Um die komplexe Dynamik von Neuronen und deren Informationsverarbeitung mittels Pulssequenzen zu verstehen, ist es wichtig, die stationäre Puls-Aktivität zu charakterisieren. Die statistischen Eigenschaften von Pulssequenzen können durch vereinfachte stochastische Neuronenmodelle verstanden werden. Eine gut ausgearbeitete Theorie existiert für die Klasse der Erneuerungsmodelle, welche die statistische Unabhängigkeit der Interspike-Intervalle (ISI) annimmt. Experimente haben jedoch gezeigt, dass viele Neuronen Korrelationen zwischen ISIs aufweisen und daher nicht gut durch einen Erneuerungsprozess beschrieben werden. Solche Korrelationen können durch Nichterneuerungs-Modelle erfasst werden, welche jedoch theoretisch schlecht verstanden sind. Diese Arbeit ist eine analytische Studie von Nichterneuerungs-Modellen, die zwei bedeutende Korrelationsmechanismen untersucht: farbiges Rauschen, welches zeitlich-korrelierten Input darstellt, und negative Puls-Rückkopplung, welche Feuerraten-Adaption realisiert. Für das "Perfect-Integrate-and-Fire" (PIF) Modell, welchen durch ein allgemeines Gauss''sches farbiges Rauschen getrieben ist, werden die Statistiken höherer Ordnung der Output-Pulssequenz hergeleitet, insbesondere der Koeffizient der Variation, der serielle Korrelationskoeffizient (SCC), die ISI-Dichte und der Fano-Faktor. Weiterhin wird die Dynamik des PIF Modells mit Puls-getriggertem Adaptionsstrom und weissem Stromrauschen im Detail analysiert. Die Theorie liefert einen Ausdruck für den SCC, der für schwaches Rauschen aber beliebige Adaptions-Stärke und Zeitskale gültig ist, sowie die lineare Antwortfunktion und das Leistungsspektrum der Pulssequenz. Ausserdem wird gezeigt, dass ein stochastischer Adaptionsstrom wie ein langsames farbiges Rauschen wirkt, was ermöglicht, die dominierende Quellen des Rauschen in einer auditorischen Rezeptorzelle zu bestimmen. Schliesslich wird der SCC für das fluktuations-getriebene Feuerregime berechnet. / To understand the complex dynamics of neurons and its ability to process information using a sequence of spikes, it is vital to characterize its stationary spontaneous spiking activity. The statistical properties of spike trains can be explained by reduced stochastic neuron models that account for various sources of noise. A well-developed theory exists for the class of renewal models, in which the interspike intervals (ISIs) are statistically independent. However, experimental studies show that many neurons are not well described by a renewal process because of correlations between ISIs. Such correlations can be captured by generalized, non-renewal models, which are, however, poorly understood theoretically. This thesis represents an analytical study of non-renewal models, focusing on two prominent correlation mechanisms: colored-noise driving representing temporally correlated inputs, and negative feedback currents realizing spike-frequency adaptation. For the perfect integrate-and-fire (PIF) model driven by a general Gaussian colored noise input, the higher-order statistics of the output spike train is derived using a weak-noise analysis of the Fokker-Planck equation. This includes formulas for the coefficient of variation, the serial correlation coefficient (SCC), the ISI density and the Fano factor. Then, the dynamics of a PIF model with a spike-triggered adaptation and a white-noise current is analyzed in detail. The theory yields an expression for the SCC valid for weak noise but arbitrary adaptation strengths and time scale, and also provides the linear response to time-dependent stimuli and the spike train power spectrum. Furthermore, it is shown that a stochastic adaptation current acts like a slow colored noise, which permits to determine the source of spiking variability observed in an auditory receptor neuron. Finally, the SCC is calculated for the fluctuation-driven spiking regime by assuming discrete states of colored noise or adaptation current.

