2.5.   Calibration of Linear Relationships

For multivariate calibration of the concentrations of one or more analytes (variables y) and the device responses (independent variables x) the simplest and the most common approach is the assumption of a linear relationship. A model for this linear relationship can be setup by a multiple linear regression (MLR), which is also known as inverse least-squares regression (ILS). This calibration can be seen as an extension of the univariate linear regression. For each response variable, a linear equation is formulated:

(5)

with b_ij as regression coefficients and e_j as residuals. Using matrix notation equation (5) can be expressed as:

(6)

 The regression coefficients can be estimated by

(7)

For the inversion of the variance-covariance matrix at least as many samples as device responses have to be measured. This inversion of the variance-covariance matrix of the independent variables, which is needed not only by MLR but also by most linear regression techniques to identify the model parameters, causes several problems. If the variables show a collinear behavior or if the variables are highly correlated, the resulting variance-covariance matrix will be ill-conditioned leading to unreliable model parameters und thus producing unstable calibration models.

Several methods have been developed to overcome these problems like principal component regression (PCR), partial least squares regression (PLS or PLSR), ridge regression (RR) and many more. Among these methods, "PLSR is the de facto standard for constructing a multivariate model" [29].