This dissertation is devoted to studying a fast and analytic approximation method, called the variational Bayesian (VB) method, and aims to give insight into its general applicability and usefulness, and explore its applications to various real-world problems. This work has three main foci: 1) The general applicability and properties; 2) Diagnostics for VB approximations; 3) Variational applications. Generally, the variational inference has been developed in the context of the exponential family, which is open to further development. First, it usually consider the cases in the context of the conjugate exponential family. Second, the variational inferences are developed only with respect to natural parameters, which are often not the parameters of immediate interest. Moreover, the full factorization, which assumes all terms to be independent of one another, is the most commonly used scheme in the most of the variational applications. We show that VB inferences can be extended to a more general situation. We propose a special parameterization for a parametric family, and also propose a factorization scheme with a more general dependency structure than is traditional in VB. Based on these new frameworks, we develop a variational formalism, in which VB has a fast implementation, and not be limited to the conjugate exponential setting. We also investigate its local convergence property, the effects of choosing different priors, and the effects of choosing different factorization scheme. The essence of the VB method relies on making simplifying assumptions about the posterior dependence of a problem. By definition, the general posterior dependence structure is distorted. In addition, in the various applications, we observe that the posterior variances are often underestimated. We aim to develop diagnostics test to assess VB approximations, and these methods are expected to be quick and easy to use, and to require no sophisticated tuning expertise. We propose three methods to compute the actual posterior covariance matrix by only using the knowledge obtained from VB approximations: 1) To look at the joint posterior distribution and attempt to find an optimal affine transformation that links the VB and true posteriors; 2) Based on a marginal posterior density approximation to work in specific low dimensional directions to estimate true posterior variances and correlations; 3) Based on a stepwise conditional approach, to construct and solve a set of system of equations that lead to estimates of the true posterior variances and correlations. A key computation in the above methods is to calculate a uni-variate marginal or conditional variance. We propose a novel way, called the VB Adjusted Independent Metropolis-Hastings (VBAIMH) method, to compute these quantities. It uses an independent Metropolis-Hastings (IMH) algorithm with proposal distributions configured by VB approximations. The variance of the target distribution is obtained by monitoring the acceptance rate of the generated chain. One major question associated with the VB method is how well the approximations can work. We particularly study the mean structure approximations, and show how it is possible using VB approximations to approach model selection tasks such as determining the dimensionality of a model, or variable selection. We also consider the variational application in Bayesian nonparametric modeling, especially for the Dirichlet process (DP). The posterior inference for DP has been extensively studied in the context of MCMC methods. This work presents a a full variational solution for DP with non-conjugate settings. Our solution uses a truncated stick-breaking representation. We propose an empirical method to determine the number of distinct components in a finite dimensional DP. The posterior predictive distribution for DP is often not available in a closed form. We show how to use the variational techniques to approximate this quantity. As a concrete application study, we work through the VB method on regime-switching lognormal models and present solutions to quantify both the uncertainty in the parameters and model specification. Through a series numerical comparison studies with likelihood based methods and MCMC methods on the simulated and real data sets, we show that the VB method can recover exactly the model structure, gives the reasonable point estimates, and is very computationally efficient.