Submitted by Daniella Santos (daniella.santos@fgv.br) on 2010-03-11T13:25:30Z No. of bitstreams: 1 Tese_Fabio_Araujo_Final.pdf: 715897 bytes, checksum: 17afb0d85c3fff397df747b1a0d56bf9 (MD5) / Approved for entry into archive by Andrea Virginio Machado(andrea.machado@fgv.br) on 2010-03-12T13:27:41Z (GMT) No. of bitstreams: 1 Tese_Fabio_Araujo_Final.pdf: 715897 bytes, checksum: 17afb0d85c3fff397df747b1a0d56bf9 (MD5) / Made available in DSpace on 2010-03-15T12:06:11Z (GMT). No. of bitstreams: 1 Tese_Fabio_Araujo_Final.pdf: 715897 bytes, checksum: 17afb0d85c3fff397df747b1a0d56bf9 (MD5) Previous issue date: 2009-08-10 / This work proposes alternative ways to consistently estimate an abstract measure, crucial to the study of intertemporal decisions, which is at the core of most macroeconomics and financial studies: the Stochastic Discount Factor (SDF). Using the Pricing Equation in a panel-data framework, is constructed a novel consistent estimator of the SDF which relies on the fact that its logarithm is pervasive to all asset returns of the economy. The resulting estimator is very simple to compute, does not dependent on strong economic assumptions, is suitable for testing different preference specifications or investigating intertemporal substitution puzzles, and can be used as basis to construct an estimator for the risk-free rate. Alternative identification strategies are applied and a parallel between it and identifications strategies based on other frameworks is drawn. Adding structure to the initial setup, two environments were the asymptotic distribution can be derived are presented. Finally, methodologies proposed are applied US and Brazilian data. Preference specifications usually found in the macro literature, as well as a class of state dependent preferences, are tested. The results for the US economy are particularly interesting, by performing formal tests, we cannot reject standard preference specifications used in the literature and estimates of the relative risk-aversion coefficient are between 1 and 2, and statistically indistinguishable from the unity. Moreover, for the class of state dependent preferences and using US quarterly data from 1972:1 and 2001:4, we estimate a highly dynamic path for the relative risk-aversion (rra) coefficient, confined to the interval [1.15, 2.05], and also reject the hypothesis of a constant level. / Este trabalho propõe maneiras alternativas para a estimação consistente de uma medida abstrata, crucial para o estudo de decisões intertemporais, o qual é central a grande parte dos estudos em macroeconomia e finanças: o Fator Estocástico de Descontos (SDF, sigla em Inglês). Pelo emprego da Equação de Apreçamento constrói-se um inédito estimador consistente do SDF que depende do fato de que seu logaritmo é comum a todos os ativos de uma economia. O estimador resultante é muito simples de se calcular, não depende de fortes hipóteses econômicas, é adequado ao teste de diversas especificações de preferência e para a investigação de paradoxos de substituição intertemporal, e pode ser usado como base para a construção de um estimador para a taxa livre de risco. Alternativas para a estratégia de identificação são aplicadas e um paralelo entre elas e estratégias de outras metodologias é traçado. Adicionando estrutura ao ambiente inicial, são apresentadas duas situações onde a distribuição assintótica pode ser derivada. Finalmente, as metodologias propostas são aplicadas a conjuntos de dados dos EUA e do Brasil. Especificações de preferência usualmente empregadas na literatura, bem como uma classe de preferências dependentes do estado, são testadas. Os resultados são particularmente interessantes para a economia americana. A aplicação de teste formais não rejeita especificações de preferências comuns na literatura e estimativas para o coeficiente relativo de aversão ao risco se encontram entre 1 e 2, e são estatisticamente indistinguíveis de 1. Adicionalmente, para a classe de preferência s dependentes do estado, trajetórias altamente dinâmicas são estimadas para a tal coeficiente, as trajetórias são confinadas ao intervalo [1,15, 2,05] e se rejeita a hipótese de uma trajetória constante.

