PLS post-processing by similarity transformation: a simple alternative to OPLS

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PLS post processing by similarity transformation (PLS+ST): a simple alternative to OPLS

Theoretical properties and proofs

Rolf Ergon

Telemark University College Porsgrunn, Norway Email: [email protected]

This Supplementary Appendix gives the details and proofs of properties and results in the paperPLS post-processing by similarity transformation (PLS+ST): A simple alternative to OPLS [9]. For the readers convenience, the OPLS algorithm [2] is also included.

Property 1 The Martens factorization (2) has the special property that all score vectors except the …rst one are orthogonal to bothyandy.^

Proof: Sincew₁ is given byw₁= _k^X_X^TT^yyk andT_2:A=XW_2:A, and sinceW^TW=I, it follows thatT^T_2:Ay=W^T_2:AX^Ty=°°X^Ty°°W_2:A^T w₁=0. From the prediction formula (3) further follows

^

y=XW¡

W^TX^TXW¢_¡1

W^TX^Tyand thus

£ t1 T2:A ¤T

^

y = T^T^y=W^TX^TXW¡

W^TX^TXW¢¡1

W^TX^Ty=W^TX^Ty

= °°X^Ty°°W^Tw1=£ °°X^Ty°° 0 ¤T

; (17)

i.e. T^T_2:A^y=0:

Property 2 The residualEin the Martens factorization (2) is also orthogonal toy.

Proof: FromX^Ty=°°X^Ty°°w₁,w^T₁w1= 1andy^TT2:A= 0followsy^TE=y^T³

X¡TW^T´

= y^TX¡y^T¡

t1w^T₁ +T2:AW^T_2:A¢

=y^TX¡y^Tt1w₁^T =y^TX¡y^TXw1w^T₁ =°°X^Ty°°w₁^T¡°°X^Ty°°w^T₁ = 0.

Property 3 The factorizations (13) and (2) are identical, i.e. TWP^TW=T. Proof: From the two well known estimator expressionsb^=W¡

W^TX^TXW¢_¡1

W^TX^Ty and

^b=W¡

P^TW¢¡1

qW =W¡

P^TW¢¡1¡

T^T_WTW¢¡1

T^T_Wy [7] follows W¡

W^TX^TXW¢¡1

W^TX^Ty=W³¡

P^TW¢T

T^T_WTWP^TW´_¡1¡

P^TW¢T

T^T_Wy; (18) i.e. TWP^TW=XW=T:

Property 4 The loading matrices in the factorizations (12) and (13) are P = £

p₁ p₂ ¢ ¢ ¢ p_A_¡₁ p_A ¤ and WW^TP = £

p₁ p₂ ¢ ¢ ¢ p_A_¡₁ w_A ¤, i.e. they are di¤erent in the last column vector only.

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Proof: The orthogonalized PLS algorithm results in an upper triangular and bi-diagonal matrix P^TW, with ones along the main diagonal [8]. We thus have

P^TWW^T = 2 66 66 66 64

1 p^T₁w₂ 0 ¢ ¢ ¢ 0 0 1 p^T₂w₃ ... ...

... ... ... ... 0

... ... 1 p^T_A¡1w_A 0 ¢ ¢ ¢ ¢ ¢ ¢ 0 1

3 77 77 77 75 2 66 66 66 4

w^T₁ w^T₂ ... ... w^T_A

3 77 77 77 5

= 2 66 66 64

w^T₁ +p^T₁w₂w₂^T w^T₂ +p^T₂w₃w₃^T

...

w^T_A¡1+p^T_A¡1w_Aw_A^T w^T_A

3 77 77 75 :

(19) From this follows that p_A in P is replaced by w_A, as stated. For a complete proof we must also show that for 2 · i · A we have w^T_i_¡₁+p^T_i_¡₁w_iw^T_i = p^T_i_¡₁, or equivalently that w_i^T = ¡

p^T_i_¡₁w_i¢¡1¡

p^T_i_¡₁¡w_i^T_¡₁¢. Forming ¡

p^T_i_¡₁w_i¢¡1¡

p^T_i_¡₁¡w^T_i_¡₁¢

w_j we …nd the following possibilities for2·i·A:

