the autosimoa project
DESCRIPTION
AUTOMATING D.E.S OUTPUT ANALYSIS:. The AutoSimOA Project. HOW MANY REPLICATIONS TO RUN. Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School WSC 07. A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation. http://www.wbs.ac.uk/go/autosimoa. Objective - PowerPoint PPT PresentationTRANSCRIPT
The AutoSimOA Project
Katy Hoad, Stewart Robinson, Ruth DaviesWarwick Business School
WSC 07
A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation.
http://www.wbs.ac.uk/go/autosimoa
Objective
To provide an easy to use method, that can be incorporated into existing simulation software, that enables practitioners to
obtain results of a specified accuracy from their discrete event simulation model.
(Only looking at analysis of a single scenario)
OUTLINE
IntroductionMethods in literatureOur AlgorithmTest Methodology & ResultsDiscussion & Summary
Underlying Assumptions
Any warm-up problems already dealt with.
Run length (m) decided upon.
Modeller decided to use multiple replications to obtain better estimate of mean
performance.
N
jjXN
X1
1Response measure
of interest
summary statistic from each replication
Perform N replications
QUESTION IS…
How many replications are needed?
Limiting factors: computing time and expense.
4 main methods found in the literature for choosing the number of replications N to perform.
1. Rule of Thumb (Law & McComas 1990)
Run at least 3 to 5 replications.
Advantage: Very simple.
Disadvantage: Does not use characteristics of model output.
No measured precision level.
2. Simple Graphical Method (Robinson 2004)
Cumulative mean graph
45
47
49
51
53
55
1 9 17 25 33 41 49 57 65 73 81 89 97 105
Number of replications (n)
Cum
ula
tive m
ean
Advantages: Simple Uses output of interest in decision.
Disadvantages: Subjective No measured precision level.
3. Confidence Interval Method (Robinson 2004, Law 2007, Banks et al. 2005).
Advantages: Uses statistical inference to determine N.
Uses output of interest in decision.
Provides specified precision.
Disadvantage: Many simulation users do not have the skills to apply approach.
Cumulative mean graph
46
48
50
52
54
56
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106
Number of replications (n)
Cum
ula
tive m
ean
4. Prediction Formula (Banks et al. 2005)
• Decide size of error ε that can be can tolerated.• Run ≥ 2 replications - estimate variance s2.• Solve to predict N.
• Check desired precision achieved – if not recalculate N with new estimate of variance.
Advantages: Uses statistical inference to determine N. Uses output of interest in decision. Provides specified precision.
Disadvantage: Can be very inaccurate especially for small number of replications.
2
1,2
st
NN
Run
Model START:
Load Input
Produce Output Results
Run Replication Algorithm
Precision criteria met?
Recommend replication number
Run one more
replication
YES
NO
AUTOMATE Confidence Interval Method: Algorithm interacts with simulation model sequentially.
2,1 nt
n
nn
nX
nt
d
s2,1
100
is the student t value for n-1 df and a significance of 1-α,
nX
sn is the estimate of the standard deviation,
calculated using results Xi (i = 1 to n) of the n current replications.
Where
n is the current number of replications carried out,
We define the precision, dn, as the ½ width of the Confidence Interval expressed as a percentage of the cumulative mean:
is the cumulative mean,
ALGORITHM DEFINITIONS
Stopping Criteria
• Simplest method:
Stop when dn 1st found to be ≤ desired precision, drequired . Recommend that number of replications, Nsol, to user.
• Problem: Data series could prematurely converge, by chance, to incorrect estimate of the mean, with precision drequired , then diverge again.
• ‘Look-ahead’ procedure: When dn 1st found to be ≤ drequired, algorithm performs set number of extra replications, to check that precision remains ≤ drequired.
0
20
40
60
80
100
120
140
3 100
137
174
211
248
285
322
359
396
433
470
replication number (n )
f(kLim
it)
kLimit=0 kLimit=5
kLimit=10 kLimit=25
‘Look-ahead’ procedure
kLimit = ‘look ahead’ value. Actual number of replications checked ahead is
Relates ‘look ahead’ period length with current value of n.
