Download presentation

Presentation is loading. Please wait.

Published byTerrence Waggoner Modified over 2 years ago

1
بنام خدا 1

2
An Introduction to multi-way analysis Mohsen Kompany-Zareh IASBS, Nov 1-3, 2010 2 Session one

3
3 The main source:

4
4 Kronecker product Khatri-Rao product Multi-way data Matricizing the data Interaction triad GG PARAFAC Panel performance Matricizing and subarray Rank Dimensionality vector Rank-deficiency in three-way arrays Tucker3 rotational freedom Unique solution Tucker2 model Tucker1 model

5
>> A=[2 3 4; 2 3 4] >> B=[3 4; 3 5] >> krnAB=[A(1,1)*B A(1,2)*B A(1,3)*B ; A(2,1)*B A(2,2)*B A(2,3)*B] krnAB = 6 8 9 12 12 16 6 10 9 15 12 20 6 8 9 12 12 16 6 10 9 15 12 20 >> kronecker product (A B) 5

6
>> A=[2 3 4; 2 3 4] >>B=[3 4; 3 5] >> p=kron(A,B) >>p= 6 8 9 12 12 16 6 10 9 15 12 20 6 8 9 12 12 16 6 10 9 15 12 20 >> 6 All columns in A see all columns in B. kronecker product

7
>> A=[2 3 4; 2 3 4] >>C=[3 4 5; 3 5 2] >>krnAC=[kron(A(:, ),C(:, ))... column 1 kron(A(:,1),C(:,2))... column 2 kron(A(:,1),C(:,3))..... kron(A(:,2),C(:,1))..... kron(A(:, ),C(:, ))..... kron(A(:,2),C(:,3))... kron(A(:,3),C(:,1))... kron(A(:,3),C(:,2))... kron(A(:, ),C(:, ))] column 9 krnAC = 6 8 10 9 12 15 12 16 20 6 10 4 9 15 6 12 20 8 6 8 10 9 12 15 12 16 20 6 10 4 9 15 6 12 20 8 >> Khatri-Rao Product kronecker product 7 1 1 22 3 3

8
>> A=[2 3 4; 2 3 4] >>C=[3 4 5; 3 5 2] krnAC = 6 8 10 9 12 15 12 16 20 6 10 4 9 15 6 12 20 8 6 8 10 9 12 15 12 16 20 6 10 4 9 15 6 12 20 8 kronecker product 8 vec(a1 b1) vec(a2 b2)vec(a3 b3) vec(a1 b2) vec(a1 b3) vec(a2 b3) vec(a2 b1) vec(a3 b1) vec(a3 b2) Interaction terms

9
>> A=[2 3 4; 2 3 4] >> B=[3 4 5; 3 5 2] khtrAB= 6 12 20 6 15 8 6 12 20 6 15 8 >> 9 No of columns in A should be the same as the number of columns in B. Khatri-Rao Product

10
10 Kronecker product Khatri-Rao product Multi-way data Matricizing the data Interaction triad GG PARAFAC Panel performance Matricizing and subarray Rank Dimensionality vector Rank-deficiency in three-way arrays Tucker3 rotational freedom Unique solution Tucker2 model Tucker1 model

11
(generalization of matrix algebra) A zero-order tensor: a scalar; a first-order tensor : a vector; a second-order tensor (a matrix) for a sample => 3 way data, for analysis a third-order tensor (three-way array) for a sample => 4 way data, for analysis a fourth-order tensor : a four-way array and so on. 11 Multi-way Data

12
12 One component, HPLC-DAD a1 b1

13
13 One component, HPLC-DAD, different concentrations (elution profile) Only the intensities are changed... These 9 matrices form a TRIAD, the simplest trilinear data

14
14 >> a1' 0.0033 0.0971 0.8131 1.9506 1.3406 0.2640 0.0149 >> b1' 0.0222 1.7650 0.4060 0.8826 0.0111 0.0000 >> c1' 1 2 3 4 5 6 7 8 9 10 11 12 A triad : X A cube of data 12x7x7 3 rd order data for one sample Obtained from Tensor product of 3 vectors a1 b1 c1

15
% A triad by outer product % X111=a1 b1 c1... for l=1:length(a1) for m=1:length(b1) for n=1:length(c1) disp([l m n]) Xtriad(l,m,n)=a1(l)*b1(m)*c1(n); end X=Xtriad;.... a1 b1 c1

