Automating the Analysis of Simulation Output Data

Katy Hoad, Stewart Robinson, Ruth Davies

SSIG Meeting, 24th October 2007

http://www.wbs.ac.uk/go/autosimoa

The Problem

• Prevalence of simulation software: ‘easy-to-develop’ models and use by non-experts.

• Simulation software generally have very limited facilities for directing/advising user how to run the model to get accurate estimates of performance.

• With a lack of the necessary skills and support, it is highly likely that simulation users are using their models poorly.

3 Main Decisions:

• How long a warm-up is needed?

• How long a run length is needed?

• How many replications should be run?

Continuing theoretical developments

BUT little put into practical use.

Why?

• Limited testing of methods

• Requirement for detailed statistical knowledge

• Methods generally not implemented in simulation software (AutoMod/AutoStat is an exception)

A solution?

Provide an automated output ‘Analyser’.

An Automated Output AnalyserSimulation

model

Warm-upanalysis

Run-lengthanalysis

Replicationsanalysis

Use replicationsor long-run?

Recommendationpossible?

Recommend-ation

Output data

Analyser

Obt

ain

mor

e ou

tput

dat

a

Analyser advises user on:

• Warm-up length

• Run-length

• Number of replications

A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation.

The AutoSimOA Project

Main Objective:

•To propose a procedure for automated output analysis of warm-up, replications and run-length

Only looking at analysis of a single scenario

The AutoSimOA Project

WORK CARRIED OUT TO DATE:

1. Creation of a representative and sufficient set of models / data output for testing chosen simulation output analysis methods.

2. Development of an automated algorithm for estimating the number of replications to run.

3. Selection and testing of warm-up methods from the literature.

Part 1.

Creation of models and data sets

AIMS:

Provide a representative and sufficient set of models / data output for use in discrete event simulation research.

Use models / data sets to test the chosen simulation output analysis methods in the AutoSimOA Project.

Categorising Output Data Sets by Shape & Characteristics

Group A

…Group NGroup B

Auto Correlation

NormalityCycling/Seasonality

Terminating

Non-terminating

Steady state

In/out of control

Transient

Model characteristics

Deterministic or random

Significant pre-determined model changes (by time)

Dynamic internal changes i.e. ‘feed-back’

Empty-to-empty pattern

Initial transient (warm-up)

Out of control trend ρ≥1

Cycle

Auto-correlation

Statistical distribution

Output data characteristics

Modelling Warm-up Period:

Shapes of Initial Bias Functions

• Mean Shift:

• Linear:

• Quadratic:

• Exponential:

• Oscillating (decreasing):Quadratic ExponentialLinear

Artificial Data: Construct data which resembles real model output with

known values for some specific attribute. Example: Known steady state mean and variance.Example data: AR(1) with N(0,1) errors & linear initial bias.

Real Models: Collect range of models created in “real circumstances”. Examples: • Swimming Pool complex: average number in system• Production Line Manufacturing Plant: through-put / hour• Fast Food Store: average queuing time

Part 2.

WORK IN PROGRESS

Automating estimation of warm-up length

The Initial Bias Problem

• Model may not start in a “typical” state.• This may cause initial bias in the output.• Many methods proposed for dealing with

initial bias: e.g. Initial steady state conditions; run model for ‘long’ time…

• This project uses: Deletion of the initial transient data by specifying a warm-up period.

Question is:

How do you estimate the length of the warm-up period

required?

5 main types of methods:

1. Graphical Methods.

2. Heuristic Approaches.

3. Statistical Methods.

4. Initialisation Bias Tests.

5. Hybrid Methods.

Literature search – 42 methods

Summary of methods and

literature references on project

web site:

http://www.wbs.ac.uk/go/autosimoa

Currently testing methods

Part 3.

Automating analysis of number of replications

Introduction• Initial Setup:

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.

• Multiple replications performed by changing the random number streams used by the model and re-running the simulation.

N

jj

NNm

NN

m

m

XN

X

X

X

XXX

XXX

XXX

1

1

21

222

21

112

11

1

ˆ

ˆ

,,,

,,,

,,,

Output data from modelResponse measure of interest

= summary statistic from rep1

= summary statistic from repN

N replications

QUESTION IS…How many replications are

needed? • Limiting factors: computing time and

expense.

If performing N replications achieves a sufficient estimate of mean performance:> N replications: Unnecessary use of computer

time and money.< N replications: Inaccurate results → incorrect

decisions.

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

ulat

ive

mea

n

Confidence Interval Method

• User decides size of error they can tolerate.• Run increasing numbers of replications, • Construct Confidence Intervals around sequential

cumulative mean of output variable until desired precision achieved.

Advantages: Relies upon statistical inference to determine

number of replications required.

Allows the user to tailor accuracy of output results to their particular requirement

or purpose for that model and result.

Disadvantage: Many simulation users do not have the skills to apply such an approach.

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 , and recommend that number of replications, Nsol, to the 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.

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 created: Left skewed, symmetric, right skewed; Varying values of relative standard deviation (stdev/mean).

• Advantage: true mean and variance known.

• Artificial data set: 100 sequences of 2000 data values.

• 8 real models selected.

• Different lengths of ‘look ahead’ period looked at:

kLimit values = 0 (i.e. no ‘look ahead’ period), 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

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

Replication Work 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.

• Inclusion into SIMUL8 package: Full explanations of algorithm and results.

Summary Of Replications Work

• 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.

PROJECT OVERVIEW

• Created set of artificial and “real” model data including warm-up bias functions.

• Created replication algorithm.

Currently:

• Testing warm-up methods.

ACKNOWLEDGMENTSThis work is part of the Automating Simulation Output Analysis

(AutoSimOA) project that is funded by the UK (EPSRC) 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.

Stewart Robinson, Katy Hoad, Ruth Davies

SSIG Meeting, 24th October 2007

http://www.wbs.ac.uk/go/autosimoa