1. The standardised parameters are given below. Remember to use the population rather than sample standard deviation.

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1 Kapitel The standardised parameters are given below. Remember to use the population rather than sample standard deviation. The graph of cross-validated error versus component number is presented below, and it appears that two components are sufficient for modelling the data. 1

2 2. Using two PLS components, the following are the predicted and true data. The root mean square error is Notice that this is most usually obtained by taking the square root of the sum of squares of the residuals divided by 10 rather than 13 since 3 degrees of freedom are lost due to the 2 PCs and due to centring. 3. The graph of predicted versus observed is given below. Although there is not a perfect correlation, it nevertheless suggests that there is a reasonable trend and so the biological property is related to chromatographic retention parameters. 2

3 The equation = b b x can be rewritten as c which shows that a 1 = 1/b 1 and a 0 = - b 0 /b 1 2. The dimensions of the matrices are as follows. ĉ is 10x1, x is 10x2, b is 2x1 xˆ is 10x1, c is 10x2, a is 2x1 The coefficients are given by and. 3. a 1 = /b 1 = 1/ = a 0 = b 0 /b 1 = (note both intercepts are close to 0). This equality is not exact since each model has different assumptions about the errors. 4. The classical model assumes that the errors are all in the x direction and the inverse model assumes they are in the c direction. Providing there are no strong outliers the equalities and also the best fit straight lines using both methods should be fairly similar The labelled scores and loadings plots are as follows. c variables 3

4 x variables The x variables are easy to interpret, the loadings simply represent a mixture triangle (see Chapter 2), with the scores indicating the blends. For example, sample 1 contains the highest proportion of milk, sample 3 the highest proportion of sugar and sample 8 cocoa. In addition, samples 5 and 6 are replicates and so identical. The sign of PC2 is inverted in the c and x PC calculations, this cannot be controlled, but otherwise the two scores plots can almost be superimposed, suggesting very good evidence that taste and texture relate closely to blends. By looking at the variables in the c loadings plot and comparing to the x loadings plot, it is easy to see which constituents result in different sensory perceptions, for example, sweetness is clustered close to sugar, and cocoa odour and colour to cocoa. 2. The predictions are below, using standardised data and 2 PLS components. If you do not obtain these results, check your calculations carefully. 4

5 Lightness Colour Cocoa-odour Smooth-txtr Milk-taste Sweetness The percentage root mean square errors of prediction are as follows. Cocoaodour Lightness Colour Smooth-text Milk-taste Sweetness The correlation coefficients are as follows. Cocoaodour Lightness Colour Smooth-text Milk-taste Sweetness The graph is given below. The two graphs are monotonically related, the higher the error the lower the correlation coefficient. Both parameters can be used as indicators of goodness-of-fit. However, many of the correlation coefficients are close to 1, despite high percentage errors, and can give a falsely optimistic answer. 5. The root mean square errors for PLS2 are as follows. Cocoaodour Smooth-text Milk-taste Sweetness Lightness Colour They are somewhat higher than for PLS1, this is often the case. 6. Simply swap the "x" and "c" blocks around. 5

6 For a model of the form ˆ = b + b x, the following coefficients are obtained c 0 1 b b Note that b 0 is simply the average concentration of A. 2. The predicted concentrations at each wavelength together with the root mean square errors are given below. Notice that the RMS Error should, ideally, be divided by the number of degrees of freedom (8) rather than the number of objects. Wavelengths 3, 4 and 5 appear best for prediction. 3. The predicted spectrum is as follows Hence the predicted concentrations are as follows giving a root mean square error of Note that it is probably best to divide by 7 rather than 8 in this case. However there are not dramatic differences according to whether 6, 7 or 8 are used for division and the quality of predictions could be assessed graphically. The prediction is slightly worse than at the three best wavelengths using univariate methods. 4. The two spectra are as follows The predicted concentrations together with root mean square errors (dividing by 6 as two components used in the model) are as follows. 6

7 A B The errors are now quite small (Error for A is 0.66, Error for B is 0.77), for compound A comparable to the best wavelength. 5. The scores are given below The concentration estimates for compound A are as follows. 1 component 2 components The RMS are 1.19 for 1 component and 0.51 for 2 components. 7. In MLR it is only necessary to know the concentrations of one compound for effective calibration, but it is best to use 2 PLS components. MLR tries to fit the entire x matrix but if only one compound is known the information of the other compound is mixed up. In PLS although it is important to know how many components are in a mixture, it is only necessary to know the concentrations of the calibrant. 7

8 The scores and loadings for the centred data are given below. Scores: Loadings: PC 1 PC 2 PC 1 PC The total sum of squares for the centred data is , the first eigenvalue is of size and the second of size 0.182, totalling altogether, or 99.81% of the variance. Notice that there are actually two true components in the mixtures, so the comparatively small size of PC2 does not mean it should be ignored. It would not be completely obvious how many components are in the data from first inspection. 2. The coefficients that relate the scores of the centred data matrix to the centred concentrations are c A = 2.49 t t 2 c B = 1.90 t t 2 The estimated concentrations are as follows. Compound A Compound B (These are quite close to the true concentrations.) 3. The matrix R is as follows The inverse R -1 of the rotation matrix is The estimated spectra are given by R -1 P' and are as follows

9 5. The two concentration vectors are correlated. Hence when PCA is performed on the centred data matrix, there is only one PC, and it is not possible to predict the concentrations, independently of two different compounds. When performing calibration experiments it is important to ensure that the two concentrations are reasonably uncorrelated The calculations are presented, in full, below. First PLS component (note X and c are mean centred first). Second PLS component, starting with residuals from first component Third PLS component, starting with residuals from second component 9

10 2. The residual sum of squares is given as follows (using centred data). 3. Three components describe the data exactly. However there are only 3 degrees of freedom in the centred dataset, because there are only four samples and the data are centred. This is an artificially small dataset, and in reality one would take many more samples. 4. The following table should be obtained, remember to add on the mean concentration. 5. The RMS error after 2 PLS components is Only one degree of freedom is left because there are 4 samples, two PLS components and the data are centred and = 1. 10

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