Die vorliegende Arbeit beschäftigt sich mit Mechanismen, die in Einzelzellmodellen zu einer frequenzabhängigen Informationsübertragung führen können. Um dies zu untersuchen, werden Methoden aus der theoretischen Physik (Statistische Physik) und der Informationstheorie angewandt. Die Informationsfilterung in mehreren stochastischen Neuronmodellen, in denen unterschiedliche Mechanismen zur Informationsfilterung führen können, werden numerisch und, falls möglich, analytisch untersucht. Die Bandbreite der betrachteten Modelle erstreckt sich von reduzierten strombasierten ’Integrate-and-Fire’ (IF) Modellen bis zu biophysikalisch realistischeren leitfähigkeitsbasierten Modellen. Anhand numerischer Untersuchungen wird aufgezeigt, dass viele Varianten der IF-Neuronenmodelle vorzugsweise Information über langsame Anteile eines zeitabhängigen Eingangssignals übertragen. Der einfachste Vertreter der oben genannten Klasse der IF-Neuronmodelle wird dahingehend erweitert, dass ein Konzept von neuronalem ’Gedächtnis’, vermittelst positiver Korrelationen zwischen benachbarten Intervallen aufeinander- folgender Spikes, integriert wird. Dieses Model erlaubt eine analytische störungstheoretische Untersuchung der Auswirkungen positiver Korrelationen auf die Informationsfilterung. Um zu untersuchen, wie sich sogenannte ’unterschwelligen Resonanzen’ auf die Signalübertragung auswirken, werden Neuronenmodelle mit verschiedenen Nichtlinearitäten anhand numerischer Computersimulationen analysiert. Abschließend wird die Signalübertragung in einem neuronalen Kaskadensystem, bestehend aus linearen und nichtlinearen Elementen, betrachtet. Neuronale Nichtlinearitäten bewirken eine gegenläufige Abhängigkeit (engl. "trade-off") zwischen qualitativer, d.h. frequenzselektiver, und quantitativer Informations-übertragung, welche in allen von mir untersuchten Modellen diskutiert wird. Diese Arbeit hebt die Gewichtigkeit von Nichtlinearitäten in der neuronalen Informationsfilterung hervor. / Neurons transmit information about time-dependent input signals via highly non-linear responses, so-called action potentials or spikes. This type of information transmission can be frequency-dependent and allows for preferences for certain stimulus components. A single neuron can transmit either slow components (low pass filter), fast components (high pass filter), or intermediate components (band pass filter) of a time-dependent input signal. Using methods developed in theoretical physics (statistical physics) within the framework of information theory, in this thesis, cell-intrinsic mechanisms are being investigated that can lead to frequency selectivity on the level of information transmission. Various stochastic single neuron models are examined numerically and, if tractable analytically. Ranging from simple spiking models to complex conductance-based models with and without nonlinearities, these models include integrator as well as resonator dynamics. First, spectral information filtering characteristics of different types of stochastic current-based integrator neuron models are being studied. Subsequently, the simple deterministic PIF model is being extended with a stochastic spiking rule, leading to positive correlations between successive interspike intervals (ISIs). Thereafter, models are being examined which show subthreshold resonances (so-called resonator models) and their effects on the spectral information filtering characteristics are being investigated. Finally, the spectral information filtering properties of stochastic linearnonlinear cascade neuron models are being researched by employing different static nonlinearities (SNLs). The trade-off between frequency-dependent signal transmission and the total amount of transmitted information will be demonstrated in all models and constitutes a direct consequence of the nonlinear formulation of the models.

This dissertation is a collection of essays on challenges in international trade and international finance, which apply econometric methods to diverse data sets and relate them to economic policy questions. In times of crises, the question, whether individual countries have the ability to pursue idiosyncratic monetary policy, is important. The degree of integration and comovement between financial markets, for instance, is critical to better assess the real threat facing a country in a crisis. Also, from a macroeconomic modeling perspective, there has recently been a renewed interest in the cyclical and long-run comovement of interest rates. Hence, in a first essay, we reinvestigate the long- and short-run comovements in the G7-countries by conducting tests for cointegration, common serial correlation and codependence with nominal and real interest rates. Overall, we only find little evidence of comovements: common trends are occasionally observed, but the majority of interest rates are not cointegrated. Although some evidence for codependence of higher order can be found in the pre-Euro area sample, common cycles appear to exist only in rare cases. We argue that some earlier, more positive findings in the literature