16
Matricizing the data 16 X111= Unfold3D(X111, 1) (in three directions) The first chemical component

17
17...and for the 2 nd and the next chemical components: X111 = a1 b1 c1 X222 a2 b2 c2 X222 = a2 b2 c2 X333 = a3 b3 c3 Each component in a separate triad (no interaction) + + X = X111 + X222 + X333 Trilinear PARAFAC

18
18 X111 = a1 b1 c1 X222 a2 b2 c2 X222 = a2 b2 c2 2b2 X121 = a1 b2 c1 + + X = X111 + X222 + X121 Non Trilinear!! Tucker In the presence of Interaction : Interaction triad

19
19 How many interaction triads? For two components in three modes: X111 = a1 b1 c1 X112 = a1 b1 c2 X121 = a1 b2 c1 X122 = a1 b2 c2 X211 = a2 b1 c1 X212 = a2 b1 c2 X221 = a2 b2 c1 X222 = a2 b2 c2 G(111)= 2 G(112)= 0 G(121)= 1 G(122)= 0 G(211)= 0 G(212)= 0 G(221)= 0 G(222)=-3 6 possible interaction triads1 interaction triads G

20
A(11x2) G(2x2x2) C(3x2) B(100 2) G(111)= 2 G(222)=-3 G(121)= 1

21
21 For three components in three modes: (3 3 3) – 3 = 24 possible interactions

22
A(15x4) G(?x?x?) C(20x2) B(100 3) How many G elements?

23
23 % Tucker3 outer product G=rand(4,3,2); for p=1:size(G,1) for q=1:size(G,2) for r=1:size(G,3) for i=1:size(A,2) for k=1:size(C,2) for m=1:size(B,2) disp([p q r i j k]) Xtriad(l,m,n)=A(i,l)*B(j,m)*C(k,n)*G(i,j,k); end X=X+Xtriad; end One triad

24
24 What about Tucker4?

25
25

26
26 % PARAFAC outer product G=zeros(3,3,3); G(1,1,1)=1;G(2,2,2)=1;G(3,3,3)=1; for p=1:size(G,1) for q=1:size(G,2) for r=1:size(G,3) for i=1:size(A,2) for k=1:size(C,2) for m=1:size(B,2) disp([p q r i j k]) Xtriad(l,m,n)=A(i,l)*B(j,m)*C(k,n)*G(i,j,k); end X=X+Xtriad; end One triad

27
A(15x3) C(20x3) B(100 3) PARAFAC Simple interpretation

28
Monitoring panel performance within and between experiments by multi-way models Rosaria Romano and Mohsen Kompany-Zareh Copenhagen Univ, 2007

29
Organic Milk of high Quality Sensory studies 2007- University of Copenhagen - Spring experiment (May, week 21 & 22) - Autumn experiment (September, week 36 & 37) Two different experiments were conducted in 2007: The objective is to establish knowledge about production of high quality organic milk with a composition and flavour different from conventionally produced milk.

30
Spring experiment data Data description: 7 varieties of milk with respect to: - 2 cow races: Holstein-Fries (HF), Jersey (JE); - 7 farms: WB, EMC, UGJ, JP, HM, OA, KI. panel: - 9 assessors, 2 sessions (focus on the second!), 3 replicates for each session. 12 descriptors: odor (green), appearance (yellow), flavor (creamy, boiled-milk, sweet, bitter, metallic, sourness, stald-feed) after taste (astringent0, fatness, astringent20). measurement scale: continuous scale anchored at 0 and 15.

31
Parafac on the spring experiment(1) Model: Parafac with two components (27.9% ExpVar), on data averaged across the samples mode HF JE high reproducibility of the replicates in both groups; big variation in the JE group: - WB is the less yellow JE milk; - UGJ seems have something in common with HF group.

32
Parafac on the spring experiment(2) Model: Parafac with two components (27.9% ExpVar), on data averaged across the samples mode Best Reliability on Multi-way Assessment (Bro and Romano, 2008)

33
33 Kronecker product Khatri-Rao product Multi-way data Matricizing the data Interaction triad GG PARAFAC Panel performance Matricizing and subarray Rank Dimensionality vector Rank-deficiency in three-way arrays Tucker3 rotational freedom Unique solution Tucker2 model Tucker1 model

34
A has full rank (if and only if ) : r(A) = min(I,J). If r(A )= R, [Schott 1997] A = t 1 p 1 + ·· ·+t R p R R rank one matrices (t r p r, components). 34 Bases are not unique: rotational freedom intensity intensity (or scale) indeterminacy. sign indeterm sign indeterminacy. Rank

35
If X (I × J ) : generated with I × J random numbers =>probability of (X has less than full rank) =0.. => measured data sets in chemistry: always full rank (mathematical rank) <= measurment noise Ex: UV spectra (100 wavelengths) ; ten different samples, each: same absorbing species at different concentrations. X (10 ×100) if Lambert–Beer law holds : rank one. 35 mathem rank = ten + measurement errors => mathem rank = ten.

