When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput /spl lambda/(n) obtainable by each node for a randomly chosen destination is /spl Theta/(W//spl radic/(nlogn)) bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is /spl Theta/(W/spl radic/An) bit-meters per second. Thus even under optimal circumstances, the throughput is only /spl Theta/(W//spl radic/n) bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.

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The capacity of wireless networks

Statistics for Spatial Data.

ilbstract. Autocorrelation is a very general statistical property of ecological variables observed across geographic space; its most common forms are patches and gradients. Spatial autocorrelation. which comes either from the physical forcing of environmental variables or from community processes, presents a problem for statistical testing because autocorrelated data violate the assumption of independence of most standard statistical procedures. The paper discusses first how autocorrelation in ecological variables can be described and measured. with emphasis on mapping techniques. Then. proper statistical testing in the presence of autocorrelation is briefly discussed. Finally. ways are presented of explicitly introducing spatial structures into ecological models. Two approaches are proposed: in the raw-data approach, the spatial structure takes the form of a polynomial of the x and .v geographic coordinates of the sampling stations; in the matrix approach. the spatial structure is introduced in the form of a geographic distance matrix among locations. These two approaches are compared in the concluding section. A table provides a list of computer programs available for spatial analysis.

https://www.jstor.org/stable/pdfplus/1939924.pdf

Spatial Autocorrelation: Trouble or New Paradigm?

Species distributional or trait data based on range map (extent-of-occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method’s implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing

/pdf/methods-to-account-for-spatial-autocorrelation-in-the-5fbbuizhqs.pdf

Methods to account for spatial autocorrelation in the analysis of species distributional data : a review

This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.

/pdf/coverage-control-for-mobile-sensing-networks-3gs75qafnm.pdf

Coverage control for mobile sensing networks

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

/pdf/spatial-tessellations-concepts-and-applications-of-voronoi-2ivn15gdqy.pdf

Spatial Tessellations: Concepts and Applications of Voronoi Diagrams

In analogy to the exact distribution of the Durbin—Watson d statistic for serial autocorrelation of regression residuals, the exact small sample distribution of Moran's I statistic (or alternatively Geary's c) can be derived. Use of algebraic results by Koerts and Abrahamse and theoretical results by Imhof, allows the authors to determine by numerical integration the exact distribution function of Moran's I for normally distributed variables. For the case in which the explanatory variables have been neglected, an upper and a lower bound can be given within which the exact distribution of Moran's I for regression residuals will lie. Furthermore, the proposed methodology is flexible enough to investigate related topics such as the characteristics of the exact distribution for distinct spatial structures as well as their different specifications, the exact power function under different spatial autocorrelation levels, and the distribution of Moran's I for nonnormal random variables.

The Exact Distribution of Moran's I

Introduction Measures of Dispersion Quadrat Analysis Measures of Dispersion Distance Methods Measures of Arrangements Summary

/pdf/point-pattern-analysis-1s0rznvtvh.pdf

Point pattern analysis

Hot spots are typically locations of abundant phenomena. In ecology, hot spots are often detected with a spatially global threshold, where a value for a given observation is compared with all values in a data set. When spatial relationships are important, spatially local definitions - those that compare the value for a given observation with locations in the vicinity, or the neighbourhood of the observation - provide a more explicit consideration of space. Here we outline spatial methods for hot spot detection: kernel estimation and local measures of spatial autocorrelation. To demonstrate these approaches, hot spots are detected in landscape level data on the magnitude of mountain pine beetle infestations. Using kernel estimators, we explore how selection of the neighbourhood size (τ) and hot spot threshold impact hot spot detection. We found that as T increases, hot spots are larger and fewer; as the hot spot threshold increases, hot spots become larger and more plentiful and hot spots will reflect coarser scale spatial processes. The impact of spatial neighbourhood definitions on the delineation of hot spots identified with local measures of spatial autocorrelation was also investigated. In general, the larger the spatial neighbourhood used for analysis, the larger the area, or greater the number of areas, identified as hot spots.

Detecting spatial hot spots in landscape ecology

In spatial statistics and spatial econometrics two coding schemes are used predominately. Except for some initial work, the properties of both coding schemes have not been investigated systematically. In this paper we do so for significant spatial processes specified as either a simulta-neous autoregressive or a moving average process. Results show that the C-coding scheme emphasizes spatial objects with relatively large numbers of connections, such as those in the interior of a study region. In contrast, the W-coding scheme assigns higher leverage to spatial objects with few connections, such as those on the periphery of a study region. To address this topology-induced heterogeneity, we design a novel S-coding scheme whose properties lie in between those of the C-coding and the W-coding schemes. To compare these three coding schemes within and across the different spatial processes, we find a set of autocorrelation parameters that makes the processes stochastically homologous via a method based on the ex...

Barry Boots

Papers

Spatial Tessellations: Concepts and Applications of Voronoi Diagrams

The Exact Distribution of Moran's I

Point pattern analysis

Detecting spatial hot spots in landscape ecology

A Variance-Stabilizing Coding Scheme for Spatial Link Matrices: