V T n n = A r n A n r n U V m m n n UT U = I V T V = I : A = A = UΣV T A T AV = VΣ T Σ : AB T = B T A T V A T A V A V T V = I 3 V A V T V = I : A AK =

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1 PLS Janes PLS PLS PCR MLR PCA singular value decomposition : m n A 3 A = U m n m m Σ m n VT n n U left singular matrix V Σ U = m m A m r Σ = m n σ σ r A m m r

2 V T n n = A r n A n r n U V m m n n UT U = I V T V = I : A = A = UΣV T A T AV = VΣ T Σ : AB T = B T A T V A T A V A V T V = I 3 V A V T V = I : A AK = K 4 Σ σ,, σ r A T A Σ 5 AA T U = UΣΣ T 6 3 U U T U = I 7 A = UΣV T A T AV = VΣ T U T UΣV t V = VΣ T Σ U T U = I, V T V = I

3 A T A = T A T A Iλ = λ 4 = 4 8 λ λ8 λ 6 = λλ = λ =, x y = y = x 3 AK = K x y x y = k = 5 x y 3 : dax = Adx dax + b T CDx + e = {Ax + b T CD + Dx + e T C T A}dx 3

4 4 CovXd, Ye = d T X T Ye d e X T Y d T d = e T e = X T Y = UΣV T = u σ v T + + u r σ r v T r d = a u + + a r u r e = b v + + b r v r d T X T Ye = d T u σ v T + + u r σ r vr T e = a u + + a r u r T u σ v T + + u r σ r vr T b v + + b r v r = a σ b + + a r σ r b r = a T Σb d T d = e T e = d T d = a u + + a r u r T a u + + a r u r = a + + a r = a T a d T d = a T a = e T e = b T b = d T d =, e T e = CovXd, Ye = d T X T Ye a T a =, b T b = CovXd, Ye = a T Σb Q = a T Σb λ a T a λ b T b 4

5 d dx xt x = x { Q d dx at x = a a = Σb λ a = Q b = Σ T a λ b = { Σb = λ a a T Σ = λ b T a T b λ! = λ { a T Σb = λ a T Σb = λ { λ = λ! = λ Σb = λa Σ T a = λb λ ΣT Σb = λb Σ T Σb = λ b b b,, b r 5

6 σ b σ rb r = λ b λ b r b,,, a d T X T Ye = a σ b + + a r σ r b r d T X T Ye σ,, σ r σ d T X T Ye a, b a =, b = d = a u + + a r u r e = b v + + b r v r d, e d = u e = v d, e 6

7 4 Moore-Penrose Moore-Penrose Moore-Penrose generalized inverse MLR PCR A # n m = V Σ = n n Σ n m m m UT σ σ r Ax = b A ˆx = A # b Ax b A ˆx = A # b x x 4 N = UΣV T U V d dt x x = c c v + c c v + + c n c n v n x m c m c m c mn c i c i c mi T 7

8 dx i dt = r r r n v v v m r r c n N NK = K K N GN = G 5 MLR 5 y x,, x n m y x y = Xa + ε y y = x x n x x n a a + ε ε y m x m x mn Q a a n ε m 8

9 5 Q = = m i= ε i m {y i a + a x i + + a n x in } i= = y Xa T y Xa Q a dq da = a dax + b T CDx + e = {Ax + b T CD + Dx + e T C T A}dx C = I, D = A, e = b dax + b T Ax + b = {Ax + b T A + Ax + b T A}dx = Ax + b T Adx Q = y Xa T y Xa dq = d{y Xa T y Xa} = y Xa T Xdx dq da = a y Xa T X = 9

10 y Xa T X = y Xa T X = X T y Xa = AB T = B T A T X T y = X T Xa a = X T X X T y 53 Moore-Penrose y y = a +a x + +a n x n Moore- Penrose â = X # y X T X X X # = X T X X T XX T X X # = X T XX T 6 PCA 6

11 cm kg t P C x, x t P C = w x + w x θ w, w t P C

12 X ˆX X ˆX T ˆX ˆXT ˆX loading x ij = x ij x j X = x x x n x x x n x m x m x mn cm kg X = t = w x + w x + + w n x n w T w = t m = x m w w 6 w x m = x m x m x mn, w = w w n

13 t = t t t n, t = Xw m t = m = m = m = m = m = i= t i m x i w i= m i= n x ij w j j= n m w j x ij j= n w j j= i= σt = m tt t = m Xw T Xw = w T m XT X = w T Vw w Lagrange w T w = σ t Qw = σ t λw T w = w T Vw λw T w 3

