1
Robustness of Multiway Methods Robustness of Multiway Methods in Relation to Homoscedastic in Relation to Homoscedastic
and Hetroscedastic Noiseand Hetroscedastic Noise
T. Khayamian T. Khayamian
Department of Chemistry , Isfahan University of Department of Chemistry , Isfahan University of
Technology, Isfahan 84154, IranTechnology, Isfahan 84154, Iran
2
OutlineOutline IntroductionIntroduction Prediction of component concentrations in Claus Prediction of component concentrations in Claus
data using PARAFAC and N-PLS multi-way methods data using PARAFAC and N-PLS multi-way methods original dataoriginal data (original + noise) data (original + noise) data denoised data (using wavelet as a denoising method)denoised data (using wavelet as a denoising method) - Homoscedastic noise (level independent method)- Homoscedastic noise (level independent method) - Hetroscedastic noise (level dependent and minimum - Hetroscedastic noise (level dependent and minimum
description length)description length)
ConclusionsConclusions
3
Noise definitionNoise definition
● Noise is any component of a signal Noise is any component of a signal which impedes observation, detection which impedes observation, detection or utilization of the information that or utilization of the information that the signal is carrying. the signal is carrying.
● Noise is measured by its standard Noise is measured by its standard deviation or peak to peak fluctuationdeviation or peak to peak fluctuation
4
Different types of noiseDifferent types of noise
HetroscedasticHetroscedastic
HomoscedasticHomoscedastic
NoiseNoise
5
Homoscedastic and Homoscedastic and Hetroscedastic NoiseHetroscedastic Noise
● Homoscedastic noise: Homoscedastic noise: Noise is independent of variable, sample and Noise is independent of variable, sample and
signal with the normal distribution and a signal with the normal distribution and a constant variance. constant variance.
● Hetroscedastic Noise:Hetroscedastic Noise: Noise is dependent on the variable, sample Noise is dependent on the variable, sample
and signal.and signal. (Noise from different variables or samples (Noise from different variables or samples
can be correlated) can be correlated)
11 22
RR 11
22
22
22
6
Least squares methodLeast squares method
m
i
n
jcalijij yyS
1 1
2)((exp)
2
Homoscedastic noise :
ij is constant, uniform and independent of the signal, variables and samples
m
i
n
j ij
calijij yyS
1 1
2
)((exp)2
Hetroscedastic noise :
ij is dependent on signal, variables or samples
m
i
n
j ij
calijij yyS
1 1
2
)((exp)2
7
Hetroscedastic noise in Univariate Hetroscedastic noise in Univariate and Multivariate Calibration and Multivariate Calibration
MethodsMethods● Zeroth order calibrationZeroth order calibration weighted linear regressionweighted linear regression● First order calibrationFirst order calibration weighted principle component weighted principle component
analysisanalysis ● Second order calibrationSecond order calibration Positive matrix factorization Positive matrix factorization Maximum likelihood PARAFACMaximum likelihood PARAFAC
2'* TPXW
2'* TPXW
8
Claus data “fluorescence Claus data “fluorescence Spectroscopy” Spectroscopy”
Analyte Analyte 11(tyrosine)(tyrosine)
Analyte 2Analyte 2(tyrptophan(tyrptophane)e)
