By: S.M. SajjadiIslamic Azad University, Parsian Branch, Parsian,Iran

Scalar Vector

a = a (I×1) =

Matrix

A (I×J) =

Three-way array

A (I×J×K) =

c1

EEM

c2

EEM

280 290 300

21.6 8.64 2.7

50.4 20.16 6.3

36 14.4 4.5

28.8 11.52 3.6

320

340

360

380

280 290 300

320

340

360

380

14.4 5.76 1.8

33.6 13.44 4.2

24 9.6 3

19.2 7.68 2.4

21.6 8.64 2.7

50.4 20.16 6.3

36 14.4 4.5

28.8 11.52 3.6

14.4 5.76 1.8

33.6 13.44 4.2

24 9.6 3

19.2 7.68 2.4

Constructing Three-way Data Array by Constructing Three-way Data Array by Stacking Two-way DataStacking Two-way Data

For two-way arrays it is useful to distinguish For two-way arrays it is useful to distinguish between special parts of the array, such as rows between special parts of the array, such as rows and columns.and columns.

What are spatial parts in the three-way array?

X( : , : , 1 ) = X1

X( : , : , 2 ) = X2

X(4×3×2)

X(2×4×3)??

X(4×2×3)??

Rows, Columns and Tubes

Row

Tube

Column

2

3

4

2

3

4xjk(4×1)

X( : , j , k )

Rows, Columns and Tubes

Column

2

3

4xjk(4×1)xik(3×1)

X( i , : , k ) X( : , j , k )

Rows, Columns and Tubes

Row Column

2

3

4xjk(4×1)

xij(2×1)

X( i , j , : )xik(3×1)

X( i , : , k ) X( : , j , k )

Rows, Columns and Tubes

Row

Tube

Column

23

4

Horizental Vertical

23

4

X( i , : , : )

Horizental

Vertical

X( : , j , : )

23

4

X( i , : , : )

Horizental

X( : , : , k )

32

4

Vertical

X( i , : , : )

Horizental

X( : , j , : )

There are five EEMs of different

samples that contain two analytes.

2

3

4

2

3

4 4

3

2

3

4

X( : , : )

4

63

2

4

4

6

permute ( X , [1 3 2] )X ( : , : )

4

2

3

???? X( : , : )

4

633

??

4

62

23

4

Matrisizing : X ( : , : )Matrisizing : X ( : , : )

3

8

2

permute ( X , [2 …)

There are five EEMs of different samples that contain two analytes.

Please construct three kinds of three-way data array, i.e., consider each EEM as frontal, horizontal and vertical slices.

Vector multiplicationVector multiplication

aaTTb = scalar:b = scalar:

Inner product = scalarInner product = scalar

=II

Outer product = MartixOuter product = Martix=

I

J

I

J

Vec-operator

Vec of matrix A is the IJ vector

AB = matrix

=

I

J

J

K

I

K

vectorized.... . . .

IJ

Kronecker product

Hadamard product

Khatri–Rao product

*

Tucker

Weighted PARAFAC

PARAFAC

3 4

7 3

5 8

4 12

A

9.45.0

6.36.1

2.14

B

A B =

3 B 4 B

7 B 3 B5 B 8 B

4 B 12 B

3×

4 1.2

1.6 3.6

0.5 4.9

4×

4 1.2

1.6 3.6

0.5 4.9

7×

4 1.2

1.6 3.6

0.5 4.9

3×

4 1.2

1.6 3.6

0.5 4.9

5×

4 1.2

1.6 3.6

0.5 4.9

8×

4 1.2

1.6 3.6

0.5 4.9

4×4 1.2

1.6 3.6

0.5 4.9

12×4 1.2

1.6 3.6

0.5 4.9

=

kron(A,B)

8.5866.192

2.432.194.144.6

4.14488.416

2.3945.245.2

8.288.12188

6.932620

7.145.13.345.3

8.108.42.252.11

6.3124.828

6.1927.145.1

4.144.68.108.4

8.4166.312

A B =

3 B 4 B

7 B 3 B5 B 8 B

4 B 12 B

=

BB

BB

BA

IJI

J

aa

aa

...

..

..

..

...

1

111

A(I×J) B(K×M),

37

5

4

43

812

A=

4

1.6

0.5

1.2

3.6

4.9

B =

A B =

3×4

1.6

0.5

7×4

1.6

0.5

5×4

1.6

0.5

4×4

1.6

0.5

1.2

3.6

4.9

4×

1.2

3.6

4.9

3×

1.2

3.6

4.9

8×

1.2

3.6

4.9

12×

=

8.582

2.434.6

4.1416

2.395.2

8.288

6.920

7.145.3

8.102.11

6.328

6.195.1

4.148.4

8.412

1 1a b

2 2a b

kron(A(:,1),B(:,1))

kron(A(:,2),B(:,2))

A A and and B B are partitioned matrices with an equal are partitioned matrices with an equal

number of partitions.number of partitions.

