Post on 01-Jan-2016
description
Rousseau Réjane – 1/02/2008
Combination of Independent Component Analysis and statistical modeling
for the identification of metabonomic biomarkers in 1H-NMR spectroscopy
Réjane Rousseau (Institut de Statistique, UCL, Belgium)
Rousseau Réjane – 1/02/2008
The metabonomics
METABOLITES
TISSUESOrgans
Biofluid
(Urine)
1H-NMR
spectroscopy
Spectral alterations
= Altered metabolites
= detection
Whithout contact
After contact
Frequency domain
Identification of biomarkers
« which part of the spectrum to examine? »
specific region = biomarker
How?
• In an experimental database
• with a methodology based on Principal Component Analysis (“PCA”)
Objective: propose a methodology combining ICA and statistical modeling
One molecule = several peaks ( 1 to 3 ) with specific positions in the frequency domain.
The concentration of a molecule is proportional to the area under the curve in its peaks.
Rousseau Réjane – 1/02/2008
Identification of biomarkers:
• Experimental data:
Biologists have a pool of n rats with a control pathological state. They collect n samples of urine . The H-NMR produces n spectra of m values = Spectral data Each sample or spectrum is described by l variables in a matrix of design Y.
One of these variables describes the characteristic related to the biomarkers: yk
• Biomarker identification= statistical methods to answer to the question:
« In these multivariate data, which are the most altered variables xj
when yk changes? »
Spectral data: X(n x m)
xj
y1 y2yl
13 32011 200
11 100
12 270
Design data Y(n x l)
y1= age of the rat
yk=y2= severity of diabetis
Rousseau Réjane – 1/02/2008
• Advantage of controlled data:
we know the spectral regions that should be identified as biomarkers.
• The controlled data : 28 spectra of 600 points: X(28 x 600) Each spectrum = a sample of urine
+ a chosen concentration of Citrate
+ a chosen concentration of Hippurate
X(28x600) Y (28x2) y1= concentration of citrate
y2= concentration of hippurate
• We need a biomarker to detect changes of the level of citrate described by y1
« Which are the spectral regions xj the most altered when the y1 changes?»
Spectral regions corresponding to Citrate = the biomarkers to identify.
Example: controlled data
citrate (y1)
hippurate (y2)
Rousseau Réjane – 1/02/2008
=
spectrum
3000 points
+
+
The biomarkers to identify.= spectral regions corresponding to Citrate
A spectrum of 600 values xj with ∑ xj = 1
Natural urine
Citrate
Hippurate
Citrate y1
Hippurate
y2
14 mixtures in 2 replications
28 samples
UrineCitrate
Hippurate
…
Rousseau Réjane – 1/02/2008
ex: Citrate plays an important role
NEW
II. Biomarker discovery through Statistical modelling
USUAL
Principal components are :
- uncorrelated
- in the direction of maximum of variance
Examination of the 2 first components:
This is only powerful if the biological question
is related to the highest variance in the dataset!
XTC = SAT
Components :
- are independent
- with a biological meaning
Examination of the ALL components:
to visualize unconnected
molecules in samples Score plot Loadings
Identificationof biomarker
Comparison of the intensities of biomarkers
between spectra from ≠ conditions
I. Reduction of the dimension:
PCA
I. Reduction of the dimension:
ICA
XC = TP
L1
L2
-0.2 -0.1 0.0 0.1 0.2 0.3
-0.3
-0.2
-0.1
0.0
0.1
0.2
PC1
PC2
Identificationof biomarker
Rousseau Réjane – 1/02/2008
The proposed methodology:
Part I : Dimension reduction
with ICA on the spectral data:
XTC= S.AT
Part II: Biomarker discovery through statistical modelling
Resulting components:
- meaningfull component - with some advantages over the principal components
Identification of biomarker
Comparison of the values
in biomarker
between spectra from different conditions
Rousseau Réjane – 1/02/2008
What is Independent component analysis (ICA)?
The idea: • Each observed vector of data is a linear combination of unknown independent (not only
linearly independent) components
• The ICA provides the independent components (sources, sk) which have created a vector of data and the corresponding mixing weights aki.
How do we estimate the sources? with linear transformations of observed signals that maximize the independence of the sources.
How do we evaluate this property of independence? Using the Central Limit Theorem (*), the independence of sources components can be reflect
by non-gaussianity.
Solving the ICA problem consists of finding a demixing matrix which maximises the non-gaussianity of the estimated sources under the constraint that their variances are constant.
Fast-ICA algorithm: - uses an objective function related to negentropy
- uses fixed-point iteration scheme.