Analyte 3Analyte 3(phenyl (phenyl alanine)alanine)
Sample Sample 11
2.7×12.7×100-6-6
00 00
Sample Sample 22
00 1.33×101.33×10--
55
00
Sample Sample 33
00 00 9.0×109.0×10-4-4
Sample Sample 44
1.6×11.6×100-6-6
5.4×105.4×10-6-6 3.55×103.55×10-4-4
Sample Sample 55
9.0×19.0×100-7-7
4.4×104.4×10-6-6 2.97×102.97×10-4-4
C. A. Andesson and R. Bro. The N-way Toolbox for MATABC. A. Andesson and R. Bro. The N-way Toolbox for MATABChemom. Intell. Lab. Sys. 2000, 52 (1), 1- 4Chemom. Intell. Lab. Sys. 2000, 52 (1), 1- 4http://www. models . kvl. dkhttp://www. models . kvl. dk
9
Fluorescence excitation and emission spectrum of five samplesFluorescence excitation and emission spectrum of five samples
10
XX44
201201
6161
==
++++aa11
bb11
cc11 cc22 cc33
aa22
bb22 bb33
aa33
Claus dataClaus data
PARAFAC: four samples were used for modeling PARAFAC: four samples were used for modeling
••••••••
••
ScoreScore (a1)(a1)
Concentration analyte 1Concentration analyte 1
••••••••
••
ScoreScore (a2)(a2)
Concentration analyte 2Concentration analyte 2
••••••••
••
ScoreScore (a3)(a3)
Concentration analyte 3Concentration analyte 3
11
Calculation of Score for a New Calculation of Score for a New SampleSample
6161
201201
Z = kr (B, C)Z = kr (B, C)Un = reshape (Un, 12261, 1)Un = reshape (Un, 12261, 1)Score Un = pinv(Z) * UnScore Un = pinv(Z) * Un
Un =Un =
12
Relative Errors of Predicted Relative Errors of Predicted Concentrations for Samples Concentrations for Samples 4 & 5 4 & 5
(without adding noise)(without adding noise)
Analyte 1Analyte 1 Analyte 2Analyte 2 Analyte 3Analyte 3
Sample 4Sample 4 -0.75-0.75 -7.3-7.3 9.49.4
Sample 5Sample 5 -0.56-0.56 -6.1-6.1 14.314.3
13
Generating of Noise Matrix
55
201201×61×61
(Claus data)(Claus data) 55 NoiseNoise
Homoscedastic nois:Homoscedastic nois:Standard deviation of noise = 2%, 5%, 10% of the maximum value in the claus dataStandard deviation of noise = 2%, 5%, 10% of the maximum value in the claus data
Hetroscedastic noise :Hetroscedastic noise :
N = N(0,1) .N = N(0,1) .* * 1/10 X 1/10 X
Element by element was multiplied by one-tenth of the claus data Element by element was multiplied by one-tenth of the claus data
201201×61×61
Claus data + NoiseClaus data + Noise
44
201201
6161unfoldingunfolding
14
0 2000 4000 6000 8000 10000 12000 14000-200
-150
-100
-50
0
50
100
150homoscedastic noise (10%) for sample one
201 X 610 2000 4000 6000 8000 10000 12000 14000
-80
-60
-40
-20
0
20
40
60
80hetroscedastic noise (10%) for sample one
201 X 61
Homoscedastic and Hetroscedastic noise were added to original dataHomoscedastic and Hetroscedastic noise were added to original data
Hetroscedastic noise (10%)Hetroscedastic noise (10%)Homoscedastic noise (10%)Homoscedastic noise (10%)
15
0 2000 4000 6000 8000 10000 12000 14000-200
0
200
400
600
800
1000signal + noise (homo) for sample one
201 X 610 2000 4000 6000 8000 10000 12000 14000
-100
0
100
200
300
400
500
600
700
800reshape of sample one
201 X 61
Reshape of Sample One Reshape of Sample One Sample one with adding Homoscedastic Sample one with adding Homoscedastic noisenoise
The effect of adding Homoscedastic noiseThe effect of adding Homoscedastic noise
16
0 2000 4000 6000 8000 10000 12000 14000-100
0
100
200
300
400
500
600
700
800reshape of sample one
