Multiway PLS

It is now a simple task to perform PLS (or indeed any other multivariate approach), as discussed above. The 30 variables are centred and the predictions of the concentrations performed when increasing number of components are used (note that three is the maximum permitted for column centred data in this case, so this example is somewhat simple). All the methods described above can be applied. 

An important aspect of three-way calibration involves scaling, which can be rather complex. The are four fundamental ways in which die data can be treated  

centre the columns in each J x K plane and then unfold with no further centring, so, for example, becomes 390-(390 + 635 + 300 + 65 + 835)/5  

combine methods 2 and 3, start with centring as in step 2, then unfold and recentre a second time. 

These four methods are illustrated in Table 5.18 for the case of the x(11, the variables in the top left-hand corner of each of the four two-way datasets. Note that methods 3 

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Kourti, T., Nomikos, P., and MacGregor, J.F., Analysis, monitoring and fault diagnosis of batch processes using multiblock and multiway PLS, J. Proc. Cont., 5, 277-284, 1995. 

Most traditional chemometrics is concerned with two-way data, often represented by matrices. However, over the past decade there has been increasing interest in three-way chemical data. Instead of organising the information as a two-dimensional array [Figure 4.38(a)], it falls into a three-dimensional tensor or box [Figure 4.38(b)], Such datasets are surprisingly common. In Chapter 5 we discussed multiway PLS (Section 5.5.3), the discussion in this section being restricted to pattern recognition. 

Saturate the surface of each internal well with BSA by adding 50 pL of BSA coating solution using a multiway pipet. Ensure that the bottom of each well is completely covered with the solution by tapping the plate or briefly agitating it on a vortex-type plate mixer. Then place the plates, with lids on, in an incubator for at least 5 h (seeNote 3). 

Serially dilute competing antigens. For accurate assays, the one in two dilution steps described here are appropriate. Larger steps could be used for preliminary tests. Place 120 pL of PBS in columns 3 to 11 of a flat-bottom 9 well microtiter plate, using a multiway pipeL Place the initial dilutions of the competing antigens in the wells of column 2. Serially dilute 120 pL vol across the plate using the multiway pipet, but do not dilute into columns 5 and 11, as these correspond to the no-competitor wells. 

The best way to overcome this problem is to use an electronically controlled multiway pipet such that, for example, 230 pL of solution can be drawn up and then 4 x 55 pL dispensed, leaving 10 pL in the tips. The plates are then mixed on a vortex-type plate mixer. An alternative method using a conventional multiway pipet is to draw up 55 pL of solution A or B, dispense it in the appropriate wells, mix the contents of tips and wells by pipeting up and down a few times, and then use new tips for the next row of wells. 

The most convenient way to overcome this problem is to use an electronically controlled multiway pipettor such that, for example, 230 pL can be drawn up and then 4 x 55 pL dispensed, leaving 10 pL in the tips. A manual pipettor should be used in the reverse pipetting mode. 

The background for the extension of two-way partial least squares regression to multiway data (fV-PLS) was provided by Bro [1996] and further elaborated on by Smilde [1997] and de Jong [1998], An improved model of X was later introduced which, however, maintains the same predictions as the original model [Bro et ol. 2001], Only the three-way version of A-PLS is considered here. It differs from the N-way (N > 3) partial least squares regression algorithm in that it has a closed-form solution for the situation with only one dependent variable. 

Bro R, Multiway calibration. Multi-linear PLS, Journal of Chemometrics, 1996,10, 47-61. 

Agrafiolis DK (2003) Stochastic proximity embedding. J Comput Chem 24 1215-1221 Xue L, Stahura PL, Bajorath J (2004) Cell-based partitioning. In Chemoinformatics concepts, methods, and tools for drug discovery. Chapter 9. Humana, Totowa Wickens TD (2009) Multiway contingency tables analysis for the social sciences. Psychology, New York... 

Principal components regression (PCR), partial least squares regression (PLS), and their multiway versions (N-PCR and N-PLS)... 

PLS discriminant analysis (PLS-DA), soft independent modeling of class analogy (SIMCA), multiway N-PLS-DA... 

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