2. Monte Carlo simulation
Special Topics
3.58
Monte Carlo simulation
Monte Carlo method

3. Two similar but different variations
Special Topics
3.59
Introduction
Concept
Inspired by ideas underlying gambling
Randomness, either in inputs or model, …
… exploited to explore range of possible outcomes
Two similar but different variations
Stochastic inputs, deterministic model, explicit time
Deterministic inputs, stochastic model, implicit time

4. Special Topics Monte Carlo simulation 3.60 Stochastically varying
initial conditions
Stochastically varying results
MC v1
Deterministic simulation
Probability distributions used to model variability in initial conditions
Often time-stepped physics-based model; time represented explicitly
Multiple runs with
run-to-run variability in results; analyzed statistically
Stochastically varying results
MC v2
Fixed initial conditions
Stochastic simulation
Specific known or given initial conditions
Probability distributions used to model variability in simuland processes; often no explicit representation of time
Multiple runs with
run-to-run variability in results; analyzed statistically v1 v2

5. Monte Carlo simulation (v1)
Special Topics
3.61
Definition
Monte Carlo simulation (v1)
Stochastically varying initial conditions input to deterministic physics or process model.
Modeling
Physics of simuland modeled using physics
Time modeled explicitly, as time-stepped continuous
Simulation
Randomly generated initial conditions …
… used to study range of possible results
Usually multiple trials and statistical analysis

6. Monte Carlo simulation (v2) Fixed initial conditions input
Special Topics
3.62
Definition
Monte Carlo simulation (v2)
Fixed initial conditions input
to stochastic physics or process model.
Modeling
Physics of simuland modeled using probability
Time modeled implicitly, no explicit time advance
Simulation
Randomly generated physics outcomes …
… used to study range of possible results
Usually multiple trials and statistical analysis

7. Monte Carlo v1 method (alternative)
Special Topics 3.63
Monte Carlo v1 method (alternative)
Defining a Monte Carlo model
Identify a set of random variables that specify the initial condition.
Select probability distribution and parameters for each.
Develop deterministic model to calculate results from a set of inputs.
Executing a Monte Carlo simulation
Repeat for each of n trials:
Randomly generate random variate for each input.
Calculate trial outcome with deterministic model.
Record trial outcome.
Statistically analyze the results.

8. Monte Carlo v2 method (alternative) Defining a Monte Carlo model
Special Topics
3.64
Monte Carlo v2 method (alternative)
Defining a Monte Carlo model
Identify a set of input variables that specify the initial condition.
Select specific values for each.
Develop stochastic model based on probability distributions to calculate results from a set of inputs.
Executing a Monte Carlo simulation
Repeat for each of n trials:
Initialize model with selected input values.
Calculate trial outcome with stochastic model.
Record trial outcome.
Statistically analyze the results.

9. Monte Carlo simulation
Special Topics
3.65
Monte Carlo simulation
Examples

10. Special Topics 3.66 Example 1 (MC v1): Calculating π
Overlay a unit circle arc over a square area.
Choose a random x and y, and place a dot there. Repeat this step several times.
Calculate π based upon this result (see next
slide) 1 1

11. Assessing simulation output Want to approximate “real” value of π
Special Topics
3.67
Assessing simulation output
Want to approximate “real” value of π 1 r 4 2
# dots inside circle 
# dots inside entire square r  r 4  
 Example: π/4 ≈ 340/500 1
Simulated π ≈ 2.72
Error of 12.1% ?
Can rerun with more samples to generate more accurate output 1

12. Special Topics Experimental results from implementation 3.68
Sample size
# inside circle
# inside square
Simulated value of π
Error 5 4
4.000
27.33% 10
1.600
49.07% 50 42
3.360
6.95% 44
3.520
12.05%
200
161
3.220
2.50%
500
387
3.096
1.45%
398
3.184
1.35%
1000
785
3.140
0.05%
795
3.100
1.32%
2000
1567
3.134
0.24%
1554
3.108
1.07%
Total
5741
7315
3.139
0.07%
Demo

13. earnings = unit price x unit sales – (variable costs + fixed costs) ?
Special Topics 3.69
Example 2 (MC v1): Product earnings
Want to predict product earning in future years
Historical data for last five years available
Can develop a probability distribution
Choose triangular distribution when given small amounts of data
Need minimum, maximum, and mode (most likely)
Use the following relation:
earnings = unit price x unit sales – (variable costs + fixed costs) ?

14. Determining minimum, likely, and maximum
Special Topics 3.70
Determining minimum, likely, and maximum
Historical data per year and “transformed” data:
Year 1
Year 2
Year 3
Year 4
Year 5
Unit price 50 52 55 57 65
Unit sales
2000
2200
2700
2500
2800
Variable cost
50,000
55,000
56,000
57,000
58,000
Fixed cost
10,000
12,000
15,000
16,000
17,000
Earnings
40,000
47,400
77,500
69,500
107,000
Min
Mode
Max
Unit price 50 55 70
Unit sales
2000
2440
3000
Variable cost
50,000
55,200
65,000
Fixed cost
10,000
14,000
20,000
Earnings
40,000
125,000
•Minimums
•Averages
•Best guesses

15. Special Topics Forming triangular distribution (b  a)(c  a) 0 3.71
Mode
Max
Unit price 50 55 70
Unit sales
2000
2440
3000
Variable cost
50,000
55,200
65,000
Fixed cost
10,000
14,000
20,000
Earnings
40,000
125,000
0.1
Probability 
2(x  a)
a  x  c
(b  a)(c  a) 
 2(b  x)
f (x)  
(b  a)(b  c)
c  x  b otherwise  0   50
(a) 55
(c) 60
Unit price 70
(b)