201 X 610 2000 4000 6000 8000 10000 12000 14000
-100
0
100
200
300
400
500
600
700
800signal + noise(het,20%) for sample one
201 X 61
Reshape of Sample One Reshape of Sample One Sample one with adding Hetroscedastic Sample one with adding Hetroscedastic noisenoise
The effect of adding Hetroscedastic noiseThe effect of adding Hetroscedastic noise
17
Wavelet can be used as a Wavelet can be used as a powerful tool for signal powerful tool for signal
denoising denoising Wavelet Denoising :Wavelet Denoising :● Wavelet decomposition of the signalWavelet decomposition of the signal● Selecting the threshold Selecting the threshold ● Applying the threshold to the wavelet Applying the threshold to the wavelet
coefficientscoefficients● Inverse transformation to the native Inverse transformation to the native
domaindomain
18
Thresholding methods :Thresholding methods :
● Global thresholdingGlobal thresholding● Level dependent thresholdingLevel dependent thresholding● Data dependent thresholdingData dependent thresholding● Cycle – spin thresholdingCycle – spin thresholding● Wavelet packet thresholdingWavelet packet thresholding
19
Universal threshold :Universal threshold :
N = length of data array
6745.0
)( ixmedian
Xi = detail part of coefficient
)ln(2 Nt
20
0 2000 4000 6000 8000 10000 12000 14000 16000 18000-100
0
100
200
300
400
500
600
700
800mirror padding for sample one
16384=214
21
Prediction of Analyte Concentrations for Prediction of Analyte Concentrations for
Samples 4 & 5 Samples 4 & 5 using using PARAFACPARAFAC
22
Comparison of Sum of the Square of Comparison of Sum of the Square of Residuals (Homoscedastic noise - Residuals (Homoscedastic noise -
PARAFAC)PARAFAC)SSRSSR
Model 1Model 1 SSRSSR
Model 2Model 2
Without noiseWithout noise 10.510.5 10.210.2
Noisy Noisy datadata
Homo. noise Homo. noise 2%2%
168.56168.56 168.57168.57
Homo. noise Homo. noise 5%5%
168.56168.56 168.57168.57
Homo. noise Homo. noise 10%10%
168.56168.56 168.57168.57
Denoised Denoised datadata
Homo. noise Homo. noise 2%2%
14.3414.34 15.0515.05
Homo. noise Homo. noise 5%5%
37.2437.24 37.2937.29
Homo. noise Homo. noise 10%10%
105.36105.36 103.97103.97
model 1 : sample 1, 2 , 3, 4 / model 2 : sample 1, 2, 3, 5model 1 : sample 1, 2 , 3, 4 / model 2 : sample 1, 2, 3, 5 Each number × 10Each number × 1055
23
Var. Var.
Model 1Model 1 Var.Var.
Model 2Model 2Without noiseWithout noise 99.9499.94 99.9599.95
Noisy Noisy datadata
Homo. noise Homo. noise 2%2% 99.0799.07 99.1999.19
Homo. noise Homo. noise 5%5% 99.0799.07 99.1999.19
Homo. noise Homo. noise 10%10% 99.0799.07 99.1999.19
Denoised Denoised datadata
Homo. noise Homo. noise 2%2%
99.9399.93 99.9299.92
Homo. noise Homo. noise 5%5%
99.8299.82 99.7999.79
Homo. noise Homo. noise 10%10%
99.4999.49 99.4299.42
Comparison of explained variation Comparison of explained variation (Homoscedastic noise - PARAFAC)(Homoscedastic noise - PARAFAC)
24
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 4 for Sample 4
( Homoscedastic noise – PARAFAC ) ( Homoscedastic noise – PARAFAC )
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.75-0.75 -7.3-7.3 9.49.4
Noisy Noisy datadata
2 % 2 % noisenoise
-0.41-0.41 -7.1-7.1 9.29.2
5 % 5 % noise noise
-0.41-0.41 -7.1-7.1 9.269.26
10 % 10 % noisenoise
-0.41-0.41 -7.1-7.1 9.269.26
