Dissertations / Theses on the topic 'Entropic uncertainty'

The uncertainty principle lies at the heart of quantum physics. It exhibits one of the key divergences between a classical and a quantum system: it is impossible to define a quantum state for which the values of two observables that do not commute are simultaneously specified with infinite precision. A paradigmatic example is given by Heisenberg’s original formulation of the uncertainty principle expressed in terms of variances of two canonically-conjugate variables, such as position x and momentum p, which was later generalized to a symplectic-invariant form by Schrödinger and Robertson. A different kind of uncertainty relations, originated by Białynicki-Birula and Mycielski, again for canonically-conjugate variables, relies on Shannon entropy instead of variances as a measure of uncertainty. In this thesis, we suggest several improvements of these entropic uncertainty relations and highlight the fact that they are better formulated in terms of entropy power, a notion borrowed from the information theory of real-valued signals. Our first novel entropic uncertainty relation takes x-p correlations into account and is consequently saturated by all pure Gaussian states in an arbitrary number of modes, improving on the original formulation by Białynicki-Birula and Mycielski. Our second main result is the derivation of an entropic uncertainty relation that holds for any n-tuples of not-necessarily canonically conjugate variables based on the matrix of their commutators. We then define a general form of the entropic uncertainty principle that combines both previous results. It expresses the incompatibility between two arbitrary variable n-uples and is saturated by all pure Gaussian states. Interestingly, we can also deduce from it the most general form of the Robertson uncertainty relation based on the covariance matrix of n variables.This line of research underlines the interest of defining an entropic uncertainty relation that is intrinsically invariant under symplectic transformations. Then, as a first attempt to reach this goal, we conjecture a symplectic-invariant uncertainty relation that is based on the joint differential entropy of the Wigner function. This conjecture is, however, only legitimate for states with a non-negative Wigner function. We also suggest a complex extension of this so-called Wigner entropy, which could provide the way towards an extension (and proof) of the above conjecture for all states. As a second attempt, we introduce the notion of multi-copy uncertainty observables, exploiting a connection with the algebra of angular momenta. Expressing the positivity of the variance of our multi-copy observable coincides with the Schrödinger-Robertson uncertainty relation, which suggests that the discrete Shannon entropy of such an uncertainty observable provides a new symplectic-invariant measure of uncertainty.Currently available separability criteria for continuous-variable systems imply a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leave many entangled non-Gaussian states undetected. In this thesis, we introduce two improved separability criteria that enable a stronger entanglement detection. The first improved condition is based on the knowledge of an additional parameter, namely the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations by Mandilara and Cerf. We exhibit families of non- Gaussian entangled states whose entanglement remains undetected by the Duan- Simon criterion. The second improved separability criterion is based on our improved entropic uncertainty relation that takes x-p correlations into account, and has the main advantage over the one proposed by Walborn et al. that it does not require any optimization procedure.
Le principe d’incertitude se situe au cœur de la physique quantique. Il représente l’une des différences majeures entre des systèmes classiques et quantiques, soit qu’il est impossible de définir un état quantique pour lequel deux observables qui ne commutent pas auraient des valeurs spécifiées simultanément et avec une précision infinie. La formulation originale du principe d’incertitude est due à Heisenberg et est exprimée en termes des variances de deux variables canoniquement conjuguées, telles que la position x et l’impulsion p. Cela fut par la suite généralisé par Schrödinger et Robertson qui ont donné au principe d’incertitude une forme invariante sous transformations symplectiques. Si l’incertitude est mesurée à l’aide de l’entropie différentielle de Shannon plutôt que des variances, il est alors possible de définir d’autres types de relations d’incertitude. Originellement introduites par Białynicki-Birula et Mycielski, elles expriment également l’incompatibilité entre deux variables canoniquement conjuguées. Dans cette thèse, nous proposons différentes améliorations de ces relations d’incertitude entropiques et mettons particulièrement l’accent sur le fait qu’elles s’expriment mieux sous forme de puissances entropiques, une notion empruntée à la théorie de l’information. En premier lieu, nous introduisons une nouvelle relation d’incertitude entropique qui tient compte des corrélations x-p et qui est par conséquent saturée par tous les états purs Gaussiens, ce qui représente une amélioration par rapport à la formulation originale de Białynicki- Birula et Mycielski. En second lieu, nous dérivons une relation d’incertitude entropique valide pour tous les n-uplets de variables non nécessairement canoniquement conjuguées et basée sur la matrice de leurs commutateurs. Nous définissons ensuite une forme plus générale du principe d’incertitude entropique qui combine les deux résultats précédents. Il exprime l’incompatibilité entre deux n-uplets arbitraires de variables et est saturé par tous les états purs Gaussiens. Notons que de ce principe d’incertitude entropique, nous pouvons déduire la forme la plus générale de la relation d’incertitude de Robertson, basée sur la matrice de covariance de n variables. Les résultats précédents soulignent un des points essentiels de notre axe de recherche: définir une relation d’incertitude entropique intrinsèquement invariante sous trans- formations symplectiques. Afin d’atteindre cet objectif, notre première tentative est de conjecturer une relation d’incertitude — invariante sous transformations symplectiques — basée sur l’entropie différentielle jointe de la fonction de Wigner. Cette conjecture n’est cependant légitime que pour des états décrits par une fonction de Wigner non-négative. Nous proposons aussi une extension complexe de cette en- tropie dite entropie de Wigner, qui pourrait ouvrir la voie vers une extension (et une preuve) de la conjecture proposée ci-dessus qui serait alors valide pour tous les états quantiques. Comme seconde tentative, en exploitant une connexion avec l’algèbre des moments angulaires, nous introduisons la notion d’observables d’incertitude agissant sur plusieurs copies d’un état. Exprimer la positivité de la variance de notre observable coïncide avec la relation d’incertitude de Schrödinger-Robertson, ce qui suggère que l’entropie discrète de Shannon d’une telle observable fournit une nouvelle mesure de l’incertitude. Cette relation d’incertitude est invariante sous transformations symplectiques.Les critères de séparabilité actuellement disponibles pour les variables continues donnent une condition nécessaire et suffisante afin qu’un état Gaussien bimodal soit séparable, mais laissent de nombreux états intriqués non-Gaussiens non détectés. Dans cette thèse, nous introduisons deux nouveaux critères de séparabilité qui permettent une meilleure détection de l’intrication. La première nouvelle condition est basée sur la connaissance d’un paramètre supplémentaire, à savoir le degré de Gaussianité de l’état, et exploite une connexion avec les relations d’incertitude de Mandilara et Cerf bornées par ce degré de Gaussianité. En particulier, nous donnons l’exemple de familles d’états intriqués non Gaussiens dont l’intrication est détectée par notre critère, mais pas par celui de Duan-Simon. Le second critère de séparabil- ité entropique que nous proposons est basé sur notre nouvelle relation d’incertitude entropique qui tient compte des corrélations x-p. Son principal avantage par rapport au critère de Walborn et al. est de ne nécessiter aucune procédure d’optimisation.
Doctorat en Sciences de l'ingénieur et technologie
info:eu-repo/semantics/nonPublished

