Clustering processes

The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for the case of a known number of clusters, consistency can be achieved under the only assumption that the joint distribution of the data is stationary ergodic (no parametric or Markovian assumptions, no assumptions of independence, neither between nor within the samples). If the number of clusters is unknown, consistency can be achieved under appropriate assumptions on the mixing rates of the processes. (again, no parametric or independence assumptions). In both cases we give examples of simple (at most quadratic in each argument) algorithms which are consistent.Comment: in proceedings of ICML 2010. arXiv-admin note: for version 2 of this article please see: arXiv:1005.0826v

Agent modelling of cluster formation processes in regional economic systems

The subject matter of this research is the processes of the spontaneous clustering in the regional economy. The purpose is the development and approbation of the modeling algorithm of these processes. The hypothesis: the processes of spontaneous clustering in the social and economic environment are supposed to proceed not linearly, but intermittently. The following methods are applied: agent imitating modeling with an application of FOREL and k-means algorithms. The modeling algorithm is realized in the Python 3 programming language. The course regularities of clustering processes in the region are revealed: 1) the clustering processes are intensifying, the production uniformity is increasing; 2) the increase of the level of production uniformity leads to the leveling of customer behavior; 3) the producers of high-differentiated production reduce the level of its differentiation or leave the cluster; 4) the stages of steady functioning are illustrative for clustering processes, their change is followed with arising of bifurcation points; 5) the activation of clustering processes in regional economy leads to the revenue increase of the cluster participants, each of producers and of consumers, and to the growth of synergetic effect values. These results testify the nonlinearity of processes of clustering and ambiguity of their effects. The following conclusions have been drawn: 1) a modeling of the processes of spontaneous clustering in regional economy has showed that they proceed not linearly, a steady progressive development is followed with leaps; 2) the clustering of regional economy leads to the growth of the efficiency indicators of activities of cluster-concerned entities; 3) initiation and activation of the clustering processes requires a certain environment

Order of magnitude time-reversible Markov chains and characterization of clustering processes

We introduce the notion of order of magnitude reversibility (OM-reversibility) in Markov chains that are parametrized by a positive parameter \ep. OM-reversibility is a weaker condition than reversibility, and requires only the knowledge of order of magnitude of the transition probabilities. For an irreducible, OM-reversible Markov chain on a finite state space, we prove that the stationary distribution satisfies order of magnitude detailed balance (analog of detailed balance in reversible Markov chains). The result characterizes the states with positive probability in the limit of the stationary distribution as \ep \to 0, which finds an important application in the case of singularly perturbed Markov chains that are reducible for \ep=0. We show that OM-reversibility occurs naturally in macroscopic systems, involving many interacting particles. Clustering is a common phenomenon in biological systems, in which particles or molecules aggregate at one location. We give a simple condition on the transition probabilities in an interacting particle Markov chain that characterizes clustering. We show that such clustering processes are OM-reversible, and we find explicitly the order of magnitude of the stationary distribution. Further, we show that the single pole states, in which all particles are at a single vertex, are the only states with positive probability in the limit of the stationary distribution as the rate of diffusion goes to zero.Comment: 22 pages, 3 figure

Fixation for Distributed Clustering Processes

We study a discrete-time resource flow in $Z^d$, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs