Multidimensional Urban Segregation - Toward A Neural Network Measure

We introduce a multidimensional, neural-network approach to reveal and measure urban segregation phenomena, based on the Self-Organizing Map algorithm (SOM). The multidimensionality of SOM allows one to apprehend a large number of variables simultaneously, defined on census or other types of statistical blocks, and to perform clustering along them. Levels of segregation are then measured through correlations between distances on the neural network and distances on the actual geographical map. Further, the stochasticity of SOM enables one to quantify levels of heterogeneity across census blocks. We illustrate this new method on data available for the city of Paris.Comment: NCAA S.I. WSOM+ 201

Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio

Simplification and Shift in Cognition of Political Difference: Applying the Geometric Modeling to the Analysis of Semantic Similarity Judgment

Perceiving differences by means of spatial analogies is intrinsic to human cognition. Multi-dimensional scaling (MDS) analysis based on Minkowski geometry has been used primarily on data on sensory similarity judgments, leaving judgments on abstractive differences unanalyzed. Indeed, analysts have failed to find appropriate experimental or real-life data in this regard. Our MDS analysis used survey data on political scientists' judgments of the similarities and differences between political positions expressed in terms of distance. Both distance smoothing and majorization techniques were applied to a three-way dataset of similarity judgments provided by at least seven experts on at least five parties' positions on at least seven policies (i.e., originally yielding 245 dimensions) to substantially reduce the risk of local minima. The analysis found two dimensions, which were sufficient for mapping differences, and fit the city-block dimensions better than the Euclidean metric in all datasets obtained from 13 countries. Most city-block dimensions were highly correlated with the simplified criterion (i.e., the left–right ideology) for differences that are actually used in real politics. The isometry of the city-block and dominance metrics in two-dimensional space carries further implications. More specifically, individuals may pay attention to two dimensions (if represented in the city-block metric) or focus on a single dimension (if represented in the dominance metric) when judging differences between the same objects. Switching between metrics may be expected to occur during cognitive processing as frequently as the apparent discontinuities and shifts in human attention that may underlie changing judgments in real situations occur. Consequently, the result has extended strong support for the validity of the geometric models to represent an important social cognition, i.e., the one of political differences, which is deeply rooted in human nature

Semiparametric Construction of Spatial Generalized Hedonic Models for Private Properties

This paper analyzes the empirical hedonic prices for non-standard condominiums and single family houses using nonparametric estimates as well as a generalized additive model for the spatial generalization of the attractiveness of all Swiss communities. We find that the assumption of log-linearity for continuous variables does not hold but can be replaced by partwise log-linear or quadratic terms. Due to the topographical segmentation of Switzerland, driving times seem to be more adequate than geographical distances to explain the price level of a village with the price level of its neighbours. We show, that using metric multidimensional scaling, the driving times between the villages can be converted into artificial coordinates with three principal axis to serve as a basis for the prediction of the macro-locations of the Swiss villagesHedonic prices; private property; Switzerland; robust regression; splines; multidimensional scaling

Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi

In several disciplines, as diverse as shape analysis, locationtheory, quality control, archaeology, and psychometrics, it can beof interest to fit a circle through a set of points. We use theresult that it suffices to locate a center for which the varianceof the distances from the center to a set of given points isminimal. In this paper, we propose a new algorithm based oniterative majorization to locate the center. This algorithm isguaranteed to yield a series nonincreasing variances until astationary point is obtained. In all practical cases, thestationary point turns out to be a local minimum. Numericalexperiments show that the majorizing algorithm is stable and fast.In addition, we extend the method to fit other shapes, such as asquare, an ellipse, a rectangle, and a rhombus by making use ofthe class of $l_p$ distances and dimension weighting. In addition,we allow for rotations for shapes that might be rotated in theplane. We illustrate how this extended algorithm can be used as atool for shape recognition.iterative majorization;location;optimization;shape analysis

The Past, Present, and Future of Multidimensional Scaling

Multidimensional scaling (MDS) has established itself as a standard tool for statisticians and applied researchers. Its success is due to its simple and easily interpretable representation of potentially complex structural data. These data are typically embedded into a 2-dimensional map, where the objects of interest (items, attributes, stimuli, respondents, etc.) correspond to points such that those that are near to each other are empirically similar, and those that are far apart are different. In this paper, we pay tribute to several important developers of MDS and give a subjective overview of milestones in MDS developments. We also discuss the present situation of MDS and give a brief outlook on its future