j < i¡1 ) ¡

p^T_i¡1w_i¢¡1¡

p^T_i¡1¡w^T_i¡1¢ w_j=¡

p^T_i¡1w_i¢¡1(0¡0) = 0

j=i¡1 ) ¡

p^T_i¡1w_i¢_¡1¡

p^T_i¡1¡w^T_i¡1¢ w_j=¡

p^T_i¡1w_i¢_¡1

(1¡1) = 0

j=i ) ¡

p^T_i_¡₁wi¢_¡1¡

p^T_i_¡₁¡w^T_i_¡₁¢ wj=¡

p^T_i_¡₁wi¢_¡1¡

p^T_i_¡₁wi¡0¢

= 1

j > i ) ¡

p^T_i_¡₁wi¢_¡1¡

p^T_i_¡₁¡w^T_i_¡₁¢ wj=¡

p^T_i_¡₁wi¢_¡1

(0¡0) = 0

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Since p_i and thus also ¡

p^T_i¡1w_i¢¡1(p_i_¡₁¡w_i_¡₁)belong to the span of w₁, w₂, ... w_A, and since ¡

p^T_i_¡₁wi¢_¡1¡

p^T_i_¡₁¡w^T_i_¡₁¢

wj = 1 for j = i and 0 for j 6= i, it …nally follows from the orthonormality ofWthat ¡

p^T_i¡1w_i¢¡1¡

p^T_i¡1¡w^T_i¡1¢

=w^T_i , and thus thatw^T_i¡1+p^T_i¡1w_iw_i^T = p^T_i_¡₁ for2·i·A.

Property 5 Using a predetermined loading weights matrix W, the de‡ation order in the algorithm resulting in the non-orthogonlized factorization (2) is of no importance for the resid- ual and predictions. A loading weights matrix W~ with permuted column vectors will thus give X=T ~~W^T+EwithT ~~W^T =TW^T, and ynew=x^T_newb^according to Eq. (3).

Proof: Since W is orthonormal the PLS algorithm giving the non-orthogonalized factorization (2) generally gives ti = (X¡P

over allj6=itjw_j^T)wi =Xwi. This is true irrespective of the order of de‡ation, i.e. T~ =X ~W. Introducing an invertible permutation matrix P~ with the property P~^¡¹=P~^Tand W~ =W ~P , the predictions according to Eq. (3) will be ynew = x^T_new~b = x^T_newW ~P³

PW~ ^TX^TXW ~P´¡1

PW~ ^TX^Ty=x^T_newW¡

W^TX^TXW¢¡1W^TX^Ty=x^T_newb.^

Property 6 Using a predetermined loading weights matrix W, the de‡ation order in the algorithm resulting in the orthogonlized factorizations (12) and (13) is of no importance for the residuals and predictions. A loading weights matrix W~ with permuted column vectors will thus give X = T~WP~^T +EW and X = T~WP~^TW ~~W^T +E respectively, with T~WP~^T = TWP^T and T~WP~^TW ~~W^T =TWP^TWW^T.

Proof: According to Property 3 the relation between the orthogonalized and non-orthogonalized PLS algorithms isTWP^TWW^T=TW^T. Use of the same algorithms with the predetermined and permuted matrixW~ must then necessarily result inT~_WP~^TW ~~W^T=T ~~W^T. SinceW ~~W^T =WW^T andT ~~W^T =TW^T (Property 5) it also follows that T~WP~^TWW^T=TW^T =TWP^TWW^T and thusT~WP~^T =TWP^T. From this follow unaltered residuals and predictions.

The OPLS algorithm Following [2], the OPLS algorithm is as follows:

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1. Seti= 1,Ei¡1=E0=X, andWortho,Tortho andPortho to empty matrices 2. w^OPLS_i = ^(E^i¡1⁾^T^y

k^(E^i¡1⁾^T^yk=w₁ 3. t^OPLS_i =E_i¡1w_i

4. p^OPLS_i = ^(E^i¡1⁾^T^t^{O P L S}ⁱ (^t^{O P L S}i )^T^t^{O P L S}i

5. w^ortho_i = ^p^{O P L S}ⁱ ^¡^wⁱ

k^p^{O P L S}i ¡wik andWortho = [ Wortho w^ortho_i ]