16. Special Topics 3.72 MATLAB code h = sqrt(rand(1,10000));
unit_price = (70-50)*h.*rand(1,10000)+55-(55-50)*h
variable_costs = ( )*h.*rand(1,10000) ( )*h;
fixed_costs = ( )*h.*rand(1,10000) ( )*h;
unit_sales = ( )*h.*rand(1,10000) ( )*h;
for i=1:10000
earnings(i) = unit_price(i)*unit_sales(i)- (variable_costs(i)+fixed_costs(i))
end

17. Earnings summary statistics
Special Topics
3.73
Earnings summary statistics
400
Parameter
Value
Mean
73,206
Median
71,215
Std Dev
16,523
Variance
273,009,529
Minimum
21,155
Maximum
137,930
300
Frequency
200
100 2 4 6 8
x104
Compare
Earnings
Min
Most likely
Max
Earnings
40,000
65,000
125,000

18. Example 3 (MC v1): Missile impacts [Zhang, 2008] Application
Special Topics
3.74
Example 3 (MC v1): Missile impacts [Zhang, 2008]
Application
Deterministic 6DOF model of missile trajectory
Used to calculate impact point given initial conditions
Measure x and y error w.r.t. aiming point
Compare model and live test x and y error variances
Two ranges: 60 Km and 100 Km
6 live tests, 800 Monte Carlo model trials each range
Monte Carlo analysis
For each trial, generate trajectory initial conditions from probability distributions
Calculate impact point
Repeat for 800 trials
Compare variances

19. Special Topics 3.75 Missile trajectory model Physics based
Organized into modules: velocity, rotation, atmospheric conditions, aerodynamics, thrust
Implemented in MATLAB Simulink
m dV  P cos  cos   X  mg sin  dt
mV  p(sin  cos  v  cos  sin  sin  v )  Y cos  v  Z sin  v  mg cos 
Velocity module equations d
mV cos  dv  P(sin  sin   cos  sin  sin  )  Y sin   Z cos 
v v v v dt
Velocity module block diagram

20. Special Topics Impact data 3.76 Trial x error s y error s n Model
526.62
85.91
800
Test
566.66
89.77 6
60 Km
Trial
x error s
y error s n
Model
921.39
111.25
800
Test
980.52
120.68 6
100 Km

21. Special Topics Monte Carlo simulation
3.77
Monte Carlo simulation
Case study: Bombing accuracy analysis (MC v2)

22. Special Topics 3.78 Bombing accuracy analysis
Bomber attack on ammunition depot
Conventional (unguided) bombs
Impacts randomly dispersed w.r.t aiming point, in both x (range) and y (azimuth) directions
Impact within ammunition dump perimeter is hit

23. Special Topics Ammunition dump, 1996 [Banks, 1996] 3.79 1 (-504, 198)y
2 (552, 18) –x x
950 m
Aiming point
(0, 0) –y
1250 m

24. Ammunition dump, 2010 [Banks, 2010]
Special Topics
3.80
Ammunition dump, 2010 [Banks, 2010]

25. Stochastic bomb impact model
Special Topics
3.81
Hits and misses
Configuration
Bomb aiming point is coordinate origin (0, 0)
Bombers approach from west (left)
Each bomber carries 10 bombs
Stochastic bomb impact model
Each impact assumed independent of others
Normally distributed around aiming point
Aiming point (0,0) is x mean, y mean x = 0, y = 0
x standard deviation x = 400
y standard deviation y = 200

26.  X   x x  y Normally distributed random variates
Special Topics
3.82
Normally distributed random variates
Random variate (rv)
Specific number from a given distribution
Randomly generated
Normal random variates
Normal distribution: mean μ and std dev σ
Standard normal distribution: mean 0 and std dev 1
Normal rv X from standard normal rv Z: X = μ + Zσ
Spreadsheet: =E4+E5*NORMSINV(RAND())
Given bomb impact coordinates X and Y normally distributed per μx, σx and μy, σy, equations for std normal r.v. Zx and Zy are
Given standard normal r.v. Zx and Zy, solve equations for X and Y
to get bomb impact coordinates
 X   x
X   y
Z  Z
X  400  Z Y  200  Z x y
x y
x  y
μx = 0, σx = 400
μy = 0, σy = 200

27. Procedure for each trial
Special Topics
3.83
Procedure for each trial
Generate standard normal random variates Zx, Zy
Calculate impact coordinates X, Y from Zx, Zy using model
Determine if impact point (X, Y) is within ammo dump
Trial Zx X Zy Y
Result 1
-0.84
-504
0.66
198
Miss 2
0.92
552
0.06 18
Hit 3
1.03
618
-0.13
-39 4
-1.82
-1,092
-1.40
-420 5
-0.16
-96
0.23 69 6
-1.78
-1,068
1.33
399 7
2.04
1,224
0.69
207 8
1.08
648
-1.10
-330 9
-1.50
-900
-0.72
-216 10
-0.42
-252
-0.60
-180

28. Simulated bombing runs [Banks, 2010]
Special Topics
3.84
Simulated bombing runs [Banks, 2010]

29. Special Topics Experiment results [Banks, 2010] 3.85
Sample 400 trial experiment results.
Each trial 10 bombs.
Frequency of number of bomb hits in a trial.

30. Special Topics Monte Carlo simulation Computing confidence intervals
3.86
Monte Carlo simulation
Computing confidence intervals
[Brase, 2009] [Petty, 2012] [Petty, 2013]

31. Example confidence interval
Special Topics
3.87
Example confidence interval
. . .
Sample of cereal boxes filled by machine
12.04
Sample size = 100 boxes
Mean weight of sample = ounces
Standard deviation of weight of sample = 0.1 ounces
Confidence interval for mean weight
▪ Interval [ , ]
Confidence level 0.95 (95%)