Section: New Results

Analysis of high dimensional data

Participants: K. Duarte, S. Ferrigno, J.-M. Monnez, A. Muller-Gueudin, S. Tindel

Online partial principal component analysis of a data stream

Consider a data stream and suppose that each data vector is a realization of a random vector whose expectation varies with time, the law of the centered data vector being stationary. Consider the principal component analysis (PCA) of this centered vector called partial PCA. In this study are defined online estimations of the first principal axes by stochastic approximation processes using a data batch at each step of the process or all the data until the current step. This extends a former result obtained by J.-M. Monnez by using one data vector at each step. This is applied to partial generalized canonical correlation analysis by defining a stochastic approximation process of the metric involved in this case using all the data until the current step. If the expectation of the data vector varies according to a linear model, a stochastic approximation process of the model parameters is used. All these processes can be performed in parallel. A forthcoming preprint by R. Bar and J.-M. Monnez will discuss those aspects.

Data analysis for cumulative exposure Index

Everyone is subject to environmental exposures from various sources, with negative health impacts (air, water and soil contamination, noise ...) or with positive effects (e.g., green space). Studies considering such complex environmental settings in a global manner are rare. In [5] we propose to use statistical factor and cluster analyses to create a composite exposure index with a data-driven approach, in view to assess the environmental burden experienced by populations. The study was carried out in the Great Lyon area (France, 1.2M inhabitants) at the census block group (BG) scale. We used as environmental indicators ambient air NO2 annual concentrations, noise levels, proximity to green spaces, to industrial plants, to polluted sites and to road traffic. Although it cannot be applied directly for risk or health effect assessment, the resulting index can help to identify hot spots of cumulative exposure, to prioritize urban policies or to compare the environmental burden across study areas in an epidemiological framework.

A simultaneous stepwise covariate selection

In supervised learning the number of values of a response variable to predict can be high. Also clustering them in a few clusters can be useful to perform relevant supervised classification analysis. On the other hand selecting relevant covariates is a crucial step to build robust and efficient prediction models, especially when too many covariates are available in regard to the overall sample size. As a first attempt to solve these problems, we had already devised in a previous study an algorithm that simultaneously clusters the levels of a categorical response variable in a limited number of clusters and selects forward the best covariates by alternate minimization of Wilks's Lambda. In the project carried out this year, we first extend the former version of the algorithm to a more general framework where Wilks's Lambda can be replaced by any model selection criterion. We also turned forward selection into stepwise selection in order to remove covariates in real time if necessary. Finally an application of our algorithm to real datasets from peanut allergy studies allowed to get confirmation of some previously published results and suggested new discoveries. The possibilities of this algorithm are promising and it is hoped to be useful for many practitioners.

Prognostic value of the Strauss estimated plasma

We describe here an application oriented study lead jointly by J.-M. Monnez and a medical team under the supervision of E. Albuisson at CHU Brabois. The objective is to assess the prognostic value of estimations of volemia, or of their variations, beyond clinical examination in a post-hoc analysis of the Eplerenone Post-Acute Myocardial Infarction (AMI) Heart Failure (HF) Efficacy and Survival Study (EPHESUS). Assessing congestion post-discharge is indeed challenging but of paramount importance to optimize patient management and prevent hospital readmissions. The analysis was performed in a subset on 4957 patients with available data (within a full dataset of 6632 patients). Study endpoint was cardiovascular death and/or hospitalization for HF between month 1 and month 3 after post-AMI HF. Estimated plasma volume variations between baseline and month 1 were estimated by the Strauss formula, which includes hemoglobin and hematocrit ratios. Other potential predictors including congestion surrogates, hemodynamic and renal variables, and medical history variables were tested. An instantaneous estimation of plasma volume at month 1, ePVS M1, was defined and also tested. Multivariate analysis was performed using stepwise logistic regression and linear discriminant analysis. In HF complicating MI, congestion assessed by the Strauss formula and an instantaneous derived measurement of plasma volume displayed an added predictive value of early cardiovascular events, beyond routine clinical assessment. Trials assessing congestion management guided by this simple tool to monitor plasma volume are warranted.

Non parametric estimation of the conditional cumulative distribution function

This project fits into the global aim of improving local regression techniques. Indeed, we propose in [21] to study the local linear estimator of the conditional distribution function. Namely, having an i.i.d. sample ${(X_i,Y_i)}_{1\le i\le n}$, we estimate the conditional distribution function $F\left(t\right|x)=\mathbb{P}(Y\le t|X=x)$ by:

where ${}^{\left(1\right)}$ denotes the order 1 of the local polynomial estimator, ${\widehat{f}}_{n,j}$ stands for a kernel estimator with order $j$ of the probability density function $f_X$ of $X$, ${\widehat{r}}_{n,j}$ estimates the distribution of the couple $(X,Y)$ and $h_n$ is a bandwidth parameter.

This estimator is a particular case of the local polynomial estimators. It is the local polynomial estimator of order $p=1$. Another simpler estimator, with order $p=0$, is well known as the Nadaraya-Watson estimator.

We are interested in showing the advantage of this estimator over the Nadaraya-Watson estimator. We show asymptotic results for our estimator (exact rate of uniform consistency), and establish also uniform asymptotic certainty bands for the conditional cumulative distribution function.

We obtain the following result under some assumptions on the cumulative distribution $F$, $f_X$, the kernel $K$ and the bandwidth $h_n$,

where

As corollaries of this result, we extend our results to other statistical functions, such as the quantiles and the regression function.

We illustrate our results with simulations and an application on foetopathologic data.

We have also started a study about the regression function in the application on foetopathologic data. We consider the nonparametric model

where $Y$ is the fetal weight , $X$ are the gestational weeks, $m$ is a smooth unknown function and $ϵ$ the error. The goal is to provide a test to detect significant features (or change points) of this regression curve. The regression curve is estimated using local polynomial kernel smoothers.