* almost any measured quantity which depends on several underlying independent factors has a Gaussian PDF
1 1 2 21
... l
i k ki i i l lik
x s a s a s a s a
Rousseau Réjane – 1/02/2008
Each spectrum is a weighted sum of the
independent spectral expressions which each one can correspond to
an independent (composite) metabolite contained in the studied sample.
(aT , weight quantity)
X (nxm) n spectra defined by m variables ex: (28x600)
XT (mxn)
XTC (mxn)
T (mxq) = XTC. P
S (mxq) = XTC. P.W
= XTC. A
XTC= S.AT
ICA
“Whitening”: PCA
Centeringmixture and sj have zero mean
1. we obtain uncorrelated scores with unit variance → demixing matrix is orthogonal
2. possibility to discard irrelevant scores = chose the number of sources to estimate
I. Dimension reduction by ICA :
Transposition
Rousseau Réjane – 1/02/2008
Example: I. Dimension reduction by ICA XTC= S.AT
s1,1 s1,6
sij
s600,1
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
39.42 %
39.42 %
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
29.61 %
29.61 %
05
10
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
27.84 %
27.84 %
-55
15
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.77 %
0.77 %
-10
010
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.28 %
0.28 %
-60
4
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.04 %
at1,1 at1,28
at6,1
at1
at2
at3
at4
at5
at6
0 5 10 15 20 25
0.00
240.
0027
Index
00 0001 01
02 0204 04
1010 11 11
12 12 20 20 2121
2222
2424 40
4042
42 4444
0 5 10 15 20 25
0.00
180.
0026
Index
00 0001 01
02 02
04 04
10 1011 11
12 12
20 20
21 2122 22
24 24
40 40
42 42
44 44
0 5 10 15 20 25
0.00
180.
0026
Index
00 00 01 01 02 02 04 04
10 10 11 11 12 12
20 20 21 21 22 2224 24
40 40 42 42 44 44
0 5 10 15 20 25
0.00
025
Index
00
0001
01
02 02
0404
10
1011
11
12
12
2020
2121
22
2224
24
40
4042 42
44
44
0 5 10 15 20 25
0.00
015
0.00
030
Index
0000
01
01 02
02 04 0410
10
11
1112
1220 20
21
21
22
22
24
24
4040
42
42
44
44
0 5 10 15 20 25
0 e
+00
00
00
01
0102 02
04 04
10
10
11
11
1212 20
20
21
21
22 22
24
24
40
4042
42
44
44
s1 s2 s3 s4 s5 s6
XTC(600 x 28) = S (600 x 6) AT
(6x28)
....
....
....
....
xTC1 xTC
28
Urine
+ citrate
+ hippurate
Rousseau Réjane – 1/02/2008
S (600 x 6) AT
Natural urine
Citrate
Hippurate
28 spectra
0 5 10 15 20 25
0.002
40.0
027
Index
00 0001 01
02 0204 04
1010 11 11
12 12 20 20 2121
2222
2424 40
4042
42 4444
0 5 10 15 20 25
0.001
80.0
026
Index
00 0001 01
02 02
04 04
10 1011 11
12 12
20 20
21 2122 22
24 24
40 40
42 42
44 44
0 5 10 15 20 25
0.001
80.0
026
Index
00 00 01 01 02 02 04 04
10 10 11 11 12 12
20 20 21 21 22 2224 24
40 40 42 42 44 44
0 5 10 15 20 25
0.000
25Index
00
0001
01
02 02
0404
10
1011
11
12
12
2020
2121
22
2224
24
40
4042 42
44
44
0 5 10 15 20 25
0.000
150.0
0030
Index
0000
01
01 02
02 04 0410
10
11
1112
1220 20
21
21
22
22
24
24
4040
42
42
44
44
0 5 10 15 20 25
0 e+
00
00
00
01
0102 02
04 04
10
10
11
11
1212 20
20
21
21
22 22
24
24
40
4042
42
44
44
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
39.42 %
39.42 %
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
29.61 %
29.61 %
05
10
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
27.84 %
27.84 %
-55
15
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.77 %
0.77 %
-10
010
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.28 %
0.28 %
-60
4
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.04 %
aTi,8
Rousseau Réjane – 1/02/2008
• Similarities: projection methods linearly decomposing multi-dimensional into components.
• Differences: The number of sources, q, has to be fixed Sources are not naturally sorted according to their importances The independence condition = the biggest advantage of the ICA:
- independent components are more meaningful than uncorrelated components
- more suitable for our question in which the component of interest are not always in the direction
with the maximum variance .