DenoiseDenoised datad data
2 % 2 % noisenoise
-0.77-0.77 -7.5-7.5 9.39.3
5 % 5 % noise noise
-0.66-0.66 -7.5-7.5 8.58.5
10 % 10 % noisenoise
-1.3-1.3 -7.1-7.1 7.37.3
25
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.75-0.75 -7.3-7.3 9.49.4
Noisy Noisy datadata
2 % 2 % noisenoise
-0.71-0.71 -6.1-6.1 14.214.2
5 % 5 % noise noise
-0.71-0.71 -6.1-6.1 14.214.2
10 % 10 % noisenoise
-0.71-0.71 -6.1-6.1 16.916.9
DenoiseDenoised datad data
2 % 2 % noisenoise
-0.55-0.55 -6.1-6.1 14.314.3
5 % 5 % noise noise
-1.7-1.7 -6.4-6.4 13.613.6
10 % 10 % noisenoise
-2.8-2.8 -5.3-5.3 11.611.6
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 5 for Sample 5
( Homoscedastic noise - PARAFAC ) ( Homoscedastic noise - PARAFAC )
26
Comparison of Sum of the Square of Comparison of Sum of the Square of Residuals (Hetroscedastic noise)Residuals (Hetroscedastic noise)
SSRSSR
Model Model 11
SSRSSR
Model 2Model 2
Without noiseWithout noise 10.510.5 10.210.2Noisy Noisy datadata
Hetro. noise Hetro. noise 10%10%
35.5235.52 39.2639.26
Hetro. noise Hetro. noise 20%20%
110.27110.27 126.17126.17
Denoised Denoised datadata
Hetro. noise Hetro. noise 10%10%
33.5233.52 36.5636.56
Hetro. noise Hetro. noise 20%20%
101.32101.32 113.78113.78(Each number * 10(Each number * 1055)) wavelet denoising (level dependent method)wavelet denoising (level dependent method)
27
Comparison of explained variation Comparison of explained variation (Hetroscedastic noise - PARAFAC)(Hetroscedastic noise - PARAFAC)
Var.Var.
Model Model 11
Var.Var.
Model 2Model 2
Without noiseWithout noise 99.9499.94 99.9599.95Noisy Noisy datadata
Hetro. noise Hetro. noise 10%10%
99.8099.80 99.8199.81
Hetro. noise Hetro. noise 20%20%
99.3999.39 99.3999.39
Denoised Denoised datadata
Hetro. noise Hetro. noise 10%10%
99.8199.81 99.8299.82
Hetro. noise Hetro. noise 20%20%
99.4499.44 99.4599.45wavelet denoising (level dependent method)wavelet denoising (level dependent method)
28
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for sample 4 for sample 4
( Hetroscedastic noise - PARAFAC) ( Hetroscedastic noise - PARAFAC)
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.7-0.7 -7.3-7.3 9.49.4
Noisy Noisy datadata
10 % 10 % noisenoise
-0.85-0.85 -7.3-7.3 9.39.3
20 % 20 % noisenoise
-0.94-0.94 -7.0-7.0 9.39.3
DenoiseDenoised datad data
10 % 10 % noisenoise
-0.86-0.86 -7.16-7.16 9.289.28
20 % 20 % noisenoise
-0.94-0.94 -7.10-7.10 9.159.15
wavelet denoising (level dependent method)wavelet denoising (level dependent method)
29
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 5 for Sample 5
( Hetroscedastic noise - PARAFAC) ( Hetroscedastic noise - PARAFAC)
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.56-0.56 -6.1-6.1 14.314.3
Noisy Noisy datadata
10 % 10 % noisenoise
-0.53-0.53 -6.1-6.1 14.214.2
20 % 20 % noisenoise
-0.45-0.45 -6.2-6.2 14.114.1
DenoiseDenoised datad data
10 % 10 % noisenoise
-0.54-0.54 -6.2-6.2 14.1514.15
20 % 20 % noisenoise
-0.47-0.47 -6.04-6.04 14.0014.00
wavelet denoising (level dependent method)wavelet denoising (level dependent method)
30
Minimum Description LengthMinimum Description Length
• The MDL is an approach to simultaneous noise suppression The MDL is an approach to simultaneous noise suppression and signal compression.and signal compression.
