Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
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Transcript of Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt1
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Chp7Statistics-
2
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt2
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt3
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt4
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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 (ri
2 = 68376361) • ri+2 = 1601 (ri+1
2 = 14160169)• ri+3 = 6320 (ri+2
2 = 2563201)
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt5
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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”
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt6
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Random Number CommandsCommand Description
Rand Generates a single uniformly distributed random number between 0 and 1.
rand(n) Generates an nX?n matrix containing uniformly distributed random numbers between 0 and 1.
rand(m,n) Generates an mX?n matrix containing uniformly distributed random numbers between 0 and 1.
s = rand(’state’) Returns a 35-element vector s containing the current state of the uniformly distributed generator.
rand(’state’,s) Sets the state of the uniformly distributed generator to s.
rand(’state’,0) Resets the uniformly distributed generator to its initial state.
rand(’state’,j) Resets the uniformly distributed generator to state j, for integer j.
rand(’state’,sum(100*clock)) Resets the uniformly distributed generator to a different state each time
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt7
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Some (psuedo)Random No.s0.30253 0.35572 0.8678 0.065315 0.98548 0.62339 0.50921 0.762670.85184 0.049047 0.37218 0.2343 0.017363 0.68589 0.07429 0.72180.75948 0.75534 0.07369 0.9331 0.81939 0.67735 0.19324 0.651640.94976 0.89481 0.19984 0.063128 0.62114 0.87683 0.3796 0.754020.55794 0.28615 0.049493 0.26422 0.56022 0.012891 0.27643 0.66316
0.014233 0.2512 0.56671 0.99953 0.24403 0.3104 0.77088 0.883490.59618 0.93274 0.12192 0.21199 0.82201 0.77908 0.31393 0.272160.81621 0.13098 0.52211 0.49841 0.26321 0.3073 0.63819 0.419430.97709 0.94082 0.11706 0.29049 0.75363 0.92668 0.98657 0.212990.22191 0.70185 0.76992 0.67275 0.65964 0.67872 0.50288 0.03560.70368 0.84768 0.37506 0.95799 0.21406 0.074321 0.9477 0.0811640.52206 0.20927 0.82339 0.76655 0.60212 0.070669 0.82803 0.85057
0.9329 0.45509 0.046636 0.66612 0.60494 0.01193 0.91756 0.34020.71335 0.081074 0.59791 0.13094 0.6595 0.22715 0.11308 0.466150.22804 0.85112 0.94915 0.095413 0.18336 0.51625 0.81213 0.913760.44964 0.56205 0.2888 0.014864 0.63655 0.4582 0.90826 0.22858
0.1722 0.3193 0.88883 0.28819 0.17031 0.7032 0.15638 0.862040.96882 0.3749 0.10159 0.81673 0.5396 0.58248 0.12212 0.65662
MATLAB Command → RandTab2 = rand(18,8);
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt8
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt9
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt10
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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.
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt11
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Process Steps1. Define the System2. Generate (psuedo)Random No.s3. 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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt12
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt13
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt14
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt15
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
MATLAB RANDOM No. PDFs MATLAB rand
command produces RNs with a Uniform Distribution• i.e., ANY Value
over [0,1] just as likely as Any OTHER
MATLAB randn, by Contrast, produces a NORMAL Distribution• i.e., The MIDDLE
Value is MORE Likely than any other
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt16
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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]
axaby
>> y =(37-19)*rand + 19
Example: Use rand to Produce Uniformly Dist Random No over [19,37]
>> y =(37-19)*rand + 19y = 36.1023>> y =(37-19)*rand + 19y = 23.1605
