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Statistical method

Partial least squares regression (PLS regression) is a statistical method that bears some relation to principal components regression; instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable variables to a new space. Because both the X and Y data are projected to new spaces, the PLS family of methods are known as bilinear factor models. Partial least squares discriminant analysis (PLS DA) is a variant used when the Y is categorical. PLS is used to find the fundamental relations between 2 matrices (X and Y), i. e. a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the Y space. PLS regression is particularly suited when the matrix of predictors has more variables than observations, and when there is multicollinearity among X values. By contrast, standard regression will fail in these cases (unless it is regularized). Partial least squares was introduced by the Swedish statistician Herman O. A. Wold, who then developed it with his son, Svante Wold. An alternative term for PLS is projection to latent structures, but the term partial least squares is still dominant in many areas. Although the original applications were in the social sciences, PLS regression is today most widely used in chemometrics and related areas. It is also used in bioinformatics, sensometrics, neuroscience, and anthropology.

Given paired random samples In the first step , the partial least squares regression searches for the normalized direction , such that the correlation is maximized Note below, the algorithm is denoted in matrix notation.

Another extension of PLS regression, named L PLS for its L shaped matrices, connects 3 related data blocks to improve predictability. In brief, a new Z matrix, with the same number of columns as the X matrix, is added to the PLS regression analysis and may be suitable for including additional background information on the interdependence of the predictor variables.

In 2015 partial least squares was related to a procedure called the three pass regression filter (3PRF). Supposing the number of observations and variables are large, the 3PRF (and hence PLS) is asymptotically normal for the "best" forecast implied by a linear latent factor model. In stock market data, PLS has been shown to provide accurate out of sample forecasts of returns and cash flow growth.

A PLS version based on singular value decomposition (SVD) provides a memory efficient implementation that can be used to address high dimensional problems, such as relating millions of genetic markers to thousands of imaging features in imaging genetics, on consumer grade hardware.

PLS correlation (PLSC) is another methodology related to PLS regression, which has been used in neuroimaging and sport science, to quantify the strength of the relationship between data sets. Typically, PLSC divides the data into two blocks (sub groups) each containing one or more variables, and then uses singular value decomposition (SVD) to establish the strength of any relationship (i. e. the amount of shared information) that might exist between the two component sub groups. It does this by using SVD to determine the inertia (i. e. the sum of the singular values) of the covariance matrix of the sub groups under consideration.