Post on 25-Feb-2016
description
23.02.03 1
Successive Bayesian EstimationSuccessive Bayesian EstimationAlexey Pomerantsev
Semenov Institute of Chemical PhysicsRussian Chemometrics Society
23.02.03 2
AgendaAgenda
1. Introduction. Bayes Theorem
2. Successive Bayesian Estimation
3. Fitter Add-In
4. Spectral Kinetics Example
5. New Idea (Method ?)
6. More Applications of SBE
7. Conclusions
23.02.03 3
1. Introduction1. Introduction
23.02.03 4
The Bayes Theorem, 1763The Bayes Theorem, 1763
Thomas Bayes (1702-1761)
Posterior Probability Prior Probabilities
L(a,2)=h(a,2)L0(a,2)
Likelihood Function
Where to takethe prior
probabilities?
23.02.03 5
Jam Sampling & Blending TheoryJam Sampling & Blending Theory
0.20 0.30 0.50
0.50 0.20 0.05
Now we know the origin ofa worm in the jam!
23.02.03 6
2.Successive Bayesian Estimation (SBE)2.Successive Bayesian Estimation (SBE)
23.02.03 7
SBE ConceptSBE Concept
y 1 X 1 y 2 X 2 . . . y k X k
. . .
. . .
Whole data set
f 1 (X 1 , a 0 , a 1 ) f 2 (X 2 , a 0 , a 2 ) f k (X k , a 0 , a k )
Data subset 1 Data subset 2 Data subset k
Posta 0 , a 1
s 12 N 1
Priora 0
s 12 N 1
Posta 0 , a 2
s 22 N 2
Priora 0
s k2 N k
Posta 0 , a k
s k2 N k
Resulta 0 , a 1 ,…, a k
s 2 N
SBE principles
1) Split up whole data set
2) Process each subset alone
3) Make posterior information
4) Build prior information
5) Use it for the next subset
How to eat away
an elephant?Slice by slice!
23.02.03 8
OLS & SBE Methods for Two SubsetsOLS & SBE Methods for Two Subsets
OLS
SBE
Quadraticapproximation
near theminimum!
23.02.03 9
Posterior & Prior InformationPosterior & Prior InformationSubset 1. Posterior Information
Rebuilding (common & partial parameters)
Subset 2. Prior Information
Make Posterior,rebuild it and apply as Prior!
23.02.03 10
Prior Information of Type IPrior Information of Type IPosterior Information Prior Information
Parameter estimates Prior parameter values b
Matrix A Recalculated matrix H
Variance estimate s2 Prior variance value s02
NDF Nf Prior NDF N0
Objective Function
The same errorvariance in theeach subset
of data!
23.02.03 11
Prior Information of Type IIPrior Information of Type II
Posterior Information Prior Information
Parameter estimates Prior parameter values b
Matrix A Recalculated matrix H
Objective Function
aDifferent errorvariances in the
each subsetof data!
23.02.03 12
SBE Main TheoremSBE Main Theorem
Different order of subsets processing
Theorem (Pomerantsev & Maksimova , 1995)
SBEagree with
OLS!