6. tôrtho_i =E_i_¡₁w_iôrtho andTortho = [ Tortho tôrtho_i ] 7. pôrtho_i =(E^{O P L S}_i¡1 )^Tt^{o r t h o}_i+1

(^t^{o r t h o}i+1 )^T^t^{o r t h o}i+1

andPortho = [ Portho p^ortho_i ] 8. E_i=X¡TorthoP^T_ortho

9. Leti=i+ 1and return to step 2 for additional orthogonal components, otherwise go to step 10

10. End.

The resultingE_iare the …lteredXdata, and a one component PLS factorization after removal ofi=A¡1components further gives

EA¡1=t^OPLS_A ¡

p^OPLS_A ¢T

+EOPLS: (21)

Note that all steps givew^OPLS_i =w1.

Property 7 The OPLS loading weights matrix may be found from the ordinary PLS loading weights matrix asWortho=¡W_2:A.

Proof: From Property 6 follows that orthogonalized PLS regression with the permuted loading weights matrix W~ =£

W_2:A w₁ ¤ gives the same …tted response vector ^y as with use of W.

Since the sign of aw_i vector has nothing to say for the productst_ip^T_i and t^ortho_i ¡

p^ortho_i ¢T

, this is true also forW~ =£

¡W2:A w1 ¤

:We use induction in the parameterirelated toWortho to show that the OPLS algorithm usesWortho =¡W2:A.

Fori= 1, i.e. oney-orthogonal component, the OPLS algorithm gives w^ortho₁ = p^OPLS₁ ¡w1

°°p^OPLS₁ ¡w₁°° = X^TXw1(w₁^TX^TXw1)^¡¹¡w1

°°X^TXw₁(w₁^TX^TXw₁)^¡¹¡w₁°°; (22) while the recursive formula for the loading weights vectors developed by Helland [7] and the prediction formula (3) give (wherey^₁ is the …tted response vector using one PLS component)

w₂ = X^T(y¡^y₁)

kX^T(y¡^y₁)k= X^T¡

y¡Xw₁(w^T₁X^TXw₁)^¡¹w₁^TX^Ty¢

°°X^T(y¡Xw₁(w₁^TX^TXw₁)^¡¹w^T₁X^Ty)°°

= w₁¡X^TXw₁(w₁^TX^TXw₁)^¡¹

°°w₁¡X^TXw₁(w₁^TX^TXw₁)^¡¹°° =¡w^ortho₁ : (23) Assuming the property to be true up tow^ortho_i_¡₁ we …nd according to the OPLS algorithm

w^ortho_i = p^OPLS_i ¡w₁

°°p^OPLS¡w₁°°; (24)

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with

Ei¡1=X¡TorthoP^T_ortho; (25)

where TorthoP^T_ortho is the factorization of thei¡1removedy-orthogonal components. From the recursive loading weights formula [7] we also …nd

w_i+1 = X^T(y¡y^_i) kX^T(y¡y^_i)k =

w₁¡p^X^T^{^}^yⁱ

y^TXX^Ty

°°

°°w₁¡p^X^T^{^}^yⁱ

y^TXX^Ty

°°

; (26)

where^y_i is the …tted response vector using a total oficomponents.

In order to show thatw_i^ortho=¡w_i+1 we …nally make use of the OPLS facts thatT^T_orthoy=0 and T^T_orthoy^=0(see [2] for proofs), i.e. E^T_i¡1y =¡

X¡TorthoP^T_ortho¢T

y= X^Ty and E^T_i¡1y^_i =

¡X¡TorthoP^T_ortho¢T

^

y=X^Ty. We then use the prediction formula (3) and the fact that OPLS^ gives the same predictions as ordinary PLS, and develop p^OPLS_i into (also using w^T₁X^Ty = w₁^Tw1

q

y^TXX^Ty= q

y^TXX^Ty)

p^OPLS_i = E^T_i_¡₁E_i_¡₁w₁¡

w^T₁E^T_i_¡₁E_i_¡₁w₁¢¡1

=E^T_i_¡₁E_i¡1w₁¡

w^T₁E^T_i_¡₁E_i¡1w₁¢¡1w^T₁E^T_i_¡₁y w₁^TE^T_i_¡₁y

= E^T_i_¡₁ ^y_i

w^T₁E^T_i_¡₁y = X^T^y_i

w^T₁X^Ty = X^Ty^_i q

y^TXX^Ty

; (27)

and insertion into Eq. (24) and comparison with Eq. (26) …nally shows thatw^ortho_i =¡w_i+1. Property 8 After the removal ofA¡1y-orthogonal components, the OPLS factorization (14) results in the same residualEOPLS=EW and the same predictions as the original orthogonalized PLS factorization (12).