are difficult to reconcile due to differing assumptions about the underlying stochastic properties of interest rates. Hence, we conclude that they cannot be generalized for all interest rates, time periods, and reasonable alternative estimation procedures. This finding indicates that scope for individual countries to pursue stabilization policy does still exist in a globalized world. Emerging economies, in general, are much more exposed and vulnerable to crises than industrialized countries. Accordingly, stabilization policy is especially important in these countries and the selection of the best monetary regime is essential. This is why, in a second essay, we contrast two different views in the debate on official dollarization: the Mundell (1961) framework of optimum currency areas and a model on boom-bust cycles by Schneider and Tornell (2004), who take account of credit market imperfections prevalent in middle income countries. We highlight the strikingly different role of the exchange rate in the two models. While in the Mundell framework the exchange rate is expected to smooth the business cycle, the second model predicts the exchange rate to play an amplifying role. We empirically evaluate both models for eight highly dollarized Central American economies. We document the existence of credit market imperfections and find that shocks from the exchange rate indeed amplify business cycles in these countries. Using a new method proposed by Cubadda (1999 and 2007), we furthermore test for cyclical comovement and reject the hypothesis that the selected countries form an optimum currency area with the United States according to the Mundell definition. In the context of the recent global crisis, globalization and vertical integration in particular were often blamed for being the cause for the severe trade crisis. For that reason, in the essay that contributes to the trade literature, we analyze the role of international supply chains in explaining the long-run trade elasticity and its short-term volatility in the context of the recent trade collapse. We adopt an empirical strategy based on two steps: first, stylized facts on long- and short-term trade elasticity are derived from exploratory analysis and formal modeling on a large and diversified sample of countries. Then, we derive observations of interrelated input-output matrices for a demonstrative sub-set of countries. We find evidence for two supply chain related factors to explain the overshooting of trade elasticity during the 2008-2009 trade collapse: the composition and the bullwhip effect. However, evidence for a magnification effect could not be found. Overall, we do not accept the hypothesis that international supply chains explain all by themselves the changes in trade-income elasticity.

Management and conservation of populations of animals requires information on where they are, why they are there, and where else they could be. These objectives are typically approached by collecting data on the animals’ use of space, relating these to prevailing environmental conditions and employing these relations to predict usage at other geographical regions. Technical advances in wildlife telemetry have accomplished manifold increases in the amount and quality of available data, creating the need for a statistical framework that can use them to make population-level inferences for habitat preference and space-use. This has been slow-in-coming because wildlife telemetry data are, by definition, spatio-temporally autocorrelated, unbalanced, presence-only observations of behaviorally complex animals, responding to a multitude of cross-correlated environmental variables. I review the evolution of techniques for the analysis of space-use and habitat preference, from simple hypothesis tests to modern modeling techniques and outline the essential features of a framework that emerges naturally from these foundations. Within this framework, I discuss eight challenges, inherent in the spatial analysis of telemetry data and, for each, I propose solutions that can work in tandem. Specifically, I propose a logistic, mixed-effects approach that uses generalized additive transformations of the environmental covariates and is fitted to a response data-set comprising the telemetry and simulated observations, under a case-control design. I apply this framework to non-trivial case-studies using data from satellite-tagged grey seals (Halichoerus grypus) foraging off the east and west coast of Scotland, and northern gannets (Morus Bassanus) from Bass Rock. I find that sea bottom depth and sediment type explain little of the variation in gannet usage, but grey seals from different regions strongly prefer coarse sediment types, the ideal burrowing habitat of sandeels, their preferred prey. The results also suggest that prey aggregation within the water column might be as important as horizontal heterogeneity. More importantly, I conclude that, despite the complex behavior of the study species, flexible empirical models can capture the environmental relationships that shape population distributions.