Note: Comparison with the usual PCA
Hippurate
Citrate
Hippurate
Citrate
Hippurate
Citrate
PCA ICA1
2
Natural urine
Rousseau Réjane – 1/02/2008
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.0020 0.0022 0.0024 0.0026
0.00
180.
0022
0.00
26
s2
s3
0000
01 010202
1010
11 111212
202021
21 2222
2424
4242
4444
Index
9.9917004 7.8194544 5.647086 3.4749012 1.302472
41.14 %
41.14 %
Index
9.9917004 7.8194544 5.647086 3.4749012 1.302472
27.79 %
27.79 %
9.9917004 7.8194544 5.647086 3.4749012 1.302472
25.95 %
-0.3 -0.1 0.0 0.1 0.2 0.3 0.4
-0.3
-0.1
0.1
0.3
0000
0101
0202
1010
1111
1212
2020
2121
2222
2424
4242
4444
PCA ICA
Loading 1
Loading 3
Loading 2
Natural urine
Citrate
Hippurate
Hippurate & Citrate
Hippurate & Citrate
PC1
PC2
s1
s2
s3
aT2
aT3
Rousseau Réjane – 1/02/2008
The proposed methodology:
Part I : Dimension reduction
with ICA on the spectral data:
XTC= S.AT
Part II: Biomarker discovery through statistical modeling - on the mixing matrix AT
- with covariates chosen in the design matrix
q sources representing the spectra of independent (unrelated) composite metabolites contained in the samples.
Identificationof biomarkers
Comparison of the intensities
in biomarkers
between spectra from different conditions
Rousseau Réjane – 1/02/2008
PART II: Biomarker discovery with statistical model The idea: Among the q recovered sj , we suppose that some sources:
- present biomarker regions for a chosen factor yk
- are interpretable as the spectra of pure or composite independent metabolite which has a concentration in the samples influenced by a chosen factor yk
- have weights influenced by a chosen factor yk
Modelisation of the relation between the weight vector and the design variables
AT (q x n)
0 5 10 15 20 25
0.00
240.
0027
Index
00 0001 01
02 0204 04
1010 11 11
12 12 20 20 2121
2222
2424 40
4042
42 4444
0 5 10 15 20 25
0.00
180.
0026
Index
00 0001 01
02 02
04 04
10 1011 11
12 12
20 20
21 2122 22
24 24
40 40
42 42
44 44
0 5 10 15 20 25
0.00
180.
0026
Index
00 00 01 01 02 02 04 04
10 10 11 11 12 12
20 20 21 21 22 2224 24
40 40 42 42 44 44
0 5 10 15 20 25
0.00
025
Index
00
0001
01
02 02
0404
10
1011
11
12
12
2020
2121
22
2224
24
40
4042 42
44
44
0 5 10 15 20 25
0.00
015
0.00
030
Index
0000
01
01 02
02 04 0410
10
11
1112
1220 20
21
21
22
22
24
24
4040
42
42
44
44
0 5 10 15 20 25
0 e
+00
00
00
01
0102 02
04 04
10
10
11
11
1212 20
20
21
21
22 22
24
24
40
4042
42
44
44
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
39.42 %
39.42 %
04
8
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
29.61 %
29.61 %
05
10
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
27.84 %
27.84 %
-55
15
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.77 %
0.77 %
-10
010
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.28 %
0.28 %
-60
4
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.04 %
Rousseau Réjane – 1/02/2008
Part I : ICA on the spectral data: XTC= S.AT
Part II: Biomarker discovery through statistical modelling
- prediction of mixing weights by the model- reconstruction with biomarkers sources- comparison between factor levels.
Step 2: Biomarker identification
Step 3: comparison of the intensities
in biomarkers between
spectra from different conditions
- apply statistical tests on the parameters of the models- selection of sources with significant effects.
PART II:
Step 1: Fit a linear model on AT
- relation between the weight vector and the covariates in design variables- different models.
Rousseau Réjane – 1/02/2008
Step 1: Fit a model on AT
The design matrix Y is rewritten into 2 separate matrixs:- Z1 the (n x p1) incidence matrix for the p1 covariates with fixed effects- Z2 the (n x p2) incidence matrix for the p2 covariates with random effects
For each of the q recovered sj , we assume a linear relation between its vector of weights and the design variables:
aj= Z1 βj+ Z2 γj +εj
Models with only fixed effects covariates : aj= Z1 β j + εj
• Case 1: categorical covariates: ANOVA
→ ex1: biomarker to discriminate 2 groups of subjects: disease & sane.