Η παρούσα διδακτορική διατριβή πραγματεύεται προβλήματα που αφορούν το χώρο της τεχνολογίας ομιλίας, με στόχο τη αναγνώριση συναισθημάτων από ομιλία με χρήση τεχνικών ψηφιακής επεξεργασίας σήματος και μηχανικής μάθησης. Πιο αναλυτικά, στα πλαίσια της διατριβής προτάθηκαν και μελετήθηκαν καινοτόμες μέθοδοι σε μια σειρά από εφαρμογές που αξιοποιούν σύστημα αναγνώρισης συναισθηματικών καταστάσεων από ομιλία. Ο βασικός στόχος των μεθόδων ήταν η αντιμετώπιση των προκλήσεων που παρουσιάζονται όταν ένα σύστημα αναγνώρισης συναισθηματικών καταστάσεων καλείται να λειτουργήσει σε πραγματικές συνθήκες, με αυθόρμητες αντιδράσεις, ανεξαρτήτως ομιλητή. Πιο συγκεκριμένα, στα πλαίσια της διατριβής, αξιολογήθηκε η συμπεριφορά ενός συστήματος αναγνώρισης συναισθημάτων σε προσποιητή ομιλία και σε διαφορετικές συνθήκες θορύβου, και συγκρίθηκε η απόδοση του συστήματος με την υποκειμενική αξιολόγηση των ακροατών. Επιπλέον, περιγράφηκε ο σχεδιασμός και η υλοποίηση βάση δεδομένων συναισθηματικής ομιλίας, όπως αυτή προκύπτει από την αλληλεπίδραση μη-έμπειρων χρηστών με ένα διαλογικό σύστημα και προτάθηκε ένα σύστημα το οποίο εντοπίζει αρνητικές συναισθηματικές καταστάσεις, στο ανεξάρτητου ομιλητή πρόβλημα, με χρήση μοντέλου Γκαουσιανών κατανομών. Η προτεινόμενη αρχιτεκτονική συνδυάζει παραμέτρους ομιλίας χαμηλού και υψηλού επιπέδου και εφαρμόζεται στα πραγματικά δεδομένα. Επίσης, αξιολογήθηκε και υλοποιήθηκε η πρακτική εφαρμογή ενός συστήματος αναγνώρισης συναισθημάτων βασισμένου σε οικουμενικό μοντέλο Γκαουσιανών κατανομών σε διαφορετικούς τύπους δεδομένων πραγματικής ζωής. Ακόμα, παρουσιάστηκε μια πρωτότυπη αρχιτεκτονική κατηγοριοποίησης για αναγνώριση συνυπαρχόντων συναισθημάτων από ομιλία προερχόμενη από αλληλεπίδραση σε πραγματικά περιβάλλοντα. Σε αντίθεση με γνωστές προσεγγίσεις, η προτεινόμενη αρχιτεκτονική μοντελοποιεί τις συνυπάρχουσες συναισθηματικές καταστάσεις μέσω της κατασκευής μιας πολυσταδιακής αρχιτεκτονικής κατηγοριοποίησης. Τα πειραματικά αποτελέσματα που διενεργήθηκαν υποδεικνύουν ότι η προτεινόμενη αρχιτεκτονική είναι πλεονεκτική για τις συναισθηματικές καταστάσεις που είναι πιο διαχωρίσιμες, γεγονός που οδηγεί σε βελτίωση της συνολικής απόδοσης του συστήματος. / In this doctoral dissertation a number of novel approaches were proposed and evaluated in different applications that utilize emotion awareness. The major target of the proposed methods was facing the difficulties existing, when an emotion recognition system is asked to operate in real-life conditions, where human speech is characterized by spontaneous and genuine formulations. In detail, within the present dissertation, the performance of an emotion recognition system was evaluated, initially, in acted speech, under different noise conditions, and this performance was compared to the one of human listeners. Further, the design and implementation of a real world emotional speech corpus is described, as this results from the interaction of naive users with a smart home dialogue system. Moreover, a system which utilizes low and high level descriptors was suggested. The suggested architecture leads to significantly better performance in some working points of the integrated system in the dialogue system. Furthermore, we propose a novel multistage classification scheme for affect recognition from real-life speech. In contrast with conventional approaches for affect/emotion recognition from speech, the proposed scheme models co-occurring affective states by constructing a multistage classification scheme. The empirical experiments performed indicate that the proposed classification scheme offers an advantage for those classes that are more separable, which contributes for improving the overall performance of the affect recognition system.

