orthogonal (that is, orthonormal) matrix or not. The final prediction will be the same for all these varieties of PLS, but the components will differ. PLS is composed of iteratively repeating the following steps k times (for k components): finding the directions of maximal covariance in input and output space performing least squares regression on the input score deflating the input and/or target PLS1 PLS1 is a widely used algorithm appropriate for the vector case. It estimates as an orthonormal matrix. (Caution: the vectors in the code below may not be normalized appropriately; see talk.) In pseudocode it is expressed below (capital letters are matrices, lower case letters are vectors if they are superscripted and scalars if they are subscripted). 1 2 3 , an initial estimate of . 4 5 6 7 8 9 10 11 12 13 14 15 16 define to be the matrix Do the same to form the matrix and vector. 17 18 19 This form of the algorithm does not require centering of the input and , as this is performed implicitly by the algorithm. This algorithm features 'deflation' of the matrix (subtraction of ), but deflation of the vector is not performed, as it is not necessary (it can be proved that deflating yields the same results as not deflating). The user-supplied variable is the limit on the number of latent factors in the regression; if it equals the rank of the matrix , the algorithm will yield the least squares regression estimates for and Extensions OPLS In 2002 a new method was published called orthogonal projections to latent structures (OPLS). In OPLS, continuous variable data is separated into predictive and uncorrelated (orthogonal) information. This leads to improved diagnostics, as well as more easily interpreted visualization. However, these changes only improve the interpretability, not the predictivity, of the PLS models. Similarly, OPLS-DA (Discriminant Analysis) may be applied when working with discrete variables, as in classification and biomarker studies. The general underlying model of OPLS is or in O2-PLS L-PLS 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 amount 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. 3PRF In 2015 partial least squares