Information entropy parameter has been applied for an expression of the result of data assessment and it is supplemented by an index of sample of data that was evaluated out of set of information. A modelling system and software have been developed that can be used and are used for practical processing of measurement data for circular raster scales.
Duomenų įverčiui išreikšti pritaikytas informacinės entropijos parametras pateiktoje rezultato išraiškoje yra papildytas rodikliu apie duomenų imtį, kuri buvo įvertinta iš visos šį objektą charakterizuojančių duomenų aibės. Sukurta modeliavimo sistema ir programinė įranga gali būti naudojama didelio skaičiaus nežinomųjų lygtims spręsti, o praktikoje naudojama rastrinių skalių matavimo duomenims apdoroti.

This paper examines the statistical mechanical and thermodynamical consequences of variable phase-space volume element $h_I=?igtriangleup x_i?igtriangleup p_i$. Varying $h_I$ leads to variations in the amount of measured entropy of a system but the maximum entropy remains constant due to the uncertainty principle. By taking $h_u ightarrow 0^+$ an infinite unobservable entropy is attained leading to an infinite unobservable energy per particle and an unobservable chemical equilibrium between all particles. The amount of heat fluxing though measurement apparatus is formulated as a function of $h_I$ for systems in steady state equilibrium as well as the number of measured particles or sub-particles so any system can be described as unitary or composite in number. Some example systems are given using variable $h_I$.