23.02.03 13
3. Fitter3. Fitter Add-InAdd-In
23.02.03 14
A B C D E F G H I J K L M N O P Q R S T12 Data3 x t y w f A B C4 13 0 0.047 1 0.047 1 0 05 13 2 0.553 1 0.56 0.125 0.448 0.42666 13 4 0.412 1 0.403 0.016 0.209 0.7757 13 6 0.304 1 0.308 0.002 0.079 0.91948 13 8 0.275 1 0.27 2E-04 0.028 0.9729 13 10 0.253 1 0.257 3E-05 0.01 0.9904
1011 Bayesian Information12 Name Value Matrix Exclude13 k1 1.07 265.3 146.5 0 0 014 k2 0.554 146.5 1117 0 0 015 0 0 0 0 0 016 0 0 0 0 0 017 0 0 0 0 0 018192021222324252627282930
FitterFitter Workspace WorkspaceA B C D E F G H I J K L M N O P Q R S T1
2 Data General3 x t y w f A B C Date 01.08.01 19:074 13 0 0.047 1 0.047 1 0 0 Data Bayes!rData5 13 2 0.553 1 0.56 0.125 0.448 0.4266 Model Bayes!ABCbayes6 13 4 0.412 1 0.403 0.016 0.209 0.775 ParametersBayes!rParam7 13 6 0.304 1 0.308 0.002 0.079 0.9194 Bayes Bayes!rBayes8 13 8 0.275 1 0.27 2E-04 0.028 0.972 Precision1E-119 13 10 0.253 1 0.257 3E-05 0.01 0.9904 Convergence0.001
10 Error typeRelative11 Bayesian Information Significance0.0512 Name Value Matrix Exclude Confidence0.9513 k1 1.07 265.3 146.5 0 0 0 PredictionLinearization14 k2 0.554 146.5 1117 0 0 015 0 0 0 0 0 0 Parameters estimation16 0 0 0 0 0 0 Name Initial Final Deviation17 0 0 0 0 0 0 k1 1.06969 1.03907 0.06050918 k2 0.55366 0.53661 0.02795419 p -3.05769 -3.05769 0.01937120 q -0.002 -0.002 0.03999421 r -1.38757 -1.38757 0.0173322223 Parameters Search Progress24 k1 1.03907 Objective value 0.013925 k2 0.53661 Completeness 100%26 p -3.05769 Objective change -1E-0727 q -0.002 Iteration 228 r -1.387572930
x =
y
A
B
C
13
0.0
0.5
1.0
0 2 4 6 8 10 t
y
y=exp(p)*A+exp(q)*B+exp(r)*CA=A0*exp(-k1*t)B=k1*A0/(k1-k2)*[exp(-k2*t)-exp(-k1*t)]+B0*exp(-k2*t)C=A0+B0+C0+A0/(k1-k2)*[k2*exp(-k1*t)-k1*exp(-k2*t)]-B0*exp(-k2*t) A0="cA0" B0="cB0" C0="cC0" k1=? k2=? p=? q=? r=?
Fitter is atool for SBE!
23.02.03 15
A B C D E F G H I J K L M1
2
3
4
5
6
7
8
9
1011
A B C D E F G H I J K L M1
2 BoxBod Data3 x y w f4 0 05 1 109 16 2 149 17 3 149 18 5 191 19 7 213 110 10 224 111
Data & Model Prepared for FitterData & Model Prepared for Fitter
A B C D E F G H I J K L M1
2 BoxBod Data Parameters3 x y w f a 1004 0 0 b 0.45 1 109 16 2 149 17 3 149 18 5 191 19 7 213 110 10 224 111
'BoxBOD modely=a*[1-exp(-b*x)] a=? b=?
A B C D E F G H I J K L M1
2 BoxBod Data Parameters3 x y w f a 213.809414 0 0 0.00 b 0.54723755 1 109 1 90.116 2 149 1 142.247 3 149 1 172.418 5 191 1 199.959 7 213 1 209.1710 10 224 1 212.9111
0
100
200
0 4 8 x
y
'BoxBOD modely=a*[1-exp(-b*x)] a=? b=?
ResponseWeight
Fitting
Predictor
ParametersEquation
CommentValues
Apply Fitter!
23.02.03 16
Model Model ff((xx,,aa))Different shapes of the same model
Explicit model y = a + (b – a)*exp(–c*x)
Implicit model 0 = a + (b – a)*exp(–c*x) – y
Diff. equation d[y]/d[x] = – c*(y –a); y(0) = b
Presentation at worksheet
'Цикл "увлажнение-сушка"M=Sor*hev(t1-t)+Des*[hev(t-t1)+imp(t-t1)]'Кинетика "увлажнения" Sor=Sor1*hev(USESor1)+Sor2*[hev(-USESor1)+imp(-USESor1)]'Кинетика "сушки" Des=Des1*hev(USEDes1)+Des2*[hev(-USEDes1)+imp(-USEDes1)]'Условие применимости асимптотик USESor1=Sor2-Sor1 USEDes1=Des1-Des2'константы и промежуточные величины t3=(t-t1)*hev(t-t1) t4=t*hev(t1-t)+t1*[hev(t-t1)+imp(t-t1)] P2=PI*PI P12=(PI)^(-0.5) R=r*(M1-M0)*exp(-r*t4) K=M1+(M0-M1)*exp(-r*t4) V0=M0-C0 V1=M1-C0'асимптотика сорбции при 0<t<tau Sor1=C0+4*P12*(d*t)^0.5*[M0-C0+(M1-M0)*beta] beta=1-exp(-z) x=r*t z=(a1*x+a2*x*x+a3*x*x*x)/(1+b1*x+b2*x*x+b3*x*x*x) a1=0.6666539250029 a2=0.0121051017749 a3=0.0099225322428 b1=0.0848006232519 b2=0.0246634591223 b3=0.0017549947958'кинетика сорбции при tau<t<t1 Sor2=K-8*S1 S1=U01/n0+U11/n1+U21/n2+U31/n3+U41/n4 n0=P2 U01=[(V0*n0*d-V1*r)*exp(-n0*d*t4)+R]/(n0*d-r) n1=P2*9 U11=[(V0*n1*d-V1*r)*exp(-n1*d*t4)+R]/(n1*d-r) n2=P2*25 U21=[(V0*n2*d-V1*r)*exp(-n2*d*t4)+R]/(n2*d-r) n3=P2*49 U31=[(V0*n3*d-V1*r)*exp(-n3*d*t4)+R]/(n3*d-r) n4=P2*81 U41=[(V0*n4*d-V1*r)*exp(-n4*d*t4)+R]/(n4*d-r)'асимптотика десорбции при t1<t<t1+tau Des1=K*[1-4*P12*(d*t3)^0.5]-8*S1'кинетика десорбции при t1+tau<t Des2=8*S2 S2=U02/n0+U12/n1+U22/n2+U32/n3+U42/n4 U02=(K-U01)*exp(-n0*d*t3) U12=(K-U11)*exp(-n1*d*t3) U22=(K-U21)*exp(-n2*d*t3) U32=(K-U31)*exp(-n3*d*t3) U42=(K-U41)*exp(-n4*d*t3)'неизвестные параметры d=? M0=? M1=? C0=? r=? t1=?
