Ordinary least squares regression values, responses

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

The transformed response values were regressed on the transformed amount values using the simple linear regression model and ordinary least squares estimation. The standard deviation of the response values (about the regression line) was calculated, and plots were formed of the transformed response values and of the residuals versus transformed amounts. 

Ordinary least squares technique, used for treatment of the calibration data, is correct only when uncertainties in the certified value of the measurement standards or CRMs are negligible. If these uncertainties increase (for example, close to the end of the calibration interval or the shelf-life), they are able to influence significantly the calibration and measurement results. In such cases, regression analysis of the calibration data should take into account that not only the response values are subjects to errors, but also the certified values. 

Y = XB -I- F (see Figure 5), with y (7 x 1) a vector of response values for many samples, Y (7 x M) a matrix of response values for many responses, X (7 x K) the spectral data for the samples, b (Kxl) a vector of regression coefficients, B (X X M) a matrix of regression coefficients for the M responses, f I x 1) a vector of residuals and F (7 X M) a vector of residuals for many responses. One may consider percentages of fat, water and protein as three different responses (M = 3). The traditional least-squares solution for b B) is called multiple linear regression or ordinary least squares. The method was used in the pioneering days of NIR, when there were only few wavelength bands available. 

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