Ce travail porte sur le développement d'algorithmes et l'application de méthodes numériques pour la résolution des problèmes inverses d'estimation de paramètres, d'identification de conditions aux limites et d'identification de sources dans un milieu poreux. Ces outils seront d'une grande utilité pour l'aide à la gestion des ressources en eaux souterraines et à leur préservation quant aux dégradations. L'objectif de cette thèse est de résoudre ces problèmes inverses en se basant sur différentes approches : Une résolution basée sur l'optimisation de forme topologique qui est la recherche de la géométrie d'un objet qui soit optimale vis à vis d'un critère donné, et ce sans aucun a priori sur sa topologie, c'est-à-dire sur le nombre de "trous" qu'il peut contenir. Sachant que ces trous représentent les puits recherchés. Pour ce faire, nous avons adopté la méthode du gradient topologique, qui consiste à étudier le comportement d'une fonction objectif lors de la création d'un petit trou à l'intérieur du domaine. Une résolution basée sur la minimisation d'une fonctionnelle d'erreur en énergie en utilisant des données surabondantes sur une partie de la frontière du domaine afin de compléter les données sur toute la frontière du domaine et de déterminer les positions, les débits et le nombre de puits existants à l'intérieur du domaine. Une résolution par le couplage de la méthode de paramétrisation adaptative qui a l'avantage de minimiser le nombre des inconnus de paramètres permettant d'interpréter au mieux les données disponibles et la méthode du gradient topologique. Ce couplage nous permet à la fois d'identifier les zones géologiques, de déterminer les valeurs de la transmissivité hydraulique dans chaque zone et de localiser les positions des puits. / This work deals with the development of algorithms and application of numerical methods for solving inverse problems of parameters estimation, identification of boundary conditions and localisation of sources in porous media. These tools will be usefull in the management of groundwater resources and their preservation as to damage. The objective of this thesis is to solve the inverse problem based on different approaches: A resolution based on topological shape optimization is to find an optimal design without any priori assumption about its topology, that is, about the number of holes it may contain. Knowing that these holes represent the searched wells. To do this, we have adopted the method of topological gradient, which is to study the behavior of an objective function when creating a small hole inside the domain. A resolution based on the minimization of a constitutive law gap functional by using overspecified data on a part of the boundary of the domain to complete the data on all the boundary of the domain and determine the positions, the flows and the number of existing wells inside the domain. A resolution by the coupling of the adaptive parameterization method which has the advantage to minimize the number of the unknowns of parameters allowing to interpret at best the available data and the method of the topological gradient. This coupling allows us at the same time to identify the geological zones, to determine the values of the hydraulic transmissivity in every zone and to locate wells' positions.

The work is focused on the issue of parameterization of the enterprise capabilities of its multifunction, stabilisation and growth by means of the non-capital measures. The basic terms and categories of this issue are characterized in the search section such as the agrarian sector and its position in the national economy of the Czech Republic, the common agricultural policy, rural development programme, the marginality, the LFA, the sustainability of agriculture, business multifunctional potential and its diversification, the financial health of the company, creditworthy and bankruptcy models. Doctoral dissertation tasks were solved on the file of legal entities in the South Bohemia region (229 farms - i.e. 1,504 statistical units) in the years 2005-2010, after the correction then on 214 agricultural holdings with the initial year 2010 and a scientific experiment was performed on 5 model enterprises with the initial year 2012. The main outputs of the dissertation are as follows: definition of zoning formulas, new theoretical defining business multifunctional potential including the methodology for determining business potential and its parameterization. Investigation was carried out on the test set of agricultural enterprises through models of non-investment measures, including subsidies; consideration was given to progressive increases and size degression. Variants of optimization (profit maximization and advantages of livestock production) were tested at scientific experiment on five chosen enterprises. The result of multifunctional potential parameterization process is also expressed in differentiation of the test set on creditworthy businesses and non-performing businesses and businesses falling into, so-called, a gray area. Solving the dissertation was connected to the state task of the Institute of Agricultural Economics and Information (IAEI, Prague) in preparation for the new programming period, the Rural Development Programme for the years 2014-2020 and the government document Strategy of Agriculture and Food Industry of the Czech Republic.

