Hybrid data regression modelling in measurement

Hybrid data regression modelling in measurement Vladimir B. Bokov∗ NPP Automatica JSC, Russia Summery. Measurement involves the determination of quantitative estimates of physical quantities from experiment, along with estimates of their associated uncertainties. Herewith an experimental system model is the key to extracting information from the experimental data. The measurement information obtained depends directly on the quality of the model. With this concern novel regression modelling techniques have been fashioned by data integration from computer-simulation and physical designed experiments. These techniques have allowed attaining the advanced level of model completeness, parsimony, and precision via approximation of the exact unknown model by mathematical product of available theoretical and appropriate empirical functions. The purpose of this approximation is to represent adequately the true model on the considered region of factor space with all advantages of theoretical modelling. This allows a further focus on the measurement science of issue. Pneumatic gauge hybrid data candidate models’ building, solving and validation reviled that such adequate models permit to attain minimum discrepancy from empirical evidence. Keywords: Physical model; Experimental design; Computer-simulation experiment 1. Introduction Computer models or codes are often used to perform computer-simulation experiments before physical experiments are undertaken. The codes may have high-dimensional inputs, which can be scalars or functions, and output may also be multivariate. In this paper interest is focused on a relatively small set of inputs or explanatory variables and on a single response. The design and analysis of computer experiments has evolved as the power of computer has grown. Sacks et al. (1989) provided a review of issues for design and technique used in the analysis of response from computer codes. Making a number of computer code runs at various input values is what they call a computer experiment. And the computer models considered here are deterministic; replicate observations from running the code with the same input values will be identical. This lack of random error makes computer experiments different from physical experiments. Traditional statistical approaches consider computer and physical experiments separately with corresponding separate designs, analyses, and results. A compelling argument can be made that better, more powerful statistical results can be obtained if we simultaneously analyze the combined data of physical and computer designed experiments. The analysis of such combined data permits the unknown coefficients in an assumed overall regression or response surface model to be estimated more precisely, thereby producing a better-fitting response surface model which is crucial in measurement. Measurement involves the determination of quantitative estimates of physical quantities from experiment, along with estimates of their associated uncertainties. In this endeavour, a mathematical model of measurement system is required in order to extract information from the experimental data (Cox, Forbes, and Harris, 2002). This implies model building; developing a mathematical model of the experimental system in terms of equations involving

∗ E-mail: bvb@uk2.net

2 parameters that describe all the relevant aspects of the system, and model solving; determining estimates of model parameters from the measured data by solving the equations constructed as part of the model. The response of many measurement systems depends on more than one variable and it is important to model the response of such systems as a function of all relevant explanatory variables or factors. Common approaches to modelling multivariate data have been reviewed by Boudjemaa et al. (2003) including approaches specific to data on a regular grid (e.g., tensor product polynomial and splines) and more general approaches (e.g., radial basis functions and support vector machines).
Furthermore, it is useful to classify the types of data arising in metrology into three categories: discrete, continuous and hybrid (Cox, Forbes, and Harris, 2002). Discrete data represent the measurement of a single response variable at a finite number of settings of factors. Continuous data define one or more attributes of the system over a continuum, while hybrid data have both discrete and continuous components. Herewith, Reese et al. (2004) noted that it is statistically efficient and desirable to fit a single common response surface model that combines the physical experimental data and the computer model output data to express the relationship between the factors and response variable. And in this paper we make use of designed experiment approach for multivariate hybrid data modelling employing data from two sources; computer-simulation and ph

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