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A number of variants of PLS exist for estimating the factor and loading matrices {{mvar|T, U, P}} and {{mvar|Q}}. Most of them construct estimates of the linear regression between {{mvar|X}} and {{mvar|Y}} as <math>Y = X \tilde{B} + \tilde{B}_0</math>. Some PLS algorithms are only appropriate for the case where {{mvar|Y}} is a column vector, while others deal with the general case of a matrix {{mvar|Y}}. Algorithms also differ on whether they estimate the factor matrix {{mvar|T}} as an orthogonal (that is, [[orthonormal matrix|orthonormal]]) matrix or not.<ref>
{{cite journal |last1=Lindgren |first1=F |last2=Geladi |first2=P |last3=Wold |first3=S |title=The kernel algorithm for PLS |journal=J. Chemometrics |volume=7 |pages=45–59 |year=1993 |doi=10.1002/cem.1180070104 |s2cid=122950427 }}</ref><ref>{{cite journal |last1=de Jong |first1=S. |last2=ter Braak |first2=C.J.F. |title=Comments on the PLS kernel algorithm |journal=J. Chemometrics |volume=8 |issue=2 |pages=169–174 |year=1994 |doi=10.1002/cem.1180080208 }}</ref><ref>{{cite journal |last1=Dayal |first1=B.S. |last2=MacGregor |first2=J.F. |title=Improved PLS algorithms |journal=J. Chemometrics |volume=11 |issue=1 |pages=73–85 |year=1997 |doi=10.1002/(SICI)1099-128X(199701)11:1<73::AID-CEM435>3.0.CO;2-# }}</ref><ref>{{cite journal |last=de Jong |first=S. |title=SIMPLS: an alternative approach to partial least squares regression |journal=Chemometrics and Intelligent Laboratory Systems |volume=18 |pages=251–263 |year=1993 |doi=10.1016/0169-7439(93)85002-X |issue=3 }}</ref><ref>{{cite journal |last1=Rannar |first1=S. |last2=Lindgren |first2=F. |last3=Geladi |first3=P. |last4=Wold |first4=S. |title=A PLS Kernel Algorithm for Data Sets with Many Variables and Fewer Objects. Part 1: Theory and Algorithm |journal=J. Chemometrics |volume=8 |issue=2 |pages=111–125 |year=1994 |doi=10.1002/cem.1180080204 |s2cid=121613293 }}</ref><ref>{{cite journal |last=Abdi |first=H. |title=Partial least squares regression and projection on latent structure regression (PLS-Regression) |journal=Wiley Interdisciplinary Reviews: Computational Statistics |volume=2 |pages=97–106 |year=2010 |doi=10.1002/wics.51 }}</ref>
The final prediction will be the same for all these varieties of PLS, but the components will differ.
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This form of the algorithm does not require centering of the input {{mvar|X}} and {{mvar|Y}}, as this is performed implicitly by the algorithm.
This algorithm features 'deflation' of the matrix {{mvar|X}} (subtraction of <math>t_k t^{(k)} {p^{(k)}}^\mathrm{T}</math>), but deflation of the vector {{mvar|y}} is not performed, as it is not necessary (it can be proved that deflating {{mvar|y}} yields the same results as not deflating<ref>{{cite journal |last1=Höskuldsson |first1=Agnar |title=PLS Regression Methods |journal=Journal of Chemometrics |date=1988 |volume=2 |issue=3 |page=219 |doi=10.1002/cem.1180020306 |s2cid=120052390 }}</ref>). The user-supplied variable {{mvar|l}} is the limit on the number of latent factors in the regression; if it equals the rank of the matrix {{mvar|X}}, the algorithm will yield the least squares regression estimates for {{mvar|B}} and <math>B_0</math>
==Extensions==
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 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.<ref>{{Cite journal
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</ref> L-PLS extends PLS regression to 3 connected data blocks.<ref>{{cite journal |last1=Sæbøa |first1=S. |last2=Almøya |first2=T. |last3=Flatbergb |first3=A. |last4=Aastveita |first4=A.H. |last5=Martens |first5=H. |title=LPLS-regression: a method for prediction and classification under the influence of background information on predictor variables |journal=Chemometrics and Intelligent Laboratory Systems |volume=91 |issue=2 |pages=121–132 |year=2008 |doi=10.1016/j.chemolab.2007.10.006 }}</ref> Similarly, OPLS-DA (Discriminant Analysis) may be applied when working with discrete variables, as in classification and biomarker studies.
In 2015 partial least squares was related to a procedure called the three-pass regression filter (3PRF).<ref>{{Cite journal|
A PLS version based on [[Singular value decomposition|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.<ref>{{Cite journal|
PLS correlation (PLSC) is another methodology related to PLS regression,<ref name=":0">{{Cite journal|
==See also==
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