Rathercomplexmodel!
23.02.03 17
4. Spectral Kinetics Modeling4. Spectral Kinetics Modeling
17
1319
2531
3743
49 0 2 4 6 8 10
23.02.03 18
Spectral Kinetic DataSpectral Kinetic Data
wavelengths wavelengths wavelengths
= +spec
ies
tim
e
spectral signal conc. pure spectra errors
tim
e
species
t
imeY C P E
Y(t,x,k)=C(t,k)P(x)+E
Y is the (NL) known data matrix
C is the (NM) known matrix depending on unknown parameters k
P is the (ML) unknown matrix of pure component spectra
E is the (NL) unknown error matrix
K constants L wavelengths M species N time points
This is largenon-linearregressionproblem!
23.02.03 19
How to Find Parameters k?How to Find Parameters k?Method Idea Dimension Problem
Full OLS(hard) K+ML >> 1 Large
dimension
Short OLS(hard)
K+MS 10
Smallprecision
WCR(hard&soft) K 10 Matrix
degradation
GRAM(soft)
K+MA 100
Just onemodel
)(ln stk
kt
ee
s1k
This is a challenge!
23.02.03 20
Simulated Example Goals Simulated Example Goals
Compare SBE estimates with ‘true’ values
Compare SBE estimates for different order
Compare SBE estimates with OLS estimates
23.02.03 21
Model. Two Step KineticsModel. Two Step Kinetics
0C0CBkdtdC
0B0BBkAkdtdB
1A0AAkdtdA
02
021
01
)(;
)(;
)(;
Ak
Bk
C1 2
‘True’ parameter values
k1=1 k2=0.5
Standard‘training’
model
23.02.03 22
Data SimulationData Simulation
C1(t) = [A](t)
C2(t) = [B](t)
C3(t) = [C](t)
P1(x) = pA (x)
P2(x) = pB (x)
P3(x) = pC (x)
Simulated concentration profiles Simulated pure component spectra
B
CA
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10time
conc
entr
atio
ns
A B C
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25 30 35 40 45 50conventional wavelengths
spec
tral
sig
nal
Y(t,x)=C(t)P(x)(I+E)
STDEV(E)=0.03
Usual way ofdata simulation
23.02.03 23
t=0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal
t=0
t=2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal
t=0
t=2
t=4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal
Simulated Data. Spectral ViewSimulated Data. Spectral View
t=0
t=2
t=4
t=6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal
t=0
t=2
t=4
t=6t=8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal
t=0
t=2
t=4
t=6t=8
t=10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 10 20 30 40 50conventional wavelengths
spec
tral
sig
nal Spectral
view of data
23.02.03 24
Simulated Data. Kinetic ViewSimulated Data. Kinetic View
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
0.4
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0.8
1
0 2 4 6 8 10time
spec
tral
sig
nal
0
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1
0 2 4 6 8 10time
spec
tral
sig
nal
0
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1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
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1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
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0.8
1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
0.4
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1
0 2 4 6 8 10time
spec
tral
sig
nal
0
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0.8
1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
0.4
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0.8
1
0 2 4 6 8 10time
spec
tral
sig
nal
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10time
spec
tral
sig
nalKinetic view
of data
23.02.03 25
One Wavelength EstimatesOne Wavelength Estimates Conventional wavelength 3
Estimates
30.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
k 1
k 2
30.0
1.0
2.0
3.0
4.0
14
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
Conventional wavelength 14
k 1
k 2
1430.0
1.0
2.0
3.0
4.0
Conventional wavelength 51
510.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
k 1
k 2
14 5130.0
1.0
2.0
3.0
4.0
k 1
k 2
14 51 O30.0
1.0
2.0
3.0
4.0Bad accuracy!
