Zbyněk Pawlas

First-spike latency in the presence of spontaneous activity

A new statistical method for the estimation of the response latency is proposed. When spontaneous discharge is present, the first spike after the stimulus application may be caused by either the stimulus itself, or it may appear due to the prevailing spontaneous activity. Therefore, an appropriate method to deduce the response latency from the time to the first spike after the stimulus is needed. We develop a nonparametric estimator of the response latency based on repeated stimulations. A simulation study is provided to show how the estimator behaves with an increasing number of observations and for different rates of spontaneous and evoked spikes. Our nonparametric approach requires very few assumptions. For comparison, we also consider a parametric model. The proposed probabilistic model can be used for both single and parallel neuronal spike trains. In the case of simultaneously recorded spike trains in several neurons, the estimators of joint distribution and correlations of response latencies are also introduced. Real data from inferior colliculus auditory neurons obtained from a multielectrode probe are studied to demonstrate the statistical estimators of response latencies and their correlations in space.

Parameters of spike trains observed in a short time window

We study the estimation of statistical moments of interspike intervals based on observation of spike counts in many independent short time windows. This scenario corresponds to the situation in which a target neuron occurs. It receives information from many neurons and has to respond within a short time interval. The precision of the estimation procedures is examined. As the model for neuronal activity, two examples of stationary point processes are considered: renewal process and doubly stochastic Poisson process. Both moment and maximum likelihood estimators are investigated. Not only the mean but also the coefficient of variation is estimated. In accordance with our expectations, numerical studies confirm that the estimation of mean interspike interval is more reliable than the estimation of coefficient of variation. The error of estimation increases with increasing mean interspike interval, which is equivalent to decreasing the size of window (less events are observed in a window) and with decreasing the number of neurons (lower number of windows).

Weak and strong convergence of empirical distribution functions from germ-grain processes

We observe randomly placed random compact sets (called grains or particles) in a bounded, convex sampling window W n of the d-dimensional Euclidean space which is assumed to expand unboundedly in all directions as n→∞. In addition, we suppose that the grains are independent copies of a so-called typical grain Ξ0, which are shifted by the atoms of a homogeneous point process Ψ in such a way that each individual grain lying within W n can be observed. We define an appropriate estimator ˆ F n (t) for the distribution function F(t) of some m-dimensional vector f(Ξ0)=(f 1(Ξ0), ˙s, f m (Ξ0)) (describing shape and size of Ξ0) on the basis of the corresponding data vectors of those grains which are completely observable in W n . As main results, we prove a Glivenko-type theorem for ˆ F n (t) and the weak convergence of the multivariate empirical processes √Ψ(W n )(ˆ F n (t)−F(t)) to an m-dimensional Brownian bridge process as n→∞.

Particle sizes from sectional data.

We propose a new statistical method for obtaining information about particle size distributions from sectional data without specific assumptions about particle shape. The method utilizes recent advances in local stereology. We show how to estimate separately from sectional data the variance due to the local stereological estimation procedure and the variance due to the variability of particle sizes in the population. Methods for judging the difference between the distribution of estimated particle sizes and the distribution of true particle sizes are also provided.

Nonparametric Testing of Distribution Functions in Germ-grain Models

Germ-grain models are random closed sets in the d-dimensional Euclidean space ℝd which admit a representation as union of random compact sets (called grains) shifted by the atoms (called germs) of a point process. In this note we consider the distribution function F of an m-dimensional random vector describing shape and size parameters of the typical grain of a stationary germ-grain model. We suggest a ratio-unbiased weighted (Horvitz-Thompson type) empirical distribution function \(\hat F_n \) to estimate F, based on the corresponding data vectors of those shifted grains which lie completely within the sampling window Wn ⊆ ℝd. Since, as Wn increases, the empirical process \(\hat F_n \)(t) − F(t) (after scaling) converges weakly to an m-parameter Brownian bridge process, it is possible for the particular case where m = 1, to examine the the goodness-of-fit of observed data to a hypothesised continuous distribution function F, analogous to the Kolmogorov-Smirnov test.

On the computation of area probabilities based on a spatial stochastic model for precipitation cells and precipitation amounts

A main task of weather services is the issuing of warnings for potentially harmful weather events. Automated warning guidances can be derived, e.g., from statistical post-processing of numerical weather prediction using meteorological observations. These statistical methods commonly estimate the probability of an event (e.g. precipitation) occurring at a fixed location (a point probability). However, there are no operationally applicable techniques for estimating the probability of precipitation occurring anywhere in a geographical region (an area probability). We present an approach to the estimation of area probabilities for the occurrence of precipitation exceeding given thresholds. This approach is based on a spatial stochastic model for precipitation cells and precipitation amounts. The basic modeling component is a non-stationary germ-grain model with circular grains for the representation of precipitation cells. Then, we assign a randomly scaled response function to each precipitation cell and sum these functions up to obtain precipitation amounts. We derive formulas for expectations and variances of point precipitation amounts and use these formulas to compute further model characteristics based on available sequences of point probabilities. Area probabilities for arbitrary areas and thresholds can be estimated by repeated Monte Carlo simulation of the fitted precipitation model. Finally, we verify the proposed model by comparing the generated area probabilities with independent rain gauge adjusted radar data. The novelty of the presented approach is that, for the first time, a widely applicable estimation of area probabilities is possible, which is based solely on predicted point probabilities (i.e., neither precipitation observations nor further input of the forecaster are necessary). Therefore, this method can be applied for operational weather predictions.