Download ENGR-25_Lec-19_Statistics-2_Simulation

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
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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 (often uniform; often not)
• Numbers are obtained completely 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
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Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random Number Generator
 John 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
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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
algorithmic operation have their own built-in
characteristics
• MATLAB uses a 35 Element “seed”
Engineering/Math/Physics 25: Computational Methods
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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.
n matrix containing
Generates an nX?
uniformly distributed random numbers
between 0 and 1.
n matrix containing
Generates an mX?
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
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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
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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, Plot, etc.
Engineering/Math/Physics 25: Computational Methods
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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 Random Numbers
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
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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
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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 (UCB)
 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
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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
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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.
• Let x be a random
No. over [0,1], then
a random number
y over [a,b]
y  b  a x  a
Engineering/Math/Physics 25: Computational Methods
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 Example: Use rand
to Produce Uniformly
Distributed Random
Numbers over the
Range [19,37]
>> y =(37-19)*rand + 19
• Example Result
>> y =(37-19)*rand + 19
y =
36.1023
>> y =(37-19)*rand + 19
y =
23.1605
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Scaled & Offset Random No.s
33.0445
26.0153
23.3504
26.2704
20.7362
21.3755
35.9569
36.2104
29.3538
20.0760
23.2260
25.3569
33.7815
19.2773
19.7744
22.0418
30.6841
32.1710
30.6594
27.1166
28.8462
24.3338
32.4045
22.4012
31.3620
22.3032
25.6327
30.2611
33.0441
20.4603
35.7289
32.9628
27.7622
26.8455
27.0421
24.5143
28.1532
28.1939
33.7173
33.3070
30.5977
25.8150
33.6084
28.5909
25.3131
35.9020
34.7670
28.9028
30.2046
29.5668
22.7394
24.4224
27.4766
23.1488
34.1976
22.5058
23.0666
22.0727
23.0980
26.8426
24.5998
35.6208
26.7437
22.3267
35.2879
36.6355
26.8997
21.0001
23.6452
26.3570
29.7081
23.7198
29.8512
31.8019
22.9914
21.1135
24.3402
24.7380
26.6350
28.1414
20.5393
23.7247
33.4183
19.5260
35.7194
32.1460
27.7950
29.4135
23.2711
27.2593
36.3356
28.8425
28.3804
23.1687
27.8002
30.2331
31.2244
26.1193
25.6139
36.7837
19.6793
34.9330
35.4392
33.3313
20.7768
23.7137
25.0364
31.2351
21.4580
31.9821
20.9217
30.7676
27.8951
33.0229
31.8707
35.2670
35.0366
25.0149
31.5774
22.5606
Engineering/Math/Physics 25: Computational Methods
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19.5497
32.3933
28.0004
27.6386
35.2850
29.9776
30.1180
34.4700
33.4988
29.3810
22.2926
23.3188
34.9572
19.5161
27.8182
22.0227
36.6163
31.8285
28.0085
27.4796
20.0731
31.2755
19.7638
20.2860
28.3897
20.7411
33.7267
33.7158
32.0039
21.6976
30.8729
28.3347
36.5135
30.6818
33.4060
27.1684
26.7830
33.8556
20.5025
21.3971
rand1937 = (3719)*rand(20,8) + 19
>> Rmax
=max(max(rand1937))
Rmax =
36.7837
>> Rmin =
min(min(rand1937))
Rmin =
19.2773
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, and σ = r
w  rv  p
Width
Offset
Engineering/Math/Physics 25: Computational Methods
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 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
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
100
RN100 = 100*rand(10000,1);
hist(RN100,100),
title('rand'),grid
Engineering/Math/Physics 25: Computational Methods
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0
-300
-200
-100
0
100
200
300
400
500
Norm100 = 100*randn(10000,1)
+ 100
hist(Norm100,100),
title('randn'), grid
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
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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
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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
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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
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 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
25
 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)
% Bruce Mayer, PE • ENGR25 • 25Oct11 • 13Apr16 • 31Oct16
% Normal Dist Task Duration on PERT Chart
% file = Monte_Carlo_Wharehouse.m
 The MATLAB
Script File
%
% 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;
Monte_Carlo_Wharehouse_Vectorized.m
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)
 See also
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 (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 2.1794 by RMS calc
Engineering/Math/Physics 25: Computational Methods
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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
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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
29
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
 A Proportionality, 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
30
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
31
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
InTerp  Pt-Slope Line Eqn
 It’s LINEAR as the Interp
Eqn can be cast into the y  y1  mx  x1 
familiar Point-Slope Eqn
 ReWorking the Interp Equation
 yhi  ylo 
yint  ylo xact  xlo

 yint  ylo  
xact  xlo 
yhi  ylo
xhi  xlo
 xhi  xlo 
 yhi  ylo 
Let mxact  
  yint  ylo  mxact xact  xlo 
 xhi  xlo 
The LOCAL slope evaluated about xact
Engineering/Math/Physics 25: Computational Methods
32
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
33
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
34
 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 a Pretty Good Estimate as we are
within the Data “Envelope”
 EXTRApolation PROJECTS Beyond the
Known Data to Predict Additional Values
• Much MORE Uncertainty in Estimated value
Engineering/Math/Physics 25: Computational Methods
35
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
36
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
37
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
38
 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
39
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt
Random No. Table
Engineering/Math/Physics 25: Computational Methods
40
Bruce Mayer, PE
[email protected] • ENGR-25_Lec-20_Statistics-2.ppt