23.02.03 26
1234
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
k 1
k 2
D0.0
0.5
1.0
1.5
Direct order
Estimates
Four Wavelengths EstimatesFour Wavelengths Estimates
53525150
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
k 1
k 2
D I0.0
0.5
1.0
1.5
Inverse order
165
298
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 time
y
k 1
k 2
D RI0.0
0.5
1.0
1.5
Random order
k 1
k 2
D ORI0.0
0.5
1.0
1.5
Bad accuracy,again!
23.02.03 27
SBE Estimates at the Different OrderSBE Estimates at the Different OrderDirect 1, 2, 3, ….
Random 16, 5, 29, ….
k 2
k 1
0.25
0.50
0.75
1.00
1.25
1.50
53 49 45 41 37 33 29 25 21 17 13 9 5 1conventional wavelengths
k 2
k 1
0.25
0.50
0.75
1.00
1.25
1.50
1 8 15 22 29 36 43 50conventional wavelengths
k 2
k 1
0.25
0.50
0.75
1.00
1.25
1.50
16 41 27 33 19 2 15 51 21 9 24 50 12 22conventional wavelengths
Inverse 53, 52, 51, ….
0.95 Confidence Ellipses
Random
Direct
Inverse
'True'
0.85
0.95
1.05
1.15
1.25
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56
k 2
k 1
Random
Direct
Inverse
'True'
0.85
0.95
1.05
1.15
1.25
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56
k 2
k 1
SBE (practically)doesn’t depend onthe subsets order!
23.02.03 28
SBE Estimates and OLS EstimatesSBE Estimates and OLS Estimates
OLS
SBE
'True'
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56
k 2
k 1
SBE estimatesare close to
OLS estimates!
23.02.03 29
Spectrum A
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
spec
tral s
igna
l
-0.2
0
0.2
0.4
0.6
accu
racy
Spectrum A
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
spec
tral s
igna
l
-0.2
0
0.2
0.4
0.6
accu
racy
Pure Spectra EstimatingPure Spectra EstimatingSpectrum B
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
spec
tral s
igna
l
-0.2
0
0.2
0.4
0.6
accu
racy
Spectrum C
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
spec
tral s
igna
l
-0.2
0
0.2
0.4
0.6
accu
racySBE gives
good spectraestimates!
23.02.03 30
Real World Example Goals Real World Example Goals
Apply SBE for real world data
Compare SBE with other known methods
23.02.03 31
DataData
Bijlsma S, Smilde AK. J.Chemometrics 2000; 14: 541-560Epoxidation of 2,5-di-tert-butyl-1,4-benzoquinoneSW-NIR spectra
-8-6-4-202468
860 865 870 875 880wavelength
spec
tral s
igna
l240 spectra1200 time points21 wavelengthsPreprocessing:Savitzky-Golay filter
-8-6-4-202468
860 865 870 875 880wavelength
spec
tral s
igna
lPreprocessedData
23.02.03 32
Progress in SBE EstimatesProgress in SBE Estimates
k 1
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880wavelength (nm)
k 2
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880wavelength (nm)
k 2
k 1
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880wavelength (nm)
SBE workswith the realworld data!
23.02.03 33
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.05 0.00 0.05 0.10 0.15 0.20 0.25
k 2
k 1
WCR
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.05 0.00 0.05 0.10 0.15 0.20 0.25
k 2
k 1
SBE and the Other MethodsSBE and the Other Methods
WCR
LM-PAR0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.05 0.00 0.05 0.10 0.15 0.20 0.25
k 2
k 1
WCR
GRAM
LM-PAR0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.05 0.00 0.05 0.10 0.15 0.20 0.25
k 2
k 1
SBE
WCR
GRAM
LM-PAR0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.05 0.00 0.05 0.10 0.15 0.20 0.25
k 2
k 1
SBE gives thelowest deviationsand correlation!
