Download Random Numbers - Chabot College

Engr/Math/Physics 25
Chp7
Statistics-2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Engineering/Math/Physics 25: Computational Methods
1
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Learning Goals
 Create HISTOGRAM Plots
 Use MATLAB to solve Problems in
• Statistics
• Probability
 Use Monte Carlo (random) Methods
to Simulate Random processes
 Properly Apply Interpolation to
Estimate values between or outside
of know data points
Engineering/Math/Physics 25: Computational Methods
2
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random Numbers (RNs)
 There is no such thing as a ‘‘random number”
• is 53 a random number? (need a Sequence)
 Definition: a SEQUENCE of statistically
INDEPENDENT numbers with a Defined
DISTRIBUTION (usually uniform)
• Numbers are obtained by chance
• They have nothing to do with the
other numbers in the sequence
 Uniform distribution → each possible
number is equally probable
Engineering/Math/Physics 25: Computational Methods
3
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random Number Generator
 von Neumann (ca. 1946) Developed the
Middle Square Method
 take the square of the previous number
and extract the middle digits
 example: four-digit numbers
•
•
•
•
ri = 8269
ri+1 = 3763 (ri2 = 68376361)
ri+2 = 1601 (ri+12 = 14160169)
ri+3 = 6320 (ri+22 = 2563201)
Engineering/Math/Physics 25: Computational Methods
4
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
PSUEDO-Random Number
 Most Computer Based Random Number
Generators are Actually PSUEDO-Random
in implementation
 Note that for the von Nueman Method
• Each number is COMPLETELY determined
by its predecessor
• The sequence is NOT random but appears to be
so statistically → pseudo-random numbers
 All random number generators based on an
arithmetic operation have their own built-in
characteristics
• MATLAB uses a 35 Element “seed”
Engineering/Math/Physics 25: Computational Methods
5
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random Number Commands
Command
Rand
rand(n)
rand(m,n)
s = rand(’state’)
rand(’state’,s)
rand(’state’,0)
rand(’state’,j)
rand(’state’,sum(100*clock))
Engineering/Math/Physics 25: Computational Methods
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Description
Generates a single uniformly
distributed random number between 0
and 1.
Generates an nX?
n matrix containing
uniformly distributed random numbers
between 0 and 1.
Generates an mX?
n matrix containing
uniformly distributed random numbers
between 0 and 1.
Returns a 35-element vector s
containing the current state of the
uniformly distributed generator.
Sets the state of the uniformly
distributed generator to s.
Resets the uniformly distributed
generator to its initial state.
Resets the uniformly distributed
generator to state j, for integer j.
Resets the uniformly distributed
generator to a different state each time
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Some (psuedo)Random No.s
0.30253 0.35572
0.8678 0.065315 0.98548 0.62339
0.85184 0.049047 0.37218
0.2343 0.017363 0.68589
0.75948 0.75534 0.07369
0.9331 0.81939 0.67735
0.94976 0.89481 0.19984 0.063128 0.62114 0.87683
0.55794 0.28615 0.049493 0.26422 0.56022 0.012891
0.014233
0.2512 0.56671 0.99953 0.24403
0.3104
0.59618 0.93274 0.12192 0.21199 0.82201 0.77908
0.81621 0.13098 0.52211 0.49841 0.26321
0.3073
0.97709 0.94082 0.11706 0.29049 0.75363 0.92668
0.22191 0.70185 0.76992 0.67275 0.65964 0.67872
0.70368 0.84768 0.37506 0.95799 0.21406 0.074321
0.52206 0.20927 0.82339 0.76655 0.60212 0.070669
0.9329 0.45509 0.046636 0.66612 0.60494 0.01193
0.71335 0.081074 0.59791 0.13094
0.6595 0.22715
0.22804 0.85112 0.94915 0.095413 0.18336 0.51625
0.44964 0.56205
0.2888 0.014864 0.63655
0.4582
0.1722
0.3193 0.88883 0.28819 0.17031
0.7032
0.96882
0.3749 0.10159 0.81673
0.5396 0.58248
0.50921 0.76267
0.07429
0.7218
0.19324 0.65164
0.3796 0.75402
0.27643 0.66316
0.77088 0.88349
0.31393 0.27216
0.63819 0.41943
0.98657 0.21299
0.50288
0.0356
0.9477 0.081164
0.82803 0.85057
0.91756
0.3402
0.11308 0.46615
0.81213 0.91376
0.90826 0.22858
0.15638 0.86204
0.12212 0.65662
MATLAB Command → RandTab2 = rand(18,8);
Engineering/Math/Physics 25: Computational Methods
7
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random No. Simulation
 Started During WWII for the
purpose of Developing
Inexpensive methods for
testing engineered systems
by IMITATING their Real
Behavior
 These Methods are Usually
called MONTE CARLO
Simulation Techniques
Engineering/Math/Physics 25: Computational Methods
8
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Simulation (1)
 The Basis for These Methods
• Develop a Computer-Based Analytical
Model, or Equation/Algorithm, that
(hopefully) Predicts System Behavior
• The Model is then Evaluated Many Times
to Produce a STATISTICAL PROBABILITY
for the System Behavior
• Each Evaluation (or Simulation) Cycle is
based on Randomly-Set Values for
System Input/Operating Parameters
Engineering/Math/Physics 25: Computational Methods
9
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo (2)
• Analytical Tools are Used to ensure that
the Random assignment of Input
Parameter Values meet the Desired
Probability Distribution Function
 The Result of MANY Random Trials
Yields a Statistically Valid Set of
Predictions
• Then Use standard Stat Tools to Analyze
Result to Pick the “Best” Overall Value
– e.g.: Mean, Median, Mode, Max, Min, etc.
Engineering/Math/Physics 25: Computational Methods
10
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Process Steps
1. Define the System
2. Generate (psuedo)Random No.s
3. Generate Random VARIABLES
•
Usually Involves SCALING and/or
OFFSETTING the RNs
4. Evaluate the Model N-Times; each
time using Different Random Vars
5. Statistical Analysis of the N-trial
Results to assess Validity & Values
Engineering/Math/Physics 25: Computational Methods
11
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo System
 The System Definition Should Include
• Boundaries (Barriers that don’t change)
• Input Parameters
• Output (Behavior) Parameters
• Processes (Architecture) that Relate the
Input Parameters to the Output Parameters
Engineering/Math/Physics 25: Computational Methods
12
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Fixed Model Architecture
 The Model is
assumed to be
Unvarying; i.e., it
behaves as a Math
FUNCTION
 Example: SPICE
• SPICE ≡ Simulation
Program with
Integrated Circuit
Emphasis
 SPICE has Monte
Carlo BUILT-IN
Engineering/Math/Physics 25: Computational Methods
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 SPICE uses
• UNchanging Physical
Laws  KVL & KCL
• IDEAL Circuit
Elements  I/V
Sources, R, C, L
Component
VALUES for
R, L, C, Vs,
and Q can
Vary
Randomly
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Summarized

Monte Carlo Method: Probabilistic
simulation technique used when a
process has a random component
1. Identify a
Probability Distribution
Function (PDF)
2. Setup intervals of
random numbers to
match probability distribution
3. Obtain the random numbers
4. Interpret the results
Engineering/Math/Physics 25: Computational Methods
14
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
MATLAB RANDOM No. PDFs
 MATLAB rand
command produces
RNs with a Uniform
Distribution
 MATLAB randn, by
Contrast, produces
a NORMAL
Distribution
• i.e., ANY Value over
[0,1] just as likely as
Any OTHER
• i.e., The MIDDLE
Value is MORE
Likely than any other
Engineering/Math/Physics 25: Computational Methods
15
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Scaling rand
 rand covers the
interval [0,1] – To
cover [a,b] SCALE &
OFFSET the
Random No.