36
X = cs’ + E = X hat + E (model of X) vector c : concns, s : pure UV spectrum of the abs species E : noise part. systematic varNoise 1. systematic variation 2. Noise (undesirable) X hat pseudo-rank =Math rank (X hat ) = one X < math rank (X). ‘chemical rank’ : number of chemical sources of variation in data. 36

37
Rank deficiency pseudo-rank < chemical rank. ( linear relations in or restrictions on the data). Ex; X = c 1 s 1 + c 2 s 2 + c 3 s 3 + E, s 1 = s 2 (linear relation) => X = (c 1 + c 2 )s 1 + c 3 s 3 + E Chem rank (X)= 3 pseudo-rank (X)= 2, rank deficient 37

38
38

39
A randomly generated 2 × 2 × 2 array to have a rank lower than three : a positive probability [Kruskal 1989]. a probability of 0.79 of obtaining a rank two array a probability of 0.21 of obtaining a rank three. probability of obtaining rank one or lower is zero. generalized to : 2 × n × n arrays [Ten Berge 1991]. 39

40
2 × 2 × 2 array: the maximum rank: three typical rank: {2, 3}, (almost all individual rank: very hard to establish. Three way rank : important in second-order calibration and curve resolution. for degrees of freedom ?? for significance testing. 40

41
X(4 × 3 × 2) Boldfaces : in the foremost frontal slice 41 Matricizing and Sub-arrays Matricizing

42
42 sub-arrays

43
Row-rank, column-rank, tube-rank two-way X : rank(X) = rank(X’) column rank= row rank :not hold for three-way arrays. three-way array X(I × J × K) : matricized in three different ways (i) row-wise, giving X(J ×IK), a two-way array (ii) column-wise, giving X(I×JK), (iii) tube-wise, giving X(K×IJ). and three more with the same ranks,not mentioned ranks of the arrays X(J×IK),X(I×JK) and X(K×IJ), = (P, Q, R): dimensionality vector of X. 43 Dimensionality vector

44
44 P, Q and R: not necessarily equal. In contrast with two-way P = Q = r(X). dimensionality vector (P, Q, R) of a three-way array X with rank S Obeys certain inequalities [Kruskal 1989]: (i) P ≤ QR ; Q ≤ PR; R ≤ PQ (ii) max(P, Q, R) ≤ S ≤ min(PQ, QR, PR)

45
These arrays have rank 4, 3, and 2. Dimensionality vector is [4 3 2] P, Q and R can be unequal. 45 Three matricized forms:

46
Pseudo-rank, rank deficiency and chemical sources of variation pseudo-rank of three-way arrays: straight generalization of the two-way definit. X = X hat + E E : array of residuals. pseudo-rank of X = minimum # PARAFAC components necessary to exactly fit X hat. 46

47
Spectrophometric acid-base titration of mixtures of three weak mono-protic acids (or Flow injection analysis + pH gradient) HA2 H + + A2 - A3 HA3 H + + A3 - HA4 HA4 H + + A4 - six components models of separate titration of the three analytes (HA2, HA3, HA4), X HA2 = c a,2 s a,2 + c b,2 s b,2 + E HA2 X HA3 = c a,3 s a,3 + c b,3 s b,3 + E HA3 X HA4 = c a,4 s a,4 + c b,4 s b,4 + E HA4 10 samples, 15 titn points, and 20 wavel.s => X(10×15×20), 47 Rank-deficiency in three-way arrays

48
X = X hat + E c a,2 + c b,2 = α(c a,3 + c b,3 ) = β(c a,4 + c b,4 ) only four independently varying concn profiles. Pseudo-rank (X(IJ K)) = four. pseudo-rank (X(3 × JK)) =three. six different ultraviolet spectra form, pseudo-rank (X(6 × KI)) =six ==>> a Tucker3 (6,4,3) model is needed to fit X. 48

49
49 3 6 4 = 72 nonzero elements !! Inequality laws: (i) P ≤ QR ; Q ≤ PR; R ≤ PQ (ii)max(3, 6, 4) ≤ S ≤ min(PQ, QR, PR) 6 ≤ S ≤ 12

50
50 three-way rank of X is ≥ 6 (six PARAFAC components fit the data) Pseudo rank (S=6) is not less than chemical rank(6) => no three-way rank deficiency. rank deficiencies in one loading matrix of a three-way array are not the same as a three-way rank deficiency.