• It is free from any parameter setting such as threshold It is free from any parameter setting such as threshold selection, which can be particularly useful for real data selection, which can be particularly useful for real data where the noise level is difficult to estimate. where the noise level is difficult to estimate.
m = filter typem = filter typellmm = the number of major coefficients retained = the number of major coefficients retained
γγjj,k,k = the vector of wavelet coefficients of transformed type m = the vector of wavelet coefficients of transformed type m
γγj,kj,k = = the vector of the contractedthe vector of the contracted wavelet coefficientswavelet coefficients
2
22
3 mlkj
mkj
NNlmlMDL ,,loglogmin),(
mlml
31
Signal Denoising with MDL Signal Denoising with MDL methodmethod
0 5000 10000 15000-200
0
200
400
600
800
1000Raw Data
0 0.5 1 1.5 2
x 104
-200
0
200
400
600
800
1000Noisy Data
0 0.5 1 1.5 2
x 104
2
4
6
8
10
12
14x 10
4 MDL
0 0.5 1 1.5 2
x 104
-200
0
200
400
600
800
1000Denoised Data
32
Comparison of Sum of the Square of Comparison of Sum of the Square of Residuals (Hetroscedastic noise - Residuals (Hetroscedastic noise -
PARAFAC)PARAFAC)SSRSSR
Model Model 11
SSRSSR
Model 2Model 2
Without noiseWithout noise 10.510.5 10.210.2Noisy Noisy datadata
Hetro. noise Hetro. noise 10%10%
35.5235.52 39.2639.26
Hetro. noise Hetro. noise 20%20%
110.27110.27 126.17126.17
Denoised Denoised datadata
Hetro. noise Hetro. noise 10%10%
35.5235.52 39.2539.25
Hetro. noise Hetro. noise 20%20%
110.26110.26 126.17126.17Each number × 10Each number × 1055 Wavelet Denoising (MDL)Wavelet Denoising (MDL)
33
Comparison of explained variation Comparison of explained variation (Hetroscedastic noise - PARAFAC)(Hetroscedastic noise - PARAFAC)
Var.Var.
Model Model 11
Var.Var.
Model 2Model 2
Without noiseWithout noise 99.9499.94 99.9599.95Noisy Noisy datadata
Hetro. noise Hetro. noise 10%10%
99.8099.80 99.8199.81
Hetro. noise Hetro. noise 20%20%
99.3999.39 99.3999.39
Denoised Denoised datadata
Hetro. noise Hetro. noise 10%10%
99.8099.80 99.8199.81
Hetro. noise Hetro. noise 20%20%
99.3999.39 99.3999.39Wavelet Denoising (MDL)Wavelet Denoising (MDL)
34
Relative Errors of Predicted Relative Errors of Predicted Concentrations for sample 4 Concentrations for sample 4
( Hetroscedastic noise - PARAFAC) ( Hetroscedastic noise - PARAFAC)
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.75-0.75 -7.3-7.3 9.49.4
Noisy Noisy datadata
10 % 10 % noisenoise
-0.85-0.85 -7.3-7.3 9.39.3
20 % 20 % noisenoise
-0.94-0.94 -7.0-7.0 9.39.3
DenoiseDenoised datad data
10 % 10 % noisenoise
-0.85-0.85 -7.28-7.28 9.39.3
20 % 20 % noisenoise
-0.94-0.94 -7.01-7.01 9.39.3
Wavelet Denoising (MDL)Wavelet Denoising (MDL)
35
Relative Errors of Predicted Relative Errors of Predicted Concentrations for Sample 5 Concentrations for Sample 5
( Hetroscedastic noise - PARAFAC ) ( Hetroscedastic noise - PARAFAC )
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-0.5-0.5 -6.1-6.1 14.314.3
Noisy Noisy datadata
10 % 10 % noisenoise
-0.5-0.5 -6.1-6.1 14.214.2
20 % 20 % noisenoise
-0.4-0.4 -6.2-6.2 14.114.1
DenoiseDenoised datad data
10 % 10 % noisenoise