Space missions increasingly require sophisticated guidance, navigation and control algorithms, the development of which is reliant on verification and validation (V&V) techniques to ensure mission safety and success. A crucial element of V&V is the assessment of control system robust performance in the presence of uncertainty. In addition to estimating average performance under uncertainty, it is critical to determine the worst case performance. Industrial V&V approaches typically employ mu-analysis in the early control design stages, and Monte Carlo simulations on high-fidelity full engineering simulators at advanced stages of the design cycle. While highly capable, such techniques present a critical gap between pessimistic worst case estimates found using analytical methods, and the optimistic outlook often presented by Monte Carlo runs. Conservative worst case estimates are problematic because they can demand a controller redesign procedure, which is not justified if the poor performance is unlikely to occur. Gaining insight into the probability associated with the worst case performance is valuable in bridging this gap. It should be noted that due to the complexity of industrial-scale systems, V&V techniques are required to be capable of efficiently analysing non-linear models in the presence of significant uncertainty. As well, they must be computationally tractable. It is desirable that such techniques demand little engineering effort before each analysis, to be applied widely in industrial systems. Motivated by these factors, this thesis proposes and develops an efficient algorithm, based on the cross entropy simulation method. The proposed algorithm efficiently estimates the probabilities associated with various performance levels, from nominal performance up to degraded performance values, resulting in a curve of probabilities associated with various performance values. Such a curve is termed the probability profile of performance (PPoP), and is introduced as a tool that offers insight into a control system's performance, principally the probability associated with the worst case performance. The cross entropy-based robust performance analysis is implemented here on various industrial systems in European Space Agency-funded research projects. The implementation on autonomous rendezvous and docking models for the Mars Sample Return mission constitutes the core of the thesis. The proposed technique is implemented on high-fidelity models of the Vega launcher, as well as on a generic long coasting launcher upper stage. In summary, this thesis (a) develops an algorithm based on the cross entropy simulation method to estimate the probability associated with the worst case, (b) proposes the cross entropy-based PPoP tool to gain insight into system performance, (c) presents results of the robust performance analysis of three space industry systems using the proposed technique in conjunction with existing methods, and (d) proposes an integrated template for conducting robust performance analysis of linearised aerospace systems.

Cooperative heuristics have traditionally been researched through the lens of standard dual-process models of cognition and from the perspective of evolutionary psychology. Despite the popularity of these approaches, research on intuitive versus extensional processing falls short in its endeavor to methodologically quantify heuristic processing and to empirically validate existing theories of social evaluation. Furthermore, several conceptualizations of the term heuristic have been proposed in the social psychology literature, leading to a lack of consensus on how cooperative heuristics function. to address these issues, the current study proposes a novel method for quantifying heuristic cognition. We propose a Bayesian cognition model of heuristics based on the free energy principle and present a framework for defining heuristics as Bayesian priors. to test our model, we ran an experiment on Amazon Mechanical Turk and used a modified version of the Prisoner’s Dilemma game. Overall, the results of experiment supported our theoretical predictions and our quantitative model of cooperative heuristics. Additionally, we found evidence to suggest that men and women respond differently to social uncertainty in cooperative exchanges.

This thesis is concerned with scheduling the use of resources, or allocating resources, so as to meet future demands for the entities produced by the resources. We consider applications in mobile communications such as scheduling users' transmissions so that the amount of transmitted information is maximized, and scenarios in the manufacturing industry where the task is to distribute work among production units so as to minimize the number of missed orders.

The allocation decisions are complicated by a lack of information concerning the future demand and possibly also about the capacities of the available resources. We therefore resort to using probability theory and the maximum entropy principle as a means for making rational decisions under uncertainty.

By using probabilities interpreted as a reasonable degree of belief, we find optimum decision rules for the manufacturing problem, bidding under uncertainty in a certain type of auctions, scheduling users in communications with uncertain channel qualities and uncertain arrival rates, quantization of channel information, partitioning bandwidth between interfering and non-interfering areas in cellular networks, hand-overs and admission control. Moreover, a new method for making optimum approximate Bayesian inference is introduced.

We further discuss reasonable optimization criteria for the mentioned applications, and provide an introduction to the topic of probability theory as an extension to two-valued logic. It is argued that this view unifies a wide range of resource-allocation problems, and we discuss various directions for further research.

