Nonparametric Nearest Neighbor Random Process Clustering

We consider the problem of clustering noisy finite-length observations of stationary ergodic random processes according to their nonparametric generative models without prior knowledge of the model statistics and the number of generative models. Two algorithms, both using the L1-distance between estimated power spectral densities (PSDs) as a measure of dissimilarity, are analyzed. The first algorithm, termed nearest neighbor process clustering (NNPC), to the best of our knowledge, is new and relies on partitioning the nearest neighbor graph of the observations via spectral clustering. The second algorithm, simply referred to as k-means (KM), consists of a single k-means iteration with farthest point initialization and was considered before in the literature, albeit with a different measure of dissimilarity and with asymptotic performance results only. We show that both NNPC and KM succeed with high probability under noise and even when the generative process PSDs overlap significantly, all provided that the observation length is sufficiently large. Our results quantify the tradeoff between the overlap of the generative process PSDs, the noise variance, and the observation length. Finally, we present numerical performance results for synthetic and real data.Comment: IEEE International Symposium on Information Theory (ISIT), June 2015, to appea

Toward a generic representation of random variables for machine learning

This paper presents a pre-processing and a distance which improve the performance of machine learning algorithms working on independent and identically distributed stochastic processes. We introduce a novel non-parametric approach to represent random variables which splits apart dependency and distribution without losing any information. We also propound an associated metric leveraging this representation and its statistical estimate. Besides experiments on synthetic datasets, the benefits of our contribution is illustrated through the example of clustering financial time series, for instance prices from the credit default swaps market. Results are available on the website www.datagrapple.com and an IPython Notebook tutorial is available at www.datagrapple.com/Tech for reproducible research.Comment: submitted to Pattern Recognition Letter

Clustering Financial Time Series: How Long is Enough?

Researchers have used from 30 days to several years of daily returns as source data for clustering financial time series based on their correlations. This paper sets up a statistical framework to study the validity of such practices. We first show that clustering correlated random variables from their observed values is statistically consistent. Then, we also give a first empirical answer to the much debated question: How long should the time series be? If too short, the clusters found can be spurious; if too long, dynamics can be smoothed out.Comment: Accepted at IJCAI 201

Reducing statistical time-series problems to binary classification

We show how binary classification methods developed to work on i.i.d. data can be used for solving statistical problems that are seemingly unrelated to classification and concern highly-dependent time series. Specifically, the problems of time-series clustering, homogeneity testing and the three-sample problem are addressed. The algorithms that we construct for solving these problems are based on a new metric between time-series distributions, which can be evaluated using binary classification methods. Universal consistency of the proposed algorithms is proven under most general assumptions. The theoretical results are illustrated with experiments on synthetic and real-world data.Comment: In proceedings of NIPS 2012, pp. 2069-207

Upper Confidence Reinforcement Learning exploiting state-action equivalence

Leveraging an equivalence property on the set of states of state-action pairs in anMarkov Decision Process (MDP) has been suggested by many authors. We takethe study of equivalence classes to the reinforcement learning (RL) setup, whentransition distributions are no longer assumed to be known, in a discrete MDP withaverage reward criterion and no reset. We study powerful similarities betweenstate-action pairs related to optimal transport. We first analyze a variant of theUCRL2 algorithm called C-UCRL2, which highlights the clear benefit of leveragingthis equivalence structure when it is known ahead of time: the regret bound scalesas ~O(D√KCT) where C is the number of classes of equivalent state-action pairsand K bounds the size of the support of the transitions. A non trivial question iswhether this benefit can still be observed when the structure is unknown and mustbe learned while minimizing the regret. We propose a sound clustering techniquethat provably learn the unknown classes, but show that its natural combination withUCRL2 empirically fails. Our findings suggests this is due to the ad-hoc criterionfor stopping the episodes in UCRL2. We replace it with hypothesis testing, whichin turns considerably improves all strategies. It is then empirically validated thatlearning the structure can be beneficial in a full-blown RL problem