Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSensitivity analysis for multidimensional and functional outputs
113
0
0.0
(
0
)
Added by
Alexandre Janon
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
Authors
Fabrice Gamboa
Ask ChatGPT about the research
No Arabic abstract
Let $X:=(X_1, ldots, X_p)$ be random objects (the inputs), defined on some probability space $(Omega,{mathcal{F}}, mathbb P)$ and valued in some measurable space $E=E_1timesldots times E_p$. Further, let $Y:=Y = f(X_1, ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $mathbb{H}$ ($mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Yinmathbb R$ ), when the output belongs to $mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.
rate research
Read More
The global sensitivity analysis of a complex numerical model often calls for the estimation of variance-based importance measures, named Sobol indices. Metamodel-based techniques have been developed in order to replace the cpu time-expensive computer code with an inexpensive mathematical function, which predicts the computer code output. The common metamodel-based sensitivity analysis methods are well-suited for computer codes with scalar outputs. However, in the environmental domain, as in many areas of application, the numerical model outputs are often spatial maps, which may also vary with time. In this paper, we introduce an innovative method to obtain a spatial map of Sobol indices with a minimal number of numerical model computations. It is based upon the functional decomposition of the spatial output onto a wavelet basis and the metamodeling of the wavelet coefficients by the Gaussian process. An analytical example is presented to clarify the various steps of our methodology. This technique is then applied to a real hydrogeological case: for each model input variable, a spatial map of Sobol indices is thus obtained.
Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts.
Network tomography has been regarded as one of the most promising methodologies for performance evaluation and diagnosis of the massive and decentralized Internet. This paper proposes a new estimation approach for solving a class of inverse problems in network tomography, based on marginal distributions of a sequence of one-dimensional linear projections of the observed data. We give a general identifiability result for the proposed method and study the design issue of these one dimensional projections in terms of statistical efficiency. We show that for a simple Gaussian tomography model, there is an optimal set of one-dimensional projections such that the estimator obtained from these projections is asymptotically as efficient as the maximum likelihood estimator based on the joint distribution of the observed data. For practical applications, we carry out simulation studies of the proposed method for two instances of network tomography. The first is for traffic demand tomography using a Gaussian Origin-Destination traffic model with a power relation between its mean and variance, and the second is for network delay tomography where the link delays are to be estimated from the end-to-end path delays. We compare estimators obtained from our method and that obtained from using the joint distribution and other lower dimensional projections, and show that in both cases, the proposed method yields satisfactory results.
Several statistics-based detectors, based on unimodal matrix models, for determining the number of sources in a field are designed. A new variance ratio statistic is proposed, and its asymptotic distribution is analyzed. The variance ratio detector is shown to outperform the alternatives. It is shown that further improvements are achievable via optimally selected rotations. Numerical experiments demonstrate the performance gains of our detection methods over the baseline approach.
Functional Magnetic Resonance Imaging (fMRI) maps cerebral activation in response to stimuli but this activation is often difficult to detect, especially in low-signal contexts and single-subject studies. Accurate activation detection can be guided by the fact that very few voxels are, in reality, truly activated and that activated voxels are spatially localized, but it is challenging to incorporate both these facts. We provide a computationally feasible and methodologically sound model-based approach, implemented in the R package MixfMRI, that bounds the a priori expected proportion of activated voxels while also incorporating spatial context. Results on simulation experiments for different levels of activation detection difficulty are uniformly encouraging. The value of the methodology in low-signal and single-subject fMRI studies is illustrated on a sports imagination experiment. Concurrently, we also extend the potential use of fMRI as a clinical tool to, for example, detect awareness and improve treatment in individual patients in persistent vegetative state, such as traumatic brain injury survivors.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