Este trabalho discutiu a relação entre o espaçamento inter-estrias , isto é, a taxa de propagação da trinca microscópica SL (local) e SP (projetada) e a taxa de propagação macroscópica , da/dN, obtido após o ensaio de propagação de trinca por fadiga. Corpos de prova compacto C(T) foram extraídos na orientação LT do centro de uma placa laminada de liga de alumínio SAE-AMS 7475 T7351 onde posteriormente foram realizados ensaios de fadiga sob carga ou &#916K constantes, à temperatura ambiente de acordo com a Norma ASTM-E647 (1993). Para o ensaio sob carga constante foram obtidos os dados do estágio II e III da curva da/dN vs. &#916K. Para o ensaio de fadiga sob &#916K constante, quatro níveis de &#916K foram utilizados com taxas de crescimento de trinca variando entre 0,15 à 2,4 &#956m/ciclo. Adicionalmente foram realizados ensaios de tenacidade à fratura, KIC nas direções LT e TL conforme a Norma ASTM-E1820 (1999). A análise fratográfica por microscopia eletrônica de varredura foi conduzida no ensaio de fadiga a &#916K constante para determinar os espaçamentos inter-estrias. As taxas de propagação macroscópica e microscópica foram comparadas e boas correlações foram obtidas para um intervalo entre 0,1 a 1,0 &#956m/ciclo. Através da técnica proposta por Berkovitz (1995), o espectro de carregamento foi estimado através da técnica de parametrização do fator K e os resultados foram comparados ao espectro de carregamento real imposto pela máquina de ensaio. Utilizando o espaçamento inter-estrias local, SL, e para valores de &#916K menores que 23 MPam 1/2 (região de Paris), a estimativa de carregamento exibiu um erro menor que 15%. As vidas estimadas em propagação fornecidas pelas taxas microscópicas de propagação de trinca foram comparadas à vida real de propagação obtida a partir da taxa de propagação de trinca macroscópica e o erro absoluto entre ambos os valores foi menor que 15,7 %. Estes resultados sugerem que a vida em fadiga e o espectro de carga podem ser adequadamente estimados utilizando o espaçamento inter-estrias local, SL. / This work discusses the relationship between striation spacing, i.e. the microscopic crack propagation rate, SL (local) e SP (projected), as measured in post-mortem fractographic inspection of fatigue fractured surfaces, and the macroscopic crack propagation rate, da/dN, as monitored during fatigue crack growth tests. Compact tensile specimens C(T) were extracted in LT orientation from the core of a laminated plate of a SAE-AMS 7475-T7351 Al alloy and subsequently fatigue tested either at constant load or at constant &#916K, at room temperature, according to ASTM-E647. For constant amplitude loading fatigue tests, da/dN- &#916K data points were obtained for Regions II and III of crack growth curve. For constant &#916K fatigue tests, four &#916K levels were selected to give crack growth rates typically ranging from 0.15 to 2.4 &#956m/cycle. Additionally, KIC fracture toughness tests were performed in both LT and TL directions, according to ASTM E1820. SEM fractographic analysis was conducted on constant &#916K fatigue specimens, in order to determine striation spacing created during the fatigue test. Macro- and microscopic crack growth rates were compared and good correlation have been obtained for data within the range from 0,1 to 1,0 &#956m/ciclo. Using the technique proposed by Berkovitz, loading history was estimated by the K-factor parametrization and the results were compared to the true loading spectrum imposed by the testing machine. Using the local striation spacing SL and for &#916K values lower than 23 MPam 1/2 (Paris region), the estimated loading exhibits an error less than 15%. Propagation life estimated by local striation spacing (microscopic crack growth rate) was compared to the true propagation life obtained from macroscopic crack growth rate values and the absolute difference between them was less than 15,7%. These results suggest that both life and loading history can be adequately estimated using the local SL striation spacing values.

Ce travail a consisté à développer un modèle de canopée (CIM), qui pourrait servir d’interface entre des modèles méso-échelles de calcul du climat urbain et des modèles micro-échelles de besoin énergétique du bâtiment. Le développement est présenté en conditions atmosphériques variées, avec et sans obstacles, en s’appuyant sur les théories précédemment proposées. Il a été, par exemple, montré que, pour être en cohérence avec la théorie de similitude de Monin-Obukhov, un terme correctif devait être rajouté au terme de flottabilité de la T.K.E. CIM a aussi été couplé au modèle méso-échelle WRF. Une méthodologie a été proposée pour profiter de leurs avantages respectifs (un plus résolu, l’autre intégrant des termes de transports horizontaux) et pour assurer la cohérence de leurs résultats. Ces derniers ont montré que ce système, en plus d’être plus précis que le modèle WRF à la même résolution, permettait, par l’intermédiaire de CIM, de fournir des profils plus résolus près de la surface. / This study consisted in the development of a canopy model (CIM), which could be use as an interface between meso-scale models used to simulate urban climate and micro-scale models used to evaluate building energy use. The development is based on previously proposed theories and is presented in different atmospheric conditions, with and without obstable. It has been shown, for example, that to be in coherence with the Monin-Obukhov Similarity Theory, that a correction term has to be added to the buoyancy term of the T.K.E. CIM has also been coupled with the meteorological meso-scale model WRF. A methodology was proposed to take advantage of both models (one being more resolved, the other one integrating horizontal transport terms) and to ensure a coherence of the results. Besides being more precise than the WRF model at the same resolution, this system allows, through CIM, to provide high resolved vertical profiles near the surface.

