Projection to latent structures PLS regression

The method of PLS, also known as Partial Least Squares, is a highly utilized regression tool in the chemometrics toolbox,1 and has been successfully used for many process analytical applications. Like the PCR method, PLS uses the exact same mathematical models for the compression of the X-data and the compression of the Y-data  

Like PCR, the compressed variables in PLS have the mathematical property of orthogonality, and the technical and practical advantages thereof. PLS models can also be built knowing only the property of interest for the calibration samples. 

The difference between PLS and PCR is the manner in which the X-data are compressed. Unlike the PCR method, where X-data compression is done solely on the basis of explained variance in X followed by subsequent regression of the compressed variables (PCs) to y (a simple two-step process), PLS data compression is done such that the most variance in both X and Y is explained. Because the compressed variables obtained in PLS are different from those obtained in PCA and PCR, they are not PCs Instead, they are often referred to as latent variables. 

Like PCR, a PLS model can be condensed into a set of regression coefficients bpLs- For the NIPALS algorithm discussed above, the regression coefficients can be calculated by the following equation  

There are some distinct advantages of the PLS regression method over the PCR method. Because Y-data are used in the data compression step, it is often possible to build PLS models that are simpler (i.e. require fewer compressed variables), yet just as effective as more complex PCR models built from the same calibration data. In the process analytical world, simpler models are more stable over time and easier to maintain. There is also a small advantage of PLS for qualitative interpretative purposes. Even though the latent variables in PLS are still abstract, and rarely express pure chemical or physical phenomena, they are at least more relevant to the problem than the PCs obtained from PCR. 

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Several steps are involved in rapid analysis method development. These include gathering appropriate calibration samples, chemical characterization of the calibration samples, developing spectroscopic methods for the rapid technique, projection-to-latent-structures (PLS) regression, validation of the PLS algorithm, and the development of QA/QC procedures.128... 

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