100,100
100,)(
nkLimitn
nkLimitkLimitf
23
25
27
29
31
33
35
37
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Replication number (n)
NsolNsol + f(kLimit)
f(kLimit)
Precision ≤ 5%X
X
95% confidence limits
Cumulative mean,
Replication Algorithm
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Replication number (n)
Precision
≤ 5%
Precision
> 5%
Precision ≤ 5%
f(kLimit)
Nsol2Nsol2 + f(kLimit)
Nsol1
• 24 artificial data sets: Left skewed, symmetric, right skewed; Varying values of relative st.dev (st.dev/mean).
• 100 sequences of 2000 data values.
• 8 real models selected.
• Different lengths of ‘look ahead’ period tested:
kLimit values = 0 (i.e. no ‘look ahead’), 5, 10, 25.
• drequired value kept constant at 5%.
TESTING METHODOLOGY
5 performance measures
1. Coverage of the true mean2. Bias3. Absolute Bias4. Average Nsol value5. Comparison of 4. with Theoretical Nsol
value
• For real models: ‘true’ mean & variance values - estimated from whole sets of output data (3000 to 11000 data points).
Microsoft Excel Worksheet
Results
• Nsol values for individual algorithm runs are very variable.
• Average Nsol values for 100 runs per model close to the theoretical values of Nsol.
• Normality assumption appears robust.
• Using a ‘look ahead’ period improves performance of the algorithm.
Mean bias significantly different to zero
Failed in coverage of true mean
Mean est. Nsol significantly different to theoretical Nsol (>3)
No ‘look-ahead’ period
Proportion of Artificial models
4/24 2/24 9/18
Proportion of Real models
1/8 1/8 3/5
kLimit = 5 Proportion of Artificial models
1/24 0 1/18
Proportion of Real models
0 0 0
% decrease in absolute mean bias
kLimit = 0 tokLimit = 5
kLimit = 5 tokLimit = 10
kLimit = 10 tokLimit = 25
ArtificialModels
8.76% 0.07% 0.26%
RealModels
10.45% 0.14% 0.33%
Impact of different look ahead periods on performance of algorithm
Number of times the Nsol value changes (out of 100 runs of the algorithm per model) because of the lengthening of the ‘look ahead’ period.
Model ID
kLimit = 0 to kLimit = 5
kLimit = 5 tokLimit = 10
kLimit = 10 to kLimit = 25
R1 0 0 0
R3 2 0 0
R5 24 0 1
R8 24 4 1
A5 30 1 3
A6 26 6 3
A15 1 0 0
A17 22 0 1
A21 25 2 1
A24 37 0 0
Model ID
kLimit Nsol Theoretical Nsol (approx)
Mean estimate significantly different to the true mean?
A9 0 4 112 Yes
5 120 No
A24 0 3 755 Yes
5 718 No
R7 0 3 10 Yes
5 8 No
R4 0 3 6 Yes
5 7 No
R8 0 3 45 Yes
5 46 No
Examples of changes in Nsol & improvement in estimate of true mean
DISCUSSION
• kLimit default value set to 5.
• Initial number of replications set to 3.
• Multiple response variables - Algorithm run with each response - use maximum estimated value for Nsol.
• Different scenarios - advisable to repeat algorithm every few scenarios to check that precision has not degraded significantly.
• Implementation into Simul8 simulation package.
SUMMARY
• Selection and automation of Confidence Interval Method for estimating the number of replications to be run in a simulation.
• Algorithm created with ‘look ahead’ period -efficient and performs well on wide selection of artificial and real model output.
• ‘Black box’ - fully automated and does not require user intervention.
ACKNOWLEDGMENTSThis work is part of the Automating Simulation Output
Analysis (AutoSimOA) project (http://www.wbs.ac.uk/go/autosimoa) that is funded by
the UK Engineering and Physical Sciences Research Council (EP/D033640/1). The work is being carried out in
collaboration with SIMUL8 Corporation, who are also providing sponsorship for the project.
Katy Hoad, Stewart Robinson, Ruth DaviesWarwick Business School
WSC 07