51
51 How it is possible to have a rank deficient three-way data?

52
52 Kronecker product Khatri-Rao product Multi-way data Matricizing the data Interaction triad GG PARAFAC Panel performance Matricizing and subarray Rank Dimensionality vector Rank-deficiency in three-way arrays Tucker3 rotational freedom Unique solution Tucker2 model Tucker1 model

53
Tucker component models Ledyard Tucker was one of the pioneers in multi-way analysis. He proposed a series of models nowadays called N-mode PCA or Tucker models [Tucker 1964- 1966] 53

54
54 TUCKER3 MODELS : nonzero off-diagonal elements in its core.

55
In Kronecker product notation the Tucker3 model 55

56
PROPERTIES OF THE TUCKER3 MODEL T A : arbitrary nonsingular matrix Such a transformation of the loading matrix A can be defined similarly for B and C, using T B and T C, respectively 56 Tucker3 rotational freedom

57
Tucker3 model has rotational freedom, But: it is not possible to rotate Tucker3 core-array to a superdiagonal form (and to obtain a PARAFAC model.! 57 The Tucker3 model : not give unique component matrices it has rotational freedom.

58
rotational freedom Orthogonal component matrices (at no cost in fit by defining proper matrices T A, T B and T C ) convenient : to make the component matrices orthogonal easy interpretation of the elements of the core- array and of the loadings by the loading plots 58

59
59 SS of elements of core-array amount of variation explained by combination of factors in different modes. variation in X: unexplained and explained by model Using a proper rotation all the variance of explained part can be gathered in core.

60
60 The rotational freedom of Tucker3 models can also be used to rotate the core-array to a simple structure as is also common in two-way analysis (will be explained).

61
Imposing the restrictions A’A = B’B = C’C = I : not sufficient for obtaining a unique solution To obtain uniqe estimates of parameters, 1. loading matrices should be orthogonal, 2. A should also contain eigenvectors of X(CC’ ⊗ BB’)X’ corresp. to decreasing eigenvalues of that same matrix; similar restrictions should be put on B and C [De Lathauwer 1997, Kroonenberg et al. 1989]. 61 Unique solution

62
62 Unique Tucker Simulated data: Two components, PARAFAC model

63
63 Unique Tucker3 component model P=Q=R=3 Only two significant elements in core

64
64 Not exactly unique!

65
65 Not exactly unique! But very similar

66
66 Kronecker product Khatri-Rao product Multi-way data Matricizing the data Interaction triad GG PARAFAC Panel performance Matricizing and subarray Rank Dimensionality vector Rank-deficiency in three-way arrays Tucker3 rotational freedom Unique solution Tucker2 model Tucker1 model

67
all three modes are reduced In tucker 3 67

68
68 Data reduction only in two dimensions... Tucker2 model

69
Tucker1 models : reduce only one of the modes. + X (and accordingly G) are matricized : 69 Tucker1 model

70
70 different models [Kiers 1991, Smilde 1997]. Threeway component models for X (I × J × K), A : the (I × P) component matrix (of first (reduced) mode, X(I×JK) : matricized X; A,B,C : component matrices; G : different matricized core-arrays ; I :superdiagonal array (ones on superdiagonal. (compon matrices, core-arrays and residual error arrays : differ for each model => PARAFAC model is a special case of Tucker3 model. PARAFAC: X (IxJK) = A G (RxRR) (C B)’ Tucker3: X (IxJK) = A G (PxQR) (C B)’ Tucker2: X (IxJK) = A G (PxQK) (I B)’ Tucker1: X (IxJK) = A G (PxJK) (I I)’

71
71 Thanks and See you in the next session...

Similar presentations

Presentation is loading. Please wait....

OK

Matrices, Vectors, Determinants.

Matrices, Vectors, Determinants.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on power sharing in india download Ppt on solar energy conservation Ppt on national education policy 1986 ford Download ppt on band pass filter Ppt on stock exchange of india Ppt on indian postal service Fun ppt on biomes of the world Ppt on polynomials download Ppt on distance-time graph and velocity time graph Ppt on switching devices for communicating