A =[a1, a2 ,…, an] B =[b1, b2 ,…, bn];

.A B = ]...[ 2211 nn bababa

Hadamard or element wise product, which is

defined for matrices A and B of equal size ( I ×

J )

IJIJII

JJ

baba

baba

...

..

..

..

...

11

111111

BA

9.40

52.26.1

6.96.3

10

7.01

8.09.0

9.45.0

6.36.1

2.14

BA

+ =

K

J

I

K

J

I

K

J

I

- =

K

J

I

K

J

I

K

J

I

K

J

I

+ EA

B

C

QGP

R

I

J

K

K

J

I

= + EA

B

C

N

N N

XkA

B2

2

=ck1

If N=2:

ck2

Horizental Slices

Vertical Slices

Frontal Slices

Xk = ADkB = ck1a1b1 + ck2a2b2

Across all slices Xk , the components ar and br remain the same, only their weights dk1 , . . . , dk2 are different.

XkA

B2

2

=

Dk

There are excitation, emission and concentration matrix of two analytes.

=

=XSensitivity Matrix

C

S

S = C+X

Calibration step:

Prediction Step:

c = S+ x

Frontal Slices

Xk = ADkB = ck1a1b1 + ·· ·+ckRaRbR

We need to estimate the parameters We need to estimate the parameters AA and and BB of the of the

calibration model, which we can then use for future calibration model, which we can then use for future

predictions.predictions.

Sample1: [c11 c12] Z(1) (4×3)Sample2: [c21 c22] Z(2) (4×3)

1.Vectorizing of Matrices

.

...

Sample3: [c31 c32] Z(3) (4×3)

2. Folding of Vectorized Matrices

Folding

3. Obtaining Sensitivity Matrix

=

S = C+X

For unknown matrix Z0calculate )( 00 ZS vecc cal

Only contribution of first component

Only contribution of another of component

Matricized

SVDSVD

a1,b1 a2,b2

K

J

I

= A

B2

2 2C

b1

IJ

a1

b2

I

J

a2

.A B = ][ 2211 baba

Alternating least squares PARAFAC algorithm

Algorithms for fitting the PARAFAC model are usually Algorithms for fitting the PARAFAC model are usually

based on alternating least squares. This is based on alternating least squares. This is

advantageous because the algorithm is simple to advantageous because the algorithm is simple to

implement, simple to incorporate constraints in, and implement, simple to incorporate constraints in, and

because it guarantees convergence. However, it is because it guarantees convergence. However, it is

also sometimes slow.also sometimes slow.

The PARAFAC algorithm begins with an initial guess of

the two loading modes

The solution to the PARAFAC model can be found by

alternating least squares (ALS) by successively assuming the

loadings in two modes known and then estimating the

unknown set of parameters of the last mode.

Determining the rank of three-way array

Suppose initial estimates of B and C loading modes are given

=

K

J

I

Matricizing

I

JK

IJK

N

N

K

NC

JK

N

Khatri-Rao

=I

JKN

I

JK

A = XZ+

N

B

N

J CB=ZA

X (I×J×K) X (J×IK)

B =X ZB+

=

J

IKN

J

IK N

Matricizing

X(J×IK) = B(J×N)(CA)T = B ZBT

X (I×J×K) X (K×IJ)

=K

IJN

K

IK N

Matricizing

C =X ZC+X(K×IJ) = C(K×N)(BA)T = C ZC

T

5. Go to step 1 until relative change in fit is small.

4-1. Reconstructing Three-way Array from obtained A and B and C profiles

4-2. Calculating the norm of residual array

2

1 1 1

( )I J K

ijk ijki j k

Rss x x

X

100)1(%

1 1 1

2

I

i

J

j

K

kijkx

Rssfit

Initialize B and C

2 A = X(I×JK ) ZA(ZAZA)−1

3 B = X(J×IK ) ZB(ZBZB)−1

4 C = X(K×JI ) ZC(ZCZC)−1

Given: X of size I × J × K

Go to step 1 until relative change in fit is small5

ZA=CB

ZB=CA

ZC=BA

Please simulate a Three-way data by these matrices.

There are excitation, emission and concentration matrix of two analytes.

Please do Khatri-Rao product of excitation and emission matrix.

Requires excessive memoryAn updating scheme by Harshman and Carroll and Chang

oSlow convergence

سپاس با