Stochastic Simulation The modelling process Bo Friis Nielsen - - PowerPoint PPT Presentation

Stochastic Simulation The modelling process

Bo Friis Nielsen

Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby – Denmark Email: bfni@dtu.dk

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Explanation: What is the problem with the Pareto distribution Explanation: What is the problem with the Pareto distribution

• Moment distributions
• For nonnegative valued random variables

Gj(x) = x

0 tjf(t)dt

∞

0 tjf(t)dt =

x

0 tjf(t)dt

E (Xj) The contribution to the j’th moment from values ≤ x. x t1f(t)dt = x

β

tk β t β −k−1 dt = x

β

k t β −k dt = β k k − 1 x

β

k − 1 β t β −k dt = βk k − 1

• 1 −

x β −k+1

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Explanation: What is the problem with the Pareto distribution Explanation: What is the problem with the Pareto distribution

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 Pareto distribution 1 - exp(-k*log(t)) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 Pareto distribution 1 - exp(-k*log(t)) 1 - exp(-(k-1)*log(t)) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 Pareto distribution 1 - exp(-k*log(t)) 1 - exp(-(k-1)*log(t)) 1 - exp(-(k-2)*log(t))

• The first moment distribution for the Pareto distribution (green)
• The second moment distribution for the Pareto distribution

(blue)

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Some numbers β = 1 Some numbers β = 1

F(t) = 1 − t−k f(t) = kt−k−1 G1(t) = 1 − t−k+1 G2(t) = 1 − t−k+2 For k = 2.05 t F(t) G1(t) G2(t) 2 0.7585 0.5170 0.0341 10 0.9911 0.9109 0.1190 100 0.9999 0.9921 0.2057 844.5 1 − 10−6 0.9992 0.2860

• Even when if we simulate 106 values we can not expect to get a

decent estimate of the variance!

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What to learn: What to learn:

• Care is needed when using simulation
• Especially if one wants to study strange or rare phenomena.
• Always use your practical, theoretical and intuitive understanding
• f the system to support the analysis by simulation.

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The modelling process The modelling process

• Problem identification
• Goal/purpose
• System analysis and data gathering
• System syntesis - model formulation
• Estimation of model parameters
• Preliminary model validation
• Program development
• Final validation
• Experimental design
• Statistical analysis

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Goal/purpose Goal/purpose

• Describe the objective of the problem
• e.g. to design an inventory control system with stable production

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Problem identification Problem identification

• Define which part of reality which is to be modelled
• A company producing one product with an ingoing and outgoing

inventory wants an adequate inventory control system

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System analysis and data gathering System analysis and data gathering

• Investigate the system identifying parts with direct impact on

the goal. Provide data on these parts.

• Demand, relations between order, production and inventory.

Insignificant: unemployment, weather.

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System syntesis System syntesis

• Define state variables. Describe dynamics and relations,

determine distributions.

• Example: Inventory at time t, determine the type and form of

the demand function.

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Estimation of model parameters Estimation of model parameters

• Determine parameter values, values for model

constants/parameters.

• Example: Estimate mean and variance of the demand function.

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Preliminary model evaluation Preliminary model evaluation

• Control of fundamental logical structures. Common sense

control of parameters

• Does the model imply inventory sizes exceeding system capacity

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Program development Program development

• Translate state definitions into data structures. Formalise logical

and physical relations. Prepare the program for debugging. Make a modular program which is easy to extend.

• Example: Inventory size is defined as an integer variable. Time is

define integer or continuous depending on the context.

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Final validation Final validation

• Run the program for input combination with known analytical
• solution. Run the program with extreme values of the
• parameters. Common sense control of output. Study animations.

Compare with real world data if possible (existing system).

• Example: Choose exponential distribution. Reduce/simplify

relations.

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Experimental design Experimental design

• Planning of sensitivity analyses.
• Realistic/interesting combinations of parameter values.
• Example: Periods of constant demand. Periods with highly

varying demand.

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Statistical analysis Statistical analysis

• Estimation of systemparameters. Confidence intervals. Variance

reduction techniques. Time series analysis.

• Example: Variance of number of orders.

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Verification and validation of simulation models

• L&K chapter 5

Verification and validation of simulation models

• L&K chapter 5
• Validation is the process of determining whether a simulation

model (as opposed to the computer program) is an accurate representation for the particular objectives in study.

• Verification is concerned with determining whether the

conceptual simulation model (model assumptions) has been correctly translated into a computer “program”.

• A simulation model and its results have Credibility if the

manager and other key project personnel accept them as “correct”.

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Validiation Validiation

• Conceptually, if a simulation model is “valid” then it can be

used to make decisions about the system similar to those that would be made if it were feasible and cost-effective to experiment with the system itself.

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Credibility Credibility

• The managers understanding and agreement with the models

assumptions.

• Demonstration that the model has been validated and verified.
• The managers ownership of and involvement with the project
• Reputation of the model developers

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What is simulation? What is simulation?

• Computer experiments with mathematical model
• General engineering technique
• Analytical/numerical solutions

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Course goal Course goal

• Topics related to scientific computer experimentation
• Specialised techniques

⋄ Variance reduction methods ⋄ Random number generation ⋄ Random variable generation ⋄ The event-by-event principle

• Simulation based statistical techniques

⋄ Markov chain Monte Carlo simulated annealing ⋄ Bootstrap

• Validition and verification of models
• Model building

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Project types Project types

• Model a system (e.g., like the ferry example) in order to assess

performance, under varying designs.

• Study a mathematical model, that is impossible or hard to

analyze

• Study any one of the techniques we have been through, more
• closely. For example, generating random numbers with the

Mersenne Twister.

• Come up with your own project type.

Simulation projects Simulation projects

• Post office (discrete event)
• Simulation and estimation in a Markov model of breast cancer

(discrete event)

• Simulation of queueing system with input generated by

interrupted Poisson processes (discrete event)

• Simulation of Levy processes (discrete event)
• Simulation of queues with Brownian motion input (discrete

event, can be developed to something more advanced)

• Microemulsions (MCMC)
• Your own suggestion to discuss

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General guidelines General guidelines

• Do not make model too complicated
• Clear objective
• Time to experiment with model
• Apply variance reduction techniques if possible

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Deliverable 2: Project report Deliverable 2: Project report

• Precision of objective
• Model validation
• Program verification
• Experimental design Standard report, the important issue

though is to have some time to experiment with your model.

• Deadline Thursday June 27th (Monday June 30th if you like)

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Registering Registering

• Register online. I have created an exercise for group hand in on

Campusnet.

• Register with a group name. Use the title/content of your

project as (part of) the group name

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Plan for the next two weeks Plan for the next two weeks

• Nicolai, Jakob and I will be available on a consultancy basis. The

availability will be communicated through Campusnet and/or the web page.

• I will be unavailable from noon monday (today) until noon

tuesday (tomorrow)

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And remember And remember

• Course evaluation (Campus net)
• Comments and suggestions, at any level of detail