A Statistical Toolbox For Mining And Modeling Spatial Data

Most data mining projects in spatial economics start with an evaluation of a set of attribute variables on a sample of spatial entities, looking for the existence and strength of spatial autocorrelation, based on the Moran’s and the Geary’s coefficients, the adequacy of which is rarely challenged, despite the fact that when reporting on their properties, many users seem likely to make mistakes and to foster confusion. My paper begins by a critical appraisal of the classical definition and rational of these indices. I argue that while intuitively founded, they are plagued by an inconsistency in their conception. Then, I propose a principled small change leading to corrected spatial autocorrelation coefficients, which strongly simplifies their relationship, and opens the way to an augmented toolbox of statistical methods of dimension reduction and data visualization, also useful for modeling purposes. A second section presents a formal framework, adapted from recent work in statistical learning, which gives theoretical support to our definition of corrected spatial autocorrelation coefficients. More specifically, the multivariate data mining methods presented here, are easily implementable on the existing (free) software, yield methods useful to exploit the proposed corrections in spatial data analysis practice, and, from a mathematical point of view, whose asymptotic behavior, already studied in a series of papers by Belkin & Niyogi, suggests that they own qualities of robustness and a limited sensitivity to the Modifiable Areal Unit Problem (MAUP), valuable in exploratory spatial data analysis

Three Simulation Algorithms for Labelled Transition Systems

Algorithms which compute the coarsest simulation preorder are generally designed on Kripke structures. Only in a second time they are extended to labelled transition systems. By doing this, the size of the alphabet appears in general as a multiplicative factor to both time and space complexities. Let $Q$ denotes the state space, $\rightarrow$ the transition relation, $\Sigma$ the alphabet and $P_{sim}$ the partition of $Q$ induced by the coarsest simulation equivalence. In this paper, we propose a base algorithm which minimizes, since the first stages of its design, the incidence of the size of the alphabet in both time and space complexities. This base algorithm, inspired by the one of Paige and Tarjan in 1987 for bisimulation and the one of Ranzato and Tapparo in 2010 for simulation, is then derived in three versions. One of them has the best bit space complexity up to now, $O(|P_{sim}|^2+|{\rightarrow}|.\log|{\rightarrow}|)$, while another one has the best time complexity up to now, $O(|P_{sim}|.|{\rightarrow}|)$. Note the absence of the alphabet in these complexities. A third version happens to be a nice compromise between space and time since it runs in $O(b.|P_{sim}|.|{\rightarrow}|)$ time, with $b$ a branching factor generally far below $|P_{sim}|$, and uses $O(|P_{sim}|^2.\log|P_{sim}|+|{\rightarrow}|.\log|{\rightarrow}|)$ bits

Bisimulations over DLTS in O(m.log n)-time

The well known Hopcroft's algorithm to minimize deterministic complete automata runs in $O(kn\log n)$-time, where $k$ is the size of the alphabet and $n$ the number of states. The main part of this algorithm corresponds to the computation of a coarsest bisimulation over a finite Deterministic Labelled Transition System (DLTS). By applying techniques we have developed in the case of simulations, we design a new algorithm which computes the coarsest bisimulation over a finite DLTS in $O(m\log n)$-time and $O(k+m+n)$-space, with $m$ the number of transitions. The underlying DLTS does not need to be complete and thus: $m\leq kn$. This new algorithm is much simpler than the two others found in the literature.Comment: Submitted to DLT'1

Ground states of the massless Dereziński–Gérard model

We consider the massless Dereziński–Gérard model introduced by Dereziński and Gérard in 1999. We give a sufficient condition for the existence of a ground state of the massless Dereziński–Gérard model without the assumption that the Hamiltonian of particles has compact resolvent