Parameter Estimation in Panels of Intercorrelated Time Series

We consider parameter estimation in panels of intercorrelated time series. By a factorisation of the conditional log-likelihood function we obtain a new estimator \hat{a}_n,T for panels of intercorrelated autoregressive time series. We generalise this model to a factor model, where a single underlying background process is responsible for the common behaviour of the time series in the panel, and derive the corresponding conditional maximum likelihood estimators. Consistency and asymptotic normality are proved for the estimators in both models. It turns out that \hat{a}_n,T is asymptotically equivalent to the estimator \hat{a}_HT given in Hjellvik and Tjøstheim (1999a) if the number of time series in the panel tends to infinity. It is more efficient if only the length of the time series increases. Furthermore the mean squared errors of the dominant terms in the stochastic expansions of these estimators have the ratio (n-1)/n, which indicates that already the small sample bias of \hat{a}_n,T is smaller than that of \hat{a}_HT . These properties are confirmed in the simulations. The second part of the thesis is concerned with robust estimation in panels of autoregressive time series. We investigate three different approaches. Firstly we robustify the above estimators in a direct way. Furthermore we generalise the robust autocovariance estimator of Ma and Genton (2000) to the panel case. We define a panel breakdown point for time series in two ways depending on the type of outliers assumed and compute its value for the panel autocovariance estimator. The estimated autocovariances are then used for the robust parameter estimation. Finally we propose an outlier test based upon the phase space representation of the time series in the panel, which can be used for eliminating outliers from the data set before using a non-robust method of estimation. We derive the asymptotic distribution of the test statistic and define a robust version of the test. For comparison we include other estimators in the analysis. The performance of the proposed robust procedures is investigated in a simulation study. For assessing the applicability of the above methods we analyse two sets of empirical data

Probabilistic stimulation mapping from intra-operative thalamic deep brain stimulation data in essential tremor

International audienceAbstract Deep brain stimulation (DBS) is a therapy for Parkinson’s disease (PD) and essential tremor (ET). The mechanism of action of DBS is still incompletely understood. Retrospective group analysis of intra-operative data recorded from ET patients implanted in the ventral intermediate nucleus of the thalamus (Vim) is rare. Intra-operative stimulation tests generate rich data and their use in group analysis has not yet been explored. 

Objective: To implement, evaluate, and apply a group analysis workflow to generate probabilistic stimulation maps (PSM) using intra-operative stimulation data from ET patients implanted in Vim. 

A group-specific anatomical template was constructed based on the Magnetic Resonance Imaging (MRI) scans of 6 ET patients and 13 PD patients. Intra-operative test data (total: n=1821) from the 6 ET patients was analyzed: patient-specific electric field simulations together with tremor assessments obtained by a wrist-based acceleration sensor were transferred to this template. Occurrence and weighted mean maps were generated. Voxels associated with symptomatic response were identified through a linear mixed model approach to form a PSM. Improvements predicted by the PSM were compared to those clinically assessed. Finally, the PSM clusters were compared to those obtained in a multicenter study using data from chronic stimulation effects in ET. 

Regions responsible for improvement identified on the PSM were in the posterior sub-thalamic area (PSA) and at the border between the Vim and ventro-oral nucleus of the thalamus (VO). The comparison with literature revealed a center-to-center distance of less than 5mm and an overlap score (Dice) of 0.4 between the significant clusters. 

Our workflow and intra-operative test data from 6 ET-Vim patients identified effective stimulation areas in PSA and around Vim and VO, affirming existing medical literature. This study supports the potential of probabilistic analysis of intra-operative stimulation test data to reveal DBS's action mechanisms and to assist surgical planning