Transcript Diapositive 1 - Chemometrics

Outer Product Analysis (OPA)
studying the relations among sets of variables
measured on the same individuals
Douglas N. Rutledge
Some publications on Outer Product Analysis
Infrared spectroscopy and outer product analysis for
quantification of fat, nitrogen, and moisture of cocoa powder
A. Vesela, A. S. Barros, A. Synytsya, I. Delgadillo, J. Copıkova, M. A. Coimbra
Analytica Chimica Acta 601 (2007) 77–86
Multi-way analysis of outer product arrays using PARAFAC
D. N. Rutledge, D. Jouan-Rimbaud Bouveresse
Chemometrics and Intelligent Laboratory Systems 85 (2007) 170–178
Image processing of outer-product matrices – a new way to
classify samples. Examples using visible/NIR/MIR spectral data
B. Jaillais, V. Morrin, G. Downey
Chemometrics and Intelligent Laboratory Systems xx (2006) xxx–xxx
Some publications on Outer Product Analysis
Variability of cork from Portugese Quercus suber studied by solid
state 13C-NMR and FTIR spectroscopies
M.H. Lopes, A.S. Barros, C. Pascoal Neto, D. Rutledge, I. Delgadillo, A. M. Gil
Biopolymers (Biospectroscopy) 62 (5) (2001) 268–277
Outer Product Analysis of electronic nose and visible spectra:
application to the measurement of peach fruit characteristics
C. di Natale, M. Zude-Sasse, A. Macagnano, R. Paolesse, B. Herold, A. D'Amico
Analytica Chimica Acta 459 (2002) 107–117
Determination of the degree of methylesterification of pectic
polysaccharides by FT-IR using an outer product PLS1
regression
A.S. Barros, I. Mafra, D. Ferreira, S. Cardoso, A. Reis, J.A. Lopes de Silva, I.
Delgadillo, D.N. Rutledge, M.A. Coimbra
Carbohydrate Polymers 50 (2002) 85–94
Some publications on Outer Product Analysis
Enhanced multivariate analysis by correlation scaling and
fusion of LC/MS and 1H NMR data
J. Forshed, R. Stolt, H. Idborg, S. P. Jacobsson
Chemometrics and Intelligent Laboratory Systems 85 (2007) 179–185
Outer-product analysis (OPA) using PCA to study the influence of
temperature on NIR spectra of water
B. Jaillais, R. Pinto, A.S. Barros, D.N. Rutledge
Vibrational Spectroscopy 39 (2005) 50–58
Outer-product analysis (OPA) using PLS regression to study the
retrogradation of starch
B. Jaillais, M.A. Ottenhof, I.A. Farhat, D.N. Rutledge
Vibrational Spectroscopy 40 (2006) 10–19
Principal Components Analysis (PCA)
Calculate the covariance matrix, C, of the original data matrix, X
Covariance Matrix : C
n
 x
p
p
Cij = cov(i,j)
Cov( x1 , x2 )  sx21 , x2  i1,i 21
i1
 x1  xi 2  x2 
n 1
Calculate the matrix of individual covariances
between variables of one data set, X
1
p
p
1
x iT
xi
C i = x iT . x i
p
p
Mutual weighting of each signal by the other:
• if intensities simultaneously high in the two domains, the product is higher;
• if intensities simultaneously low in the two domains, the product is lower;
• if one intensity high and the other low, the product tends to an intermediate value
Calculate all the individual covariance matrices
of a single matrix, X
For n samples, one gets n Outer Product matrices
Group them together one under the other in the form of a cube of
individual matrices of covariances among variables
p
1 1
.
.
n
1
p
p
1
1
.
n
1
p
A cube of symmetrical matrices
Decomposition of the « Mean » OP matrix by SVD
≡ Principal Components Analysis
Calculate the mean of the individual covariance matrices to have a :
matrix of mean covariances
p
1 1
.
