Regression and Calibration

Having two random variables, x and y, e.g. values measured independently from each other and each of them being normally distributed, then the following questions may be of interest  

Whereas the first question is answered by correlation analysis, the second one is subject of regression analysis. 

Correlation analysis investigates stochastic relationships between random variables on the basis of samples. The interdependence of two variables x 

For the calculation of measures in correlation and regression analysis the following sums are of relevance  

As the last term in Eq. (6.3) shows, the correlation coefficient corresponds to the covariance of x and y, covfx, y) = sxy, divided by the standard deviations sx and sy. 

The family to which/is supposed to belong to is defined via some set of parameters which are coupled in a particular way with x. Therefore, it is reasonable to extend the above mentioned functional relationship to y=J x,p), where the parameter p is variable. Some prominent families of regression models are  

Of course, the linear model is a special case of the polynomial model. Generally, a model is called quasilinear when / is a linear function of p. This does not exclude the case that/is nonlinear in X, In particular, the polynomial model is quasilinear although the functional dependence on, v may be quadratic, as in Fig. 4 b. Given the data pairs (.i y,), the parameter p yielding the best approximation of / to all these data pairs is found by minimizing the sum of the squares of the deviations between the measured values r, and their modeled counterparts 

The advantage of quasilinear models is that the exact solution for p is easily obtained by solving a system of linear equations (see Section 3.11.1). In the simple case of linear models, one can even directly indicate the explicit solution  

The coefficients p, and p2 refer to the intercept and slope of the straight line fitting the data points (see Fig. 4 a). Being based on the random variables X and y, the coefficients p, and p2 are outcomes of random variables themselves. The true coefficients can be covered by the following 95 % confidence intervals  

In contrast to the polynomial model, the multiexponential model is not quasilinear. This makes the determination of the parameters p more complicated. Then iterative methods have to be employed in order to approach a solution. This issue falls into the framework of nonlinear regression analysis. For more details, see [5]. 

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In case of correlated parameters, the corresponding covariances have to be considered. For example, correlated quantities occur in regression and calibration (for the difference between them see Chap. 6), where the coefficients of the linear model y = a + b x show a negative mutual dependence. 

Weighted Averaging. WA regression and calibration is a robust, computationally simple, and straightforward method for reconstructing environmental variables. It provides a more accurate and precise inference... 

Figure 12. Graphs of observed versus diatom-inferred total phosphorus concentrations (TP) and observed minus diatom-inferred TP (i.e., a residual analysis) are based on weighted averaging regression and calibration models and classical deshrinking. The large circles indicate two coincident values. This analysis is discussed in detail in reference 46.

S. Burke, Regression and calibration, LC-GC Europe Online Supplement Statistics and Data Analysis (2001), 13-18. 

K. Baumann, Regression and calibration for analytical separation techniques. Part II Validation, weighted and robust regression. Process Control and Quality 10 (1997), 75-112. 

For regression and calibration work, it has been shown that transformation of the data is sometimes necessary and the above-mentioned transformations have been used frequently to build and improve calibration models using two-way data. 

Brown P, Measurement, Regression and Calibration, Clarendon Press, Oxford, 1993. 

Davidian, M. and Haaland, P.D. Regression and calibration with nonconstant error variance. Chemometrics and Intelligent Laboratory Systems 1990 9 231-248. 

Brereton RG (2003) Chemometrics Data Analysis for the Laboratory and Chemical Plant. Chichester Wiley. Brown PJ (1993) Measurement, Regression and Calibration. Oxford Clarendon Press. 

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