William H. Lawton

Self Modeling Curve Resolution

This paper presents a method for determining the shapes of two overlapping functions f 1(x) and f 2(x) from an observed set of additive mixtures, [α i f 1(x) + β i f 2(x); i = 1, …, n), of the two functions. This type of problem arises in the fields of spectrophotometry, chromatography, kinetic model building, and many others. The methods described by this paper are based on the use of principal component techniques, and produce two bands of functions, each of which contains one of the unknown, underlying functions. Under certain mild restrictions on the fj (x), each band reduces to a single curve, and the fi (x) are completely determined by the analysis.

Curve Resolution Using a Postulated Chemical Reaction

The paper presents a method for resolving additive mixtures of overlapping curves by combining nonlinear regression and principal component analysis. The method can be applied to spectroscopy, chromatography, etc. The method makes use of the postulated chemical reaction, and allows one to check the reaction and estimate chemical rate and equilibrium constants.

Principal modes of variation for processes with continuous sample curves

Analysis of a process with continuous sample curves can be carried out in a manner similar to principal components analysis of vector processes. By appropriate definition of a best linear model in the continuous case, we show that principal modes of variation consist of eigenfunctions of the process covariance function C(s, t). Procedures for estimation of these eigenfunctions from a finite sample of observed curves are given, and results are compared with principal components analysis of the same data.

Self Modeling Nonlinear Regression

The paper is concerned with parametric models for populations of curves; i.e. models of the form yi (Z) = f(θ i ; x) + error, i = I, 2, …, n. The shape invariant model f(θ i ; x) = θ0i + θ1i g([x – θ2i /θ3i ) is introduced. If the function g(x) is known, then the θ i may be estimated by nonlinear regression. If g(x) is unknown, then the authors propose an iterative technique for simultaneous determination of the best g(x) and θ i . Generalizations of the shape invariant model to curve resolution are also discussed. Several applications of the method are also presented.

Statistical Comparison of Multiple Analytic Procedures: Application to Clinical Chemistry

The basic sciences all require an ability to measure the amounts of substances under study. With new methods of measurement constantly being proposed there is a need for techniques for comparing these methods in terms of their precision and accuracy. Of particular interest is the case in which none of the individual methods are known to measure “truth”. A multiple methods comparison technique for this case is proposed in this paper, and is illustrated by an example from the field of clinical chemistry. Estimates of the components of variance for each method are developed, and some of their properties explored.

Crowded Emulsions: Granularity Theory for Multilayers*

A crowded photographic emulsion is viewed as a sandwich of stacked, crowded monolayers. An earlier renewal model of granularity in a crowded monolayer, combined with a new analysis of the general way in which granularity propagates through layers, leads to predictions of the granularity of the multilayer sandwich as a function of the number of layers. For a fixed concentration of grains per unit projected area in the sandwich, rms density fluctuations increase as the number of layers decreases because rms transmittance fluctuations decrease at a slower rate than mean transmittance. These changes are similar to the entropy decrease of grain configurations in the emulsion. For sandwiches consisting of at least 15 layers having a maximum density not greater than 2, the change of rms density fluctuations vs mean density for an exposure series is accurately predicted by the honest random-dot model. Any discrepancy between the theoretical predictions of the honest random-dot model and experimental data for normal emulsions cannot be attributed solely to the neglect of crowding constraints by that model.

Some probability problems associated with cross-impact analysis

Abstract Various probability models are discussed which may be applied to Cross-Impact analysis. The bulk of this article is concerned with what is currently the most commonly employed model that utilizes marginal probabilities and either simple conditional probabilities or ”impacts“. Requirements for admissible probability sets are given and some misgivings about current practices are discussed. Bounds are given for the joint occurrence or non-occurrence of a number of events which could replace the Monte Carlo generation currently in use. It is concluded that the multitude of probability problems associated with Cross-Impact analysis are such as to preclude its use as a quantitative technique in its present form although its use as a vehicle for promoting discussion can still be recommended and some of the results can serve as a rough guide in forecasting.

Comarison of ANOVA and Harmonic Components of Variance

Statistical spectral analysis is much like a Model II analysis of variance in that the total variability of a data set is broken down into a number of components, the amount of variability associated with various bands of frequencies. Using the spectral density, it is possible to define a collection of what we shall call harmonic variance components. It is hoped that harmonic variance components will make the interpretation and analysis of spectra somewhat easier for those who are familiar with the ANOVA techniques, but have done little work in spectral analysis. In this paper it will be shown that an ANOVA model may be stated in terms of harmonic variance components. It will also be shown that while the variance components obtained from the ANOVA and the harmonic variance components obtained from the spectral density are not numerically equal, they may be transformed one into the other if the ANOVA assumptions are valid. The relationship between the two sets of components may be used to check on the vali...

The Spectrum of a Model II Nested ANOVA and Its Applications

Parzen [4] and others have noted the similarity between probability densities, spectral densities, and their sample estimates. In the familiar x 2 goodness-of-fit test one seeks to determine whether an observed sample distribution could reasonably have resulted from sampling some specified probability distribution. One could ask a similar question concerning the spectral density of a process. That is, could the estimated spectral density ĝ(ω) of some sampled process have reasonably resulted from a process whose true spectrum is f(ω)? Durbin [l] considers a problem of this type in testing for randomness. The present paper considers a process {X(t), t = 1, 2, · · ·} where X(t) is an asymptotically stationary random process with spectrum g(ω). In Sections 1 and 2 it is shown that, if X(t) is generated by a Model II nested ANOVA, then X(t) is asymptotically stationary and possesses an asymptotic spectrum S(ω). This spectrum completely characterizes the process and is a function of the ANOVA variance component...

Colour Granularity: A Layered Model

AbstractWe show how the classical random-dot model of photographic granularity may be modified to yield quantitative predictions of the phenomenon reported by Zwick that it is possible “to attain an image of finite density that is essentially grainless” with dye systems. Our analysis also shows that substantial errors can arise in assuming that the variance of density fluctuations of sublayers is additive for layers that do not scatter light.