.
n
1
p
p
1
p
1
1
.
n
1
p
Decomposition of the column-mean matrix by SVD
 Principal Components Analysis
1
p
SVD applied to the initial data matrix, X
5
10
15
20
25
30
20
40
60
80
100
X(n,p)
S:
V:
U*S :
120
140
160
=
U(n,r)
S(r,r)
diagonal matrix of singular values
loadings matrix
scores matrix
VT(r,p)
SVD applied the covariance matrix, XTX
or column-means of the Outer Product cube
20
40
60
80
100
120
140
160
20
40
60
80
100
120
140
160
XTX(p,p)
=
S2 :
V:
X*V :
diagonal matrix of eigenvalues
loadings matrix
scores matrix
V(p,r)
S2(r,r)
VT(r,p)
Application of « Mean » Outer Product Analysis to real data (1)
Lignin-starch mixtures by TD-NMR
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
50
100
150
D.N. Rutledge, Food Control, (2001) 12(7), 437-445
200
250
Column-means of Outer Products
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
150
200
250
250
50
100
1
100
200
0.9
250
150
0.8
200
100
0.7
150
50
0.6
250
0.5
200
50
150
0.4
100
0.3
50
0.2
0
0
0.1
0.1
Decomposition of the matrix by SVD
(Principal Components Analysis)
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
0
-0.1
-0.2
-0.2
50
100
150
200
250
-0.4
-0.6
0
40
X*V 30
Scores on PC2, PC3 & PC4
10
20
50
60
V
Loadings on PC2, PC3 & PC4
Application of « Mean » Outer Product Analysis to real data (2)
Retrogradation of starch by X-ray diffraction
500
400
300
200
100
0
-100
0
50
100
150
200
250
300
350
Diffraction Rayons X
B. Jaillais, M.A. Ottenhof, I.A. Farhat, D.N. Rutledge, Vib. Spec. (2006), 40, 10–19.
Column-means of Outer Products
450
400
350
300
250
200
150
100
200
250
300
100
150
150
200
200
250
250
300
300
50
450
150
100
400
100
50
350
300
300
250
250
200
200
150
50
100
150
50
50
0
0
100
50
Decomposition of the matrix by SVD
Principal Components Analysis
5
0.15
4
0.1
3
2
0.05
1
0
0
-1
-2
-0.05
-3
-0.1
-4
-5
1
2
3
4
5
6
7
X*V
Scores on PC2
8
9
50
100
150
200
250
V
Loadings on PC2
300
« Unfold » Outer Product Analysis
Analyse the unfolded individual covariance matrices
Unfold the cube to form a matrix
p
1
.
.
n
1
p
p
1
1
1
1
.
n
n
1
n-PLS, n-PCA
(ANOVA) …
p
pxp
Different data unfolding schemes
3 x PCA
X1
X2
p
n
q
n
X3
p
X
p
Application of unfolded OP to real data (1)
Lignin-starch mixtures by TD-NMR
unfolded OP matrix (X1)
pxp
5
10
15
20
25
30
35
40
45
50
n
55
1
2
3
4
5
6
4
x 10
Decompose the unfolded OP matrix (X1) by SVD
(Unfold-PCA)
p
8
6
50
4
2
100
0
-2
150
-4
200
-6
-8
-10
0
10
20
30
40
50
60
U*S
Scores of X1 on PC2, PC3 & PC4
p
250
50
100
150
200
250
V
Refolded Loadings of X1
on PC2, PC3 & PC4
Decompose the unfolded OP matrices (X1 & X2) by SVD
(Unfold-PCA)
8
3
6
2
4
2
1
0
0
-2
-1
-4
-2
-6
-3
-8
-4
-10
0
10
20
30
40
50
60
U*S
Scores of X1 on PC2, PC3 & PC4
50
100
150
200
250
U*S
Scores of X2 on PC2, PC3 & PC4
Decomposition of the matrix by SVD
(Column-mean PCA)
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
0
-0.1
-0.2
-0.2
50
100
150
200
250
-0.4
-0.6
0
40
X*V 30
Scores on PC2, PC3 & PC4
10
20
50
60
V
Loadings on PC2, PC3 & PC4
Decompose the unfolded OP matrices (X1 & X2) by SVD