 Example: Use rand
to Produce
Uniformly Dist
Random No over
[19,37]
• Let x be a random
>> y =(37-19)*rand + 19
No. over [0,1], then a
• Example Result
random number y
>> y =(37-19)*rand + 19
over [a,b]
y  b  a x  a
Engineering/Math/Physics 25: Computational Methods
16
y =
36.1023
>> y =(37-19)*rand + 19
y =
23.1605
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
rand vs randn – scaled and offset
 rand
 randn
rand
randn
140
350
120
300
100
250
80
200
60
150
40
100
20
50
0
0
10
20
30
40
50
60
70
80
90
RN100 = 100*rand(10000,1);
hist(RN100,100),
title('rand')
Engineering/Math/Physics 25: Computational Methods
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100
0
-300
-200
-100
0
100
200
300
400
500
Norm100 = 100*randn(10000,1)
+ 100
hist(Norm100,100),
title('randn')
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Scaling randn
 randn Produces a
Normal Dist. with
µ = 0, and σ = 1
• Let v be a normal
random No. with µ=0
& σ=1, then a
random number w
with
µ=p&σ=r
w  rv  p
Engineering/Math/Physics 25: Computational Methods
18
 Example: Use
randn to Produce
Normal Dist with
µ = –17 & σ = 2.3
>> w =(2.3)*randn - 17
• Example Result
>> w =(2.3)*randn - 17
w =
-20.8308
>> w =(2.3)*randn - 17
w =
-16.7117
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (1)
 Build a Wharehouse from PreCast
Concrete (a Tilt-Up) Per PERT Chart
A
2
B
E
1. Project Start
4
C
3
F
5
G
6
H
1. Project End
7
D
 PERT  Program Evaluation and
Review Technique
• A Scheduling Tool Developed for
the USA Space Program
Engineering/Math/Physics 25: Computational Methods
19
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (2)
A
2
B
E
1. Project Start
4
C
3
F
5
G
6
H
7
1. Project End
D
 In This Case The Schedule Elements
A. Excavate Foundation
B. Construct Foundation
C. Fabricate PreCast
Components
D. Ship PreCast Parts
to Building Site
Engineering/Math/Physics 25: Computational Methods
20
E. Install PreCast Parts
on Foundation
F. Build Roof
G. Finish Interior
and Exterior
H. Inspect Result
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (3)
 Task Durations → Normal Random Variables
• Assume Normally Distributed
Task
ID
Task Description
Mean Duration Std Dev
(days)
(days)
A
Foundation Excavation
3.5
1
B
Pour Foundation
2.5
0.5
C
Fab PreCast Elements
5
1
D
Ship PreCast Parts
0.5
0.5
E
Tilt-Up PreCast Parts
5
1.5
F
Roofing
2
1
G
Finish Work
4
1
Expected Duration = 17 Days
Engineering/Math/Physics 25: Computational Methods
21
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (4)
 Analytical Model
• Foundation-Work and PreCasting
Done in PARALLEL
– One will be The GATING Item before Tilt-Up
• Other Tasks Sequential
 Mathematical Model
tbld  max A  B, C  D E  F  G
Early GATE
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (5)
Task-A Task-B Task-C Task-D Task-E Task-F Task-G Task-Sum
4.82
2.47
6.32
0.61
2.86
1.85
3.75
15.74
1.77
2.29
4.39
0.86
5.51
2.88
4.14
17.78
3.35
2.46
5.29
1.08
6.21
0.64
4.06
17.29
3.28
2.79
4.70
1.07
4.73
0.53
4.70
16.03
3.94
2.78
4.31
0.92
1.64
3.10
4.00
15.46
1.89
1.89
5.21
0.61
3.49
1.55
4.70
15.56
3.04
2.52
5.80
0.95
5.21
0.38
3.18
15.51
3.93
2.13
5.56
-0.19
5.69
2.63
4.19
18.57
2.49
2.33
5.44
-0.30
3.95
0.92
4.50
14.52
5.23
2.85
4.25
0.61
4.66
1.15
4.17
18.07
3.61
1.93
4.32
0.36
5.98
0.75
3.70
15.98
3.02
2.99
6.76
1.21
6.37
2.33
4.03
20.71
2.00
2.29
4.98
0.61
3.49
1.34
4.28
14.70
2.31
2.11
5.27
1.27
3.42
2.63
3.60
16.19
3.19
2.20
6.26
0.93
1.84
1.64
4.28
14.95
1.94
2.40
4.57
0.75
3.69
2.08
3.74
14.83
2.09
2.31
4.54
0.35
4.55
0.41
4.55
14.40
4.82
2.44
4.26
0.61
4.40
2.06
3.12
16.85
5.19
2.66
5.72
1.30
1.90
1.26
4.09
15.10
4.03
2.22
5.30
-0.11
4.72
1.70
4.48
17.14
Engineering/Math/Physics 25: Computational Methods
23
 Run-1
• µ = 16.27
Days
• σ = 1.61
Days
 See some
Negative
Durations!