Les bâtiments représentent 40 pourcents de la consommation finale d'énergie. Ils sont ainsi le fer de lance des politiques de réduction des dépenses énergétiques. Récemment, des systèmes de modèles climatiques qui incluent un modèle atmosphérique régional et des paramétrisations urbaines sophistiquées ont été développés. Ils considèrent la complexité de l’îlot de chaleur urbain et ses interactions avec les besoins énergétiques des bâtiments. Dans quelle mesure ces systèmes constituent-ils des outils d’aide à la décision pour les autorités locales ? Cette étude menée sur le territoire de l'Eurodistrict (Strasbourg - Kehl) en 2010, puis en 2030, à l’aide du système de modèles de climat WRF/ARW-BEP+BEM a démontré que si le système de modèles estimait de manière fiable les besoins en chauffage des bâtiments, ces derniers étaient davantage sensibles aux caractéristiques intrinsèques des bâtiments qu'aux formes urbaines et à l'îlot de chaleur urbain induit par ces formes. / Buildings represent 40 percent of the end-use energy. Thus, they constitute a key point of the energy saving policies. Recently, climate modeling systems that include a mesoscale atmospheric model, sophisticated urban parameterizations have been developed to account for the complexity of the urban climate and its interactions with the building energy loads. This study aims to assess the capability of such climate modeling systems to provide climate and energy guidelines to urban planners. For this, we used the research collaborative WRF/ARW-BEP+BEM climate modeling system and performed sensitivity tests considering the territory of the Eurodistrict in 2010, and then in 2030. The results reveal that the climate modeling system achieves estimating the building energy needs over the study area, but also indicate that the building energy needs are more sensitive to the building intrinsic properties and occupant behavior than to the urban forms and their induced urban heat island.

En viabilité hivernale, la prévision de l’état de surface des infrastructures s’avère indispensable, et permet une anticipation, une meilleure coordination et une efficacité d’intervention des services d’exploitation. La majorité des pays dispose de modèles de prévision de la température de surface d'infrastructures et des routes en particulier (TSR). La complexité de ces outils d’aide à la décision est croissante, pour servir au mieux les usagers et l’exploitant. Le microclimat urbain influence le bilan énergétique de surface selon différents processus : radiatifs, aérodynamiques et hydrologiques. Néanmoins, d’autres processus physiques anthropiques influencent cette TSR, tel que le trafic. Des travaux ont été menés par le passé concernant l’apport énergétique du trafic dans le bilan thermique de la ville. Celui-ci a fait l’objet d’études sur les périodes estivales et les îlots de chaleur urbains associés. Cependant, dans les cas de dégradations des conditions hivernales de circulation, ces apports énergétiques ont été intégrés de façon marginale dans la modélisation des paramètres de surface de la route. L’absence de cette contribution du trafic dans la modélisation du bilan énergétique de surface explique, dans une certaine limite, la prévision imparfaite de l’état de surface de la route. La bibliographie recense plusieurs études conduites afin d’identifier et de quantifier ces effets du trafic. Elles n'ont pas ou peu traité la perte ou le gain d'énergie causé par le passage des véhicules sur le bilan énergétique de surface, ou sur la modélisation de la TSR. Dans la présente étude, deux approches ont été proposées pour paramétrer le trafic dans le modèle numérique Town Energy Balance (TEB), l'une globale et la seconde détaillée. Leur analyse comparée indique que la seconde a significativement amélioré les résultats de la modélisation de la TSR. Les apports thermiques du trafic ont augmenté la TSR de 2 à 4°C pour la rapprocher des mesures expérimentales (écart de 0.5 à 1°C en moyenne). Elle est le résultat de l’effet cumulatif des différents processus physiques associés au trafic, et varie en fonction de ses paramètres (densité, vitesse de circulation, fluidité, etc.). Une étude de sensibilité a été menée afin d’apprécier les processus physiques responsables de l’amélioration de la TSR. Les résultats indiquent que l’effet turbulent, la densité de flux radiatif ainsi que la densité de flux due aux frottements des pneumatiques contribuent le plus à l'augmentation la TSR. Néanmoins la contribution énergétique de chacun de ces processus dépend à la fois des paramètres du trafic et des conditions météorologiques. Cette étude a présenté également une analyse préliminaire de l’influence de la lame d’eau présente en surface sur la TSR. Cette dernière décroit en fonction de l’épaisseur de la lame d’eau. Les facteurs anthropiques tels que le trafic et l’adjonction de fondants routiers sur la lame d’eau présente en surface sont décrits et discutés, et une paramétrisation proposée en perspectives / In winter maintenance, forecasting the infrastructure surface status is mandatory, to allow anticipation, better coordination and efficiency of services. The majority of countries have forecast models of the infrastructure surface temperature and especially roads one (RST). The complexity of these decision tools is growing, to best serve the users and managers. The urban microclimate influences the surface energy balance according to various processes: radiative, aerodynamic and hydrologic. However, other anthropogenic physical processes influence this RST, such as traffic. Studies have