Unfold-PCA
4
3
3
2.5
2
2
1.5
1
1
0
0.5
-1
0
-2
-0.5
-1
5
10
15
20
25
30
35
40
45
50
U*S
Scores of X1 on PC4
55
50
100
150
200
U*S
Scores of X2 on PC4
250
Decompose the unfolded OP matrix (X2) by SVD
Unfold-PCA
p
3
5
2.5
10
2
15
1.5
20
1
25
0.5
30
0
35
-0.5
40
-1
45
50
100
150
200
250
n
50
55
U*S
Scores of X2 (X3) on PC4
V 200
250
Refolded Loadings of X2 (X3) on PC4
50
100
150
Application of unfolded OP to real data (2)
Starch retrogradation by XRD
unfolded OP matrix (X1)
pxp
1
2
3
4
5
6
7
8
n
9
1
2
3
4
5
6
7
8
9
10
11
4
x 10
Decompose the unfolded OP matrix (X1) by SVD
Unfold-PCA
p
150
50
100
100
50
150
0
200
-50
250
-100
-150
300
1
2
3
4
5
6
7
U*S
Scores of X1 on PC2
8
9
p
50
100
150
200
250
300
V
Refolded Loadings of X1 on PC2
Decompose the unfolded OP matrices (X1 & X2) by SVD
Unfold-PCA
150
20
100
15
10
50
5
0
0
-5
-50
-10
-100
-15
-150
1
2
3
4
5
6
7
U*S
Scores of X1 on PC2
8
9
-20
50
100
150
200
250
U*S
Scores of X2 on PC2
300
Decompose the unfolded OP matrix (X2) by SVD
Unfold-PCA
p
20
1
15
2
10
3
4
5
5
0
6
-5
7
-10
8
-15
n
-20
50
100
150
200
250
U*S
Scores of X2 on PC2
300
9
50
100
150
200
250
300
V
Refolded Loadings of X2 on PC2
« Multi-way » Outer Product Analysis
Group them together, one under the other, in the form of a cube
of individual matrices of covariances among variables
p
1
.
.
n
1
p
p
1
1
1
.
n
1
p
Decomposition of the cube  PARAFAC
PARAFAC – Parallel Factor Analysis
3-way data X (n,q,p)
:
F
xijk =  aifbjfckf + eijk
f=1
F is the number of Factors used in the PARAFAC model.
This model minimises the sum of squared residuals.
q
q
1
=
n
1
q
1
+
n
1
…
+
n
R. Bro, Chemometrics and Intelligent Laboratory Systems, (1997), 38, 149-171
F
Cube of individual covariances matrices
PARAFAC applied to OP cube
Starch retrogradation by XRD
25
500
400
20
300
15
1, 2
200
100
10
0
5
-100
-200
1
2
3
4
5
Sample
6
7
8
9
Loadings on the 1° mode (samples)
0
1
2
3
4
5
Time
6
7
8
9
Starch Data : PARAFAC Model
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
1, 2
1, 2
Loadings on the 2° mode (XRD)
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
50
100
150
200
Variable
250
300
-0.08
50
100
150
200
Variable
250
300
Loadings on the 3° mode (XRD)
Comparaison PARAFAC / SVD
SVD on XTX= PCA
OP-PARAFAC
3000
400
2500
300
2000
200
1500
100
1000
0
500
-100
0
1, 2
500
-200
1
2
3
4
5
Sample
6
7
8
0.12
9
-500
1
2
3
4
5
6
7
8
9
0.2
0.1
0.15
0.08
0.1
0.06
1, 2
0.04
0.05
0.02
0
0
-0.02
-0.05
-0.04
-0.1
-0.06
-0.08
50
100
150
200
Variable
250
300
-0.15
0
50
100
150
200
250
300
350
Calculate the matrix of individual covariances
between variables of 2 different matrices X & Y
1
yi
1,1
q
1, q
1
C i = x iT . y i
xi
p
p, q
Visualisation of the Outer Product cube
n
n
Signal 2
Signal 1

n
q
=
p
p
q
Calculate the matrix of covariances of 2 matrices X & Y
For n samples, one gets n Outer Product matrices
Group them together in the form of a “cube”
Calculate the column-mean of the individual covariance matrices to give
the matrix of covariances between the 2 groups of variables
Apply SVD
p
1
q
1 1
.