• May want
to Adjust
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (6)
Task-A Task-B Task-C Task-D Task-E Task-F Task-G Task SUM
15.13
4.56
1.86
3.67
0.77
3.02
2.26
2.79
14.10
4.08
-0.79
3.18
0.99
6.63
1.59
4.98
15.28
3.95
1.34
5.11
0.00
3.46
2.76
2.12
17.85
4.73
-1.10
5.84
0.63
7.77
2.76
3.47
18.16
4.46
2.49
6.85
0.05
4.26
2.43
1.94
15.16
3.56
1.21
4.18
0.28
3.37
3.33
2.87
20.51
3.76
3.51
5.12
0.50
6.02
2.73
5.40
15.00
4.57
0.44
4.32
0.46
4.49
2.26
3.40
15.54
4.14
1.63
2.79
0.69
6.29
2.13
3.73
17.76
3.92
2.45
4.73
-0.23
5.34
2.21
4.45
17.84
3.89
2.03
3.41
1.30
7.21
1.80
3.44
16.08
4.18
2.43
2.87
0.11
5.49
2.69
3.92
20.98
4.75
2.88
6.65
-0.14
5.75
3.27
3.42
16.67
3.84
1.25
5.56
1.09
4.39
2.70
3.32
18.59
4.38
2.89
4.73
0.76
4.44
2.85
3.75
16.11
4.63
0.79
3.92
-0.07
4.64
2.62
4.16
14.36
3.41
0.61
3.08
0.87
6.39
2.43
2.52
17.76
3.78
1.49
6.38
0.00
4.06
2.51
3.61
20.45
4.60
1.98
7.21
0.00
3.24
2.49
4.16
16.56
4.53
3.00
3.47
0.85
4.27
1.40
4.16
Engineering/Math/Physics 25: Computational Methods
24
 Run-2
• µ = 16.99
Days
• σ = 2.05
Days
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (7)
% Program Demo_Monte_Carlo_WhareHouse
% Normal Dist Task Duration on PERT Chart
% Bruce Mayer, PE ENGR25 27Sep05
%
% Use 20 Random No.s for Simulation
% Set 20-Val Row-Vectors for Task Durations
%
for k = 1:20;
tA(k) = 1*randn + 3.5;
tB(k) = 0.5*randn + 2.5;
tC(k) = 1*randn + 5;
tD(k) = 0.5*randn + 0.5;
tE(k) = 1.5*randn + 5;
tF(k) = 1*randn + 2;
tG(k) = 0.5*randn + 4;
end
%
% Calc Simulated Durations per Model
for k = 1:20;
tSUM(k) = max((tA(k)+tB(k)),(tC(k)+tD(k)))+tE(k)+tF(k)+tG(k);
end
%
% Put into Table for Display Purposes
%
t_tbl =[tA',tB',tC',tD',tE',tF',tG',tSUM']
%
tmu = mean(tSUM)
 The MATLAB
Script File
Engineering/Math/Physics 25: Computational Methods
25
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Monte Carlo Example (8)
 Just for Fun Try 1000 Random
Simulation Cycles
A
2
B
E
1. Project Start
4
C
3
F
5
G
6
H
7
1. Project End
D
 µ1000 = 17.3730 days
• Expected 17
 σ1000 = 2.1603 days
• Expected 2.345 by RMS calc
Engineering/Math/Physics 25: Computational Methods
26
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear Interpolation (1)
 During a Hardness Testing Lab in
ENGR45 we measure the HRB at 67.3
on a ½” Round Specimen
 The Rockwell Tester was Designed for
FLAT specimens, so the Instruction
manual includes a TABLE for ADDING
an amount to the Round-Specimen
Measurement to Obtain the
CORRECTED Value
Engineering/Math/Physics 25: Computational Methods
27
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear Interpolation (2)
 From the Rockwell Tester Manual
67.3
• To Apply LINEAR interpolation Need to
Find Only the Data Surrounding:
– The Independent (Measured) Variable
– The Corresponding Dependent Variable Values
Engineering/Math/Physics 25: Computational Methods
28
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear Interpolation (3)
 Then the Linear Interpolation Eqn
yint  ylo xact  xlo

yhi  ylo
xhi  xlo
 Where
• xact  actual
MEASURED value
• yint  Unknown
INTERPOLATED value
• xlo  TABULATED
Value Just Below xact