ON THE USE OF NON-EUCLIDEAN ISOTROPY IN GEOSTATISTICS

This paper investigates the use of non-Euclidean distances to characterize isotropic spatial dependence for geostatistical related applications. A simple example is provided to demonstrate there are no guarantees that existing covariogram and variogram functions remain valid (i.e.\ positive definite or conditionally negative definite) when used with a non-Euclidean distance measure. Furthermore, satisfying the conditions of a metric is not sufficient to ensure the distance measure can be used with existing functions. Current literature is not clear on these topics. There are certain distance measures that when used with existing covariogram and variogram functions remain valid, an issue that is explored. No new theorems are provided, rather links between existing theorems and definitions related to the concepts of isometric embedding, conditionally negative definiteness, and positive definiteness are used to demonstrate classes of valid norm dependent isotropic covariogram and variogram functions, results most of which have yet to appear in mainstream geostatistical literature or application. These classes of functions extend the well known classes by adding a parameter to define the distance norm. In practice, this distance parameter can be set a priori to represent, for example, the Euclidean distance, or kept as a parameter to allow the data to choose the distance norm. Applications of the latter are provided for demonstration

Mixed Tree and Spatial Representation of Dissimilarity Judgments

Whereas previous research has shown that either tree or spatial representations of dissimilarity judgments may be appropriate, focussing on the comparative fit at the aggregate level, we investigate whether there is heterogeneity among subjects in the extent to which their dissimilarity judgments are better represented by ultrametric tree or spatial multidimensional scaling models. We develop a mixture model for the analysis of dissimilarity data, that is formulated in a stochastic context, and entails a representation and a measurement model component. The latter involves distributional assumptions on the measurement error, and enables estimation by maximum likelihood. The representation component allows dissimilarity judgments to be represented either by a tree structure or by a spatial configuration, or a mixture of both. In order to investigate the appropriateness of tree versus spatial representations, the model is applied to twenty empirical data sets. We compare the fit of our model with that of aggregate tree and spatial models, as well as with mixtures of pure trees and mixtures of pure spaces, respectively. We formulate some empirical generalizations on the relative importance of tree versus spatial structures in representing dissimilarity judgments at the individual level.Multidimensional scaling;tree models;mixture models;dissimilarity judgments

Non-stationary patterns of isolation-by-distance: inferring measures of local genetic differentiation with Bayesian kriging

Patterns of isolation-by-distance arise when population differentiation increases with increasing geographic distances. Patterns of isolation-by-distance are usually caused by local spatial dispersal, which explains why differences of allele frequencies between populations accumulate with distance. However, spatial variations of demographic parameters such as migration rate or population density can generate non-stationary patterns of isolation-by-distance where the rate at which genetic differentiation accumulates varies across space. To characterize non-stationary patterns of isolation-by-distance, we infer local genetic differentiation based on Bayesian kriging. Local genetic differentiation for a sampled population is defined as the average genetic differentiation between the sampled population and fictive neighboring populations. To avoid defining populations in advance, the method can also be applied at the scale of individuals making it relevant for landscape genetics. Inference of local genetic differentiation relies on a matrix of pairwise similarity or dissimilarity between populations or individuals such as matrices of FST between pairs of populations. Simulation studies show that maps of local genetic differentiation can reveal barriers to gene flow but also other patterns such as continuous variations of gene flow across habitat. The potential of the method is illustrated with 2 data sets: genome-wide SNP data for human Swedish populations and AFLP markers for alpine plant species. The software LocalDiff implementing the method is available at http://membres-timc.imag.fr/Michael.Blum/LocalDiff.htmlComment: In press, Evolution 201

Structuring time in human lateral entorhinal cortex

Episodic memories consist of event information linked to spatio-temporal context. Notably, the hippocampus is involved in the encoding, representation and retrieval of temporal relations that comprise a context, but it remains largely unclear how coding for elapsed time arises in the hippocampal-entorhinal region. The entorhinal cortex (EC), the main cortical input structure of the hippocampus, has been hypothesized to provide temporal tags for memories via contextual drift and recent evidence demonstrates that time can be decoded from population activity in the rodent lateral EC. Here, we use fMRI to show that the anterior-lateral EC (alEC), the human homologue region of rodent lateral EC, maps the temporal structure of events. Participants acquired knowledge about temporal and spatial relationships between object positions - dissociated via teleporters - along a fixed route through a virtual city. Multi-voxel pattern similarity in alEC changed through learning to reflect elapsed time between event memories. Furthermore, we reconstructed the temporal structure of object relationships from alEC pattern similarity change. In contrast to the hippocampus, which maps the subjective time between event memories in this task, the temporal map in alEC reflected the objective time elapsed between events. Our findings provide evidence for the notion that alEC represents the temporal structure of memories, putatively derived from slowly-varying population signals during learning. Further, our findings suggest a dissociation between objective and subjective temporal maps in EC and hippocampus; thereby providing novel evidence for the role of the hippocampal-entorhinal region in representing time for episodic memory