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A multiple filter test for the detection of rate changes in renewal processes with varying variance
(2014)
The thesis provides novel procedures in the statistical field of change point detection in time series.
Motivated by a variety of neuronal spike train patterns, a broad stochastic point process model is introduced. This model features points in time (change points), where the associated event rate changes. For purposes of change point detection, filtered derivative processes (MOSUM) are studied. Functional limit theorems for the filtered derivative processes are derived. These results are used to support novel procedures for change point detection; in particular, multiple filters (bandwidths) are applied simultaneously in oder to detect change points in different time scales.
Poster presentation from Twentieth Annual Computational Neuroscience Meeting: CNS*2011 Stockholm, Sweden. 23-28 July 2011. In statistical spike train analysis, stochastic point process models usually assume stationarity, in particular that the underlying spike train shows a constant firing rate (e.g. [1]). However, such models can lead to misinterpretation of the associated tests if the assumption of rate stationarity is not met (e.g. [2]). Therefore, the analysis of nonstationary data requires that rate changes can be located as precisely as possible. However, present statistical methods focus on rejecting the null hypothesis of stationarity without explicitly locating the change point(s) (e.g. [3]). We propose a test for stationarity of a given spike train that can also be used to estimate the change points in the firing rate. Assuming a Poisson process with piecewise constant firing rate, we propose a Step-Filter-Test (SFT) which can work simultaneously in different time scales, accounting for the high variety of firing patterns in experimental spike trains. Formally, we compare the numbers N1=N1(t,h) and N2=N2(t,h) of spikes in the time intervals (t-h,t] and (h,t+h]. By varying t within a fine time lattice and simultaneously varying the interval length h, we obtain a multivariate statistic D(h,t):=(N1-N2)/V(N1+N2), for which we prove asymptotic multivariate normality under homogeneity. From this a practical, graphical device to spot changes of the firing rate is constructed. Our graphical representation of D(h,t) (Figure 1A) visualizes the changes in the firing rate. For the statistical test, a threshold K is chosen such that under homogeneity, |D(h,t)|<K holds for all investigated h and t with probability 0.95. This threshold can indicate potential change points in order to estimate the inhomogeneous rate profile (Figure 1B). The SFT is applied to a sample data set of spontaneous single unit activity recorded from the substantia nigra of anesthetized mice. In this data set, multiple rate changes are identified which agree closely with visual inspection. In contrast to approaches choosing one fixed kernel width [4], our method has advantages in the flexibility of h.
Der Bolthausen-Sznitman Koaleszent ist ein zeitstetiger Markovprozess mit Werten in der Menge der Partitionen der natürlichen Zahlen. Der Prozess startet in Singletons und seine Dynamik erlaubt lediglich Übergänge in gröbere Partitionen. In dieser Arbeit wird der Bolthausen-Sznitman Koaleszent zum Zeitpunkt seines letzten Übergangs analysiert. Das Hauptresultat ist ein Grenzwertsatz, welcher eine gemeinsame Aussage sowohl über die Blockanzahl als auch über die Blockgrößen des Koaleszenten zu diesem Zeitpunkt macht. Dafür wird der Koaleszent durch ein gewisses Abholzverfahren zufälliger rekursiver Bäume modelliert, wobei diese Bäume wiederum anhand von Yule-Prozessen generiert werden.