You are welcome to the Seminar in Statistics!
Date: 9 May 2017 (Tuesday)
Time: 13:00 – 15:00
Place: K1073
Lecturer: Krzysztof Podgórski (Lund University)
Title:
Statistics at random events -- theory and applications
Abstract:
The talk intends to provide general introduction to a method of analyzing statistical
distributions of variables observed at random events of a stochastic process or field.
The methodology is aiming at very practical problems and its applicability will be
shown on several examples. Nevertheless, it also roots in some purely mathematical
considerations going back to the Banach Indicatrix theorem (1925) that later has
been generalized in multidimensional formulation through the area-coarea theorem
in geometrical measure theory. Our approach draws also on Kac (1943) and Rice's
(1944-45) results on the average number of zeros of a random signal. After
motivation and a survey of historical aspects of the theory, we turn to the
generalized Rice formula approach to deriving long-run distributions of
characteristics defined at random events of a stochastic process or field. We
illustrate it through a number of practical contexts in which the methodology proves
to be useful. In particular, we show applications to sea surface dynamical models
and analyses of stochastic vehicle responses.
Short break
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Lecturer: Josef Wilzén (Linköping University)
Title:
Physiological Gaussian Process Priors for the Hemodynamics in fMRI Analysis
(Jointly with Anders Eklund and Mattias Villani).
Abstract:
Inference from fMRI data faces the challenge that the hemodynamic system that
relates the underlying neural activity to the observed BOLD fMRI signal is not
known. We propose a new Bayesian model for task fMRI data with the following
features: (i) joint estimation of brain activity and the underlying hemodynamics, (ii)
the hemodynamics is modeled nonparametrically with a Gaussian process (GP) prior
with physiological information and (iii) predicted BOLD is not necessarily a linear
time-invariant (LTI) system. We place a GP prior directly on the predicted BOLD
time series, rather than on the hemodynamic response function as in previous
literature. This allows us to incorporate physiological information via the GP prior
mean in a flexible way. The prior mean function may be a standard LTI system
based on a canonical hemodynamic response function, or a more elaborate
physiological model such as the Balloon model. This gives us the nonparametric
flexibility of the GP, but allows the posterior to fall back on the physiologically
based prior when the data are weak. Results on simulated data show that even if we
use an erroneous prior for the GP, the proposed model is still able to discriminate
between active and non-active voxels in a satisfactory way and is able to discover
the true functional form of the underlying hemodynamics. The proposed model is
applied to real fMRI data, where our Gaussian process model finds activity where a
baseline model with fixed hemodynamics does not. The learned hemodynamics
show a time varying behavior, that can not easily be captured with by an LTI
system, even when the hemodynamic response function is estimated.
Lecturer: Anders Hildeman (Chalmers University of Technology)
Title:
Bayesian level set methods for point process data.
Abstract:
The level set method is a popular approach to inverse problems where interfaces
between regions of differing properties are to be identified.
Recent research has reformulated such problems in to a Bayesian context where
regularization is introduced by the choice of priors. We develop this approach for
point process data in spatial statistics. The observed data is assumed to be a point
pattern over a spatial domain consisting of several regions with fundamentally
different point pattern behavior, and where the classification of the spatial domain in
to these regions is unknown. The aim of the analyst might be either to classify the
regions, perform Kriging predictions or derive some field parameter properties from
one or several of the point pattern classes.
The proposed level set method can be viewed as an extension to the log-Gaussian
Cox process model, where the latent Gaussian random field is replaced by a latent
level set mixture of Gaussian fields.