Frank O'Brien

A Crowding Index for Finite Populations

The classical definition of population density used in social science disciplines as a measure of density or physical crowding has severe shortcomings since actual spatial orientation is disregarded. A formula is proposed to measure physical crowding or population density which corrects the deficiency of the conventional formulation. As a generalization of the Euclidean distance formula, the proposed index measures relative population density as a function of spatial orientation and size of geometric area. Three mathematical properties of the index stated as conjectures have been supported by Monte Carlo simulations. The formula is evaluated and justified on three key scientific criteria: parsimony, mathematical soundness and empirical verifiability. Crowding is an obvious problem in the small compartmentalized workspaces of U.S. Navy nuclear submarines. The measurement of physical crowding by use of the proposed formula in Concepts of Operations Experiments conducted at the Newport Laboratory of the Naval Underwater Systems Center is reported and discussed.

APPROXIMATION METHODS IN DISCRETE SPATIAL DENSITY ANALYSIS FOR FINITE POPULATIONS

The authors three-parameter square-root model for the measurement of discrete spatial density in human populations was previously derived under the assumption that exact coordinate locations of the density points were available. The model, called the population density index (PDI) model, has been expanded to include a set of routines for calculating two-dimensional spatial density measures based upon in situ geometric approximations of the interobject Euclidean distance measure for any finite sample size. The derivation and specification of the algorithm for the abbreviated calculation routines are presented and exemplified. The author has been able to apply the methods of the PDI model to submarine environments at the U.S. Naval Underwater Systems Center, resulting in several U.S. Patent applications.

Generalization of the Population Density Index Model

The authors population density index (PDI) model was previously derived for the measurement of population density in studies employing 100 density points or less. The PDI model is now generalized through a derived equation which allows its basic component, the unit lattice average Euclidean distance, to be calculated for any finite lattice rectangle or square in a population density study.

The Moi Formula for Improper Exponential Definite Integrals

A general formula is developed for solving a type of improper exponential definite integral of order n in the number plane. Termed the Moi Formula, it is shown to produce substantially simpler derivations of the finite moments of a probability distribution employed for assessing stochastic randomness, such as recently published by the authors. Other applications of the integral formula are discussed.

COMPARISON OF TWO TESTS OF PLANAR RANDOMNESS

Two statistical methods were derived recently for testing a randomness hypothesis in ecological spatial studies. The relative performance of each method is investigated in this methodological study with synthetically constructed random distributions.

A TEST FOR RANDOMNESS IN THREE DIMENSIONS

A mathematical method based on a nearest neighbor spatial Poisson process is described for assessing stochastic randomness in three-dimensional Euclidean space. The classical central limit theorem is invoked to obtain a normal approximation formula for testing the hypothesis of randomness. The performance of the method is evaluated with Monte Carlo simulations. A brief description is given of the software employed for implementation of the method in practice.

Adaptive filtering in underwater tracking with correlated measurement noise

The authors describe a multistage, hierarchical adaptive filtering technique derived for a complex, real-time nonlinear state estimation problem where the measurement noise is markedly non-Gaussian. The relative efficiency of the derived mathematical model was demonstrated through a Monte Carlo simulation experiment employing noisy angle-of-arrival measurements for undersea tracking.<<ETX>>

Counting your M and N'S

This paper introduces into the population density index model the measure AO which is derived to summarize the occupancy rate of a rectangular grid in the interpretation of spatial density information of finite populations. An approximation of the new measure is also given.

POISSON ENSEMBLES IN THREE DIMENSIONS

The authors recently derived a method for assessing stochastic randomness in three dimensional Euclidean space. The method was derived from a nearest neighbor spatial Poisson process. An alternative probability model based on a box-counting method derived from a partial sum of a Poisson series is presented in this paper. The performance of the method is evaluated through Monte Carlo simulations with synthetically constructed random distributions. A comparison between the new discrete distribution method and the initial distance method showed that a greater likelihood of detecting randomness existed among populations with the box-counting method.

A NONMETRIC MEASURE OF EUCLIDEAN DISTANCE

The results of a large scale simulation study to test the fundamental properties of the authors finite population spatial density model are reported. A conjecture is given relating metric distance to a nonmetric estimate in two-dimensional space.