Cluster Analysis: An Introduction

Abstract

Various methods and diierent (linear or not, simple linear, or multivariate) models have been adopted in industry to address the calibration problem. In practice, most of the models attempt to deal with the simple linear calibration technique, mostly applied in chemical applications, especially when some instruments are to be calibrated (exam-ples include pH meters, NIR instruments, and establishing calibration graphs in chromatography). e early work of Shukla () put forward the problem on the real statistical dimensions, and even early on it was realized that when a non-linear model describes the phenomenon (Schwartz), a linear approximation is eventually adopted. But even so, in the end we come to a nonlinear function to be estimated as best as possible (Kitsos and Muller). When the variance of the measurement is due to many sources of variability, different techniques are used. Statistical calibration has been reviewed by Osborn (), who provides a list of pertinent references; when a robust approach might be appropriate, see Kitsos and Muller (). Certainly, to consider the variance constant and to follow a statistical quality control method (see Statistical Quality Control), Hochberg and Marom () might be helpful, but not in all cases. For the multivariate case, see the compact book of Brown (), Brereton (), and for an application Oman and Wax (). Moreover, diierent methods have been used on the development of the calibration problem like cross-validation (see Clark). Next we brieey introduce the statistical problem and the optimal design approach is adopted in the sequence to tackle the problem. Consider the simple regression model with n = E(yu) = θ + θ u u ∈ U = [−, ,] where U is the design space, which can always be transformed to [−, ,]. Moreover, the involved error is assumed to be from the normal distribution with mean zero and variance σ >. e aim is to estimate the value of u = u given n = C, i.e., u = (C − θ))θ which is a nonlinear function of the involved linear parameters , as we have already emphasized above. e most well-known competitive estimators of u when y is provided are the so-called "classical predictor" C (u) = ¯ x + S xx S xy (˙ y − ¯ y) and the "inverse predictor" I (u) = ¯ u + S xy S yy (˙ y − ¯ y) with: S tr = (ti − ¯ t)(r i − ¯ r) were by ˙ y we mean the average of the possible k observations taken at the prediction stage (or experimental condition) and ¯ y as usually the average of the collected values. e comparison of C(u) and I(u) is based on the values of the sample size n and the proportion σθ under the assumption that x belongs to the experimenter area.