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PLS

S quares. L east. P artial. PLS. A Standard Tool for :. Multivariate R e g r e s s i o n. Regression :. Modeling dependent variable(s): Y. Chemical property Biological activity. By predictor variables: X. Chem. composition Chem. structure (Coded). MLR.

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PLS

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  1. Squares Least Partial PLS A Standard Tool for : Multivariate R e g r e s s i o n

  2. Regression : Modeling dependent variable(s): Y • Chemical property • Biological activity By predictor variables: X • Chem. composition • Chem. structure (Coded)

  3. MLR Traditional method: If X-variables are: • few ( # X-variables < # Samples) • Uncorrelated (Full Rank X) • Noise Free ( when some correlation exist)

  4. But ! Data … Instruments Spectrometers Chromatographs Sensor Arrays • Numerous • Correlated • Noisy • Incomplete

  5. Correlated X : Independent Variables Predictor

  6. PLSRModels: The relationbetween two Matrices X and Y By a LinearMultivariate Regression 1 2 The Structureof both X and Y Richer results than MLR

  7. PLSR is able to analyze Data with: PLSRis ageneralizationof MLR • Noise • Collinearity(Highly Correlated Data) • Numerous X-variables(> # samples) • Incompletenessin both X and Y

  8. History HermanWold (1975): Modeling of chain matrices by: NonlinearIterativePArtialLeastSquares Regression between : - a variable matrix - a parameter vector Other parameter vector Fixed

  9. SvanteWold & H. Martens (1980): Completion and modification of Two-blocks ( X, Y) PLS (simplest) Herman Wold (~2000): Projection to Latent Structures As a more descriptive interpretation

  10. AQSPRexample : OneY-variable: a chemical property The Free Energy of unfolding of a protein Quant. description of variation in chem. structure Seven X-variables: 19 different AminoAcids in position 49 of protein Highly Correlated

  11. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 PIE 0.23 -0.48 -0.61 0.45 -0.11 -0.51 0.00 0.15 1.20 1.28 -0.77 0.90 1.56 0.38 0.00 0.17 1.85 0.89 0.71 PIF 0.31 -0.60 -0.77 1.54 -0.22 -0.64 0.00 0.13 1.80 1.70 -0.99 1.23 1.79 0.49 -0.04 0.26 2.25 0.96 1.22 DGR -0.55 0.51 1.20 -1.40 0.29 0.76 0.00 -0.25 -2.10 -2.00 0.78 -1.60 -2.60 -1.50 0.09 -0.58 -2.70 -1.70 -1.60 SAC 254.2 303.6 287.9 282.9 335.0 311.6 224.9 337.2 322.6 324.0 336.6 336.3 366.1 288.5 266.7 283.9 401.8 377.8 295.1 MR 2.126 2.994 2.994 2.933 3.458 3.243 1.662 3.856 3.350 3.518 2.933 3.860 4.638 2.876 2.279 2.743 5.755 4.791 3.054 Lam -0.02 -1.24 -1.08 -0.11 -1.19 -1.43 0.03 -1.06 0.04 0.12 -2.26 -0.33 -0.05 -0.31 -0.40 -0.53 -0.31 -0.84 -0.13 Vol 82.2 112.3 103.7 99.1 127.5 120.5 65.0 140.6 131.7 131.5 144.3 132.3 155.8 106.7 88.5 105.3 185.9 162.7 115.6 DDGTS 8.5 8.2 8.5 11.0 6.3 8.8 7.1 10.1 16.8 15.0 7.9 13.3 11.2 8.2 7.4 8.8 9.9 8.8 12.0 data X Y

  12. Symmetrical Distribution Transformation 12.5 4235 0.2 546 100584 1.097 3.627 -0.699 2.737 5.002 log

  13. More weights for more informative X-variables Scaling No Knowledge about importance of variables Auto Scaling • Scale to unit variance (xi /SD). • Centering (xi – xaver). Same weights for all X-variables

