A method for the detection of changes in the expectation in univariate sequences is provided. Moving sum processes are studied. These rely on the selection of a tuning bandwidth. Here, a framework to overcome bandwidth selection is presented - the bandwidth adjusts gradually. For that, moving sum processes are made dependent on both time and the bandwidth: the domain becomes a triangle. On the triangle, paths are constructed which systematically lead to change points. An algorithm is provided that estimates change points by subsequent consideration of paths. Strong consistency for the number and location of change points is shown. Simulations support estimation precision. A companion R-package mscp is made available on CRAN.

A method for change point detection is proposed. We consider a univariate sequence of independent random variables with piecewise constant expectation and variance, apart from which the distribution may vary periodically. We aim to detect change points in both expectation and variance. For that, we propose a statistical test for the null hypothesis of no change points and an algorithm for change point detection. Both are based on a bivariate moving sum approach that jointly evaluates the mean and the empirical variance. The joint consideration helps improve inference as compared to separate univariate approaches. We infer on the strength and the type of changes with confidence. Nonparametric methodology supports the analysis of diverse data. Additionally, a multi-scale approach addresses complex patterns in change points and effects. We demonstrate the performance through theoretical results and simulation studies. A companion R-package jcp (available on CRAN) is discussed.

An approach is presented that combines a statistical test for peak detection with the estimation of peak positions in time series. Motivated by empirical observations in neuronal recordings, we aim at investigating peaks of different heights and widths. We use a moving window approach to compare the differences of estimated slope coefficients of local regression models. We combine multiple windows and use the global maximum of all different processes as a test statistic. After rejection, a multiple filter algorithm combines peak positions estimated from multiple windows. Analysing neuronal activity recorded in anaesthetized mice, the procedure could identify significant differences between two brain states concerning peak occurrences and intermediate down states showing no peaks. This suggests that the method can be useful in the analysis of time series showing variability of peak shapes. The method is implemented in the-package(available on CRAN).

We present a model for point processes with gamma distributed increments. We assume a piecewise constant latent process controlling shape and scale of the distribution. For the discrete number of states of the latent process we use a non-parametric assumption by utilizing a Chinese restaurant process (CRP). For the inference of such inhomogeneous gamma processes with an unbounded number of states we do Bayesian inference using Markov Chain Monte Carlo. Finally, we apply the inference algorithm to simulated point processes and to empirical spike train recordings, which inherently possess non-stationary and non-Poissonian behavior.

A framework for the detection of change points in the expectation in sequences of random variables is presented. Specifically, we investigate time series with general distributional assumptions that may show an unknown number of change points in the expectation occurring on multiple time scales and that may also contain change points in other parameters. To that end we propose a multiple filter test (MFT) that tests the null hypothesis of constant expectation and, in case of rejection of the null hypothesis, an algorithm that estimates the change points. The MFT has three important benefits. First, it allows for general distributional assumptions in the underlying model, assuming piecewise sequences of i.i.d. random variables, where also relaxations with regard to identical distribution or independence are possible. Second, it uses a MOSUM type statistic and an asymptotic setting in which the MOSUM process converges weakly to a functional of a Brownian motion which is then used to simulate the rejection threshold of the statistical test. This approach enables a simultaneous application of multiple MOSUM processes which improves the detection of change points that occur on different time scales. Third, we also show that the method is practically robust against changes in other distributional parameters such as the variance or higher order moments which might occur with or even without a change in expectation. A function implementing the described test and change point estimation is available in the R package MFT.

Non-stationarity of the rate or variance of events is a well-known problem in the description and analysis of time series of events, such as neuronal spike trains. A multiple filter test (MFT) for rate homogeneity has been proposed earlier that detects change points on multiple time scales simultaneously. It is based on a filtered derivative approach, and the rejection threshold derives from a Gaussian limit process $L$ which is independent of the point process parameters. Here we extend the MFT to variance homogeneity of life times. When the rate is constant, the MFT extends directly to the null hypothesis of constant variance. In the presence of rate change points, we propose to incorporate estimates of these in the test for variance homogeneity, using an adaptation of the test statistic. The resulting limit process shows slight deviations from $L$ that depend on unknown process parameters. However, these deviations are small and do not considerably change the properties of the statistical test. This allows practical application, e.g.~to neuronal spike trains, which indicates various profiles of rate and variance change points.

