Concept# Statistique descriptive

Résumé

La statistique descriptive est la branche des statistiques qui regroupe les nombreuses techniques utilisées pour décrire un ensemble relativement important de données.
Description statistique
L'objectif de la statistique descriptive est de décrire, c'est-à-dire de résumer ou représenter, par des statistiques, les données disponibles quand elles sont nombreuses.
Les données disponibles
Toute description d'un phénomène nécessite d'observer ou de connaître certaines choses sur ce phénomène.

- Les observations disponibles sont toujours constituées d'ensemble d'observations synchrones. Par exemple : une température, une pression et une mesure de densité à un instant donné dans une cuve précise. Ces trois variables synchrones peuvent être observées plusieurs fois (à plusieurs dates) en plusieurs lieux (dans plusieurs cuves).
- Les connaissances disponibles sont quant à elles constituées de formules qui relient certaines variables. Par exemple, la loi des gaz parf

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The reliability of new overhead electric and telecommunication lines depends principally on the quality of their support structures. These structures are generally made of wood, metal or concrete. The complexity of a natural substance such as wood requires a thorough analysis of the various factors that influence its overall quality. In the case of wood poles, such factors include initial forest growth pattern, the species of wood and its preservative treatment, ageing characteristics, and its various mechanical defects such as knots, cracks etc. The accumulation of knowledge on the effect of the various variables that contribute to the overall quality of a wood support structure permits an optimum use of such a resource. For example, less variability and higher strength of wood support structures permits optimum loading and spacing between structures, thus reducing the number needed in a specific length of an overhead line. If one assumes that in Western Europe 1 wood pole is employed for every 2 inhabitants, and that this proportion increases in less densely populated countries such as the US and Scandinavia, the economics of optimum use of wood as a resource soon become apparent. In less developed countries, the proportions and the economics vary depending on the natural resources such as wood that they employ. The goal of this research is to establish, thanks to non destructive evaluations, a general ageing probabilistic law of the wooden pole based on two distinguished laws: one on the new pole in studying the influence of a grading of the bad elements based on a normal law: "left-truncation of a normal distribution", point 1; and another one based on the in-field wooden pole in exploiting the different parameters such as: the age of the pole, its chemical treatment, its species, its knots etc. in order to define the pole's damage law, point 2. Statistical distribution law of the new wooden pole after grading by non destructive sorting (ultrasounds) of the high mechanical performances supports: This new distribution law is a Gaussian law or evolves to a Log or Weibull's law with 3 parameters according to the inspected species. This grading allows a revalorization of the properties of the new poles and of the design values while guaranteeing an index of reliability required by the design standards, or in improving directly this nominal reliability (economic gain and reliability gain). Statistical distribution law of an aged in-field population (20-50 years old) approached by a bi-modal law which depends on: The distribution law of the new component (see point 1) and its minimal extreme law, which is asymmetrical, for an observation on 50 years. The statistical distribution at the time t of the residual mechanical performances of a group of supports making a local net, evaluated by non destructive methods. The non destructive evaluation is based on the measurements of physical variables (density, biological moisture content) and some descriptive variables from natural origins (diameter, knots, cracks...) and from accidental origins (diameter reduction, lightning cracks...). The statistical distribution at the time t is then obtained on the basis of a model of multivariate non destructive evaluation, generalized to the whole of species and treatments. This model is the other concrete goal to reach in this thesis. As a conclusion, the research demonstrates the influence and the interaction of the new pole grading (distribution at t0) on the modelisation of the distribution at ti (multivariate non destructive model). The data used for the mentioned modelisations come from a significant international database with a large amount of inspected wood poles and with studied cases. This database is the synthesis of about 15 years of research and development leaded by IBOIS-EPFL and its international partners. The probabilistic approaches are then validated by a huge database allowing thus to be directly exploitable. On this basis, all the standards dealing with the new poles and dealing with the controls and maintenances of a wooden pole networks, could be re-examined for a double gain: Concerning the economy: by increasing the capacity of the new poles profiting of an objective quality assurance, and by increasing the life time of the in-field pole, in knowing how to purge only the ones which are under the critical threshold of damage Concerning the reliability: by increasing the reliability of the network from the stage "new pole", by eliminating the weakest components, and by maintaining this reliability during all the life time of the network thanks to a cyclic preventive maintenance (every 5 to 8 years) and the replacement of only the weakened poles.

The energy transition in Switzerland specifies reduction of the country’s primary energy consumption and greenhouse gas emissions. In the building sector, one way to achieve this is through adoption of efficient and sustainable technologies. With this aim, the building energy systems are gaining complexity. Modelling tools are becoming increasingly important in the design and evaluation of such systems. Two modelling methods are commonly used, simulation and optimization, which substantially differ in their approach and purpose. Simulation is a descriptive tool used to virtually represent systems’ behaviour under given conditions and operation strategies. Optimization approaches explore the possible scenarios under given system limits; they determine the best solution by optimizing an objective function described in mathematical form. In this project, the interactions and complementary use of these methods are investigated through evaluation of the newly installed energy systems of multi-familiy houses built in Zurich. The systems include renewable sources from solar and geothermal energy. The main components are a heat pump coupled with a borehole heat exchanger (BHE) and solar photovoltaic panels (PV) or hybrid photovoltaic panels (PV/T) for electricity production. Two system variants including different rates of borehole regeneration resulting from free cooling of the houses or injection of heat produced by PV/T are evaluated. The systems are divided into interlinked blocks representing the main components of the system: the borehole heat exchanger and surrounding ground, the heat pump, the storage tanks, the piping systems and the photovoltaic installations. The blocks are simulated with an hourly time scale. Energy consumption, self-consumption of on site produced electricity as well as performance degradation due to the long term operation are some of the results obtained from simulation. Monitoring data are used for calibration and validation of the simulation model. Separate optimization models of the evaluated energy systems are then developed. They are improved based on system characteristics obtained from the simulation model. In this way, optimal operation strategies which take into account specific operational limits are identified. Different levels of precision of parameters integration from the simulation results (constant over the year and hourly defined) are implemented and the influences on the optimization results are investigated. The results of the optimization are subsequently implemented in the simulation model. The results of the simulation found that the boreholes are sustainably exploited due the conservative design of the system according the simulated operation conditions. The influence of the regeneration is noticeable, however the simulation of the long term operation of a hypothetical variant without regeneration induces only a limited efficiency decrease and can also be considered as sustainable. The optimization results showed that the self consumption of electricity produced on site can be significantly improved by adapting the heat pump production profile with the electricity production; this was achieved through judicious management of the storage tanks. The analysis of the different level of parameter integration in the optimization model revealed that the optimization results are sensitive to the level of precision of the parameters. The iterative process allowed to combine the strengths of both modelling methods: the simulation was used to precisely describe the systems behaviour. Operational profiles from optimization were subsequently used in the simulation. The outcome of this process helped to better understand the system, identify optimal operating strategies while considering system limitations, as well as presenting

2017Covariance operators play a fundamental role in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on the space of such operators, as well as with their intrinsic infinite-dimensionality.
We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric.
We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

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