Le contrôle explicite de la convergence des sommes convenablement normalisées de variables aléatoires, ainsi que l'étude du principe de déviations modérées associé à ces sommes constituent les thèmes centraux de cette thèse. Nous étudions principalement deux types de processus. Premièrement, nous nous intéressons aux processus indexés par un arbre binaire, aléatoire ou non. Ces processus ont été introduits dans la littérature afin d'étudier le mécanisme de la division cellulaire. Au chapitre 2, nous étudions les chaînes de Markov bifurcantes. Ces chaînes peuvent être vues comme une adaptation des chaînes de Markov "usuelles'' dans le cas où l'ensemble des indices à une structure binaire. Sous des hypothèses d'ergodicité géométrique uniforme et non-uniforme d'une chaîne de Markov induite, nous fournissons des inégalités de déviations et un principe de déviations modérées pour les chaînes de Markov bifurcantes. Au chapitre 3, nous nous intéressons aux processus bifurcants autorégressifs d'ordre p (). Ces processus sont une adaptation des processus autorégressifs linéaires d'ordre p dans le cas où l'ensemble des indices à une structure binaire. Nous donnons des inégalités de déviations, ainsi qu'un principe de déviations modérées pour les estimateurs des moindres carrés des paramètres "d'autorégression'' de ce modèle. Au chapitre 4, nous traitons des inégalités de déviations pour des chaînes de Markov bifurcantes sur un arbre de Galton-Watson. Ces chaînes sont une généralisation de la notion de chaînes de Markov bifurcantes au cas où l'ensemble des indices est un arbre de Galton-Watson binaire. Elles permettent dans le cas de la division cellulaire de prendre en compte la mort des cellules. Les hypothèses principales que nous faisons dans ce chapitre sont : l'ergodicité géométrique uniforme d'une chaîne de Markov induite et la non-extinction du processus de Galton-Watson associé. Au chapitre 5, nous nous intéressons aux modèles autorégressifs linéaires d'ordre 1 ayant des résidus corrélés. Plus particulièrement, nous nous concentrons sur la statistique de Durbin-Watson. La statistique de Durbin-Watson est à la base des tests de Durbin-Watson, qui permettent de détecter l'autocorrélation résiduelle dans des modèles autorégressifs d'ordre 1. Nous fournissons un principe de déviations modérées pour cette statistique. Les preuves du principe de déviations modérées des chapitres 2, 3 et 4 reposent essentiellement sur le principe de déviations modérées des martingales. Les inégalités de déviations sont établies principalement grâce à l'inégalité d'Azuma-Bennet-Hoeffding et l'utilisation de la structure binaire des processus. Le chapitre 5 est né de l'importance qu'a l'ergodicité explicite des chaînes de Markov au chapitre 3. L'ergodicité géométrique explicite des processus de Markov à temps discret et continu ayant été très bien étudiée dans la littérature, nous nous sommes penchés sur l'ergodicité sous-exponentielle des processus de Markov à temps continu. Nous fournissons alors des taux explicites pour la convergence sous exponentielle d'un processus de Markov à temps continu vers sa mesure de probabilité d'équilibre. Les hypothèses principales que nous utilisons sont : l'existence d'une fonction de Lyapunov et d'une condition de minoration. Les preuves reposent en grande partie sur la construction du couplage et le contrôle explicite de la queue du temps de couplage. / The explicit control of the convergence of properly normalized sums of random variables, as well as the study of moderate deviation principle associated with these sums constitute the main subjects of this thesis. We mostly study two sort of processes. First, we are interested in processes labelled by binary tree, random or not. These processes have been introduced in the literature in order to study mechanism of the cell division. In Chapter 2, we study bifurcating Markov chains. These chains may be seen as an adaptation of "usual'' Markov chains in case the index set has a binary structure. Under uniform and non-uniform geometric ergodicity assumptions of an embedded Markov chain, we provide deviation inequalities and a moderate deviation principle for the bifurcating Markov chains. In chapter 3, we are interested in p-order bifurcating autoregressive processes (). These processes are an adaptation of $p$-order linear autoregressive processes in case the index set has a binary structure. We provide deviation inequalities, as well as an moderate deviation principle for the least squares estimators of autoregressive parameters of this model. In Chapter 4, we dealt with deviation deviation inequalities for bifurcating Markov chains on Galton-Watson tree. These chains are a generalization of the notion of bifurcating Markov chains in case the index set is a binary Galton-Watson tree. They allow, in case of cell division, to take into account cell's death. The main hypothesis that we do in this chapter are : uniform geometric ergodicity of an embedded Markov chain and the non-extinction of the associated Galton-Watson process. In Chapter 5, we are interested in first-order linear autoregressive models with correlated errors. More specifically, we focus on the Durbin-Watson statistic. The Durbin-Watson statistic is at the base of Durbin-Watson tests, which allow to detect serial correlation in the first-order autoregressive models. We provide a moderate deviation principle for this statistic. The proofs of moderate deviation principle of Chapter 2, 3 and 4 are essentially based on moderate deviation for martingales. To establish deviation inequalities, we use most the Azuma-Bennet-Hoeffding inequality and the binary structure of processes. Chapter 6 was born from the importance that explicit ergodicity of Markov chains has in Chapter 2. Since explicit geometric ergodicity of discrete and continuous time Markov processes has been well studied in the literature, we focused on the sub-exponential ergodicity of continuous time Markov Processes. We thus provide explicit rates for the sub-exponential convergence of a continuous time Markov process to its stationary distribution. The main hypothesis that we use are : existence of a Lyapunov fonction and of a minorization condition. The proofs are largely based on the coupling construction and the explicit control of the tail of the coupling time.

