In estimating the number of failures using the truncated data for the Weibull model, we often encounter a case that the estimate is smaller than the true one when we use the likelihood principle to conditional probability. In infectious disease predictions, the SIR model described by simultaneous ordinary differential equations are often used, and this model can predict the final stage condition, i.e., the total number of infected patients, well, even if the number of observed data is small. These two models have the same condition for the observed data: truncated to the right. Thus, we have investigated whether the number of failures in the Weibull model can be estimated accurately using the ordinary differential equation. The positive results to this conjecture are shown.

1. Hideo Hirose
Department of Systems Design and Informatics, Kyushu Institute of Technology
Fukuoka, 820-8502 Japan
1
Parameter Estimation for the Truncated Weibull Model
Using the Ordinary Differential Equation
twitter@hirosehideo

2. 2
failure, pandemic, disaster
t
Prediction: failures, pandemic, disaster
trade-off between importance and accuracy
prediction
accuracy
prediction
importance
If the prediction
is done at early
stage, it is
valuable, but it
has less accuracy
Statistical truncated model
SIR model
Agent-based model
It is very important to predict the pandemic
as early as possible.
If the prediction
is done at late
stage, it is not so
valuable, but it
has accuracy
simulation by scenario
early detection
final analysis

3. 3
electric board failures
lot
time
#offailures
133
2997
failures
shipped
?
2997
problem:
133
60
90
120
150
30
0
cumulativenumberoffailures
133
60
90
120
150
30
0
estimated final value using the
trunsored model
Weibull distribution
147
133
173
likelihood ratio 95CI
cumulativenumberoffailures
3-dimensional
frequency dist.

4. 4
electric board failures
lot
time
#offailures
133
2997
failures
shipped
?
2997
problem:
133
60
90
120
150
30
0 Until July, 2000, 42 additional failures were observed;
175 failures were finally observed in total.
175
Even at almost final stage,
the estimated value is small.
cumulativenumberoffailures
133
60
90
120
150
30
0
estimated final value using the
trunsored model
Weibull distribution
147
133
173
likelihood ratio 95CI
cumulativenumberoffailures
objective: even in early stage,
we want to estimate accurate
final number of failures
doubt: Statistics can predict the
final stage at early stage.

5. observedcumulativeinfectedpopulation
0
500
1000
1500
2000
2500
3000
0 20 40 60 80 100
early
stage
5
Observed
in Hong Kong in 2003
1755
final value
days from 3/17/2003
3/17/2003 4/6 4/26 5/16 6/5 6/25 day
mid
stage
late
stage
observed cumulative
infected population
SARS
infected
popula1on

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感染者数
!""#$#$%&からの経過日数
$月%日
)月)日
6
SARS
infected
popula1on
logistic distribution
truncated grouped data
observed
truncated model
SARS spread prediction
in Hong Kong in 2003
predictedpopulationforinfectedpopulation
March 22
April 1
April 10
April 22
Using the data of 3/17 - 4/6, the truncated
model gives underestimated final result.
days from 3/17/2003
days from 3/17/2003
3/17/2003 4/6 4/26 5/16 6/5 6/25 day
early
stage
1755
final observed value
predicted
846
final estimated value

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(月#)日
(月#%日
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$月#%日
$月#*日
7
SARS
infected
popula1on
logistic distribution
truncated grouped data
using data
by the truncation timeobserved
truncated model
predicted
SARS spread prediction
in Hong Kong in 2003
predictedpopulationforinfectedpopulation
March 22
April 1
April 10
April 22
days from 3/17/2003
days from 3/17/2003
3/17/2003 4/6 4/26 5/16 6/5 6/25 day
early
stage
mid
stage
late
stage
Even in mid stage,
the number of
predicted patients are
small comparing to the
final observed value of
1755.
The truncated model
gives underestimated
final result except the
late stage prediction.
1755
final observed value

8. 10
20
30
40
50
60
70
80
観測値
days
observed
0
500
1000
1500
2000
2500
3000
0 20 40 60 80 100
系列1
系列2
系列3
系列4
系列5
系列6
系列7
系列8
系列9
observed
10
20
30
40
50
60
70
80
観測値
8
predicted
0
500
1000
1500
2000
2500
3000
0 20 40 60 80 100
observed
predictedpopulationforinfectedpopulation
1755
final observed value
days from 3/17/2003
3/17/2003 4/6 4/26 5/16 6/5 6/25 day
SARS
infected
popula1on
SIR model SARS spread prediction in Hong Kong in 2003
early
stage
mid
stage
late
stage
The SIR model does NOT provide
the underestimated final result
even in early stage estimation.
2219
final estimated value
For example, using data 3/17 - 4/6, early stage,
the SIR model predicts,
2219