In this thesis, we consider a robust state estimation problem for discrete-time, homogeneous, first-order, finite-state finite-alphabet hidden Markov models (HMMs). Based on Kolmogorov's Theorem on the existence of a process, we first present the Kolmogorov model for the HMMs under consideration. A new change of measure is introduced. The statistical properties of the Kolmogorov representation of an HMM are discussed on the canonical probability space. A special Kolmogorov measure is constructed. Meanwhile, the ergodicity of two expanded Markov chains is investigated. In order to describe the uncertainty of HMMs, we study probability distance problems based on the Kolmogorov model of HMMs. Using a change of measure technique, the relative entropy and the relative entropy rate as probability distances between HMMs, are given in terms of the HMM parameters. Also, we obtain a new expression for a probability distance considered in the existing literature such that we can use an information state method to calculate it. Furthermore, we introduce regular conditional relative entropy as an a posteriori probability distance to measure the discrepancy between HMMs when a realized observation sequence is given. A representation of the regular conditional relative entropy is derived based on the Radon-Nikodym derivative. Then a recursion for the regular conditional relative entropy is obtained using an information state method. Meanwhile, the well-known duality relationship between free energy and relative entropy is extended to the case of regular conditional relative entropy given a sub-[special character]-algebra. Finally, regular conditional relative entropy constraints are defined based on the study of the probability distance problem. Using a Lagrange multiplier technique and the duality relationship for regular conditional relative entropy, a finite horizon robust state estimator for HMMs with regular conditional relative entropy constraints is derived. A complete characterization of the solution to the robust state estimation problem is also presented.

Stochastic (or random) processes are inherent to numerous fields of human endeavour including engineering, science, and business and finance. This thesis presents multiple novel methods for quickly detecting and estimating uncertainties in several important classes of stochastic processes. The significance of these novel methods is demonstrated by employing them to detect aircraft manoeuvres in video signals in the important application of autonomous mid-air collision avoidance.

The key objective of this thesis is the outline of an alternative capital market modeling framework, the Volatility Asset Pricing Model, VAPM, inspired by the innovative dual approach of Mandelbrot and Hudson using the method based on synthesis of two seemingly antagonistic factors -- the volatility of market prices and their serial dependence determining the capital markets' dynamics. The pilot tests of this model in various periods using the market index as well as a portfolio of selected securities delivered generally satisfactory results. Firstly, the work delivers a brief recapitulation regarding the concepts of a consumer/investor choice under general conditions of hypothetical certainty. Secondly, this outline is then followed by a description of the "classical" methodologies in the risky environment of uncertainty, with assessment of their corresponding key models, i.e. the CAPM, SIM, MIM, APTM, etc., notwithstanding results of the related testing approaches. Thirdly, this assessment is based on evaluation of the underlying doctrine of Efficient Market Hypothesis in relation to the so called Random Walk Model. Fourthly, in this context the work also offers a brief exposure to a few selected tests of these contraversial concepts. Fifthly, the main points of conteporary approaches such as the Fractal Dimension and the Hurst Exponent in the dynamic framework of information entropy are subsequently described as the theoretical tools leading to development of the abovementioned model VAPM. The major contribution of this thesis is considered its attempt to apply the abovementioned concepts in practice, with the intention to possibly inspire a further analytical research.

Many complex systems arising from nature and human society can be described as complex networks. In this dissertation, on the basis of complex network theory, we pay attention to the topological structure of complex network and the dynamics on it. We established models to investigate the influences of the structure on the dynamics of networks and to shed light on some peculiar properties of complex systems. This dissertation includes four parts. In the first part, the empirical properties (degree distribution, clustering coefficient, diameter, and characteristic path length) of urban road network of Le Mans city in France are studied. The degree distribution shows a double power-law which we studied in detail. In the second part, we propose two models to investigate the possible mechanisms leading to the deviation from simple power law. In the first model, probabilistic addition of nodes and links, and rewiring of links are considered; in the second one, only random and preferential link growth is included. The simulation results of the modelling are compared with the real data. In the third part,the probabilistic uncertainty behavior of double power law distribution is investigated. The network optimization and optimal design of scale free network to random failures are discussed from the viewpoint of entropy maximization. We defined equilibrium network ensemble as stationary ensembles of graphs by using some thermodynamics like notions such as "energy", "temperature", "free energy" for network. In the forth part, an union-division model is established to investigate the time evolution of certain networks like cultural or economical networks. In this model, the nodes represent, for example, the cultures. Several quantities such as richness, age, identity, ingredient etc. are used to parameterize the probabilistic evolution of the network. The model offers a long term view on the apparently periodic dynamics of an ensemble of cultural or economic entities in interaction.