• Example Result
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt17
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Scaled & Offset Random No.s
33.0445 28.8462 30.5977 24.5998 20.5393 19.6793 19.5497 20.073126.0153 24.3338 25.8150 35.6208 23.7247 34.9330 32.3933 31.275523.3504 32.4045 33.6084 26.7437 33.4183 35.4392 28.0004 19.763826.2704 22.4012 28.5909 22.3267 19.5260 33.3313 27.6386 20.286020.7362 31.3620 25.3131 35.2879 35.7194 20.7768 35.2850 28.389721.3755 22.3032 35.9020 36.6355 32.1460 23.7137 29.9776 20.741135.9569 25.6327 34.7670 26.8997 27.7950 25.0364 30.1180 33.726736.2104 30.2611 28.9028 21.0001 29.4135 31.2351 34.4700 33.715829.3538 33.0441 30.2046 23.6452 23.2711 21.4580 33.4988 32.003920.0760 20.4603 29.5668 26.3570 27.2593 31.9821 29.3810 21.697623.2260 35.7289 22.7394 29.7081 36.3356 20.9217 22.2926 30.872925.3569 32.9628 24.4224 23.7198 28.8425 30.7676 23.3188 28.334733.7815 27.7622 27.4766 29.8512 28.3804 27.8951 34.9572 36.513519.2773 26.8455 23.1488 31.8019 23.1687 33.0229 19.5161 30.681819.7744 27.0421 34.1976 22.9914 27.8002 31.8707 27.8182 33.406022.0418 24.5143 22.5058 21.1135 30.2331 35.2670 22.0227 27.168430.6841 28.1532 23.0666 24.3402 31.2244 35.0366 36.6163 26.783032.1710 28.1939 22.0727 24.7380 26.1193 25.0149 31.8285 33.855630.6594 33.7173 23.0980 26.6350 25.6139 31.5774 28.0085 20.502527.1166 33.3070 26.8426 28.1414 36.7837 22.5606 27.4796 21.3971
rand1937 = (37-19)*rand(20,8) + 19
>> Rmax =max(max(rand1937))Rmax = 36.7837
>> Rmin = min(min(rand1937))Rmin = 19.2773
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt18
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
prvw
>> w =(2.3)*randn - 17
Example: Use randn to Produce Normal Dist with µ = –17 & σ = 2.3
>> w =(2.3)*randn - 17w = -20.8308>> w =(2.3)*randn - 17w = -16.7117
• Example Result
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt19
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
rand vs randn – scaled and offset rand
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
120
140rand
RN100 = 100*rand(10000,1);hist(RN100,100), title('rand')
randn
Norm100 = 100*randn(10000,1) + 100hist(Norm100,100), title('randn')
-300 -200 -100 0 100 200 300 400 5000
50
100
150
200
250
300
350randn
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt20
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (1) Build a Wharehouse from PreCast
Concrete (a Tilt-Up) Per PERT Chart
1. Project Start
2
3
4 5 6 7 1. Project End
A B
C D
E F G H
PERT Program Evaluation and Review Technique• A Scheduling Tool Developed for
the USA Space Program
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt21
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (2)
1. Project Start
2
3
4 5 6 7 1. Project End
A B
C D
E F G H
In This Case The Schedule ElementsE. Install PreCast Parts
on FoundationF. Build RoofG. Finish Interior
and ExteriorH. Inspect Result
A. Excavate FoundationB. Construct FoundationC. Fabricate PreCast
ComponentsD. Ship PreCast Parts
to Building Site
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt22
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (3) Task Durations → Normal Random Variables
• Assume Normally DistributedTask
ID Task Description Mean Duration(days)
Std Dev(days)
A Foundation Excavation 3.5 1B Pour Foundation 2.5 0.5C Fab PreCast Elements 5 1D Ship PreCast Parts 0.5 0.5E Tilt-Up PreCast Parts 5 1.5F Roofing 2 1G Finish Work 4 1
Expected Duration = 17 Days
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt23
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
GFEDCBAtbld ,max
Early GATE
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt24
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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 15.741.77 2.29 4.39 0.86 5.51 2.88 4.14 17.783.35 2.46 5.29 1.08 6.21 0.64 4.06 17.293.28 2.79 4.70 1.07 4.73 0.53 4.70 16.033.94 2.78 4.31 0.92 1.64 3.10 4.00 15.461.89 1.89 5.21 0.61 3.49 1.55 4.70 15.563.04 2.52 5.80 0.95 5.21 0.38 3.18 15.513.93 2.13 5.56 -0.19 5.69 2.63 4.19 18.572.49 2.33 5.44 -0.30 3.95 0.92 4.50 14.525.23 2.85 4.25 0.61 4.66 1.15 4.17 18.073.61 1.93 4.32 0.36 5.98 0.75 3.70 15.983.02 2.99 6.76 1.21 6.37 2.33 4.03 20.712.00 2.29 4.98 0.61 3.49 1.34 4.28 14.702.31 2.11 5.27 1.27 3.42 2.63 3.60 16.193.19 2.20 6.26 0.93 1.84 1.64 4.28 14.951.94 2.40 4.57 0.75 3.69 2.08 3.74 14.832.09 2.31 4.54 0.35 4.55 0.41 4.55 14.404.82 2.44 4.26 0.61 4.40 2.06 3.12 16.855.19 2.66 5.72 1.30 1.90 1.26 4.09 15.104.03 2.22 5.30 -0.11 4.72 1.70 4.48 17.14