  14. Numerically More Stable Auto Scaling

  15. Weights (usuallylinear) Base of PLSRModel A few “new” variables : X-scoresta(a=1,2, …,A) Modelers of X Predictors of Y Orthogonal &Linear Combinationof X-variables : T=XW*

  16. loadings ta(a=1,2, …,A) T(X-scores) Are: X =TP’+ E • Modelers of X: • Predictors of Y: Y=TQ’+F PLS-Regression Coefficients(B) Y = XW*Q’+F

  17. Estimation of T : By stepwise subtraction of each component (tap’a) from X X = TP’ + E X - TP’ = E Residual after subtraction of ath component X - tapa’ = Xa

  18. X1 X2 X3 Xa-1 X0=t1p1 +t2p2+ t3p3+ t4p4+… + tapa+ E Xa

  19. Stepwise “Deflation” of X-matrix t1 = X0w1 X1= X – t1p1’ t2 = X1w2 X2= X1 – t2p2’ t3 = X2w3 . . . . . . Xa-1= Xa-2 – ta-1p’a-1 ta = Xa-1wa Xa= Xa-1 – tapa’= E

  20. Geometrical Interpretation t,s are modelers of X and predictors of Y

  21. PLS-1 PLS-2 PCA OnePLS-2model PLS-1 models MultivariableY or ? One y at a time all in a single model Y Rankof Y ( #PCs) If #PCs << # Y variables : If #PCs =< # Y variables :

  22. No of PLS components !! Underfitting If proper : Overfitting GOODpredictionability

  23. X Y Pred. Pred. Calibr. Pred. Cross Validation: Predictive REsidual Sum of Squares

  24. Different # components in the model Different PRESS values Model withproper # components is The model with minPRESS value

  25. PLS Algorithm Common and simple Nonlinear Iterative PArtial Least Squares Initially : Transformation, Scaling and Centering of X and Y

  26. P = X’T X Utilizing X-model T P T = XP X = TP’ + E Base : Y = UQ’ + F = TQ’ + F

  27. Having: X Y • Autoscaled • Not deflated Fora=1toA is (X0, or X1, …, or Xa-1) is (Y0, or Y1, …, or Ya-1) A Getting u(temporary Y-score): One of Y columns For using as X-score

  28. Makew’awa=1 B Calculating wa( X-weights ) Temp. X-loadings Xa-1=uawa’ + E wa= X’a-1ua/u’aua

  29. C Calculating ta (X-scores): Xa-1=tawa’ + E ta= Xa-1wa Scores for both X and Y

  30. D Calculating pa ( X-loading) and qa(Y-loading) Xa-1 = tapa’ + E pa = Xa-1ta/ta’ta Ya-1 = taqa’ + F qa = Ya-1ta/ta’ta

  31. (ta)new- ta / (ta)new < 10-7 E Testing desireness of ua: By calculating taagain (ua )new = Ya-1qa/ qa’qa wa= X’a-1ua/u’aua (ta)new= Xa-1wa Performing convergence test on it.

  32. Goto Using (ua)new B Xa = Xa-1 - tapa’ Ya = Ya-1 - taqa’ B Or : a=a+1 and Goto F If No convergence: G If convergence : Calculating new X and Y for the next cycle Next a

  33. B = W(P’W)-1Q’ H Last Step (when a = A) Y = X B+B0 PLS-Regression Coefficients (B)

  34. summary Scores X0 Y0 Loadings X1 = X0 – t1p1’ Y1 = Y0 – t1q1’ t1 u1 q1 p1 w1

  35. Scores X1 Y1 Loadings X2 = X1 – t2p2’ Y2 = Y1 – t2q2’ t2 u2 q2 p2 w2

  36. Scores Xa-1 Ya-1 Loadings Xa = Xa-1 – tapa’= E Ya = Ya-1 – taqa’ =F ta ua qa pa wa

  37. T U X Y P W Q + A , E, andF

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