-0.5-0.5 -6.1-6.1 14.214.2
20 % 20 % noisenoise
-0.4-0.4 -6.2-6.2 14.114.1
Wavelet Denoising (MDL)Wavelet Denoising (MDL)
36
Prediction of Analyte Concentrations for Prediction of Analyte Concentrations for
Samples 4 & 5 Samples 4 & 5 using using N-PLSN-PLS
37
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 4 for Sample 4
( Homoscedastic noise – NPLS model)( Homoscedastic noise – NPLS model)
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-2.88-2.88 -0.44-0.44 4.874.87
Noisy Noisy datadata
2 % 2 % noisenoise
-2.91-2.91 -0.39-0.39 4.914.91
5 % 5 % noise noise
-2.97-2.97 -0.33-0.33 4.954.95
10 % 10 % noisenoise
-3.07-3.07 -0.22-0.22 4.994.99
DenoiseDenoised datad data
2 % 2 % noisenoise
-2.91-2.91 -0.44-0.44 4.924.92
5 % 5 % noise noise
-3.00-3.00 -0.20-0.20 4.674.67
10 % 10 % noisenoise
-3.29-3.29 0.360.36 4.364.36
X-block > 99X-block > 99 Y-block > 99Y-block > 99
38
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 5 for Sample 5
( Homoscedastic noise – NPLS model ) ( Homoscedastic noise – NPLS model )
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-1.67-1.67 0.440.44 11.2411.24
Noisy Noisy datadata
2 % 2 % noisenoise
-1.65-1.65 0.350.35 11.3811.38
5 % 5 % noise noise
-1.62-1.62 0.220.22 11.5711.57
10 % 10 % noisenoise
-1.57-1.57 -0.01-0.01 11.8611.86
DenoiseDenoised datad data
2 % 2 % noisenoise
-1.69-1.69 0.310.31 11.3411.34
5 % 5 % noise noise
-2.08-2.08 0.270.27 11.5611.56
10 % 10 % noisenoise
-2.89-2.89 0.930.93 10.6610.66
X-block > 99X-block > 99 Y-block > 99Y-block > 99
39
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-2.88-2.88 -0.44-0.44 4.874.87
Noisy Noisy datadata
10 % 10 % noisenoise
-3.00-3.00 -0.28-0.28 4.894.89
20 % 20 % noisenoise
-3.12-3.12 -0.12-0.12 4.914.91
DenoiseDenoised datad data
10 % 10 % noisenoise
-3.00-3.00 -0.28-0.28 4.914.91
20 % 20 % noisenoise
-3.12-3.12 -0.12-0.12 4.914.91
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 4 for Sample 4
( Hetroscedastic noise – NPLS model) ( Hetroscedastic noise – NPLS model)
Wavelet Denoising (MDL)Wavelet Denoising (MDL)X-block > 99X-block > 99 Y-block > 99Y-block > 99
40
Comparison of Sum of the Square of Comparison of Sum of the Square of Residuals (Hetroscedastic noise - Residuals (Hetroscedastic noise -
PARAFAC)PARAFAC)SSRSSR
Model Model 11
SSRSSR
Model 2Model 2
Without noiseWithout noise 10.510.5 10.210.2Noisy Noisy datadata
Hetro. noise Hetro. noise 10%10%
35.5235.52 39.2639.26
Hetro. noise Hetro. noise 20%20%
110.27110.27 126.17126.17
Denoised Denoised datadata
Hetro. noise Hetro. noise 10%10%
33.5233.52 36.5636.56
Hetro. noise Hetro. noise 20%20%
101.32101.32 113.78113.78Each number × 10Each number × 1055
41
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-2.88-2.88 -0.44-0.44 4.874.87
Noisy Noisy datadata
10 % 10 % noisenoise
-3.00-3.00 -0.28-0.28 4.894.89
20 % 20 % noisenoise
-3.12-3.12 -0.12-0.12 4.914.91
DenoiseDenoised datad data
10 % 10 % noisenoise
-3.00-3.00 -0.36-0.36 4.884.88
20 % 20 % noisenoise
-3.12-3.12 -0.23-0.23 5.785.78
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 4 for Sample 4
( Hetroscedastic noise – NPLS model) ( Hetroscedastic noise – NPLS model)
42
