Statistical Analysis of Constrained Data SetsPractical, comprehensive computer-based methods for analyzing data sets are developed. Methods for calculating data adjustment, parameter values, variance-covariance of adjusted data and parameters and for detecting aberrant data are presented. A simple calculation algorithm and application of the methods for design of experimental measurements are proposed.
JAY
Research and Development
Napenille, IL 60540Least squares analysis, commonly used to fit regression equations to sets of experimental data, also provides powerful techniques for analyzing the measured data themselves, when these data can be interrelated through constraining physical laws. Least squares parameter estimation problems are typically "parameter-rich in the sense that the data set is constrained by a single regression equation. (In this article, the term "parameter" is used in the engineering sense of a quantity to be estimated from other data, rather than in the statistical sense, wherein it applies to all estimated quantities.)In another frequently met situation of particular interest in this work, a large set of data is interrelated by physical laws such as heat balances, material balances or kinetic equations. This situation may be thought of as being "comtraint-rich", and is typical of many laboratory experiments, pilot unit tests, and commercial unit performance tests. Constraints, together with estimates of the variances of the measured data, can be used to adjust data to more accurate values and to draw conclusions about their credibility.The data estimation techniques discussed here greatly facilitate analysis of constrained data sets by providing maximum likelihood estimates of the meawred data and any parameters, by assessing the probability that there are extraordinary errors, and by providing error information about the calculated quantities for use in subsequent analysis. Further, the techniques can be used to develop experiments producing data with improved accuracy. The method can be thought of as an extension of the averaging process to situations, in which each measured quantity may enter into one or a number of physical constraints, and by which alternative values can be inferred. It can be applied, if necessary, to detect gross measurement errors and to isolate any such by using data redundancies, much as a skilled analy$t would. And, replicates or near replicates are not required to judge credibility of the data values.The net effect can be significantly increased efficiency of experimentation, offering the happy choice of either obtaining more accurate data for a given cost, or of achieving the same accuracy in the final results with less experimental cost. Because the techniques are general, they can handle data sets described as a typical linear or nonlinear regression problem, a parameter-free data adjustment problem, or any combination of these.The basic data correction algorithm was first derived for and applied to surveying problems (Demming 1946). Data adjustment techniq...