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SEIR Differential Equation Model
9
infection rate
removal rate
transmission rate
SIRSEIR
SEIR
SEIR
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when parameters are given

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!'月!(日 !)月!#日 !*月!+日 $!月!+日 $$月!&日 $%月!&日 !$月!%日 !%月!$日 !&月!&日 !+月!%日 !#月!%日
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206142
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16532694
2月1日
27819991
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22483176
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32956724
6/26 - 7/15
A(H1N1) flu prediction in Japan 2009
observed
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numberofcumulativepatients
10
prediction
Using the data from June 26 to
July 15 2009, the SIR model
predicted the final number of
patients to be 32,000,000.
The observed final number of
patients was 21,000,000.
Taking into account of the
vaccination and the school
closing, the estimation is not so
bad.
32,000,000
21,000,000
!"
#!!"
$!!!"
$#!!"
%!!!"
%#!!"
&!" &#" '!" '#" #!" ##" (!" (#" )!"
final observed
final predicted
0.0E+00
5.0E+04
1.0E+05
1.5E+05
2.0E+05
2.5E+05
3.0E+05
3.5E+05
4.0E+05
4.5E+05
5.0E+05
0 50 100 150 200 250 300
5月1日
5月2日
5月3日
5月4日
5月5日
5月6日
5月7日
5月8日
5月9日
5月10日
5月11日
5月12日
5月13日
5月14日
5月15日
5月16日
5月17日
5月18日
5月19日
5月20日
5月21日
5月22日
5月23日
5月24日
5月25日
5月27日
5月29日
6月1日
6月3日
6月5日
6月8日
6月10日
6月12日
6月15日
6月17日
6月19日
6月22日
6月24日
6月26日
6月29日
7月1日
実測値
5月!日からの日数
除外者の人数（人）
truncated model
SIR model
using data
6/26 - 7/15
500,000
final predicted
using data
6/26 - 7/1
February AprilDecemberOctoberAugust
2009 2010
500,000

11. 11
Foot-and-mouth disease (FMD) spread prediction in Miyazaki Prefecture in 2010
FMD spread prediction in Japan 2010
using data
by the truncation
time
predicted
observed
cattle
to be killed
40,000
5/224/20 7/1 day6/10
vaccinated
truncated model
10,000
5,000
In earlier stages, the
computational final results
by the truncated model are
smaller than the observed
final data. It gives
underestimated predicted
values.
0
15000
30000
45000
60000
0 30 60 90 120 150days
SIR model
days
5/224/20 7/1 day6/10
cattle
to be killed
60,000
final observed value
predicted final value
20,000
predicted
observed
In earlier stages, the SIR
model gives some
overestimated final
results, but finally it gives
the reasonable
estimates.
Removal rate is set to 7 days.
The size of animals is 300,000, which
is the maximum size in Miyazaki
Prefecture.
40,000
final observed value
vaccinated predicted final value
60,000
when S0=65,000

12. 12
we only observe I(t)
I(t)
R(t)
S(t)
we assume
the duration
basic equations
we can assume R(t)
we can calculate the parameters
forward difference equation
removal rate
backward difference equation
SIR Parameter Estimation (assimilation)

13. 13
CumulativeNumber
2003/3/17
!"
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$#!!"
%!!!"
%#!!"
&!!!"
&#!!"
'!!!"
0 10 20 30 40 50 60 70 80
前進差分 後退差分 観測値
observed
forward difference equation
backward difference equation
backward difference equation
final value
!"
#!!"
$!!!"
$#!!"
%!!!"
%#!!"
&!!!"
&#!!"
!" $!" %!" &!" '!" #!" (!" )!" *!"
forward difference equation
final value
observed
removal rate = 1
SIR Parameter Estimation (assimilation)
SARS
final value
final value
using data

14. 14
we only observe I(t)
I(t)
R(t)
S(t)
we assume
the duration
basic equations
we can assume R(t)
we can calculate the parameters
removal rate
best backward solution
1. set parameters, λ0, γ0
2. solve the SIR differential equations backward
3. improve parameters so that S0 becomes smaller
4. if S0 is stationary, then stop, otherwise goto 1
proposal
simplex method
SIR Parameter Estimation (assimilation)
best backward solution

15. 2003/3/17 Days
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%!!!"
%#!!"
&!!!"
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0 10 20 30 40 50 60 70 80
(()法 観測値
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$!!!"
$#!!"
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%#!!"
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!" $!" %!" &!" '!" #!" (!" )!" *!"
15
CumulativeNumber
2003/3/17
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$#!!"
%!!!"
%#!!"
&!!!"
&#!!"
'!!!"
0 10 20 30 40 50 60 70 80
前進差分 後退差分 観測値
observed
forward difference equation
backward difference equation
backward difference equation
final value
!"
#!!"
$!!!"
$#!!"
%!!!"
%#!!"
&!!!"
&#!!"
!" $!" %!" &!" '!" #!" (!" )!" *!"
forward difference equation
final value
observed
best backward solution
best backward solution
removal rate = 1
Among the three methods,
the BBS shows the best performance.
SIR Parameter Estimation (assimilation)
best backward solution
final value
final value
using data
SARS

