Harold J. Larson

Least squares estimation of linear splines with unknown knot locations

Abstract Many papers have appeared in the literature over the past 30 years concerning the least squares estimation of spline functions. All of these previously published procedures rely on numerical search techniques when faced with splines which have knots at unknown locations. In the special case of linear splines with unknown knot locations, such numerical searches are unnecessary, as shown in this paper. If one specifies a desired (known) number of knots using linear splines, the least squares solution(s) for the minimizing location(s) can be explicitly located. Some data generated by a computer simulation of a war game is used to illustrate the procedure.

Least Squares Estimation of the Components of a Symmetric Matrix

Y = XB + E (1) where Y and X are both s X t (s > t), of rank t, each of whose components are real and known (experimentally determined), B is the t X t symmetric flexibility matrix whose components are to be estimated and E is the matrix of discrepancies. This note exhibits the form of the symmetric matrix B. which minimizes the sum of squares of the components of E and gives a short numerical example illustrating an intuitional pitfall.

Graphical displays of synchronization of tactical units

To maximize combat readiness, the U.S. Army employs highly instrumented combat ranges for training troops under the most realistic possible conditions. Many of these ranges accommodate force-on-force battles, including simulated firings of weapons and of kills against the opponent; the physical variables used in these simulated kills (times of events, locations of players) are then available for computer replays, as well as investigations of the effects of changes to some battle details. This paper describes the use of player and event time-location data to visually portray some aspects of synchronization of forces, one of the classic (and current) battlefield tenets.

Computer visualization of battlefield tenets

The Battle Enhanced Analysis Methodologies (BEAM) project was designed to investigate the use of computer graphics in describing the performance of battalion-sized units in simulated combat. These descriptions were to be data-based and objective, providing useful critiques of actual performance according to standard Army doctrine. They would be natural candidates for use at the Armys Combat Training Centers. The project was conducted in two phases. In the first, objective graphic displays were derived which portray the destructive potential of direct fire weapons (the shooter must be able to see the target) in the defense. These displays allow straightforward objective comparisons of different defensive alignments and, from simulated battle runs, of defensive fire control strategies. These references also describe simple uncluttered displays that portray the movements and interactions of company (or higher) sized units throughout a battle. This paper describes results of the second phase of the BEAM project. A major result is the derivation of displays which portray the destructive potential of indirect fire weapons (the shooter normally cannot see the target) in the defense using the same units as the direct fire displays. This allows separate and joint examination of the direct and indirect fire destructive potential, providing, among other things, objective measures of the synchronization and agility of a force, as well as indicators of its intelligence function.

Identification of Failing Mechanical Systems through Spectrometric Oil Analysis

The results obtained from spectrometric analyses may be viewed as being stochastic in nature, so that decisions based upon them are necessarily accompanied by uncertainties. A class of rules is developed for making decisions concerning whether a mechanical system may be failing, based upon spectroscopic analyses of the systems oil over a period of time. Some considerations that went into the development of these rules, including studies of past analysis records and experiments, are presented. Identification procedures of the type suggested should perform well in connection with a computerized analysis system, at least insofar as routinely monitoring the well behaved systems, while calling the attention of appropriate personnel to possibly discrepant systems.

Introduction to Probability Theory and Statistical Inference.

Set Theory. Probability. Random Variables and Distribution Functions. Some Standard Probability Laws. Jointly Distributed Random Variables. Descriptive and Inferential Statistics. Estimation of Parameters. Tests of Hypotheses. Least Squares and Regression. Nonparametric Methods. Bayesian Methods. Appendices. Answers to Exercises. Index.