François Bavaud

On the Schoenberg Transformations in Data Analysis: Theory and Illustrations

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A distance-based discriminant algorithm and a robust multidimensional centroid estimate illustrate the theory, closely connected to the Gaussian kernels of Machine Learning.

Testing spatial autocorrelation in weighted networks: the modes permutation test

In a weighted spatial network, as specified by an exchange matrix, the variances of the spatial values are inversely proportional to the size of the regions. Spatial values are no more exchangeable under independence, thus weakening the rationale for ordinary permutation and bootstrap tests of spatial autocorrelation. We propose an alternative permutation test for spatial autocorrelation, based upon exchangeable spatial modes, constructed as linear orthogonal combinations of spatial values. The coefficients obtain as eigenvectors of the standardized exchange matrix appearing in spectral clustering and generalize to the weighted case the concept of spatial filtering for connectivity matrices. Also, two proposals aimed at transforming an accessibility matrix into an exchange matrix with a priori fixed margins are presented. Two examples (inter-regional migratory flows and binary adjacency networks) illustrate the formalism, rooted in the theory of spectral decomposition for reversible Markov chains.

Interpolating between random walks and shortest paths: a path functional approach

General models of network navigation must contain a deterministic or drift component, encouraging the agent to follow routes of least cost, as well as a random or diffusive component, enabling free wandering. This paper proposes a thermodynamic formalism involving two path functionals, namely an energy functional governing the drift and an entropy functional governing the diffusion. A freely adjustable parameter, the temperature, arbitrates between the conflicting objectives of minimising travel costs and maximising spatial exploration. The theory is illustrated on various graphs and various temperatures. The resulting optimal paths, together with presumably new associated edges and nodes centrality indices, are analytically and numerically investigated.

Spatial Weights: Constructing Weight-Compatible Exchange Matrices from Proximity Matrices

Exchange matrices represent spatial weights as symmetric probability distributions on pairs of regions, whose margins yield regional weights, generally well-specified and known in most contexts. This contribution proposes a mechanism for constructing exchange matrices, derived from quite general symmetric proximity matrices, in such a way that the margin of the exchange matrix coincides with the regional weights. Exchange matrices generate in turn diffusive squared Euclidean dissimilarities, measuring spatial remoteness between pairs of regions. Unweighted and weighted spatial frameworks are reviewed and compared, regarding in particular their impact on permutation and normal tests of spatial autocorrelation. Applications include tests of spatial autocorrelation with diagonal weights, factorial visualization of the network of regions, multivariate generalizations of Moran’s I, as well as “landscape clustering,” aimed at creating regional aggregates both spatially contiguous and endowed with similar features.

Generalized Factor Analyses for Contingency Tables

Quotient dissimilarities constitute a broad aggregation-invariant family; among them, f-dissimilarities are Euclidean embeddable (Bavaud, 2002). We present a non-linear principal components analysis (NPA) applicable to any quotient dissimilarity, based upon the spectral decomposition of the central inertia. For f-dissimilarities, the same decomposition yields a non-linear correspondence analysis (NCA), permitting us to modulate as finely as wished the contributions of positive or negative deviations from independence. The resulting coordinates exactly reproduce the original dissimilarities between rows or between columns; however, Huygens’s weak principle is generally violated, as measured by a quantity we call ’eccentricity’.

Flow-Based Dissimilarities: Shortest Path, Commute Time, Max-Flow and Free Energy

Random-walk based dissimilarities on weighted networks have demonstrated their efficiency in clustering algorithms. This contribution considers a few alternative network dissimilarities, among which a new max-flow dissimilarity, and more general flow-based dissimilarities, freely mixing shortest paths and random walks in function of a free parameter—the temperature. Their geometrical properties and in particular their squared Euclidean nature are investigated through their power indices and multidimensional scaling properties. In particular, formal and numerical studies demonstrate the existence of critical temperatures, where flow-based dissimilarities cease to be squared Euclidean. The clustering potential of medium range temperatures is emphasized.

Spectral Clustering and Multidimensional Scaling: A Unified View

Spectral clustering is a procedure aimed at partitionning a weighted graph into minimally interacting components. The resulting eigen-structure is determined by a reversible Markov chain, or equivalently by a symmetric transition matrix F. On the other hand, multidimensional scaling procedures (and factorial correspondence analysis in particular) consist in the spectral decomposition of a kernel matrix K. This paper shows how F and K can be related to each other through a linear or even non-linear transformation leaving the eigen-vectors invariant. As illustrated by examples, this circumstance permits to define a transition matrix from a similarity matrix between n objects, to define Euclidean distances between the vertices of a weighted graph, and to elucidate the “flow-induced” nature of spatial auto-covariances.

Multi-labelled Image Segmentation in Irregular, Weighted Networks: A Spatial Autocorrelation Approach.

Image segmentation and spatial clustering both face the same primary problem, namely to gather together spatial entities which are both spatially close and similar regarding their features. The parallelism is partic- ularly obvious in the case of irregular, weighted networks, where methods borrowed from spatial analysis and general data analysis (soft K-means) may serve at segmenting images, as illustrated on four examples. Our semi-supervised approach considers soft memberships (fuzzy clustering) and attempts to minimize a free energy functional made of three ingredients : a within-cluster features dispersion (hard K-means), a network partitioning objective (such as the Ncut or the modularity) and a regularizing entropic term, enabling an itera- tive computation of the locally optimal soft clusters. In particular, the second functional enjoys many possible formulations, arguably helpful in unifying various conceptualizations of space through the probabilistic selec- tion of pairs of neighbours, as well as their relation to spatial autocorrelation (Moran’s I).

Geographical Exploration and Analysis Extended to Textual Content (Short Paper)

Textual and socio-economical regional features can be integrated and merged by linearly combining the between-regions corresponding dissimilarities. The scheme accommodates for various squared Euclidean socio-economical and textual dissimilarities (such as chi2 or cosine dissimilarities derived from document-term matrix or topic modelling). Also, spatial configuration of the regions can be represented by a weighted unoriented network whose vertex weights match the relative importance of regions. Association between the network and the dissimilarities expresses in the multivariate spatial autocorrelation index delta, generalizing Morans I, whose local version can be cartographied. Our case study bears on the Wikipedia notices and socio-economic profiles for the 2251 Swiss municipalities, whose weights (socio-economical or textual) can be freely chosen.

Correspondence Analysis, Cross-Autocorrelation and Clustering in Polyphonic Music

This paper proposes to represent symbolic polyphonic musical data as contingency tables based upon the duration of each pitch for each time interval. Exploratory data analytic methods involve weighted multidimensional scaling, correspondence analysis, hierarchical clustering, and general autocorrelation indices constructed from temporal neighborhoods. Beyond the analysis of single polyphonic musical scores, the methods sustain inter-voices as well as inter-scores comparisons, through the introduction of ad hoc measures of configuration similarity and cross-autocorrelation. Rich musical patterns emerge in the related applications, and preliminary results are encouraging for clustering tasks.