.
n
n OP (p, q) matrices
p
1
p
1
1
q
1
.
n
1
q
1 “cube” (n, p, q)
1 mean matrix (p, q)
Decompose the « Mean » OP matrix by SVD
≡ Tucker Analysis
Analyse the links between 2 tables of data, X & Y
Singular Value Decomposition of the matrix of covariances
between the 2 groups of variables (1/n)XTY
That decomposition of the matrix (1/n)XTY corresponds to looking for
successive pairs of variables (th = Xah , uh = Ybh ) where :
- covariance between th et uh maximal,
- axes ah orthogonal
- axes bh orthogonal
L. Tucker, Psychometrika, (1958), 23, 111-136
SVD applied to the covariance matrix, XTY
or column-means of the Outer Product cube
20
40
60
80
100
120
140
160
20
40
60
80
100
120
XTY =
S:
VX et VY :
X*VX :
Y*VYT :
140
160
VX
S
diagonal matrix of singular values
X & Y loadings matrices
scores of X
scores of Y
VYT
Application of « Mean » Outer Product Analysis to real data (3)
Complexation between TPP & Cu
TD-NMR
Vis
Spectres RMN
Spectres UV
1
2.2
0.9
2
0.8
1.8
1.6
0.7
1.4
0.6
1.2
0.5
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0
1000
2000
3000
4000
5000
6000
550
600
650
700
D.N. Rutledge, A.S. Barros, F. Gaudard, Mag. Res. in Chemistry, 35 (1997), 13–21
Column-means of Outer Products
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
120
140
40
60
60
80
80
100
100
120
120
140
140
160
160
20
40
160
1
100
40
0.9
80
20
0.8
60
160
0.7
140
0.6
120
0.5
100
0.4
80
20
60
0.3
40
0.2
20
0.1
0
0.2
SVD on matrix of column-means of Outer Product
(Tucker Analysis)
[uRMN, sRMN_Vis, vVis] = svd (meanRMN_Vis,'econ');
-0.05
0
-0.1
-0.1
-0.15
-0.2
-0.2
-0.02
0.15
-0.04
0.1
-0.06
0.05
-0.08
0
-0.1
-0.05
-0.12
50
100
150
50
100
150
0.1
-0.1
50
100
150
0.15
0.1
0
-0.1
-0.1
-0.2
-0.2
100
150
50
100
150
0.1
0.1
0
50
0.05
0.05
0
0
-0.05
-0.05
-0.1
-0.1
50
100
150
uRMN
50
100
150
50
100
150
vVis
SVD on matrix of column-means of Outer Product
(Tucker Analysis)
sRMN = RMN x uRMN / (uRMN' x uRMN);
sVis = Vis x vVis / (vVis' x vVis);
-2
-3
1
0
0.5
-0.1
0
-0.2
-0.5
-0.3
-1
-0.4
5
0
-4
-5
-6
0
10
20
30
40
0.2
0
-1.5
0
10
20
30
40
-0.5
0.05
30
0
20
-0.05
10
-0.1
0
-0.15
-10
-5
0
10
20
30
40
0
10
20
30
40
0
10
20
30
40
200
100
-0.2
0
-0.4
-0.6
-10
0
10
20
30
40
-0.2
0
sRMN
10
20
30
40
-20
-100
0
10
20
30
40
sVis
-200
Application of unfolded OP to real data (3)
Complexation between TPP & Cu
unfolded OP matrix (X1)
pxq
5
10
15
20
25
30
n
0.5
1
1.5
2
2.5
4
x 10
Decompose the unfolded OP matrix (X1) by SVD
Unfold-PCA
q
40
20
20
40
0
60
-20
80
-40
100
-60
120
-80
-100
140
p
160
0
5
10
15
20
25
30
U*S
Scores of X1 on PC1 & PC2
35
20
40
60
80
100
120
140
160
V
Loadings of X1 on PC1 & PC2
Decompose the unfolded OP matrices (X1 & X2) by SVD
Unfold-PCA
40
10
0
20
-10
0
-20
-20
-30
-40
-40
-50
-60
-60
-80
-70
-100
0
5
10
15
20
25
30
U*S
Scores of X1 on PC1 & PC2