Proof: SinceWortho =¡W2:Athe OPLS factorization is equivalent with the factorization ob- tained by the standard PLS NIPALS algorithm with predetermined and permuted loading weights vectors in the orderw2,w3, ... ,wA andw1. From Property 6 thus follows that the residuals and the predictions are the same.

Result 1 The second similarity transformationTorthoP^T_ortho=TorthoP^T_orthoW2:A¡

P^T_orthoW2:A¢_¡1

P^T_ortho results in the transformed OPLS score matrixT_orthoP^T_orthoW_2:A=T_2:A.

Proof: According to Property 3 the two factorizationsX=TW^T+EandX=TWP^TWW^T+ Eare identical, i.e. TWP^TW=T. Using a permuted loading weights matrixW~ =£

W2:A w1 ¤ we correspondlingly haveT~=£

T_2:A t₁ ¤

=T~WP~^TW, and that is independent of the number~ of components used. As the OPLS algorithm givesTorthoP^T_ortho by use ofXandWortho=¡W2:A

(Property 7) in exactly the same way as we …nd the …rstA¡1components inT~WP~^T, this will necessarily give

TorthoP^T_orthoW_2:A=T_2:A: (28)

Property 9 The last OPLS componentt^OPLS_A ¡

p^OPLS_A ¢T multiplied withWW^T becomes t^OPLS_A ¡

p^OPLS_A ¢T

WW^T =t^OPLS_A w^T₁. Proof: We have

¡p^OPLS_A ¢T

WW^T =¡

p^OPLS_A ¢T¡

w₁w^T₁ +W_2:AW^T_2:A¢

; (29)

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where

¡p^OPLS_A ¢T

w₁= w^T₁E^T_A_¡₁EA¡1

w^T₁E^T_A_¡₁EA¡1w1

w₁= 1 (30)

and

¡p^OPLS_A ¢T

W2:A = w^T₁E^T_A_¡₁EA¡1

w^T₁E^T_A_¡₁E_A_¡₁w₁W2:A

= w^T₁E^T_A_¡₁ w^T₁E^T_A_¡₁E_A_¡₁w₁

¡X¡TorthoP^T_ortho¢ W2:A

= w^T₁E^T_A_¡₁ w^T₁E^T_A_¡₁EA¡1w1

¡T2:A¡TorthoP^T_orthoW2:A¢

=0; (31) where we in the …nal equality make use of Result 1.

Property 10 After the removal ofA¡1y-orthogonal components, the modi…ed OPLS factorization (15) results in the same residualEand the same predictions as the modi…ed PLS factorization (13).

Proof: SinceWortho =¡W2:Athe OPLS factorization is equivalent with the factorization ob- tained by the standard PLS NIPALS algorithm with predetermined and permuted loading weights vectors in the orderw2,w3, ... ,wAandw1. From Property 6 and Eqs. (13) and (14) thus follows

X = TWP^TWW^T+E=³

TorthoP^T_ortho+t^OPLS_A ¡

p^OPLS_A ¢T´

WW^T +E

= TorthoP^T_orthoWW^T +t^OPLS_A w^T₁ +E; (32)

where the …nal equality making use of Property 9 results in equality with Eq. (15).

Result 2 The …nal modi…ed OPLS component is identical with the …rst PLS+ST component, i.e. t^OPLS_A w^T₁ =t^PLS+ST₁ w₁^T.

Proof: WhenA¡1y-orthogonal components are subtracted fromX;it follows from the OPLS algorithm that the remaining score vector is

t^OPLS_A =¡

X¡TorthoP^T_ortho¢

w1=EA¡1w1: (33) Using the standard prediction formula (3) we further …nd

^

y=E_A_¡₁w₁¡

w^T₁E^T_A_¡₁E_A_¡₁w₁¢¡1

w^T₁E^T_A_¡₁y=E_A_¡₁w₁d=t^OPLS_A d; (34) wheredis a scalar. This con…rms thatt^OPLS_A is in the direction ofy, which according to Property^ 6 is also identical with the …tted response vector using ordinary PLS regression.

