1 5. Multiway calibration Quimiometria Teórica e Aplicada Instituto de Química - UNICAMP

2 Multiway regression problems e.g. batch reaction monitoring Process measurements Product quality X process variable time batch product quality Y

3 Multiway regression problems e.g. tandem mass spectrscopy X5X5 X4X4 X3X3 X2X2 X1X1 samples parent ion m/z daughter ion m/z sample compound MS-MS spectra Compound concentrations

4 Some terminology Univariate calibration (OLS – ordinary least squares) Multivariate calibration (ridge regression, PCR, PLS etc.) Second-order advantage (PARAFAC, restricted Tucker, GRAM, RBL etc.) zero-order first-order second-order Cannot handle interferents Can handle interferents if they are present in the training set Can handle unknown interferents (although see work of K.Faber) N-PLS(?)

5 Multiway calibration methods PARAFAC (already discussed on first day) (Unfold-PLS) Multiway PCR N-PLS MCovR (multiway covariates regression) (see work of Smilde & Gurden) GRAM, NBRA, RBL (see work of Kowalski et al.)

6 Unfold-PLS Matricize (or ‘unfold’) the data and use standard two- way PLS: X J K I X1X1...XIXI I JK But if a multiway structure exists in the data, multiway methods have some important advantages!! M Y I

7 Two-way PCR Standard PCR for X (I  J) and y (I  1). 1.Calculate PCA model of X: X = TP T + E 2.Use PCA scores for ordinary regression: y = Tb + E b = (T T T) -1 T T y 3.Make predictions for new samples: T new = X new P y new = T new b Y b 1.Calculate PCA model of X: X = TP T + E 2.Use PCA scores for ordinary regression: y = Tb + E b = (T T T) -1 T T y X E PTPT T + = 1.Calculate PCA model of X: X = TP T + E

8 Multiway PCR Multiway PCR for X (I  J  K) and y (I  1). 1.Calculate multiway model: X = A(C|  |B) T + E 2.Use scores for regression: y = A b PCR + E b PCR = (A T A) -1 A T y 3.Make predictions for new samples: A new = X new P(P T P) -1 where P = (C|  |B) y new = A new b PCR Y b PCR 1.Calculate multiway model: X = A(C|  |B) T + E 2.Use scores for regression: y = A b PCR + E b PCR = (A T A) -1 A T y BTBT A + = CTCT XE 1.Calculate multiway model: X = A(C|  |B) T + E

9 N-PLS N-PLS is a direct extension of standard two-way PLS for N-way arrays. The advantages of N-PLS are the same as for any multiway analysis: –a more parsimonious model –loadings which are easier to plot and interpret

10 N-PLS The standard two-way PLS algorithm (see ‘Multivariate Calibration’ by Martens and Næs): The N-PLS algorithm (R.Bro) uses PARAFAC-type loadings, but is otherwise very similar

11 N-PLS graphic (taken from R.Bro)

12 Other methods Multiway covariates regression (MCovR) –different to PLS-type models –choice of structure on X (PARAFAC, Tucker, unfold etc.) –sometimes loadings are easier to interpret – standard, N mixture, N + M Restricted Tucker, GRAM, RBL, NBRA etc. –for more specialized use –second-order advantage, i.e. able to handle unknown interferents 11 00 N M restricted loadings, A

13 Conclusions There are a number of different calibration methods for multiway data. N-PLS is a extension of two-way PLS for multiway data. All the normal guidelines for multivariate regression still apply!! –watch out for outliers –don’t apply the model outside of the calibration range

14 Outliers are objects which are very different from the rest of the data. These can have a large effect on the regression model and should be removed. Outliers (1) Remove outlier bad experiment

15 Outliers (2) Outliers can also be found in the model space or in the residuals Scores PC 1 Scores PC 2

16 Model extrapolation... Univariate example: mean height vs age of a group of young children A strong linear relationship between height and age is seen. For young children, height and age are correlated. Moore, D.S. and McCabe G.P., Introduction to the Practice of Statistics (1989).

17... can be dangerous! Linear model was valid for this age range......but is not valid for 30 year olds!