Test de Kolmogorov-Smirnov | EPFL Graph Search

En statistiques, le test de Kolmogorov-Smirnov est un test d'hypothèse utilisé pour déterminer si un échantillon suit bien une loi donnée connue par sa fonction de répartition continue, ou bien si deux échantillons suivent la même loi. Le test de Kolmogorov-Smirnov est par exemple utilisé pour tester la qualité d'un générateur de nombres aléatoires. Principe gauche|vignette|Fonction de répartition empirique Fn (en bleu) et fonction de répartition théorique F (rouge). On modélise une loi donnée par sa fonction de répartition F (voir courbe rouge sur la figure). La fonction de répartition associe à tout x (en abscisses), la probabilité qu'une variable suivant cette loi soit plus petite que x (en ordonnées). Le test consiste alors à comparer la fonction de répartition de la loi (fonction de répartition théorique) à la fonction de répartition empirique Fn (en bleu). Cette dernière est une fonction de répartition obtenue en attribuant la probabilité 1/n à chacun des n élément

Metrizing Fairness

Daniel Kuhn, Yves Rychener, Bahar Taskesen

We study supervised learning problems for predicting properties of individuals who belong to one of two demographic groups, and we seek predictors that are fair according to statistical parity. This means that the distributions of the predictions within the two groups should be close with respect to the Kolmogorov distance, and fairness is achieved by penalizing the dissimilarity of these two distributions in the objective function of the learning problem. In this paper, we showcase conceptual and computational benefits of measuring unfairness with integral probability metrics (IPMs) other than the Kolmogorov distance. Conceptually, we show that the generator of any IPM can be interpreted as a family of utility functions and that unfairness with respect to this IPM arises if individuals in the two demographic groups have diverging expected utilities. We also prove that the unfairness-regularized prediction loss admits unbiased gradient estimators if unfairness is measured by the squared L2-distance or by a squared maximum mean discrepancy. In this case the fair learning problem is susceptible to efficient stochastic gradient descent (SGD) algorithms. Numerical experiments on real data show that these SGD algorithms outperform state-of-the-art methods for fair learning in that they achieve superior accuracy-unfairness trade-offs—sometimes orders of magnitude faster. Finally, we identify conditions under which statistical parity can improve prediction accuracy.

2022

Applying the Multivariate Time-Rescaling Theorem to Neural Population Models

Joao Emanuel Felipe Gerhard

Statistical models of neural activity are integral to modern neuroscience. Recently interest has grown in modeling the spiking activity of populations of simultaneously recorded neurons to study the effects of correlations and functional connectivity on neural information processing. However, any statistical model must be validated by an appropriate goodness-of-fit test. Kolmogorov-Smirnov tests based on the time-rescaling theorem have proven to be useful for evaluating point-process-based statistical models of single-neuron spike trains. Here we discuss the extension of the time-rescaling theorem to the multivariate (neural population) case. We show that even in the presence of strong correlations between spike trains, models that neglect couplings between neurons can be erroneously passed by the univariate time-rescaling test. We present the multivariate version of the time-rescaling theorem and provide a practical step-by-step procedure for applying it to testing the sufficiency of neural population models. Using several simple analytically tractable models and more complex simulated and real data sets, we demonstrate that important features of the population activity can be detected only using the multivariate extension of the test.

Massachusetts Institute of Technology Press2011

Statistical tests for neural population models - The multivariate time rescaling theorem

Joao Emanuel Felipe Gerhard

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

Statistical tests for neural population models - The multivariate time rescaling theorem

Joao Emanuel Felipe Gerhard

2010