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-1.67-1.67 0.440.44 11.2411.24
Noisy Noisy datadata
10 % 10 % noisenoise
-1.61-1.61 0.430.43 11.2311.23
20 % 20 % noisenoise
-1.56-1.56 0.400.40 11.2111.21
DenoiseDenoised datad data
10 % 10 % noisenoise
-1.61-1.61 0.420.42 11.1911.19
20 % 20 % noisenoise
-1.56-1.56 0.390.39 11.1511.15
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 5 for Sample 5
( Hetroscedastic noise - NPLS model )( Hetroscedastic noise - NPLS model )
43
Analyte Analyte 11
Analyte Analyte 22
Analyte Analyte 33
0 % 0 % noisenoise
-1.67-1.67 0.440.44 11.2411.24
Noisy Noisy datadata
10 % 10 % noisenoise
-1.61-1.61 0.430.43 11.2311.23
20 % 20 % noisenoise
-1.56-1.56 0.400.40 11.2111.21
DenoiseDenoised datad data
10 % 10 % noisenoise
-1.61-1.61 0.430.43 11.2111.21
20 % 20 % noisenoise
-1.56-1.56 0.400.40 11.2111.21
Relative Errors of Predicted Concentrations Relative Errors of Predicted Concentrations for Sample 5 for Sample 5
( Hetroscedastic noise – NPLS model )( Hetroscedastic noise – NPLS model )
Wavelet Denoising (MDL)Wavelet Denoising (MDL)
44
X-Block X-Block
Model Model 11
Y-Block Y-Block
Model 1Model 1
X-Block X-Block
Model Model 22
Y-Block Y-Block
Model 2Model 2
noise noise 0%0% 99.9499.94 99.8599.85 99.9399.93 99.9499.94
Noisy Noisy datadata
noise noise 2%2% 99.8099.80 99.8599.85 99.7999.79 99.9499.94
noise noise 5%5% 99.0699.06 99.8599.85 99.0499.04 99.9499.94
noise noise 10%10% 96.5096.50 99.8499.84 96.5196.51 99.9499.94
DenoisDenoised dataed data
noise noise 2%2%
99.8599.85 99.9799.97 99.9199.91 99.8599.85
noise noise 5%5%
99.6799.67 99.9799.97 99.7799.77 99.8599.85
noise noise 10%10%
99.1099.10 99.9799.97 99.3299.32 99.8899.88
Comparison of explained variation Comparison of explained variation (Homoscedastic noise – NPLS model)(Homoscedastic noise – NPLS model)
45
Comparison of explained variation Comparison of explained variation (Hetroscedastic noise – NPLS model)(Hetroscedastic noise – NPLS model)
X-Block X-Block
Model Model 11
Y-Block Y-Block
Model 1Model 1
X-Block X-Block
Model Model 22
Y-Block Y-Block
Model 2Model 2
noise noise 0%0% 99.9499.94 99.8599.85 99.9399.93 99.9499.94
Noisy Noisy datadata
noise noise 10%10% 99.8099.80 99.8599.85 99.7899.78 99.9499.94
noise noise 20%20% 99.3899.38 99.8599.85 99.3499.34 99.9499.94
DenoisDenoised dataed data
noise noise 10%10%
99.8199.81 99.8599.85 99.7999.79 99.9499.94
noise noise 20%20%
99.4399.43 99.8699.86 99.3999.39 99.9499.94
46
Comparison of explained variationComparison of explained variation (Hetroscedastic noise – NPLS model) (Hetroscedastic noise – NPLS model)
X-Block X-Block
Model Model 11
Y-Block Y-Block
Model 1Model 1
X-Block X-Block
Model Model 22
Y-Block Y-Block
Model 2Model 2
noise noise 0%0% 99.9499.94 99.8599.85 99.9399.93 99.9499.94
Noisy Noisy datadata
noise noise 10%10% 99.8099.80 99.8599.85 99.7899.78 99.9499.94
noise noise 20%20% 99.3899.38 99.8599.85 99.3499.34 99.9499.94
DenoisDenoised dataed data
noise noise 10%10%
99.8099.80 99.8599.85 99.7899.78 99.9499.94
noise noise 20%20%
99.3899.38 99.8599.85 99.3499.34 99.9499.94
Wavelet Denoising (MDL)Wavelet Denoising (MDL)
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