Linear regression matrix notation

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

For a linear fitting exercise, e.g. the calculation of the emission spectra A, we assume to know the lifetimes t and hence the matrix Csim, which we used for the generation of the measurement. The linear regression has to be performed individually at each wavelength. This is due to the fact that at each wavelength Xj the appropriate vector (Ty j is different and each weighted matrix Cw and its pseudo-inverse, needs to be computed independently. There is no equivalent of the elegant A=C Y notation. 

Thus, y is related to a linear combination of the x-variables, plus an additive error term. The difference to simple regression is that for each additional a-variable a new regression coefficient is needed, resulting in the unknown coefficients b0, b, ..., bm for the m regressor variables. It is more convenient to formulate Equation 4.35 in matrix notation. Therefore, we use the vectors y and e like in Equation 4.19, but define a matrix X of size nx (m+ 1) which includes in its first column n values of 1,... 

As in simple linear regression, the same assumptions are made s is normally distributed, uncorrelated with each other and have mean zero with variance u2. In addition, the covariates are measured without error. In matrix notation then, the general linear model can be written as... 

For the calculation of the coefficients in our polynomial model we use linear regression, i.e. a least squares projection of the data onto the model. This is very easy to write down in matrix notation. Our model becomes ... 

With linear least-squares regression, we chose many concentrations to calculate a good curve. The same should be done for the binary mixture, so that there are many mixtures of different concentrations, all measured at the two wavenumber positions. Rather than write a long series of equations to represent all the mixmres, we can employ a shorthand notation by which we represent all the terms in Eq. 9.7 in matrix notation that is. 

Note that this equation describes the relationship between concentration C of the component and the sensor response X. It is purposely written backwards by comparison with the usual notation used with linear sensors (e.g., optical, amperometric, etc.) discussed earlier. This convention helps to define P as the matrix of regression coefficients. 

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