From the PLS+ST factorization (5) follows t^PLS+ST₁ = q₁^¡¹^y, where q1 is found as the …rst component in q = ¡

W^TX^TXW¢_¡1

W^TX^Ty: Since y is orthogonal to both T2:A (Property 1) andE(Property 2) we may also …ndq1 by use of the PLS+ST factorization (6) and

y^TX = y^T³

t^PLS+ST₁ w₁^T+T2:A¡

P^PLS+ST_2:A ¢T

+E´

=y^Tt^PLS+ST₁ w^T₁

= y^Tq₁^¡1^yw^T₁ = q₁^¡¹y^T^y q

y^TXX^Ty

y^TX; (35)

i.e. p^q^¡1¹ ^y^T^y^{^}

y^TXX^Ty = 1and

t^PLS+ST₁ =q₁^¡1^y= q

y^TXX^Ty

y^T^y ^y: (36)

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Since y^TTortho = 0 we …nd y^TEA¡1 = y^T¡

X¡TorthoP^T_ortho¢

= y^TX = q

y^TXX^Tyw^T₁, and from the t^PLS+ST₁ expression (36) using y^TXw1 =

q

y^TXX^Tyw₁^Tw1 = q

y^TXX^Ty and y^ according to Eq. (34) thus follows

t^PLS+ST₁ = q

y^TXX^Ty y^T^y y^=

q

y^TXX^TyE_A_¡₁w₁d y^TEA¡1w1d

= q

y^TXX^Tyt^OPLS_A y^TXw1

=t^OPLS_A : (37)

Result 3 The …rst modi…ed and then transformed OPLS loading matrix is identical with the PLS+ST loading matrix, i.e. WW^TPortho¡

W^T_2:APortho¢_¡1

=P^PLS+ST_2:A .

Proof: According to Property 3 the two factorizationsX=TW^T+EandX=TWP^TWW^T+ Eare identical withTWP^TW=T. According to Property 10 these factorizations are also identical with the modi…ed OPLS factorization X = TorthoP^T_orthoWW^T +t^OPLS_A w^T₁ +E and thus the transformed factorizationX=Tortho¡

P^T_orthoW_2:A¢ ¡

P^T_orthoW_2:A¢¡1P^T_orthoWW^T+t^OPLS_A w^T₁+E, while the PLS+ST method givesX=T_2:A¡

P^PLS+ST_2:A ¢T

+t^PLS+ST_A w^T₁ +E. Since Result 1 shows that Tortho¡

P^T_orthoW_2:A¢

=T_2:A, while Result 2 shows that t^OPLS_A w^T₁ =t^PLS+ST_A w^T₁, it follows that¡

P^T_orthoW_2:A¢_¡1

P^T_orthoWW^T =¡

P^PLS+ST_2:A ¢T

.

Property 11 The modi…ed loading matrix WW^TPortho is di¤erent from Portho in the last column vector only, withp^ortho_A_¡₁ replaced by¡

p^ortho_A_¡₁¢T

w1w1¡wA.

Proof: The ordinary PLS algorithm results in an upper triangular and bi-diagonal matrixP^TW, with 1 along the main diagonal [8]. SincePortho in the OPLS algorithm according to Property 7 is found from Wortho =¡W2:A in the same way as Pis found from W, the matrixP^T_orthoW2:A

must also be bi-diagonal with -1 along the main diagonal. We thus have (withp~i=p^ortho_i )

P^T_ortho£

w₁ W_2:A ¤ · w^T₁ W^T_2:A

¸

= 2 66 66 66 64

~

p^T₁w₁ ¡1 ~p^T₁w₃ 0 ¢ ¢ ¢ 0

~

p^T₂w₁ 0 ¡1 ~p^T₂w₄ ... ...

... ... . .. ... ... 0

~

p^T_A¡2w₁ ... ... ¡1 ~p^T_A¡2w_A

~

p^T_A¡1w₁ 0 ¢ ¢ ¢ ¢ ¢ ¢ 0 ¡1 3 77 77 77 75 2 66 66 66 4

w^T₁ w^T₂ ... ... w^T_A

3 77 77 77 5

= 2 66 66 64

~

p^T₁w₁w₁^T¡w^T₂ +~p^T₁w₃w^T₃

~

p^T₂w1w₁^T¡w^T₃ +~p^T₂w4w^T₄ ...