ex2: biomarker to discriminate 3 groups of subjects: disease1, disease2 & sane
• Case 2: quantitative covariates : linear regression
→ ex: biomarker to explore the severity of an illness, the concentration of a drug
Models with only random effects covariates : aj= Z2 γj + εj
→ ex: biomarker to explore variance component (machines, subjects, laboratories)
Models with fixed and random effects covariates : Mixed model: aj= Z1 β j+ Z2 γj +εj
Rousseau Réjane – 1/02/2008
Step 1: Fit a model: example
• For each of the q = 6 recovered sj, we construct a multiple linear regression model
with 2 quantitative covariates ( p = 3) and no interaction:
aj= βj0 + βj1 y1 + βj2 y2 +εj
with βj0 = the intercept
y1= the citrate concentration in mg (quantitative)
y2= the hippurate concentration in mg (quantitative)
β1= the effect of citrate on the mean aj for a fixed value of hippurate
β2= the effect of hippurate on the mean aj for a fixed value of citrate
εj= the vector of independent random error ~ N (0,σ2)
• For each of the q recovered sj, the fitted model by least square technique is :
âj= bj0 + bj1 y1 + bj2 y2
• In this example, we want to identify biomarkers for the concentration of citrate.
The covariate of interest yk= y1
Rousseau Réjane – 1/02/2008
0 100 200 300 400 500 600
05
10
15
s 3
0 50 100 150 200 250 300
0.0
01
80
.00
24
Citrate doses in mg
a 3
0 100 200 300 400 500 6000
48
12
s 2
0 50 100 150 200 250 300
0.0
01
80
.00
24
Citrate doses in mg
a 2
Step 1: Fit a model: examples3:Hippurate
s2: Citrate
0 100 200 300 400 500 600
05
1015
s 3
0 50 100 150 200 250 300
0.00
180.
0024
Citrate doses in mg
a 3
300
200
0.00200
0.00225
0.00250
1000
0.00275
100 0200300
AC
Citrate
Hippurate
0 100 200 300 400 500 600
04
812
s 2
0 50 100 150 200 250 300
0.00
180.
0024
Citrate doses in mg
a 2
Citrate (y1)
a2
hippurate (y2)
(y1) (y1)
300
2000.0018
0.0021
0.0024
1000
0.0027
100 0200300
AH
Citrate
Hippurate
Citrate (y1)hippurate (y2)
a3
Rousseau Réjane – 1/02/2008
Goal: • Among the q sources , we want to select the ones presenting a significant effect of the
chosen covariate yk on their weights. • These “discriminant sources” represent the spectrum of an independent metabolite with a concentration depending on the chosen covariate = biomarkers.
For each source sj, test the significance of the parameter βjk of the covariate of interest yk :
(ex: research of biomarkers for the dose of citrate y1, we test each of the 6 βj1)
• H0: βjk = 0 vs H1: βjk ≠ 0
• compute the following statistic: tj = bjk / s(bjk) ~ t(n-p)
• take the corresponding p-value: pj= P( t (n-p) |tj| )
We are in a multiple tests situation:
the selection of a significant set of r coefficients βjk based on q pj obtained from q individual tests.
→ Bonferroni correction: select, in a (m x r) matrix S*, the r sources with pj < 0.05/q
Step 2: Biomarker identification
Rousseau Réjane – 1/02/2008
1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
Sources
P-v
alue
Step 2: Biomarker identification: example
9.18 x 10-13 1.84x10-15
s 2 s 3s 1
α = 0.05/6
Sources
P-v
alue
s
We research of biomarkers for the dose of citrate y1=yK
→ we test each of the 6 βj1
2.86 x 10-31
Rousseau Réjane – 1/02/2008
Goal: comparison of the effects on the biomarker caused by ≠ changes in yk.
Choose 3 or more values of yk:
• yk1 : a first value of reference of yk
• yk2 : a new value of interest of yk
• yk3 : a second new value of interest of yk
Compute:
• The effect on the biomarker of the change of yk from yk1 to yk
2 :
C1= S* βk* (yk
2- yk1 )
• The effect on the biomarker of the change of yk from yk1 to yk
3 :
C2= S* βk* (yk
3- yk1 )
Step 3: Comparison of the intensities in biomarkers
Rousseau Réjane – 1/02/2008
Hippurate
Step 3: example
-0.0
02
0.0
02
0.0
06
C1
9.9917004 5.647086 2.3804354
-0.0
02
0.0
02
0.0
06
C2
9.9917004 5.647086 2.3804354
-0.0
02
0.0
02
0.0
06
C3
9.9917004 5.647086 2.3804354
Citrate yk
yk1
yk2
yk3
yk4
Rousseau Réjane – 1/02/2008
Conclusions:• With the presented methodology combining ICA with statistical modeling,
we visualize the independent metabolites contained in the studied biofluid (through the sources) and their quantity (through the mixing weights)
we identify biomarkers or spectral regions changing according to a chosen factor by a selection of source.
we compare the effects on this spectral biomarkers caused by different changes of this factor.
