Statistical Methods for Detection and Quantification of Environmental Contamination

physics and relate to geometrical concepts. The arithmetic mean is discussed as the state of equilibrium in a chemical balance setting. Chapter 2, “Abstract Geometrical and Mechanical Representations,” discusses the interpretation of likelihood function as potential energy function. The author treats the case of iid observations, although it is not clariŽ ed in the text. In this context, test of hypotheses about parameter constraints are interpreted as detecting differences in energy levels. The equivalence of the representation of a set of observations as lying in prediction, observation, and parameter space is discussed. A geometrical analogy for the correlation coefŽ cient and line Ž tting “by eye” is discussed as an introduction to the line-Ž tting problem. In Chapter 3, “Mechanical Models for Multidimensional Medians,” the median is presented as being the solution that balances the forces in a system of strings. This solution is presented based also on vector representation of forces. Mechanical representations of the median under linear, curvilinear, and regional constraints are discussed. Generalizations, such as models with non-Euclidean distances, are brie y discussed. The problem of identifying a system of roads, that connect a given set of locations with the shortest total path length is presented. In one of the book’s most interesting sections, the problem of minimizing a sum of areas instead of sum of lengths (“soap bubble models”—a problem in the calculus of variations) and the related problem of Oja’s spatial median are discussed. Chapter 4, “Method of Least Squares Deviations,” presents mechanical models for one and two-dimensional means, based on “ideal” springs. The constrained parameter estimation problems, discussed in Chapter 3, are examined for the least squares case. A mechanical model for simple linear regression is described. Mechanical models for weighted least squares, in uential observations, ridge regression, and hypotheses tests in the context of simple linear regression are brie y discussed. Orthogonal least squares are also brie y discussed, as a problem of balance of rotational forces model. Chapter 5, “Method of Least Absolute Deviation,” addresses Ž rst the problem of identifying a median as an arbitrary point of “balance of forces” in a ring of n strings. An analogous model for higher-dimensional medians is brie y discussed. A mechanical model for minimizing the weighted sum of the absolute deviations for simple linear regression is described. Linear constraints on the parameters and testing are brie y discussed. A geometrical solution for plotting the optimality function is discussed. Edgeworth’s double median, a linear programming formulation to the least sum of absolute deviations, and an application to voting in committees are also discussed. Set characterizations of the least absolute deviations and the weighted least squares solutions are given. A prediction space representation is also given for the regression problem. In Chapter 6, “Minimax Absolute Deviation Method,” the author reconsiders the Ž tting problems of earlier chapters, but uses the largest absolute deviation as a Ž tting criterion. A mechanical model based on strings and blocks is presented for identifying the midpoint of an interval of minimal length, which covers all n observations. A geometric representation of the problem of Ž nding a circle of minimum radius, that contains a set of given lines in a two-dimensional parameter space is discussed. A linear programming interpretation is given. Chapter 7, “Method of Least Median of Squared Deviations,” reexamines the same Ž tting problems, but with the median or middlemost absolute deviation used as a Ž tting criterion. Oneand two-dimensional means, simple linear regression, and geometrical solutions are discussed. A related mechanical model with strings is described. A nonlinear programming description of the problem is given. A generalization to minimum volume median ellipsoids and ellipsoidal cylinders is brie y discussed. Chapter 8, “Mechanical Models of Metric Graphs,” presents analyses of sets of data represented as metric graphs (i.e., a set of nodes and a set of paths of known lengths connecting pairs of nodes). A special problem in which a panel of voters is required to combine their individual pairwise preference orderings of a set of candidates into a single overall pairwise preference ordering is discussed. A generalized case of choosing a point in the system of tubes that minimizes the sum of the squared lengths of the paths is also brie y discussed. Chapter 9 discusses oneand two-way classiŽ cation for categorical data. A hydrostatic model of the side-by-side bar chart is described. A gas pressure model is given for an “end-on-end” bar chart and for a mechanical representation of the log-likelihood function. In Chapter 10, “Method of Averages and Curve Fitting by Splines,” mechanical models for multivariate means are discussed. The problem of Ž nding a line that joins the centroids of the extreme groups of a set of observations is considered. Smoothing by linear splines is discussed as a problem of identifying the minimum potential energy level of a system of rods and springs. Cubic splines are also brie y discussed as a better way of identifying an underlying curvilinear relationship between two variables; an analogous mechanical model is given. The cases of replacing the centroid of groups of observations by their median and multidimensional medians are brie y discussed. In Chapter 11, “Multivariate Generalizations of the Method of Least Squares,” the problem of orthogonal least squares (principal components) is discussed. The problem of determining a system of concentric elliptical contours that best Ž t the observed bivariate data is discussed as a realization of principal components. The problem of Ž tting a circle to a set of n points (Ž nding the least sum of squared deviations circle) and a corresponding mechanical model is presented. The orthogonal Procrustes rotation (rotating a second set of observations about its centroid to approach the position of the Ž rst set, after determining the centroid of each set) is presented; a mechanical model is given. A mechanical model for Ž tting spherical data is also described. A mechanical model with springs for the problem of multidimensional scaling is brie y discussed. The merit of such mechanical models, besides their historical value (see Farebrother 1999), is the ability to manipulate them and rediscover what algebra and calculus provide for an answer. Although the material in the book covers a limited area of statistical concepts and formal deŽ nitions are avoided, students with a background in mechanics can beneŽ t from understanding some standard statistical methods. However, as the author points out, such models alone do not provide good intuition for multivariate cases, or more complex problems, such as in the case of non-Euclidean distance metrics or grids that are not Cartesian, as in Ž tting of free-form surfaces (see Girod, Greiner, and Niemann 2000). In conclusion, Visualizing Statistical Models and Concepts is addressed to any students and research workers who have followed a conventional course in mathematical statistics. It can provide supplemental material for broadening an understanding of basic statistical concepts and techniques.