9.5 Methods for Detection of Data Collinearity

The general mathematical model for parameter estimation (for use in the least squares method or regression) can be written as

(9.25)

Here, the regressors x _j, j = 1, 2, ..., n are the state and input variables or their combinations, y is the dependent variable and ? ₀, ..., ? _n are unknown parameters.

Using measured data for y and x, eq. (9.25) can be written as

(9.26)

Here, Y is the measurement vector, X the matrix of regressors and 1 s (1 s are to account for the constant term in any regression equation), and ?, the unknown parameter vector. The least squares estimates of the parameters ? can be obtained using

(9.27)

Generally, the regressors X are centred and scaled to unit length. If Xj ^# denotes the columns of the normalised matrix, collinearity means that for a set of constants k _j not all equal to zero

(9.28)

Collinearity could cause computational problems due to ill-conditioning of the matrix in eq. (9.27) and this would result in inaccurate estimates of the parameters. Three commonly used methods for assessing the collinearity among regressors are discussed next [2].