proposed methods in various applications, e.g., computational genomics and spatial-temporal climatological data. In addition, we also design specialized sparse learning methods for text mining applications, including ranking and latent semantic analysis. In the last part of the thesis, we also present the future direction of the high-dimensional structured sparse learning from both computational and statistical aspects.

Ce mémoire traite de l'acoustique de la clarinette et du rôle de l'anche, résumant des travaux menés entre 2001 et 2018 sur divers sujets :I) Étude de modèles analytiques élémentaire focalisée sur : 1) le rôle des pertes. 2) les cartes itérées, mettant en évidence divers régimes de fonctionnement, utiles aussi pour la pédagogie instrumentale. II) Étude de caractérisation des anches : 1) Étude dynamique des résonances de l'anche réalisée par holographie. Elle conduit à un modèle de matériau viscoélastique expliquant certaines différences observées dans les fréquences des 15 premiers modes de l'anche. 2) Étude statique des caractéristiques mécaniques et aérauliques de l'excitateur (anche+bec+lèvre). La méthode mesure précisément la quantité d'air entrant dans l'instrument en fonction de la pression de lèvre et d'air. III) Synthèse sonore par modèle physique en temps réel : 1) Modélisation mécanique et aéraulique de l'anche, d'après mesure. Le modèle de ressort raidissant non linéaire proposé autorise une simulation dynamique efficace. 2) Estimation modale de l'impédance d'entrée (mesurée) des instruments à vent. On montre les techniques de conception de filtres numériques précis et passifs à toute fréquence. 3) Modélisation et simulation instruments à vent au moyen de guide-ondes, par estimation modale, implémentée dans un logiciel en C++. IV) Une étude de jouabilité d'un panel de 40 anches par analyse canonique des corrélations révèle des liens statistiquement solides entre mesures physiques, évaluations subjectives et synthèse sonore. Elle autorise une caractérisation des anches pouvant être réalisé par le fabricant, selon au moins 4 facteurs indépendants. / This thesis deals with the acoustics of the clarinet and the role of the reed, summarizing studies carried out between 2001 and 2018 on various topics : I) Study of elementary analytical models, focused on 1) role of losses. 2) iterated maps, highlighting various operating regimes, which are also useful for the instrumental pedagogy. II) Reed characterization study : 1) Dynamic study of the reed resonances, by holography. It leads to a model of viscoelastic material explaining some differences observed in the frequencies of the first 15 modes of the reed. 2) Static study of the mechanical and aeraulic characteristics of the exciter (reed + mouthpiece + lip). The method accurately measures the airflow entering the instrument as a function of lip and air pressure. III) Sound synthesis by physical model in real time : 1) Mechanical and aeraulic modeling of the reed, according to measurements. The proposed nonlinear stiffening spring model allows for an efficient dynamic simulation. 2) Modal estimation of the (measured) input impedance of wind instruments. Design techniques for accurate digital filters, passive at any frequency, are described. 3) Modal estimation and simulation of wind instruments by waveguides, implemented in C ++ software. IV) A playability study of a panel of 40 reeds by canonical correlation analysis reveals statistically strong links between physical measurements, subjective evaluations and sound synthesis. It allows a characterization of the reeds that can be made by the manufacturer, according to at least 4 independent factors.

Over the years, extensive research has been devoted to the study of people's head pose due to its relevance in security, human-computer interaction, advertising as well as cognitive, neuro and behavioural psychology. One of the main goals of this thesis is to estimate people's 3D head orientation as they freely move around in naturalistic settings such as parties, supermarkets etc. Head pose classification from surveillance images acquired with distant, large field-of-view cameras is difficult as faces captured are at low-resolution with a blurred appearance. Also labelling sufficient training data for headpose estimation in such settings is difficult due to the motion of targets and the large possible range of head orientations. Domain adaptation approaches are useful for transferring knowledge from the training source to the test target data having different attributes, minimizing target data labelling efforts in the process. This thesis examines the use of transfer learning for efficient multi-view head pose classification. Relationship between head pose and facial appearance from many labelled examples corresponding to the source data is learned initially. Domain adaptation techniques are then employed to transfer this knowledge to the target data. The following three challenging situations is addressed (I) ranges of head poses in the source and target images is different, (II) where source images capture a stationary person while target images capture a moving person with varying facial appearance due to changing perspective, scale and (III) a combination of (I) and (II). All proposed transfer learning methods are sufficiently tested and benchmarked on a new compiled dataset DPOSE for headpose classification. This thesis also looks at a novel signature representation for describing object sets for covariance descriptors, Covariance Profiles (CPs). CP is well suited for representing a set of similarly related objects. CPs posit that the covariance matrices, pertaining to a specific entity, share the same eigen-structure. Such a representation is not only compact but also eliminates the need to store all the training data. Experiments on images as well as videos for applications such as object-track clustering and headpose estimation is shown using CP. In the second part, Human-gaze for interest point detection for video retargeting is explored. Regions in video streams attracting human interest contribute significantly to human understanding of the video. Being able to predict salient and informative Regions of Interest (ROIs) through a sequence of eye movements is a challenging problem. This thesis proposes an interactive human-in-loop framework to model eye-movements and predicts visual saliency in yet-unseen frames. Eye-tracking and video content is used to model visual attention in a manner that accounts for temporal discontinuities due to sudden eye movements, noise and behavioural artefacts. Gaze buffering, for eye-gaze analysis and its fusion with content based features is proposed. The method uses eye-gaze information along with bottom-up and top-down saliency to boost the importance of image pixels. Our robust visual saliency prediction is instantiated for content aware Video Retargeting.