~

p^T_A_¡₂w1w^T₁ ¡w_A^T_¡₁+~p^T_A_¡₂wAw^T_A

~

p^T_A_¡₁w1w₁^T¡w^T_A

3 77 77 75

: (38)

From this follows that~p^T_A¡1 inP^T_ortho is replaced byp~^T_A¡1w₁w₁^T¡w_A^T, as stated. For a complete proof we must also show that for3·i·Awe have ~p^T_i_¡₂w₁w^T₁ ¡w^T_i_¡₁+p~^T_i_¡₂w_iw^T_i =~p^T_i_¡₂, or equivalently thatw^T_i =¡

~

p^T_i¡2w_i¢¡1¡

¡p~^T_i¡2w₁w₁^T+w^T_i¡1+p~^T_i¡2¢. Forming

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¡~p^T_i¡2wi¢¡1¡

¡~p^T_i¡2w1w^T₁ +w_i¡1^T +~p^T_i¡2¢

wj we …nd the following possibilities for3·i·A:

j= 1 ) ¡

~

p^T_i_¡₂w_i¢¡1

(¢)w_j =¡

~

p^T_i_¡₂w_i¢¡1¡

¡~p^T_i_¡₂w₁+ 0 +p~^T_i_¡₂w₁¢

= 0 1< j < i¡1 ) ¡

~

p^T_i¡2w_i¢¡1(¢)w_j=¡

~

p^T_i¡2w_i¢¡1(¡0 + 0 + 0) = 0

j=i¡1 ) ¡

~

p^T_i¡2w_i¢¡1(¢)w_j=¡

~

p^T_i¡2w_i¢¡1(¡0 + 1¡1) = 0

j=i ) ¡

~

p^T_i¡2wi¢_¡1

(¢)wj=¡

~

p^T_i¡2wi¢_¡1¡

¡0 + 0 +~p^T_i¡2wi¢

= 1

j > i ) ¡

~

p^T_i_¡₂wi¢_¡1

(¢)wj=¡

~

p^T_i_¡₂wi¢_¡1

(¡0 + 0 + 0) = 0

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For ordinary PLS we know thatp_i belongs to the span ofw₁,w₂, ... w_A, and from Property 7 and the OPLS algorithm then follows that this must be the case also for¡p~^T_i_¡₂w₁w₁+w_i¡1+~p_i¡2. Since

¡~p^T_i_¡₂wi¢¡1¡

¡~p^T_i_¡₂w1w^T₁ +w_i^T_¡₁+~p^T_i_¡₂¢

wj= 1forj=iand0forj 6=i, it …nally follows that

¡~p^T_i¡2w_i¢¡1¡

¡~p^T_i¡2w₁w^T₁ +w_i¡1^T +~p^T_i¡2¢

=w^T_i , and thus that~p^T_i¡2w₁w^T₁ ¡w^T_i¡1+~p^T_i¡2w_iw_i^T =

~ p^T_i¡2.

Result 4 For a single y-relevant component the relation between the post-processing PCP method [5] and PLS+ST is that t^PCP₁ ! t^ST₁ = t^OPLS_A and w₁^PCP ! w1 when ^y ! y, i.e.

with good predictions.

Proof: PCP uses the factorization (with normalized loadings) X=t^PCP₁ ¡

w^PCP₁ ¢T

+EPCP; (40)

witht^PCP₁ =

p^y^TXX^T^y

^

y^T^y ^yinstead oft^PLS+ST₁ =

py^TXX^Ty

y^T^y y^as in Eq. (36) andw^PCP₁ = p ^X^T^y^{^}

^ y^TXX^T^y

instead ofw₁=p ^X^T^y

y^TXX^Ty as in the PLS algorithms.

References

[1] Svensson O, Kourti T, MacGregor JF. An investigation of orthogonal signal correction algorithms and there characteristics.J. Chemometrics 2002;16: 176-188.

[2] Trygg J, Wold S. Orthogonal projections to latent structures, O-PLS.J. Chemometrics 2002;

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