35
-80
0
20
40
60
80
100
120
140
U*S
Scores of X2 on PC1 & PC2
160
Decompose the unfolded OP matrix (X2) by SVD
Unfold-PCA
p
10
0
5
-10
10
-20
-30
15
-40
20
-50
25
-60
-70
-80
30
0
20
40
60
80
100
120
140
U*S
Scores of X2 on PC1 & PC2
160
n
20
40
60
80
100
120
140
160
V
Loadings of X2 on PC1 & PC2
Decompose the unfolded OP matrices (X1 & X3) by SVD
Unfold-PCA
40
10
5
20
0
0
-5
-10
-20
-15
-40
-20
-25
-60
-30
-80
-35
-100
0
5
10
15
20
25
30
U*S
Scores of X1 on PC1 & PC2
35
-40
20
40
60
80
100
120
140
160
U*S
Scores of X3 on PC1 & PC2
Decompose the unfolded OP matrix (X3) by SVD
Unfold-PCA
q
10
5
5
0
-5
10
-10
15
-15
-20
20
-25
25
-30
-35
30
-40
20
40
60
80
100
120
140
160
U*S
Scores of X3 on PC1 & PC2
n
20
40
60
80
100
120
140
160
V
Loadings of X3 on PC1 & PC2
« Multi-way » Outer Product Analysis
For n samples, one gets n Outer Product matrices
Group them together, one under the other, in the form of a cube
of individual matrices of covariances among variables
p
1
.
.
n
1
q
p
1
1
1
.
n
1
q
Decomposition of the cube  PARAFAC
Cube of individual covariances matrices
PARAFAC applied to OP cube of real data (2)
Complexation between TPP & Cu
120
350
facteur 1
facteur 2
Loadings sur le mode 1
Cu
TPP
Cu + TPP
100
300
80
250
60
200
40
150
20
100
0
50
-20
0
5
10
15
20
25
30
35
N° éch.
« Loadings » on the 1° mode (samples)
0
0
5
10
15
20
25
Concentrations
30
35
PARAFAC applied to OP cube of real data (2)
Complexation between TPP & Cu
0.25
0.14
0.12
0.2
0.1
0.15
0.08
0.06
0.1
0.04
0.05
0.02
20
40
60
80
100
120
140
160
Loadings on the 2° mode (TD-NMR)
20
40
60
80
100
120
140
Loadings on the 3° mode (Vis)
160
PARAFAC applied to OP cube of real data (3)
Fructose solutions
Spectres IR
Spectres RMN
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
1200 1400 1600 1800 2000 2200 2400
1000
2000
3000
4000
5000
D.N. Rutledge, A.S. Barros, R. Giangiacomo,
Magnetic Resonance in Food Science—A View to the Future, RSC, 2001, pp. 179–192
Cube of individual covariances matrices
PARAFAC model
30
60
facteur 1
facteur 2
25
50
20
Conc. en fructose
40
15
10
30
20
5
10
0
-5
0
5
10
15
20
Loadings on the 1° mode
25
0
0
5
10
15
N° de l'échantillon
Concentrations
20
25
PARAFAC model
0.4
facteur 1
facteur 2
facteur 1
0.16
facteur 2
0.35
0.14
0.3
0.12
0.25
0.1
0.2
0.08
0.15
0.06
0.1
0.04
0.05
0
0.02
0
0
1000
2000
3000
Longueurs d'onde
4000
5000
6000
Loadings on the 2° mode (TD-NMR)
1200
1400
1600
1800
Longueur d'onde
2000
2200
2400
Loadings on the 3° mode (NIR)
PARAFAC on 4-D OP hypercube MIR  NMR  XRD (1)
(9 x 157 x 40 x 341)
Starch retrogradation
2.5
1.1
500
1
2
400
0.9
0.8
1.5
300
0.7
0.6
1
200
0.5
100
0.4
0.5
0.3
0
0.2
0
0
5
10
15
20
25
30
Log(TD-NMR)
35
40
0.1
0
20
40
60
80
MIR
100
120
140
160
-100
0
50
100
150
200
XRD
250
300
350