This thesis deals with differences between individual states of the United States from the demographic point of view, and searches for causes of these differences. United States are composed of several disparate areas, which are different from each other in their location, size, number and composition of the population, and historical evolution. All this affects the demographic characteristics of those territorial units. Probably the main cause of differentiation of individual states is racial composition, since the intensity of demographic events is specific to each race or ethnicity. The states differ in levels of fertility, mortality, marriage or education. To confirm these assumptions, the statistical method canonical correlation was used. Using cluster analysis has revealed that there are groups of states that have similar demographic profile. Most notably, it shows the influence of ethnic and racial composition in the south of the United States, where the highest proportion of populations is composed by black race and hispanic origin. The differentiation of levels of infant mortality and life expectancy are mainly influenced. Key words: United States of America, race, ethnicity, black population, hispanic population, total fertility rate, life expectancy, canonical correlation, cluster analysis

This thesis investigates the extension of the well-known canonical correlation analysis for random elements on abstract real measurable Hilbert spaces. One focus is on the application of this extension to the calculation of information-theoretical quantities on finite time intervals. Analytical approaches for the calculation of the mutual information and the information density between Gaussian distributed random elements on arbitrary real measurable Hilbert spaces are derived. With respect to mutual information, the results obtained are comparable to [4] and [1] (Baker, 1970, 1978). They can also be seen as a generalization of earlier findings in [20] (Gelfand and Yaglom, 1958). In addition, some of the derived equations for calculating the information density, its characteristic function and its n-th central moments extend results from [45] and [44] (Pinsker, 1963, 1964). Furthermore, explicit examples for the calculation of the mutual information, the characteristic function of the information density as well as the n-th central moments of the information density for the important special case of an additive Gaussian channel with Gaussian distributed input signal with rational spectral density are elaborated, on the one hand for white Gaussian noise and on the other hand for Gaussian noise with rational spectral density. These results extend the corresponding concrete examples for the calculation of the mutual information from [20] (Gelfand and Yaglom, 1958) as well as [28] and [29] (Huang and Johnson, 1963, 1962).:Kurzfassung Abstract Notations Abbreviations 1 Introduction 1.1 Software Used 2 Mathematical Background 2.1 Basic Notions of Measure and Probability Theory 2.1.1 Characteristic Functions 2.2 Stochastic Processes 2.2.1 The Consistency Theorem of Daniell and Kolmogorov 2.2.2 Second Order Random Processes 2.3 Some Properties of Fourier Transforms 2.4 Some Basic Inequalities 2.5 Some Fundamentals in Functional Analysis 2.5.1 Hilbert Spaces 2.5.2 Linear Operators on Hilbert Spaces 2.5.3 The Fréchet-Riesz Representation Theorem 2.5.4 Adjoint and Compact Operators 2.5.5 The Spectral Theorem for Compact Operators 3 Mutual Information and Information Density 3.1 Mutual Information 3.2 Information Density 4 Probability Measures on Hilbert Spaces 4.1 Measurable Hilbert Spaces 4.2 The Characteristic Functional 4.3 Mean Value and Covariance Operator 4.4 Gaussian Probability Measures on Hilbert Spaces 4.5 The Product of Two Measurable Hilbert Spaces 4.5.1 The Product Measure 4.5.2 Cross-Covariance Operator 5 Canonical Correlation Analysis on Hilbert Spaces 5.1 The Hellinger Distance and the Theorem of Kakutani 5.2 Canonical Correlation Analysis on Hilbert Spaces 5.3 The Theorem of Hájek and Feldman 6 Mutual Information and Information Density Between Gaussian Measures 6.1 A General Formula for Mutual Information and Information Density for Gaussian Random Elements 6.2 Hadamard’s Factorization Theorem 6.3 Closed Form Expressions for Mutual