• xhi  TABULATED
Value Just Above xact
• ylo  TABULATED Value
Corresponding to xlo
• yhi  TABULATED Value
Corresponding to xhi
Engineering/Math/Physics 25: Computational Methods
29
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear InTerp PorPortionality
yint  ylo xact  xlo

yhi  ylo
xhi  xlo
 i.e.;
yint−ylo is to yhi−ylo
AS
xact−xlo is to xhi−xlo
Engineering/Math/Physics 25: Computational Methods
30
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear Interpolation Example
 From the Rockwell Tester Manual
67.3
 The
Interp
Eqn
xlo
ylo
xhi
yhi
yint  3.5 67.3  60

 yint  3.135
3.0  3.5
70  60
Engineering/Math/Physics 25: Computational Methods
31
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Linear Interp With MATLAB
 Use the interp1
Command to find yint
>> Xtab = [60, 70];
% = [xlo, xhi]
>> Ytab = [3.5, 3.0];
% = [ylo, yhi]
>> yint =
interp1(Xtab, Ytab,
67.3)
yint =
3.1350
Engineering/Math/Physics 25: Computational Methods
32
 interp2 Does
Linear Interp in 2D
zint =
interp2(x,y,z,xint,yint)
Used to linearly interpolate a
function of two variables:
z  f (x, y). Returns a linearly
interpolated vector zint at
the specified values xint
and yint, using (tabular)
data stored in x, y, and z.
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Interpolation vs Extrapolation
 Class Q: Who can Explain the
DIFFERENCE?
 INTERpolation Estimates Data Values
between KNOWN Discrete Data Points
• Usually Pretty Good Estimate as we are
within the Data “Envelope”
 EXTRApolation PROJECTS Beyond the
Known Data to Predict Additional Values
• Much MORE Uncertainty in Est. value
Engineering/Math/Physics 25: Computational Methods
33
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
INterp vs. Extrap Graphically
Extrapolation
Known Data ENDS
Interpolation
Engineering/Math/Physics 25: Computational Methods
34
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Cubic Spline Interpolation
 If the Data exhibits
significant
CURVATURE,
MATLAB can
Interpolate with
Curves as well using
the spline form
Linear
Spline Curve
yint = spline(x,y,xint)
Computes a cubic-spline interpolation where x and y are vectors
containing the data and xint is a vector containing the values of the
independent variable x at which we wish to estimate the dependent
variable y. The result yint is a vector the same size as xint
containing the interpolated values of y that correspond to xint
Engineering/Math/Physics 25: Computational Methods
35
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
All Done for Today
Consider
the
Source
Engineering/Math/Physics 25: Computational Methods
36
 Most Engineering Data
is NOT Sufficiently
ACCURATE nand/nor
PRECISE to Justify
Anything But LINEAR
Interpolation
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Engr/Math/Physics 25
Appendix
f x   2 x  7 x  9 x  6
3
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Engineering/Math/Physics 25: Computational Methods
37
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random No. Table
Engineering/Math/Physics 25: Computational Methods
38
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt