Information de Fisher | EPFL Graph Search

En statistique, l'information de Fisher quantifie l'information relative à un paramètre contenue dans une distribution. Elle est définie comme l'espérance de l'information observée, ou encore comme la variance de la fonction de score. Dans le cas multi-paramétrique, on parle de matrice d'information de Fisher. Elle a été introduite par R.A. Fisher. Définition Soit f(x ; θ) la distribution de vraisemblance d'une variable aléatoire X (qui peut être multidimensionnelle), paramétrée par θ. Le score est défini comme la dérivée partielle de la log-vraisemblance par rapport au paramètre θ : \frac{\partial}{\partial \theta} \log f(X;\theta)=\frac{1}{f(X;\theta)}\frac{\partial f(X;\theta)}{\partial \theta}. L'information de Fisher est alors définie comme le moment d'ordre deux de la fonction de score : I(\theta)=E\left[ \left. \left(\frac{\partial}{\partial \theta} \log f(X;\theta) \right)^2 \right| \theta \right]

Semi-supervised Extraction of Audio-Visual Sources

Patricia Calatayud Martinez

This report presents a semi-supervised method to jointly extract audio-visual sources from a scene. It consist of applying a supervised method to segment the video signal followed by an automatic process to properly separate the audio track. This approach starts with the user interaction to select the audio-visual target we want to cut. This labels the problem into two parts. One is the foreground, the object we aim to extract. The background is the remaining content which sets the other part. The segmentation method will be performed over the video signal to split its visual content into foreground and background. It is based on describing the video information as a volume. It is composed of pixels and relationships among them which set the features of the problem. This 3D structure can be interpreted as a graph. Therefore the segmentation is achieved by applying a Graph Cuts algorithm which provides the suitable results. In this work the audio separation process is an automatic task. Due to this we need to extract prior information from the audio-visual data available. Therefore a link between audio and video channels is deﬁned by an audio-visual motion map. This is a video sequence which provides the amount of synchronous motion with the audio track. An audio-based video diﬀusion method have been developed to obtain this kind of signal. Applying the video segmentation labelling to the audio-visual motion map we obtain the distribution of the amount motion into foreground and background. According to the range of these values it is possible to asses when the corresponding audio is on/oﬀ. At this point, audio samples of sources can be extract when the audio activity is due to only one. This is the prior information to learn audio models by means of spectral GMM estimation to assess the audio signals. Finally the separation method is applied. It consist of looking for the more suitable couple of states given the mixture spectrum of the total audio track. Therefore we achieve the audio separation goal. In this work we will study in detail all the stages towards the ﬁnal aim of our method, analyzing the performances of our approach.

2010

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

In this thesis, we treat robust estimation for the parameters of the Ornstein–Uhlenbeck process, which are the mean, the variance, and the friction. We start by considering classical maximum likelihood estimation. For the simulation study, where we also investigate the choice of the time lag, we use the method of moment (MoM) estimator as initial estimator for the friction parameter of the maximum likelihood estimator (MLE). However, in several aspects the MLE is not robust. For robustification, we first derive elementary M-estimates by extending the method of M-estimation from Huber (1981). We use an intuitively robustified MoM estimate as initial estimate and compare by means of simulation the M-estimate with the MLE. This approach is, however, only ad-hoc since Huber’s minimum Fisher information and minimax asymptotic variance theory remains incomplete for simultaneous location and scale, and does not cover more general models (as for example the Ornstein–Uhlenbeck process). A more general robustness concept due to Kohl et al. (2010), Rieder (1994), and Staab (1984) is based on local asymptotic normality (LAN), asymptotically linear (AL) estimates, and shrinking neighborhoods. We then apply this concept to the Ornstein–Uhlenbeck process. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. For two kind of neighborhoods (average and average square neighborhoods) we obtain optimally robust ICs. In case of average neighborhoods, their graph exhibits surprising, redescending behavior. For average square neighborhoods the graph is between the one of the elementary M-estimates and the MLE. Finally, we discuss the estimator construction, that is, the problem of constructing an estimator from the family of optimal ICs. We carry out in our context the One-Step construction dating back to LeCam and use both an intuitively robustified MoM estimate and the elementary M-estimate as initial estimate. This results in optimally AL estimates (for average and average square neighborhoods). By means of simulation we then compare the different estimators: MLE, elementary M-estimates, and optimally AL estimates. In addition, we give an application to electricity prices.

EPFL2012