Run-1• µ = 16.27
Days• σ = 1.61
Days
See some Negative Durations!• May want
to Adjust
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt25
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (6) Run-2
• µ = 16.99 Days
• σ = 2.05Days
Task-A Task-B Task-C Task-D Task-E Task-F Task-G Task SUM2.79 2.26 3.02 0.77 3.67 1.86 4.56 15.134.98 1.59 6.63 0.99 3.18 -0.79 4.08 14.102.12 2.76 3.46 0.00 5.11 1.34 3.95 15.283.47 2.76 7.77 0.63 5.84 -1.10 4.73 17.851.94 2.43 4.26 0.05 6.85 2.49 4.46 18.162.87 3.33 3.37 0.28 4.18 1.21 3.56 15.165.40 2.73 6.02 0.50 5.12 3.51 3.76 20.513.40 2.26 4.49 0.46 4.32 0.44 4.57 15.003.73 2.13 6.29 0.69 2.79 1.63 4.14 15.544.45 2.21 5.34 -0.23 4.73 2.45 3.92 17.763.44 1.80 7.21 1.30 3.41 2.03 3.89 17.843.92 2.69 5.49 0.11 2.87 2.43 4.18 16.083.42 3.27 5.75 -0.14 6.65 2.88 4.75 20.983.32 2.70 4.39 1.09 5.56 1.25 3.84 16.673.75 2.85 4.44 0.76 4.73 2.89 4.38 18.594.16 2.62 4.64 -0.07 3.92 0.79 4.63 16.112.52 2.43 6.39 0.87 3.08 0.61 3.41 14.363.61 2.51 4.06 0.00 6.38 1.49 3.78 17.764.16 2.49 3.24 0.00 7.21 1.98 4.60 20.454.16 1.40 4.27 0.85 3.47 3.00 4.53 16.56
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt26
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (7) The
MATLAB Script File
% Bruce Mayer, PE • ENGR25 • 25Oct11% Normal Dist Task Duration on PERT Chart% file = Monte_Carlo_Wharehouse.m
% % 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 Modelfor 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)
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt27
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Monte Carlo Example (8) Just for Fun Try 1000 Random
Simulation Cycles
µ1000 = 17.3730 days• Expected 17
σ1000 = 2.1603 days• Expected 2.1794 by RMS calc
1. Project Start
2
3
4 5 6 7 1. Project End
A B
C D
E F G H
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Linear Interpolation (3) Then the Linear Interpolation Eqn
lohi
loact
lohi
loint
xxxx
yyyy
A Proportionality, Where• xact actual
MEASURED value• xlo TABULATED
Value Just Below xact
• xhi TABULATED Value Just Above xact
• yint Unknown INTERPOLATED value
• ylo TABULATED Value Corresponding to xlo
• yhi TABULATED Value Corresponding to xhi
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Linear InTerp PorPortionality
lohi
loact
lohi
loint
xxxx
yyyy
i.e.; yint−ylo is to yhi−ylo
AS xact−xlo is to xhi−xlo
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
InTerp Pt-Slope Line Eqn It’s LINEAR as the Interp
Eqn can be cast into the familiar Point-Slope Eqn
ReWorking the Interp Equation
11 xxmyy
loactlohi
lohiloint
lohi
loact
lohi
loint xxxxyyyy
xxxx
yyyy
loactxlointlohi
lohix xxmyy
xxyym
actact
Let
The LOCAL slope evaluated about xact
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Linear Interpolation Example From the Rockwell Tester Manual
67.3
xlo
xhi yhi
ylo
The InterpEqn
135.36070603.67
5.30.35.3
intint yy
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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.
zint = interp2(x,y,z,xint,yint)
interp2 Does Linear Interp in 2D
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
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
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
INterp vs. Extrap Graphically
Interpolation
Extrapolation
Known Data ENDS
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Cubic Spline Interpolation If the Data
exhibits significant CURVATURE, MATLAB can Interpolate with Curves as well using the spline form
Linear
Spline Curveyint = 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
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[email protected] • ENGR-25_Lec-20_Statistics-2.ppt38
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
All Done for Today
Considerthe
Source Most Engineering Data is NOT Sufficiently ACCURATE nand/nor PRECISE to Justify Anything But LINEAR Interpolation
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Appendix 6972 23 xxxxf
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Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Random No. Table