16. 16
days
from 21th day,
stably predicted
infected
observed
predicted
!""
#!!""
$!!!""
$#!!""
%!!!""
%#!!""
!" %!" &!" '!" (!" $!!"
days
from 35th day,
stably predicted
MLE
0
500
1000
1500
2000
2500
0 20 40 60 80 100
observed
predicted
infected
Overall, we can see that the best-backward solution method provide more accurate final
value at earlier stage than the truncated model does.
truncated model SIR model
Observed & Final Estimated Values (SARS)

17. 17
Simulation Study (mimic SARS)
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forward difference equation
backward difference equation
50 30
40 15
days
days
cumulativeinfectedcumulativeinfected
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days
cumulativeinfected
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!" $!" %!" &!" '!" #!" (!" )!" *!" +!" $!!"
days
cumulativeinfected
truncated model
SIR model
best backward solution

18. 18
Failure prediction using ODE
objective: even in early stage,
we want to estimate accurate
final number of failures
133
60
90
120
150
30
0
estimated final value using the
trunsored model
Weibull distribution
147
133
173
likelihood ratio 95CI
H. Hirose,
IEEE Trans.
Reliability, 2005
failure model: grouped right
truncated data in Weibull
distribution

19. 19
Failure prediction using ODE
objective: even in early stage,
we want to estimate accurate
final number of failures
failure model: grouped right
truncated data in Weibull
distribution
truncated model
differential equation
model
statistical

20. 20
Simulation Study (mimic electric board)
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133
60
90
120
150
30
0
175

21. 21
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3:584-30,"2;,06
,=>0:081-6"0?5-1;8"2;,06
When the number of data is large,
that is, the number of failures in a day is given by a
real number, not by a discrete number,
then, the truncated model and the differential equation
model provide accurate estimate for the final number of
failures even the truncation time is in early stage.
175
Simulation Study (mimic electric board)

22. 22
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%!!""
%#!""
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!" %!" '!" (!" )!" $!!"
*+,-
!
!
./012+.3*"45*36truncated model
differential equation model If we use the daily grouped data,
then, the differential equation
model provide rather accurate
estimated value for the final
number of failures even the
truncation time is in mid stage,
and the truncated model provide
underestimated value for the final
number of failures even the
truncation time is in mid stage
175
Simulation Study (mimic electric board)

23. 23
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4256 4256
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7263"8 7263")
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Other 9 simulation cases show the
similar results.
Using the differential equation method,
we can predict the final number of
failures earlier than the method of the
truncated model.
Simulation Study (mimic electric board)

24. 24
• In estimating the number of failures using the truncated data for the
Weibull model, we often observe that the estimate is smaller than the
true one when we use the likelihood principle to conditional
probability.
• To overcome this deficiency, we have proposed to use the differential
equation method, because we have experienced that the SIR model
described by simultaneous ordinary differential equations can predict
the final stage condition, i.e., the total number of infected patients,
well, even if the number of observed data is small, contrary to the
truncated model approach.
• We have investigated, in this paper, whether the number of failures in
the Weibull model can be estimated accurately using the ordinary
differential equation, and have found that the proposed method has
an ability to predict the final stages with rather small samples, i.e.,
with the data in earlier stages.
Conclusions

25. 25
• R. Anderson and R. May, Infectious diseases of humans: Dynamics and control,
Oxford University Press, 1991.
• H. Hirose, The trunsored model and its applications to lifetime analysis: unified
censored and truncated models, IEEE Transactions on Reliability, 54, pp.11-21 (2005) .
• H. Hirose, The mixed trunsored model with applications to SARS, Mathematics and
Computers in Simulation, 74, pp.443-453 (2007).
• H. Hirose, The mixed trunsored model with applications to SARS, Mathematics and
Computers in Simulation, 74, pp.443-453 (2007).
• H, Hirose, K. Matsukuma, T. Sakumura, Infectious disease spread prediction models
and consideration, IPSJ SIG Technical Report 2010-MPS-81, 15, pp.1-6 (2010).
• W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of
epidemics-III. Further studies of the problem of endemicity, Proceedings of the Royal
Society, 141A, pp.94-122 (1933).
• J.P. Klein and M.L. Moeschberger, Survival Analysis, New York: Springer, 1997.
references

26. 26
Parameter Estimation for the Truncated Weibull Model
Using the Ordinary Differential Equation
thank you