14 dax + b T CDx + e = {Ax + b T CD + Dx + e T C T A}dx dw T Vw = w T Vdw dw T w = w T dw dqw = w T Vdw λw T dw = w T V = λw T Vw = λw V V = V T w V σ t = w T Vw = λ σ t V λ 63 i 4

15 = = σ t i σ t + σ t + + σ t n λ i λ + λ + + λ n σ t i i i i σ t i V i i i 64 t i = Xw i T = P = t t t k p p p k T = XP X = TP T 65 loading X = U Σ V T = T P T UΣ = T, V = P T P loading 5

16 66 3 cm kg X 3 X T X 4 X T X PCR 7 X X # a a X X 7 Moore-Penrose B Y = X B = T P T B X = TP T 6

17 ˆB P CR = PT # Y Y = X ˆB P CR 8 PLS PCR PLS PLS X Y T,U T U X,Y T,U X X T X X Y X T Y X Y PLS PLS X Y Höskuldsson 988 X T Y w c w c Xw Yc Höskuldsson 988 w T w = c T c = w c } t = Xw 3 t u u = Yc 4 t T t = u T u = t u 5 u = bt b b = u T t u T = bt T u T t = bt T t u T t = b 7

18 } p = X T t 6 p,q p,q q = Y T PCA loading u PCA loading 7 X tp T Y uq T PLS 8 X Y PLS X Y loadings P Q B P Q p q B b Y = UQ T = TBQ T = XP T # BQ T ˆT = XP T # = XB PLS B PLS = P T # BQ T ˆT = XP T # P = X T T P T = T T X TP T = TT T X ˆT = XP T # 8 EGF NGF ERK 3 c-fos c-jun ERK t ERK t ERK t3 EGF 4 NGF 4 4 c-fos 3min c-jun 3min EGF NGF ERK c-fos c-jun PLS PLS 8

19 MAPK IEG X Y X Y X Ȳ X Y X Ȳ X Y 3 X T Y 4 X T Y w c w T w = c T c = w c 5 t = Xw u = Yc PLS t u t T t = u T u = t u 6 PLS u = bt b b = u T t 7 p = X T t q = Y T u PLS loadings p q 8 PLS n X tp T X Y uq T Y 3 X tp T Y uq T 9 PLS Y = XB PLS B PLS a P T P T = UΣV T b P T # = VΣ U T P T # c B PLS = P T # BQ T B PLS Y Ȳ = X XB PLS c-jun ERK ERK c-fos c-jun PLS loadings p q 8 X = 4 4 4, Y = 9

20 3 X = X T Y = =, Y = 4 X T Y = UΣV T U = w w, V = c c V {X T Y T X T Y}V = VΣ T Σ = λ λ = λ 4 = λ 4λ = λ = 4, x y = 4 x = y x = k y x y k = k = c =

21 U {X T YX T Y T }U = UΣΣ T = λ λ λ = λ λ + 4λ = λλ 4λ λ = λλ 4λ = x y z x = z, y = x y z = k k = k = w = λ = 4,, = 4 x y z

22 5 t = Xw t = = t = u = Yc u = = 6 b = u T t b = = 7 p = X T t p = = q = Y T u

23 q = = 8 X tp T = 9 B PLS a p T p T = p T pu = UΣΣ T 4U = UΣΣ T 4 λ = λ = 4, u = pp T = λ λ λ = λ λ + 4λ = λλ 4λ λ = λλ 4λ = λ = 4,, 3

24 x = z, y = x y z v = k = 4 x y z v = p T p T = b p T p T # A # = VΣ U T p T # = = c B PLS B PLS = P T # BQ T B PLS = = 4 4

25 Fos EGF Fos NGF Y Ȳ = X XB PLS Jun EGF Jun NGF Y = X B PLS XB PLS + Ȳ = ERK t EGF ERK t EGF ERK t3 EGF ERK t NGF ERK t NGF ERK t3 NGF p = Fos EGF = 4 ERKt EGF + ERK t3 EGF Jun EGF = 4 ERKt EGF + ERK t3 EGF + Fos NGF = 4 ERKt NGF + ERK t3 NGF Jun NGF = 4 ERKt NGF + ERK t3 NGF + X 3 t, u q = Y u ERK X c-fos c-jun Y t t Janes and Yaffe Metrics stimuli Cell responses Science 9 R Mevik and Wehrens The pls Package: Principal Component and Parital Least Squares Regression in R, J Stat Soft 8:, 7 5

26 93 Janes and Yaffe PLS Further reading Abdi, H 7 Partial least square regression PLS regression In NJ Salkind ed: Encyclopedia of Measurement and Statistics Thousand Oaks, CA, pp PLS PDF Abdi herve/ Jane and Yaffe 6

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