Loadings on the 1° mode (Samples)
Loadings on the 2° mode (MIR)
300
0.3
factor 1
factor 2
data1
data2
250
0.25
200
Loadings on the first mode
0.2
150
0.15
100
0.1
50
0.05
0
0
-50
-100
1
2
3
4
5
Sample number
6
7
8
-0.05
9
0
20
40
60
80
100
120
140
160
0.12
0.35
data1
data2
factor 1
factor 2
0.1
0.3
0.08
0.25
Loadings on the fourth mode
0.06
0.2
0.15
0.04
0.02
0
-0.02
0.1
-0.04
0.05
-0.06
0
-0.08
0
5
10
15
20
25
30
35
40
Loadings on the 3° mode (NMR)
0
5
10
15
20
25
30
35
40
Loadings on the 4° mode (XRD)
Comparison of 2D-Correlation Spectroscopy
and unfold OP PCA
38 NIR spectra of water; acquired in the region 1300-1600 nm; from 6 to 80 ºC
1.20
Absorbance
0.90
0.60
0.30
0.00
1300
-0.30
80ºC
1350
6ºC
1400
1450
1500
-0.60
Wavelength (nm)
V.H. Segtnan et al., Anal. Chem. (2001), 73, 31-53
B. Jaillais et al., Vib. Spec., (2005), 39, 1, 50-58
1550
1600
2D-COS Sync vs PC1 Loadings
1412
Synchronous Correlation
3.0E-04
2.0E-04
1.0E-04
1491
0.0E+00
1300
1350
1400
1450
1500
1550
1600
1493
-1.0E-04
1412
-2.0E-04
Wavelength (nm)
2D-COS Async vs PC2 Loadings
6.0E-06
Asynchronous Correlation
1404
3.0E-06
1428
0.0E+00
1300
1350
1400
1450
1500
1550
1600
-3.0E-06
1446
-6.0E-06
Wavelength (nm)
ss 2D-COS vs. Loadings of PCA
on transposed row-normalised, column-centred spectra
2.5E-04
Synchronous correlation
6ºC 80ºC
1.5E-04
5.0E-05
-5.0E-05 0
10
20
30
40
50
60
70
80
-1.5E-04
-2.5E-04
Sample temperature (ºC)
Sync
PC1
ss 2D-COS vs Loadings of PCA
on transposed row-normalised, column-centred spectra
4.0E-06
(80,6)
(54,80)
Asynchronous correlation
(6,38)
2.0E-06
0.0E+00
0
10
20
30
40
50
60
70
80
-2.0E-06
80ºC
6ºC
-4.0E-06
Sample temperature (ºC)
ASync
PC2
Unfold PCT-OP-PCA (or PLS) algorithm
for huge X & Y
Step
Computation
Comments
1
2
X,Y
[TX, PX]  PCA(X)
input of X and Y matrices
full rank PCA on X
3
[TY, PY]  PCA(Y)
full rank PCA on Y
4
K = OP(TX, TY)
unfolded outer product
[T, PPCT] = PCA(K)
between TX and TY
PCA (or PLS etc.) on K
5
T = scores in the X-space & PC-space
PPCT = loadings in PC-space
6
for a=1:n
refold PPCT(a)
P(a) = PY PPCT(a) PTX
end for
rebuild PC loadings in X-space
A.S. Barros & D.N. Rutledge, Chemom. Intell. Lab. Syst., (2004) 73 245– 255
A.S. Barros & D.N. Rutledge, Chemom. Intell. Lab. Syst., (2005), 78, 125–137
Segmented PCT-PCA (or PLS)
X1
X2
X
Xq
PCA
T1
T2
Tq
P1
P2
Pq
T1 T2 ... T1
PCA/PLS
TPCT
PX1
PX2
...
PXq
PTPCT
TPCT = TX
OPA vs. 2D-COS, PCA and Tucker Analysis
• Why use the mean ! ?
(PCA & TA are in a sense compromises)
• Not limited to two data sets
• Cube analysable by unfolding or by multi-way methods
• Multi-way methods extract Factors
• Unfold-OPA can reveal relations between variables
not limited to two matrices
no need to sort samples (unlike 2D-COS)
• No memory problem with (segmented) PCT-OPA
Рәхмәт спасибо