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P artial

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

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P artial

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

  2. Regression : Modeling dependent variable(s): Y • Chemical property • Biol. 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 ! • Numerous • Correlated • Noisy • Incomplete Instruments Instruments Spectrometers Chromatographs Sensor Arrays Data …

  5. X : Independent Variables Correlated Predictor

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

  7. PLSR is able to analyze Data with: PLSR is a generalization of MLR • Noise • Collinearity (Highly Correlated Data) • Numerous X-variables(> # samples) • incompleteness in both X and Y

  8. History Herman Wold (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 One Y-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 Table1 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. Increase in weights of 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. Auto Scaling Numerically More Stable

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

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

  18. X= X1 + X2 + X3+ X4 + … + XA X= t1p1 +t2p2+ t3p3+ t4p4+… + tapa E1 E2 E3 Ea-1

  19. Stepwise “Deflation” of X-matrix t1 = Xw1 E1= X – t1p1’ t2 = E1w2 E2= E1 – t2p2’ t3 = E2w3 . . . . . . Ea-1= Ea-2 – ta-1p’a-1 ta = Ea-1wa

  20. W

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