У дисертації визначено теоретико-методологічні засади здійснення ФП в умовах невизначеності. Уточнено сутність та змістове наповнення ФП, науково-методичні та організаційно-економічні засади прогнозування розвитку економіки України. Сформульовано принципи ФП розвитку економіки в умовах невизначеності. Представлено теоретичні підходи до оцінювання впливу невизначеності фінансових процесів на структурну організацію економіки та системи класифікації невизначеності фінансових процесів. Сформовано методичні засади ФП розвитку економіки в умовах невизначеності на основі визначення ентропійності фінансових характерис-тик. Підтверджено гіпотезу про існування взаємозв’язку між спрямованістю розвитку економіки та виробництвом ентропії її фінансовими характеристи-ками. Запропоновано та формалізовано поняття фінансового гомеостазу розвитку економіки. Визначено стани фінансового гомеостазу для інфляційних процесів та фінансових характеристик зовнішнього сектора економіки України. Розроблено підходи до визначення прогнозних характеристик фінансових інструментів, необхідних для досягнення стану фінансового гомеостазу. The thesis determines theoretical and methodological basis of financial forecasting under conditions of uncertainty. The essence and content of financial forecasting, scientific and methodological as well as economic and organizational foundations of forecasting of Ukraine’s economic development have been specified. The principles of financial forecasting of economic development under conditions of uncertainty have been defined. The classification systems of uncertainty of financial processes have been presented in this work. The methodological foundations of financial forecasting of economic development under conditions of uncertainty on the basis of the entropy of financial properties have been formed. The hypothesis of interdependency of the stages of economic system’s development and entropy production of financial characteristics for Ukrainian economy have been proved. The notion of financial homeostasis of economic development has been proposed and formalized. The states of financial homeostasis for inflationary processes and financial characteristics of the external sector of the economy have been defined. The approaches to define the forecast properties of financial instruments which are essential to achieve the state of financial homeostasis have been developed.

碩士
中原大學
應用物理研究所
97
Heisenberg uncertainty relation plays a very important role in the quantum mechanics. However, the Heisenberg uncertainty has several drawbacks and may not be a good description of uncertainty relation among non-commuting observables. On the other hand the entropic uncertainty that uses the Shannon entropy as its definition is widely accepted as a measure of uncertainty in quantum mechanics. For any two non-commuting observables we can have a good estimate on the lower bound of the entropy sum by using the Krauss inequality. However, there is no such inequality for more than two non-commuting observables. In this thesis we discussed the entropy sum of the three non-commuting spin operators Sx, Sy and Sz and calculated the maximum and the minimum of the entropy sum. We found that the quantum states that give the optimal values of the entropy sum are also eigenstates of some of the symmetries, such as the mirror symmetries or the permutation symmetries, of the entropy sum. In this thesis we also define a novel measure of entropic uncertainty and then redo the calculation of the entropy sum for the three spin observables.

Uncertainty is one of the most essential and fundamental issues that requires full attention in almost all spatial models and applications. Evidently, the quality of uncertainty modelling plays a critical role in resultant outcomes of geographical models and applications with an inevitable effect on decision- making processes. Therefore, up to now, uncertainty assessment and modelling has gained extensive attention in the field of spatial sciences. Considering the growing importance of this issue, this thesis investigates uncertainty modelling that applies in spatial science along with practical strategies to deal with them. To this end, three definitions of uncertainty are adopted, including Type A in which the uncertainties are derived from series of repeated observations, Type B , such as ambiguity and/or vagueness, with the uncertainties calculated by means other than the statistical analysis of series of observations (i.e. general knowledge of the behaviour and properties of phenomena) and Type C that is uncertainties in form of randomness. Proposed strategies to deal with each type of uncertainty is also exemplified in this thesis. In terms of Type A uncertainty, in this thesis, entropy is the key term representing uncertainty. Considering the fact that repeated observations are more often aimed at implementation of predictive spatial models, three examples are described (1) assessing uncertainty of several machine-learning classification algorithms that are repeatedly applied in implementation of spatial predictive models, (2) improving performance of a classification scheme using uncertainty assessment and (3) applying an optimised algorithm using uncertainty assessment in land-use change simulation as a popular example. To this end, in Chapter 2, a strategy is proposed to evaluate the uncertainty of two machine-learning algorithms (i.e. random forest vs deep neural network) that are applied for land-use classification. These two algorithms are highly popular in implementation of predictive spatial models. In Chapter 3, a strategy to improve the performance of predictive algorithms using uncertainty models is proposed in the context of image classification. A comprehensive practical illustration is then applied for improving the quality of a land-use change model in Chapter 4. With reference to Type B uncertainty, two different examples are implemented for drought sensitivity and landslide susceptibility mapping, where fuzzy If-Then rules and fuzzy sets are used to improve the quality of spatial models in Chapters 5 and 6, respectively. Finally, Chapter 7 is focused on applying randomness for modelling Type C uncertainty that is frequently encountered in spatial models, especially land-use change models. Although randomness is an inevitable component of land-use change models, it is sometimes overlooked. In this chapter, randomness is applied as a primary component of land-use change models, but it is also applied to achieve the optimised neighbourhood setting as a key requirement of the model calibration phase. This thesis, therefore, is a comprehensive illustration of different types of uncertainty in a range of important spatial models. Here, the thesis case studies are selected, so that they can be either applied to model the same spatial phenomenon within different case studies or can be generalised to new applications in the spatial domain.

