Kolmogorov–Smirnov test

Summary

In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). In essence, the test answers the question "How likely is it that we would see a collection of samples like this if they were drawn from that probability distribution?" or, in the second case, "How likely is it that we would see two sets of samples like this if they were drawn from the same (but unknown) probability distribution?". It is named after Andrey Kolmogorov and Nikolai Smirnov. The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The nu

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Applying the Multivariate Time-Rescaling Theorem to Neural Population Models

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Massachusetts Institute of Technology Press2011

The coordinated, collective spiking activity of neuronal populations encodes and processes information. One approach towards understanding such population based computation is to fit statistical models to simultaneously recorded spike trains and use these models to make inferences about correlations, functional connectivity, and so forth. Any statistical model must, prior to making inferences from it, be validated by an appropriate goodness of fit measure, but as yet there is no commonly agreed upon statistical sufficiency test for neuronal population models. For single neuron spike trains, the time rescaling theorem provides a goodness of fit test consistent with their point process nature. Interspike intervals (ISIs) are rescaled, as a function of the model's conditional intensity function (time and history dependent spike probability) to be independent and exponentially distributed if the model is accurate. A Kolmogorov Smirnov (KS) test between the rescaled ISIs and the exponential distribution is then used to check goodness of fit. The time rescaling theorem applies to models of single spike trains, however some researchers have used it to validate population models, by applying it separately to each neuron in the population. Here we demonstrate that this approach is insufficient and that tests based upon single neuron statistics can not validate a population model. If interactions between neurons are ignored univariate tests may be erroneously passed, even when the interactions are strong. To remedy this we present the multivariate version of the time rescaling theorem and develop a practical statistical test for implementing it. The test relies on the fact that if the conditional intensity functions for all neurons are modeled correctly then the rescaled ISIs should be independent not only within a single spike train, but also across spike trains. Testing for such independence across spike trains determines if the collective activity of the neuronal population is captured by the model. Using Generalized Linear Models (GLMs) fitted to both simulated spike trains and spike trains recorded experimentally in monkey V1 cortex we demonstrate that population models are much more reliably validated by the multivariate test. This suggests that neuronal populations must be considered collectively as important features of their dynamics can only be detected from a multivariate viewpoint.

2010