been carried out in the past about the traffic heat input in the town heat balance. These were conducted on the summer periods and associated urban heat islands. However, in case of traffic in adverse winter conditions, these energy contributions were marginally integrated into the modeling of the road surface parameters. The absence of this traffic's contribution in the surface energy balance modeling explains, to a given limit, the imperfect forecasting of road surface status. The literature identifies several studies conducted to identify and to quantify these effects of traffic. They have insufficiently or not treated the loss or gain on energy caused by circulating vehicles on the surface energy balance, or on modeling the RST. In this study, two approaches have been proposed to parameterize the traffic in the Town Energy Balance (TEB) numerical model, this first one being overall and the second one detailed. Their comparative analysis indicates that the second significantly improved the results of the RST modeling. The traffic heat inputs increased RST by 2 to 4°C, results being then closer to experimental measurements (average difference of 0.5 to 1°C). It is the result of the cumulative effect of the various traffic physical processes, and varies according to its parameters (density, vehicle velocity, fluidity, etc.). A sensitivity analysis was conducted to assess the physical processes responsible for the improvement of the RST. The results indicate that the turbulent effect, the radiative heat flux and flux density due to tires friction represent the greatest contribution to RST increase. Nevertheless the energy contribution of each of these processes depends both traffic parameters and weather conditions. This study also presented a preliminary analysis of the influence of a water layer over the surface on the RST. The latter decreases as a function of the thickness of the water layer. Anthropogenic factors such as traffic and the addition of de-icing products into the water layer present on the surface are described and discussed, and a parameterization proposed as a perspective

This work deals with a design of hypokinetic dysarthria analysis system. Hypokinetic dysarthria is a speech motor dysfunction that is present in approx. 90 % of patients with Parkinson’s disease. Next there is described Parkinson's disease and change of the speech signal by this disability. The following describes the symptoms, which are used for the diagnosis of Parkinson's disease (FCR, VSA, VAI, etc.). The work is mainly focused on parameterization techniques that can be used to diagnose or monitor this disease as well as estimate its progress. A protocol of dysarthric speech acquisition is described in this work too. In combination with acoustic analysis it can be used to estimate a grade of hypokinetic dysarthria in fields of faciokinesis, phonorespiration and phonetics (correlation with 3F test). Regarding the parameterization, new features based on method RASTA. The analysis is based on parametrization sentences complicated for articulation. Experimental dataset consists of 101 PD patients with different disease progress and 53 healthy controls. For classification with feature selection have selected method mRMR.

This work is motivated by the question: how much complexity is appropriate for a cloud parameterization used in general circulation models (GCM). To approach this question, cloud parameterizations across the complexity range are explored using general circulation models and theoretical Monte-Carlo simulations. Their results are compared with high-resolution satellite observations and simulations that resolve the GCM subgrid-scale variability explicitly. A process-orientated evaluation is facilitated by GCM forecast simulations which reproduce the synoptic state. For this purpose novel methods were develop to a) conceptually relate the underlying saturation deficit probability density function (PDF) with its saturated cloudy part, b) analytically compute the vertical integrated liquid water path (LWP) variability, c) diagnose the relevant PDF-moments from cloud parameterizations, d) derive high-resolution LWP from satellite observations and e) deduce the LWP statistics by aggregating the LWP onto boxes equivalent to the GCM grid size. On this basis, this work shows that it is possible to evaluate the sub-grid scale variability of cloud parameterizations in terms of cloud variables. Differences among the PDF types increase with complexity, in particular the more advanced cloud parameterizations can make use of their double Gaussian PDF in conditions, where cumulus convection forms a separate mode with respect to the remainder of the grid-box. Therefore, it is concluded that the difference between unimodal and bimodal PDFs is more important, than the shape within each mode. However, the simulations and their evaluation reveals that the advanced parameterizations do not take full advantage of their abilities and their statistical relationships are broadly similar to less complex PDF shapes, while the results from observations and cloud resolving simulations indicate even more complex distributions. Therefore, this work suggests that the use of less complex PDF shapes might yield a better trade-off. With increasing model resolution initial weaknesses of simpler, e.g. unimodal PDFs, will be diminished. While cloud schemes for coarse-resolved models need to parameterize multiple cloud regimes per grid-box, higher spatial resolution of future GCMs will separate them better, so that the unimodal approximation improves.