23.02.03 34
5. New Idea5. New Idea
23.02.03 35
y=a1x1+a2x2+a3x3
Bayesian Step Wise Regression Bayesian Step Wise Regression Ordinarily Step Wise Regression Bayesian Step Wise Regression
Objective function
BSWR accountscorrelations of
variables in step wise estimation
23.02.03 36
BSW Regression & Ridge RegressionBSW Regression & Ridge Regression
BSWR is RR witha moving center
and non-Euclideanmetric
23.02.03 37
Example. RMSEC & RMSEPExample. RMSEC & RMSEP
BSWR givestypical U-shape ofthe RMSEP curve
23.02.03 38
Linear Model. RMSEC & RMSEPLinear Model. RMSEC & RMSEP
y=a1x1+a2x2+a3x3+a4x4+a5x5
0.0
0.1
0.2
0.3
0.4
0.5
PLS PCR OLS SWR BSWR
RMSEC
RMSEP
BSWR is notworse then PLS or PCR and betterthen SWR
23.02.03 39
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PLS PCR OLS SWR BSWR
RMSEC
RMSEP
Non-Linear Model. RMSEC & RMSEPNon-Linear Model. RMSEC & RMSEP
0.0
0.2
0.4
0.6
0.8
1.0
1.2
PLS PCR OLS SWR BSWR
RMSEC
RMSEP
5544332211 xk5
xk4
xk3
xk2
xk1 aaaaay eeeee
For non-linearmodel BSWR is
better then PLS or PCR
23.02.03 40
Variable selectionVariable selection
BSWR is just an idea, not
the method soany criticism is welcomed now!
23.02.03 41
6. More Practical Applications of SBE6. More Practical Applications of SBE
23.02.03 42
Antioxidants Activity by DSCAntioxidants Activity by DSCDSC Data Oxidation Initial Temperature (OIT)
C=0.1
C=0.05
C=0.025
470
490
510
530
550
570
0 5 10 15 20Heating rate v , grad/min
OIT
T,K
20
1510
52
-5-4-3-2-101234
460 470 480 490 500 510Temperature, K
DSC
sig
nal
To testantioxidants!
23.02.03 43
Network Density of Shrinkable PE by TMANetwork Density of Shrinkable PE by TMA
4
21
3
5
1.1
1.2
1.3
1.4
1.5
0 10 20 30 40 50 60 70 80 90
Time, min
Elon
gatio
n L/
Lo
AB
C
D 2D 1 D 30
2
4
6
8
10
12
14
0 5 10 15 20 25Dose, MRad
Che
mic
al m
odul
us, g
mm
2
TMA Data Network density
To solvetechnological
problem!
23.02.03 44
PVC Isolation Service Life by TGAPVC Isolation Service Life by TGA
Critical Level
0.0
0.1
0.2
0.3
0 5 10 15 20 25Time, yr
Con
cent
ratio
n
T=20C, F=2.0, P=0.95 T=30C, F=1.5, P=0.95
Service Life 2110
0.90
0.92
0.94
0.96
0.98
1.00
0 10 20 30 40 50Time t , min
Mas
s ch
amge
, y
370
410
450
490
Tem
pera
ture
T, K
TGA Data Service life prediction
To predictdurability!
23.02.03 45
Tire Rubber StorageTire Rubber StorageElongation at break Tensile strength
T=140 C T=125 C T=110 C
T=20 C Criticallevel
26
0
1
2
3
4
5
6
0 20 40 60Time, hr
Elon
gatio
n @
bre
ak
0 15 30 45Time, yr
T=140 C T=125 C T=110 C
T=20 CCritical
level
23
0
5
10
15
20
25
30
0 20 40 60Time, hr
Tens
ile, K
Pa
0 15 30 45Time, yr
To predictreliability!
23.02.03 46
7. Conclusions7. Conclusions
1 SBE is of general nature and it can be used for any model
2 SBE agrees with OLS
3 SBE gives small deviations and correlations
4 SBE uses no subjective a priori information
5 SBE may be useful for non-linear modeling (BWSR?)
Thanks!