The Tsallis entropy is applied to derive both 1-D and 2-D velocity distributions in an open channel cross section. These distributions contain a parameter m through which the Tsallis entropy becomes a generalization of the Shannon entropy. Different m parameter values are examined to determine the best value for describing the velocity distribution.Two Lagrangian parameters that are involved in the final form of 1-D velocity distribution equation are determined from observations of mean velocity and the maximum velocity at the water surface. For channels which are not wide and where the maximum velocity does not occur at the water surface, a 2-D velocity distribution is more appropriate. The Tsallis entropy is applied to derive 2-D velocity distributions. A new parameter M is introduced which represents the hydraulic characteristics of the channel. The derived velocity distributions are verified using both field data and experimental data. The advantages are found by comparing with Parandtl-von Karman, power law and Chiu’s velocity distributions.

JEL: C14, C22, C32, G14
Este trabalho apresenta uma análise da aplicação de medidas da teoria da informação aos mercados bolsistas, sendo de evidenciar a análise de incerteza e da dependência não-linear. De modo mais específico, é apresentada a entropia como medida de incerteza para sucessões cronológicas financeiras e comparado o seu comportamento com o das principais medidas de risco utilizadas em finanças: o desvio-padrão e os Betas resultantes do modelo CAPM. Em termos financeiros, os resultados obtidos neste estudo indicam que a entropia é sensível ao efeito diversificação e apresenta um comportamento algo semelhante ao do desvio-padrão, contudo é uma medida mais geral e incorpora mais informação acerca da distribuição de probabilidade, com vantagens óbvias no caso de excesso de curtose e assimetria. É desenvolvido um teste à independência entre variáveis baseado na informação mútua, que tem a vantagem de captar a dependência linear e não-linear sem ser necessário assumir quaisquer pressupostos quanto à distribuição de probabilidade teórica e especificação de modelos de dependência. O teste de independência e o coeficiente de correlação global (baseados na informação mútua) são utilizados para avaliar a dependência temporal global das taxas de rendibilidade de sete índices bolsistas e comparados os resultados com outros modelos normalmente utilizados na abordagem não-linear. Para além disso, este teste é também utilizado para avaliar a relação entre determinados factores macroeconómicos e o comportamento do mercado bolsista de acções português, sendo comparados os seus resultados com os obtidos por outras abordagens lineares e não-lineares. Os resultados permitem concluir que existem não-linearidades nas sucessões cronológicas macroeconómicas e financeiras e que a tradicional abordagem linear pode não ser suficiente para avaliar a relação existente entre o mercado bolsista português e algumas variáveis macroeconómicas e financeiras.
This thesis analyses the application of measures of the information theory to the stock market, highlighting the analysis of uncertainty and nonlinear dependence. More specifically, the entropy is presented as a measure of uncertainty for financial time series and we compare its behaviour with that of the main risk measures commonly used in finance: the standard deviation and the resulting Betas from the CAPM. In financial terms, the results obtained in this study indicate that the entropy is sensitive to the effect of diversification and behaves similarly to the standard deviation. However, the entropy is a more general measure and incorporates more information about the probability distribution, with evident advantages in the case of excess kurtosis and skewness. An independence test based on the mutual information is constructed. One important advantage of this approach resides precisely in the ability to account for nonlinear dependencies with no need to specify a theoretical probability distribution or to use a mean-variance model framework. The independence test and the global correlation coefficient (based on the mutual information) are applied to measure serial correlation of some stock market indexes and we found the presence of nonlinear dependence in all cases. Besides, the mutual information test is used as a measure of global dependence between the Portuguese stock market and a set of macroeconomic factors and we show that this measure can overcome some of the weaknesses of the traditional linear approaches commonly used in this context. Globally, our results indicate that some explanatory variables appear to have a statistically significant influence on the excess return and thus may constitute good proxies for this variable.