Configural Frequency Analysis: Methods, Models, and Applications

This book focuses on providing a physical context to a variety of statistical concepts through the use of geometrical and mechanical models. The material is presented in a concise and organized fashion, following a progression of complexity in the statistical settings evaluated. The author intends this book to be used as a reference source for statistical researchers, as the basis of an introductory course in statistics for physical science and engineering students, and as auxiliary reading for students of mathematical statistics. All are appropriate uses of this book, with the caveat that readers who have had exposure to some basic concepts in physics, such as force, torque, potential energy, and pressure, will have a more complete appreciation of the intuition this book provides into statistical methods that are often taught solely from the academic mathematical tradition of de nition, theorem, and proof. The statistical models considered in this book all involve parameter estimation through the optimization of a criterion function. The fundamental basis of the physical representation of these models is that the potential energy of a mechanical system is proportional to this criterion function. Therefore, the solution to the mathematical statement of the statistical problem naturally arises as the combination of parameters that minimizes potential energy. This intuition is extended to statistical hypothesis testing, where the technique of testing a null hypothesis is visualized by thinking in terms of comparing the minimum potential energy of a mechanical system that implements the restrictions of the null hypothesis to the minimum potential energy of an unconstrained system. The most basic statistical setting analyzed involves identifying point(s) of central tendency in a dataset consisting of observations in one or more dimensions. This is accomplished by minimizing some function of the Euclidean distances between each observation and a candidate central point. The criterion functions minimized for this purpose are the following: sum of distances, sum of squared distances, maximum distance, and median distance. As an example, suppose a point that minimizes the sum of distances criterion of a set of observations in two dimensions is to be found. This setting can be formulated mechanically by drilling holes through a board at the location of the observations on a superimposed Cartesian coordinate system. Strings of common length are passed through each hole and attached to unit weights, while the other end of each string is attached to a ring that is allowed to slide over the board. The location at which the ring settles corresponds to the point at which the potential energy of the system is minimized, characterized by the balancing of horizontal and vertical unit forces induced by the unit weights. This point also solves the mathematical optimization problem originally posed. The point minimizing the sum of squared distances is found by replacing the strings with springs of unit modulus. Modi cations of the basic mathematical formulation can also be worked into the physical system. For example, a wire constraining the motion of the ring represents a linear or curvilinear restriction on the solution. Adjusting the weights attached to the strings or the moduli of the springs allows weighted data to be accommodated. Regression problems are considered at length in the text. The same criterion functions utilized in the problem of estimating a point of central tendency are applied to estimating a “best- tting” line (hyperplane), with distance from each observation to the proposed regression line measured conventionally (parallel to the y axis) or orthogonally. In the case of simple linear regression, each spring is attached to a separate ring that is restricted to slide on a rigid rod representing the proposed regression line. The spring is further constrained to run parallel to the y axis in the conventional case. The solution to the least squares problem is characterized mechanically as the parameter values balancing the forces tending to pull the line in opposite directions and balancing the torques about an arbitrarily selected point on the line. Similar considerations, with strings and weights in place of springs, lead to a mechanical representation of the least absolute deviations problem. The text contains many excellent geometrical approaches to deriving solutions to the optimization problems considered. In simple linear regression settings, solutions are often more easily envisioned by shifting from the “observation space” to the “parameter space. ” In observation space, the axes correspond to the variables and observations are plotted as points. In parameter space, the axes correspond to linear intercept and slope parameters and observations are plotted as lines. Similarly, lines in observation space are represented as points in parameter space. For example, a regression line minimizing the maximum absolute deviation is obtained in observation space by calculating the parallel bisector of a pair of parallel lines, separated by minimum distance, that cover the entire set of observations. In parameter space, this regression line is more easily determined by plotting the lines corresponding to each of the observations and  nding the value of the slope parameter that minimizes the separation of the bounding lines. The intercept parameter is calculated as the midpoint of the corresponding bounding values. The book also delves into mechanical models for more complicated statistical applications such as metric graphs, which feature a geometrical analysis of pairwise and transitive preference orderings among a group of candidates, categorical data analysis, curve  tting with linear and cubic splines, and multivariate applications including principal components and multidimensional scaling. The chapter on categorical data analysis is especially interesting. The traditional chi-squared and likelihood ratio tests of independence are derived as differences in potential energy levels of hydrostatic and gas pressure models, respectively. The reader of this book will bene t from the visualization of statistical models that is a consequence of framing statistical optimization problems in terms of physical systems. In addition, the author provides an excellent account of the complete development of each problem, replete with historical and contemporary references. However, this book is not intended as a primary reference for computational solutions to these problems. It provides brief outlines of linear or nonlinear programming implementations, with the expectation that additional reference material will generally be required to implement algorithmic solutions for practical applications. The book contains some minor typographical errors that do not detract from the presentation. Overall, this is an excellent treatment at a very basic level of subject matter that is potentially of great bene t to students and practitioners of statistical methods.