• In comparison with the PCA, ICA: gives more biologically meaningful and natural representations of this data.
Rousseau Réjane – 1/02/2008
Hippurate
Citrate 18 spectra of 600 values 1 characteristic in Y
X(18x600) Y (18x1) y1= disease group of the rat (qualitative)
We want biomarkers for group of disease described in y1.
→ a model with qualitative covariates
Example2: the data
Group 1= disease 1
Group 2= disease 2
Group 3= no disease
Rousseau Réjane – 1/02/2008
Example2 :Part I. Dimension reduction by ICA XTC= S.AT
S (600 x 5) AT (5x18)
5 10 15
0.00
245
0.00
260
Index
5 10 15
0.00
180.
0022
0.00
26
Index
5 10 15
0.00
180.
0022
0.00
26Index
5 10 15
0.00
048
0.00
054
0.00
060
Index
5 10 15-0.0
0030
-0.0
0020
-0.0
0010
02
46
812
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
40.74 %
40.74 %
02
46
812
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
27.97 %
27.97 %
05
10
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
27.42 %
27.42 %
-50
510
15
Index
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
1.88 %
1.88 %
-50
510
9.9917004 8.897418 7.8194544 6.7249886 5.647086 4.5528036 3.4749012 2.3804354 1.302472 0.2082506
0.28 %
Rousseau Réjane – 1/02/2008
Example 2: Part II: biomarkers discovery through statistical modeling
Step 1: Fit a model on AT
Models with only a categorical covariate with fixed effects: ANOVA I
aj= Z1 β j + εj
Step 2: Biomarker identification: For each of the q recovered sj, test the effect of y1 → Fj statistics→ pj
Bonferroni correction: select, in a (m x r) matrix S*, the r sources with pj < 0.05/q
0.0
0.1
0.2
0.3
0.4
Sources
P-v
alu
e
s 1 s 2s 3
0.0002412604 0.005710213
0.009797431
Rousseau Réjane – 1/02/2008
Goal: comparison of the effects on the biomarker caused by ≠ changes in yk.
Choose 3 or more values of yk:
• yk1 : a first value of reference of yk
• yk2 : a new value of interest of yk
• yk3 : a second new value of interest of yk
Compute:
• The effect on the biomarker of the change of yk from yk1 to yk
2 :
C1= S* βk* (yk
2- yk1 )
• The effect on the biomarker of the change of yk from yk1 to yk
3 :
C2= S* βk* (yk
3- yk1 )
Step 3: Comparison of the intensities in biomarkers
Rousseau Réjane – 1/02/2008
-0.0
06-0
.002
0.002
0.006
C1
9.9917004 5.647086 2.3804354
-0.0
06-0
.002
0.002
0.006
C2
9.9917004 5.647086 2.3804354
-0.0
06-0
.002
0.002
0.006
C3
9.9917004 5.647086 2.3804354
Hippurate
Citrate
Step 3: Comparison of the intensities in biomarkers
Goal: comparison of the effects on the biomarker caused by the changes of group.
Rousseau Réjane – 1/02/2008
For a first chosen value of interest of yk (not necessary observed): yk0
Choose a value of reference for the other factors: y0≠k
→ Z0
1 vector of values : (1, yk0
, y0≠k)
For each of the r source selected in S*, use the model to predict weights for the biomarkers: âj
* (yk0) = bj
* Z01
→ â* (yk0) a vector of r new weights
Reconstruction of the values to expect in the biomarkers: S*â*(yk
0)
Example 1 : Step 4: Comparison of the intensities in biomarkers
Rousseau Réjane – 1/02/2008
0.0
00
0.0
10
0.0
20
0.0
30
0 mg
9.9917004 3.4749012
0.0
00
0.0
10
0.0
20
0.0
30
75 mg
9.9917004 3.4749012
0.0
00
0.0
10
0.0
20
0.0
30
150 mg
9.9917004 3.4749012
0.0
00
0.0
10
0.0
20
0.0
30
300 mg
9.9917004 3.4749012
Rousseau Réjane – 1/02/2008
Example 2: the reconstructed spectra 0.
000.
010.
020.
03
9.9917004 6.7249886 3.4749012 0.2082506
0.00
0.01
0.02
0.03
9.9917004 6.7249886 3.4749012 0.2082506
0.00
0.01
0.02
0.03
9.9917004 6.7249886 3.4749012 0.2082506