by: s.m. sajjadi islamic azad university, parsian branch, parsian,iran
Post on 19-Dec-2015
219 views
TRANSCRIPT
![Page 1: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/1.jpg)
![Page 2: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/2.jpg)
By: S.M. SajjadiIslamic Azad University, Parsian Branch, Parsian,Iran
![Page 3: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/3.jpg)
![Page 4: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/4.jpg)
![Page 5: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/5.jpg)
Scalar Vector
a = a (I×1) =
Matrix
A (I×J) =
Three-way array
A (I×J×K) =
![Page 6: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/6.jpg)
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
![Page 7: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/7.jpg)
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)??
![Page 8: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/8.jpg)
Rows, Columns and Tubes
Row
Tube
Column
2
3
4
![Page 9: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/9.jpg)
2
3
4xjk(4×1)
X( : , j , k )
Rows, Columns and Tubes
Column
![Page 10: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/10.jpg)
2
3
4xjk(4×1)xik(3×1)
X( i , : , k ) X( : , j , k )
Rows, Columns and Tubes
Row Column
![Page 11: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/11.jpg)
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
![Page 12: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/12.jpg)
23
4
Horizental Vertical
![Page 13: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/13.jpg)
23
4
X( i , : , : )
Horizental
![Page 14: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/14.jpg)
Vertical
X( : , j , : )
23
4
X( i , : , : )
Horizental
![Page 15: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/15.jpg)
X( : , : , k )
32
4
Vertical
X( i , : , : )
Horizental
X( : , j , : )
![Page 16: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/16.jpg)
There are five EEMs of different
samples that contain two analytes.
![Page 17: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/17.jpg)
2
3
4
![Page 18: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/18.jpg)
2
3
4 4
3
![Page 19: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/19.jpg)
2
3
4
X( : , : )
4
63
![Page 20: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/20.jpg)
2
4
4
6
permute ( X , [1 3 2] )X ( : , : )
4
2
3
???? X( : , : )
4
633
??
4
62
![Page 21: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/21.jpg)
23
4
Matrisizing : X ( : , : )Matrisizing : X ( : , : )
3
8
2
permute ( X , [2 …)
![Page 22: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/22.jpg)
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.
![Page 23: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/23.jpg)
Vector multiplicationVector multiplication
aaTTb = scalar:b = scalar:
Inner product = scalarInner product = scalar
=II
Outer product = MartixOuter product = Martix=
I
J
I
J
![Page 24: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/24.jpg)
Vec-operator
Vec of matrix A is the IJ vector
AB = matrix
=
I
J
J
K
I
K
vectorized.... . . .
IJ
![Page 25: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/25.jpg)
Kronecker product
Hadamard product
Khatri–Rao product
*
Tucker
Weighted PARAFAC
PARAFAC
![Page 26: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/26.jpg)
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)
![Page 27: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/27.jpg)
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
=
![Page 28: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/28.jpg)
BB
BB
BA
IJI
J
aa
aa
...
..
..
..
...
1
111
A(I×J) B(K×M),
![Page 29: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/29.jpg)
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))
![Page 30: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/30.jpg)
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
![Page 31: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/31.jpg)
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
![Page 32: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/32.jpg)
+ =
K
J
I
K
J
I
K
J
I
![Page 33: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/33.jpg)
- =
K
J
I
K
J
I
K
J
I
![Page 34: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/34.jpg)
K
J
I
+ EA
B
C
QGP
R
I
J
K
![Page 35: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/35.jpg)
K
J
I
= + EA
B
C
N
N N
XkA
B2
2
=ck1
If N=2:
ck2
![Page 36: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/36.jpg)
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
![Page 37: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/37.jpg)
There are excitation, emission and concentration matrix of two analytes.
![Page 38: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/38.jpg)
=
=XSensitivity Matrix
C
S
S = C+X
Calibration step:
Prediction Step:
c = S+ x
![Page 39: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/39.jpg)
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.
![Page 40: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/40.jpg)
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)
![Page 41: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/41.jpg)
2. Folding of Vectorized Matrices
Folding
3. Obtaining Sensitivity Matrix
=
S = C+X
For unknown matrix Z0calculate )( 00 ZS vecc cal
![Page 42: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/42.jpg)
Only contribution of first component
Only contribution of another of component
Matricized
SVDSVD
a1,b1 a2,b2
K
J
I
= A
B2
2 2C
![Page 43: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/43.jpg)
b1
IJ
a1
b2
I
J
a2
.A B = ][ 2211 baba
![Page 44: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/44.jpg)
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.
![Page 45: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/45.jpg)
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
![Page 46: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/46.jpg)
Suppose initial estimates of B and C loading modes are given
=
K
J
I
Matricizing
I
JK
IJK
N
N
![Page 47: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/47.jpg)
K
NC
JK
N
Khatri-Rao
=I
JKN
I
JK
A = XZ+
N
B
N
J CB=ZA
![Page 48: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/48.jpg)
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
![Page 49: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/49.jpg)
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
![Page 50: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/50.jpg)
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
![Page 51: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/51.jpg)
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
![Page 52: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/52.jpg)
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.
![Page 53: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/53.jpg)
Requires excessive memoryAn updating scheme by Harshman and Carroll and Chang
oSlow convergence
![Page 54: By: S.M. Sajjadi Islamic Azad University, Parsian Branch, Parsian,Iran](https://reader035.vdocuments.us/reader035/viewer/2022062421/56649d375503460f94a10ae4/html5/thumbnails/54.jpg)
سپاس با