On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $\Phi$-convexity. Under general conditions on the class of functions $\Phi$, the Krein-Milman-Ky Fan theorem asserts then, that every compact $\Phi$-convex subset of a Hausdorff space, is the $\Phi$-convex hull of its $\Phi$-extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of $\Phi$-extremal points by the smaller subset of $\Phi$-exposed points. We establish under general conditions on the class of functions $\Phi$, that every $\Phi$-convex compact metrizable subset of a Hausdorff space, is the $\Phi$-convex hull of its $\Phi$-exposed points. As a consequence we obtain that each convex weak compact metrizable (resp. convex weak$^*$ compact metrizable) subset of a Banach space (resp. of a dual Banach space), is the closed convex hull of its exposed points (resp. the weak$^*$ closed convex hull of its weak$^*$ exposed points). This result fails in general for compact $\Phi$-convex subsets that are not metrizable

Pontryagin principle for a Mayer problem governed by a delay functional differential equation

We establish Pontryagin principles for a Mayer's optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation

Interpretable Aircraft Engine Diagnostic via Expert Indicator Aggregation

Detecting early signs of failures (anomalies) in complex systems is one of the main goal of preventive maintenance. It allows in particular to avoid actual failures by (re)scheduling maintenance operations in a way that optimizes maintenance costs. Aircraft engine health monitoring is one representative example of a field in which anomaly detection is crucial. Manufacturers collect large amount of engine related data during flights which are used, among other applications, to detect anomalies. This article introduces and studies a generic methodology that allows one to build automatic early signs of anomaly detection in a way that builds upon human expertise and that remains understandable by human operators who make the final maintenance decision. The main idea of the method is to generate a very large number of binary indicators based on parametric anomaly scores designed by experts, complemented by simple aggregations of those scores. A feature selection method is used to keep only the most discriminant indicators which are used as inputs of a Naive Bayes classifier. This give an interpretable classifier based on interpretable anomaly detectors whose parameters have been optimized indirectly by the selection process. The proposed methodology is evaluated on simulated data designed to reproduce some of the anomaly types observed in real world engines.Comment: arXiv admin note: substantial text overlap with arXiv:1408.6214, arXiv:1409.4747, arXiv:1407.088

Impact of information cost and switching of trading strategies in an artificial stock market

This paper studies the switching of trading strategies and its effect on the market volatility in a continuous double auction market. We describe the behavior when some uninformed agents, who we call switchers, decide whether or not to pay for information before they trade. By paying for the information they behave as informed traders. First we verify that our model is able to reproduce some of the stylized facts in real financial markets. Next we consider the relationship between switching and the market volatility under different structures of investors. We find that there exists a positive relationship between the market volatility and the percentage of switchers. We therefore conclude that the switchers are a destabilizing factor in the market. However, for a given fixed percentage of switchers, the proportion of switchers that decide to buy information at a given moment of time is negatively related to the current market volatility. In other words, if more agents pay for information to know the fundamental value at some time, the market volatility will be lower. This is because the market price is closer to the fundamental value due to information diffusion between switchers.Comment: 15 pages, 9 figures, Physica A, 201

On the discretization of backward doubly stochastic differential equations

In this paper, we are dealing with the approximation of the process (Y,Z) solution to the backward doubly stochastic differential equation with the forward process X . After proving the L2-regularity of Z, we use the Euler scheme to discretize X and the Zhang approach in order to give a discretization scheme of the process (Y,Z)

Detecting changes in the fluctuations of a Gaussian process and an application to heartbeat time series

The aim of this paper is first the detection of multiple abrupt changes of the long-range dependence (respectively self-similarity, local fractality) parameters from a sample of a Gaussian stationary times series (respectively time series, continuous-time process having stationary increments). The estimator of the $m$ change instants (the number $m$ is supposed to be known) is proved to satisfied a limit theorem with an explicit convergence rate. Moreover, a central limit theorem is established for an estimator of each long-range dependence (respectively self-similarity, local fractality) parameter. Finally, a goodness-of-fit test is also built in each time domain without change and proved to asymptotically follow a Khi-square distribution. Such statistics are applied to heart rate data of marathon's runners and lead to interesting conclusions

A statistical network analysis of the HIV/AIDS epidemics in Cuba

The Cuban contact-tracing detection system set up in 1986 allowed the reconstruction and analysis of the sexual network underlying the epidemic (5,389 vertices and 4,073 edges, giant component of 2,386 nodes and 3,168 edges), shedding light onto the spread of HIV and the role of contact-tracing. Clustering based on modularity optimization provides a better visualization and understanding of the network, in combination with the study of covariates. The graph has a globally low but heterogeneous density, with clusters of high intraconnectivity but low interconnectivity. Though descriptive, our results pave the way for incorporating structure when studying stochastic SIR epidemics spreading on social networks

Limited operators and differentiability

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \R$ and every point $y\in Y$, $f\circ T$ is Fr\'echet differentiable at $y\in Y$ whenever $f$ is G\^ateaux differentiable at $T(y)\in X$

Anomaly Detection Based on Aggregation of Indicators

Automatic anomaly detection is a major issue in various areas. Beyond mere detection, the identification of the origin of the problem that produced the anomaly is also essential. This paper introduces a general methodology that can assist human operators who aim at classifying monitoring signals. The main idea is to leverage expert knowledge by generating a very large number of indicators. A feature selection method is used to keep only the most discriminant indicators which are used as inputs of a Naive Bayes classifier. The parameters of the classifier have been optimized indirectly by the selection process. Simulated data designed to reproduce some of the anomaly types observed in real world engines.Comment: 23rd annual Belgian-Dutch Conference on Machine Learning (Benelearn 2014), Bruxelles : Belgium (2014

A theoretical framework for trading experiments

A general framework is suggested to describe human decision making in a certain class of experiments performed in a trading laboratory. We are in particular interested in discerning between two different moods, or states of the investors, corresponding to investors using fundamental investment strategies, technical analysis investment strategies respectively. Our framework accounts for two opposite situations already encountered in experimental setups: i) the rational expectations case, and ii) the case of pure speculation. We consider new experimental conditions which allow both elements to be present in the decision making process of the traders, thereby creating a dilemma in terms of investment strategy. Our theoretical framework allows us to predict the outcome of this type of trading experiments, depending on such variables as the number of people trading, the liquidity of the market, the amount of information used in technical analysis strategies, as well as the dividends attributed to an asset. We find that it is possible to give a qualitative prediction of trading behavior depending on a ratio that quantifies the fluctuations in the model