Information and Related Quantities 6.4 The Discrete-Time Case 6.5 The Continuous-Time Case 6.6 Approximation Error 7 Additive Gaussian Channels 7.1 Abstract Channel Model and General Definitions 7.2 Explicit Expressions for Mutual Information and Related Quantities 7.2.1 Gaussian Random Elements as Input to an Additive Gaussian Channel 8 Continuous-Time Gaussian Channels 8.1 White Gaussian Channels 8.1.1 Two Simple Examples 8.1.2 Gaussian Input with Rational Spectral Density 8.1.3 A Method of Youla, Kadota and Slepian 8.2 Noise and Input Signal with Rational Spectral Density 8.2.1 Again a Method by Slepian and Kadota Bibliography / Diese Arbeit untersucht die Erweiterung der bekannten kanonischen Korrelationsanalyse (canonical correlation analysis) für Zufallselemente auf abstrakten reellen messbaren Hilberträumen. Ein Schwerpunkt liegt dabei auf der Anwendung dieser Erweiterung zur Berechnung informationstheoretischer Größen auf endlichen Zeitintervallen. Analytische Ansätze für die Berechnung der Transinformation und der Informationsdichte zwischen gaußverteilten Zufallselementen auf beliebigen reelen messbaren Hilberträumen werden hergeleitet. Bezüglich der Transinformation sind die gewonnenen Resultate vergleichbar zu [4] und [1] (Baker, 1970, 1978). Sie können auch als Verallgemeinerung früherer Erkenntnisse aus [20] (Gelfand und Yaglom, 1958) aufgefasst werden. Zusätzlich erweitern einige der hergeleiteten Formeln zur Berechnung der Informationsdichte, ihrer charakteristischen Funktion und ihrer n-ten zentralen Momente Ergebnisse aus [45] und [44] (Pinsker, 1963, 1964). Weiterhin werden explizite Beispiele für die Berechnung der Transinformation, der charakteristischen Funktion der Informationsdichte sowie der n-ten zentralen Momente der Informationsdichte für den wichtigen Spezialfall eines additiven Gaußkanals mit gaußverteiltem Eingangssignal mit rationaler Spektraldichte erarbeitet, einerseits für gaußsches weißes Rauschen und andererseits für gaußsches Rauschen mit einer rationalen Spektraldichte. Diese Ergebnisse erweitern die entsprechenden konkreten Beispiele zur Berechnung der Transinformation aus [20] (Gelfand und Yaglom, 1958) sowie [28] und [29] (Huang und Johnson, 1963, 1962).:Kurzfassung Abstract Notations Abbreviations 1 Introduction 1.1 Software Used 2 Mathematical Background 2.1 Basic Notions of Measure and Probability Theory 2.1.1 Characteristic Functions 2.2 Stochastic Processes 2.2.1 The Consistency Theorem of Daniell and Kolmogorov 2.2.2 Second Order Random Processes 2.3 Some Properties of Fourier Transforms 2.4 Some Basic Inequalities 2.5 Some Fundamentals in Functional Analysis 2.5.1 Hilbert Spaces 2.5.2 Linear Operators on Hilbert Spaces 2.5.3 The Fréchet-Riesz Representation Theorem 2.5.4 Adjoint and Compact Operators 2.5.5 The Spectral Theorem for Compact Operators 3 Mutual Information and Information Density 3.1 Mutual Information 3.2 Information Density 4 Probability Measures on Hilbert Spaces 4.1 Measurable Hilbert Spaces 4.2 The Characteristic Functional 4.3 Mean Value and Covariance Operator 4.4 Gaussian Probability Measures on Hilbert Spaces 4.5 The Product of Two Measurable Hilbert Spaces 4.5.1 The Product Measure 4.5.2 Cross-Covariance Operator 5 Canonical Correlation Analysis on Hilbert Spaces 5.1 The Hellinger Distance and the Theorem of Kakutani 5.2 Canonical Correlation Analysis on Hilbert Spaces 5.3 The Theorem of Hájek and Feldman 6 Mutual Information and Information Density Between Gaussian Measures 6.1 A General Formula for Mutual Information and Information Density for Gaussian Random Elements 6.2 Hadamard’s Factorization Theorem 6.3 Closed Form Expressions for Mutual Information and Related Quantities 6.4 The Discrete-Time Case 6.5 The Continuous-Time Case 6.6 Approximation Error 7 Additive Gaussian Channels 7.1 Abstract Channel Model and General Definitions 7.2 Explicit Expressions for Mutual Information and Related Quantities 7.2.1 Gaussian Random Elements as Input to an Additive Gaussian Channel 8 Continuous-Time Gaussian Channels 8.1 White Gaussian Channels 8.1.1 Two Simple Examples 8.1.2 Gaussian Input with Rational Spectral Density 8.1.3 A Method of Youla, Kadota and Slepian 8.2 Noise and Input Signal with Rational Spectral Density 8.2.1 Again a Method by Slepian and Kadota Bibliography