multiple filter test for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process G asymptotically takes the form G ∼ L, while L is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is to describe the behavior of G under the alternative hypothesis of rate changes and potential simultaneous variance changes. We derive the approximation G ∼Δ·( Λ + L) , with deterministic functions Δ and Λ . The function Λ accounts for the systematic deviation of G in the neighborhood of a change point. When only the rate changes, Λ is hat shaped. When also the variance changes, Λ takes the form of a shark’s fin. In addition, the parameter estimates required in practical application are not consistent in the neighborhood of a change point. Therefore, we derive the factor Δ termed here the distortion function. It accounts for the lack in consistency and describes the local parameter estimating process relative to the true scaling of the filtered derivative process.

The statistical analysis of neuronal spike trains by models of point processes often relies on the assumption of constant process parameters. However, it is a well-known problem that the parameters of empirical spike trains can be highly variable, such as for example the firing rate. In order to test the null hypothesis of a constant rate and to estimate the change points, a Multiple Filter Test (MFT) and a corresponding algorithm (MFA) have been proposed that can be applied under the assumption of independent inter spike intervals (ISIs). As empirical spike trains often show weak dependencies in the correlation structure of ISIs, we extend the MFT here to point processes associated with short range dependencies. By specifically estimating serial dependencies in the test statistic, we show that the new MFT can be applied to a variety of empirical firing patterns, including positive and negative serial correlations as well as tonic and bursty firing. The new MFT is applied to a data set of empirical spike trains with serial correlations, and simulations show improved performance against methods that assume independence. In case of positive correlations, our new MFT is necessary to reduce the number of false positives, which can be highly enhanced when falsely assuming independence. For the frequent case of negative correlations, the new MFT shows an improved detection probability of change points and thus, also a higher potential of signal extraction from noisy spike trains.

Empirical point processes such as neuronal spike trains can show changes in the rate at multiple time scales. In order to test the null hypothesis of constant rate and to estimate the change points, a multiple filter test (MFT) has been proposed by Messer et al. (2014) that uses filtered derivative processes and that can be applied under the assumption of independent life times. Here we extend the MFT to point processes with ergodic life times associated with short range dependencies. For practical application, we derive local parameter estimators under m-dependence. Our simulations illustrate the necessity of incorporating correlation estimators within the MFT as well as good performance when local parameter estimators are used. We also discuss principles of estimating m in practice, suggesting that it can be preferable to neglect small serial correlations for the sake of variability reduction when using finite analysis windows in practice. Finally, we estimate m and apply the extended MFT to an empirical data set of spike trains that show serial correlations of low order.

Neuronal spike trains can show variability with respect to process parameters such as the rate or variability of inter spike intervals. These changes can occur on fast and slow time scales, including also simultaneous and separate changes in different process parameters. Building up on results of [1] we present a multiple filter technique (MFT) that detects change points in the rate and variance of point processes on multiple time scales simultaneously. In particular, we use a filtered derivative process and its limit behavior under stationarity. The method also extends to higher order moments. The separate detection of rate and variance changes requires two techniques: First, rate changes need to be detected, irrespective of potential variance changes. To this end, our approach allows the identification of rate changes in point processes with a certain variability in their lifetimes. Second, the identified rate changes need to be considered when analyzing variance changes. We investigate the empirical properties of our asymptotic MFT method in simulations and apply the MFT to spike trains recorded from auditory cortex of behaving mice, illustrating rate and variability dynamics during the task.

Nonstationarity of the event rate is a persistent problem in modeling time series of events, such as neuronal spike trains. Motivated by a variety of patterns in neurophysiological spike train recordings, we define a general class of renewal processes. This class is used to test the null hypothesis of stationary rate versus a wide alternative of renewal processes with finitely many rate changes (change points). Our test extends ideas from the filtered derivative approach by using multiple moving windows simultaneously. To adjust the rejection threshold of the test, we use a Gaussian process, which emerges as the limit of the filtered derivative process. We also develop a multiple filter algorithm, which can be used when the null hypothesis is rejected in order to estimate the number and location of change points. We analyze the benefits of multiple filtering and its increased detection probability as compared to a single window approach. Application to spike trains recorded from dopamine midbrain neurons in anesthetized mice illustrates the relevance of the proposed techniques as preprocessing steps for methods that assume rate stationarity. In over 70% of all analyzed spike trains classified as rate nonstationary, different change points were detected by different window sizes.

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)|