eprintid: 18619
rev_number: 11
eprint_status: archive
userid: 1778
dir: disk0/00/01/86/19
datestamp: 2015-04-20 10:20:10
lastmod: 2015-04-21 08:22:15
status_changed: 2015-04-20 10:20:10
type: doctoralThesis
metadata_visibility: show
creators_name: Dueck, Johannes
title: Dependencies in Complex Systems
subjects: 510
divisions: 110400
adv_faculty: af-11
abstract: A task in statistics is to find meaningful associations or dependencies between multivariate random variables or in multivariate, time-dependent stochastic processes. Hawkes (1971) introduced the powerful multivariate point process model of mutually 
exciting processes (Hawkes model) to explain causal structure in data. Therefore, we discuss 
several causality concepts and show that causal structure is fully encoded in the 
corresponding Hawkes kernels. Hence, for causal inference and for establishing graphical models 
induced by causality it is necessary to estimate the Hawkes kernels. We provide a nonparametric, consistent and asymptotically normal
estimator of the Hawkes kernels depending on the increments on a time scale with mesh $\Delta$
using methods from infinite order regression and time series analysis. To illustrate our results we apply our method to EEG data 
from the spinal dorsal horn of a rat. 

To tackle the problem for random samples of random vectors we examine a new dependence measure, namely distance correlation (Sz\'ekely, Rizzo and Bakirov; 2007). Distance correlation provides a strikingly simple sample version in order to test for independence between two random vectors of arbitrary dimensions and finite first moments. However, distance correlation is not well understood on the population side and it fails to be invariant under the group of all invertible affine transformations. Hence, we introduce the affinely invariant distance correlation and compute the analytic usual distance correlation and affinely invariant distance correlation in various settings: for multivariate normal distributions and for Lancaster probabilities (e.g. the bivariate gamma distribution) explicitly. Furthermore, we generalize an integral which is at the core of distance correlation.   
date: 2015
id_scheme: DOI
id_number: 10.11588/heidok.00018619
ppn_swb: 1656138670
own_urn: urn:nbn:de:bsz:16-heidok-186192
date_accepted: 2015-04-13
advisor: HASH(0x564e1a220a08)
language: eng
bibsort: DUECKJOHANDEPENDENCI2015
full_text_status: public
citation:   Dueck, Johannes  (2015) Dependencies in Complex Systems.  [Dissertation]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/18619/1/PhD-Thesis_Dueck.pdf