23.02.03 1
Successive Bayesian EstimationSuccessive Bayesian Estimation
Alexey PomerantsevSemenov Institute of Chemical Physics
Russian 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
Post
a 0 , a 1
s 12 N 1
Prior
a 0
s 12 N 1
Post
a 0 , a 2
s 22 N 2
Prior
a 0
s k2 N k
Post
a 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 Information
Subset 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 I
Posterior 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 T1
2 Data
3 x t y w f A B C
4 13 0 0.047 1 0.047 1 0 0
5 13 2 0.553 1 0.56 0.125 0.448 0.4266
6 13 4 0.412 1 0.403 0.016 0.209 0.775
7 13 6 0.304 1 0.308 0.002 0.079 0.9194
8 13 8 0.275 1 0.27 2E-04 0.028 0.9729 13 10 0.253 1 0.257 3E-05 0.01 0.990410
11 Bayesian Information
12 Name Value Matrix Exclude
13 k1 1.07 265.3 146.5 0 0 0
14 k2 0.554 146.5 1117 0 0 0
15 0 0 0 0 0 0
16 0 0 0 0 0 0
17 0 0 0 0 0 0
18
19
2021
22
23
24
25
2627
28
29
30
FitterFitter Workspace WorkspaceA B C D E F G H I J K L M N O P Q R S T1
2 Data General
3 x t y w f A B C Date 01.08.01 19:07
4 13 0 0.047 1 0.047 1 0 0 Data Bayes!rData
5 13 2 0.553 1 0.56 0.125 0.448 0.4266 Model Bayes!ABCbayes
6 13 4 0.412 1 0.403 0.016 0.209 0.775 ParametersBayes!rParam
7 13 6 0.304 1 0.308 0.002 0.079 0.9194 Bayes Bayes!rBayes
8 13 8 0.275 1 0.27 2E-04 0.028 0.972 Precision1E-11
9 13 10 0.253 1 0.257 3E-05 0.01 0.9904 Convergence0.00110 Error typeRelative11 Bayesian Information Significance0.05
12 Name Value Matrix Exclude Confidence0.95
13 k1 1.07 265.3 146.5 0 0 0 PredictionLinearization
14 k2 0.554 146.5 1117 0 0 0
15 0 0 0 0 0 0 Parameters estimation
16 0 0 0 0 0 0 Name Initial Final Deviation
17 0 0 0 0 0 0 k1 1.06969 1.03907 0.060509
18 k2 0.55366 0.53661 0.027954
19 p -3.05769 -3.05769 0.019371
20 q -0.002 -0.002 0.039994
21 r -1.38757 -1.38757 0.017332
22
23 Parameters Search Progress
24 k1 1.03907 Objective value 0.0139
25 k2 0.53661 Completeness 100%
26 p -3.05769 Objective change -1E-07
27 q -0.002 Iteration 2
28 r -1.38757
29
30
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
10
11
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
= +sp
ec
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 >> 1Large
dimension
Short OLS(hard)
K+MS 10
Smallprecision
WCR(hard&soft)
K 10Matrix
degradation
GRAM(soft)
K+MA 100
Just onemodel
)(ln
stk
kt
e
e
s
1k
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
0C0CBkdt
dC
0B0BBkAkdt
dB
1A0AAkdt
dA
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
con
cen
trat
ion
s
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 50
conventional wavelengths
sp
ec
tra
l sig
na
lt=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 50
conventional wavelengths
sp
ec
tra
l sig
na
lt=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 50
conventional wavelengths
sp
ec
tra
l sig
na
l
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 50
conventional wavelengths
sp
ec
tra
l sig
na
lt=0
t=2
t=4
t=6
t=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 50
conventional wavelengths
sp
ec
tra
l sig
na
lt=0
t=2
t=4
t=6
t=8t=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 50
conventional wavelengths
sp
ec
tra
l sig
na
l Spectralview 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 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
time
sp
ec
tra
l sig
na
l
Kinetic viewof 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
29
8
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 1
conventional 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 22
conventional 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
sp
ec
tra
l sig
na
l
-0.2
0
0.2
0.4
0.6
ac
cu
rac
y
Spectrum A
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
sp
ec
tra
l sig
na
l
-0.2
0
0.2
0.4
0.6
ac
cu
rac
y
Pure Spectra EstimatingPure Spectra Estimating
Spectrum B
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
sp
ec
tra
l sig
na
l
-0.2
0
0.2
0.4
0.6
ac
cu
rac
y
Spectrum C
0
0.2
0.4
0.6
0.8
1
1.2
1 11 21 31 41 51conventional wavelength
sp
ec
tra
l sig
na
l
-0.2
0
0.2
0.4
0.6
ac
cu
rac
ySBE givesgood 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: 541-560
Epoxidation of 2,5-di-tert-butyl-1,4-benzoquinone
SW-NIR spectra
-8
-6
-4
-2
0
2
4
6
8
860 865 870 875 880
wavelength
sp
ec
tra
l sig
na
l
240 spectra
1200 time points
21 wavelengths
Preprocessing:
Savitzky-Golay filter
-8
-6
-4
-2
0
2
4
6
8
860 865 870 875 880
wavelength
sp
ec
tra
l sig
na
l
PreprocessedData
23.02.03 32
Progress in SBE EstimatesProgress in SBE Estimates
k 1
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880
wavelength (nm)
k 2
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880
wavelength (nm)
k 2
k 1
0.0
0.1
0.2
0.3
0.4
860 862 864 866 868 870 872 874 876 878 880
wavelength (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
5
2
-5
-4
-3
-2
-1
0
1
2
3
4
460 470 480 490 500 510Temperature, K
DS
C s
ign
al
To testantioxidants!
23.02.03 43
Network Density of Shrinkable PE by TMANetwork Density of Shrinkable PE by TMA
4
2
1
3
5
1.1
1.2
1.3
1.4
1.5
0 10 20 30 40 50 60 70 80 90
Time, min
Elo
ng
ati
on
L/L
o
AB
C
D 2D 1 D 30
2
4
6
8
10
12
14
0 5 10 15 20 25Dose, MRad
Ch
em
ica
l mo
du
lus
, gm
m2
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
Co
nc
en
tra
tio
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 50
Time t , min
Ma
ss
ch
am
ge
, y
370
410
450
490
Te
mp
era
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 CCritical
level
26
0
1
2
3
4
5
6
0 20 40 60Time, hr
Elo
ng
ati
on
@ b
rea
k
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
Te
ns
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!