Cette thèse est dévolue à l'étude de certaines propriétés asymptotiques du processus autorégressif d'ordre p. Ce dernier qualifie communément une suite aléatoire $(Y_{n})$ définie sur $\dN$ ou $\dZ$ et entièrement décrite par une combinaison linéaire de ses $p$ valeurs passées, perturbée par un bruit blanc $(\veps_{n})$. Tout au long de ce mémoire, nous traitons deux problématiques majeures de l'étude de tels processus : l'\textit{autocorrélation résiduelle} et la \textit{stationnarité}. Nous proposons en guise d'introduction un survol nécessaire des propriétés usuelles du processus autorégressif. Les deux chapitres suivants sont consacrés aux conséquences inférentielles induites par la présence d'une autorégression significative dans la perturbation $(\veps_{n})$ pour $p=1$ tout d'abord, puis pour une valeur quelconque de $p$, dans un cadre de stabilité. Ces résultats nous permettent d'apposer un regard nouveau et plus rigoureux sur certaines procédures statistiques bien connues sous la dénomination de \textit{test de Durbin-Watson} et de \textit{H-test}. Dans ce contexte de bruit autocorrélé, nous complétons cette étude par un ensemble de principes de déviations modérées liées à nos estimateurs. Nous abordons ensuite un équivalent en temps continu du processus autorégressif. Ce dernier est décrit par une équation différentielle stochastique et sa solution est plus connue sous le nom de \textit{processus d'Ornstein-Uhlenbeck}. Lorsque le processus d'Ornstein-Uhlenbeck est lui-même engendré par une diffusion similaire, cela nous permet de traiter la problématique de l'autocorrélation résiduelle dans le processus à temps continu. Nous inférons dès lors quelques propriétés statistiques de tels modèles, gardant pour objectif le parallèle avec le cas discret étudié dans les chapitres précédents. Enfin, le dernier chapitre est entièrement dévolu à la problématique de la stationnarité. Nous nous plaçons dans le cadre très général où le processus autorégressif possède une tendance polynomiale d'ordre $r$ tout en étant engendré par une marche aléatoire intégrée d'ordre $d$. Les résultats de convergence que nous obtenons dans un contexte d'instabilité généralisent le \textit{test de Leybourne et McCabe} et certains aspects du \textit{test KPSS}. De nombreux graphes obtenus en simulations viennent conforter les résultats que nous établissons tout au long de notre étude. / This thesis is devoted to the study of some asymptotic properties of the $p-$th order \textit{autoregressive process}. The latter usually designates a random sequence $(Y_{n})$ defined on $\dN$ or $\dZ$ and completely described by a linear combination of its $p$ last values and a white noise $(\veps_{n})$. All through this manuscript, one is concerned with two main issues related to the study of such processes: \textit{serial correlation} and \textit{stationarity}. We intend, by way of introduction, to give a necessary overview of the usual properties of the autoregressive process. The two following chapters are dedicated to inferential consequences coming from the presence of a significative autoregression in the disturbance $(\veps_{n})$ for $p=1$ on the one hand, and then for any $p$, in the stable framework. These results enable us to give a new light on some statistical procedures such as the \textit{Durbin-Watson test} and the \textit{H-test}. In this autocorrelated noise framework, we complete the study by a set of moderate deviation principles on our estimates. Then, we tackle a continuous-time equivalent of the autoregressive process. The latter is described by a stochastic differential equation and its solution is the well-known \textit{Ornstein-Uhlenbeck process}. In the case where the Ornstein-Uhlenbeck process is itself driven by an Ornstein-Uhlenbeck process, one deals with the serial correlation issue for the continuous-time process. Hence, we infer some statistical properties of such models, keeping the parallel with the discrete-time framework studied in the previous chapters as an objective. Finally, the last chapter is entirely devoted to the stationarity issue. We consider the general autoregressive process with a polynomial trend of order $r$ driven by a random walk of order $d$. The convergence results in the unstable framework generalize the \textit{Leybourne and McCabe test} and some angles of the \textit{KPSS test}. Many graphs obtained by simulations come to strengthen the results established all along the study.