ENGR-25_Lec-20_Statistics-2.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engr/Math/Physics 25 Chp7 Statistics-2

ENGR-25_Lec-20_Statistics-2.ppt 2 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

ENGR-25_Lec-20_Statistics-2.ppt 3 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

ENGR-25_Lec-20_Statistics-2.ppt 4 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 r i = 8269 r i+1 = 3763 (r i 2 = ) r i+2 = 1601 (r i+1 2 = ) r i+3 = 6320 (r i+2 2 = )

ENGR-25_Lec-20_Statistics-2.ppt 5 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”

ENGR-25_Lec-20_Statistics-2.ppt 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Random Number Commands

ENGR-25_Lec-20_Statistics-2.ppt 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Some (psuedo)Random No.s MATLAB Command → RandTab2 = rand(18,8);

ENGR-25_Lec-20_Statistics-2.ppt 8 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

ENGR-25_Lec-20_Statistics-2.ppt 9 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

ENGR-25_Lec-20_Statistics-2.ppt 10 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.

ENGR-25_Lec-20_Statistics-2.ppt 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-20_Statistics-2.ppt 12 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

ENGR-25_Lec-20_Statistics-2.ppt 13 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

ENGR-25_Lec-20_Statistics-2.ppt 14 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

ENGR-25_Lec-20_Statistics-2.ppt 15 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

ENGR-25_Lec-20_Statistics-2.ppt 16 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] >> y =(37-19)*rand + 19  Example: Use rand to Produce Uniformly Dist Random No over [19,37] >> y =(37-19)*rand + 19 y = >> y =(37-19)*rand + 19 y = Example Result

ENGR-25_Lec-20_Statistics-2.ppt 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Scaled & Offset Random No.s rand1937 = (37- 19)*rand(20,8) + 19 >> Rmax =max(max(rand1937)) Rmax = >> Rmin = min(min(rand1937)) Rmin =

ENGR-25_Lec-20_Statistics-2.ppt 18 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 >> w =(2.3)*randn - 17  Example: Use randn to Produce Normal Dist with µ = –17 & σ = 2.3 >> w =(2.3)*randn - 17 w = >> w =(2.3)*randn - 17 w = Example Result

ENGR-25_Lec-20_Statistics-2.ppt 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods rand vs randn – scaled and offset  rand RN100 = 100*rand(10000,1); hist(RN100,100), title('rand')  randn Norm100 = 100*randn(10000,1) hist(Norm100,100), title('randn')

ENGR-25_Lec-20_Statistics-2.ppt 20 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 Project End A B C D EFGH  PERT  Program Evaluation and Review Technique A Scheduling Tool Developed for the USA Space Program

ENGR-25_Lec-20_Statistics-2.ppt 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Monte Carlo Example (2) 1. Project Start Project End A B C D EFGH  In This Case The Schedule Elements E.Install PreCast Parts on Foundation F.Build Roof G.Finish Interior and Exterior H.Inspect Result A.Excavate Foundation B.Construct Foundation C.Fabricate PreCast Components D.Ship PreCast Parts to Building Site

ENGR-25_Lec-20_Statistics-2.ppt 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Monte Carlo Example (3)  Task Durations → Normal Random Variables Assume Normally Distributed Task ID Task Description Mean Duration (days) Std Dev (days) A Foundation Excavation B Pour Foundation C Fab PreCast Elements 5 1 D Ship PreCast Parts 0.5 E Tilt-Up PreCast Parts F Roofing 2 1 G Finish Work 4 1 Expected Duration = 17 Days

ENGR-25_Lec-20_Statistics-2.ppt 23 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 Early GATE

ENGR-25_Lec-20_Statistics-2.ppt 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Monte Carlo Example (5)  Run-1 µ = Days σ = 1.61 Days  See some Negative Durations! May want to Adjust

ENGR-25_Lec-20_Statistics-2.ppt 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Monte Carlo Example (6)  Run-2 µ = Days σ = 2.05 Days

ENGR-25_Lec-20_Statistics-2.ppt 26 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 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)

ENGR-25_Lec-20_Statistics-2.ppt 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Monte Carlo Example (8)  Just for Fun Try 1000 Random Simulation Cycles  µ 1000 = days Expected 17  σ 1000 = days Expected by RMS calc 1. Project Start Project End A B C D EFGH

ENGR-25_Lec-20_Statistics-2.ppt 28 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

ENGR-25_Lec-20_Statistics-2.ppt 29 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

ENGR-25_Lec-20_Statistics-2.ppt 30 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Linear Interpolation (3)  Then the Linear Interpolation Eqn  A Proportionality, Where x act  actual MEASURED value x lo  TABULATED Value Just Below x act x hi  TABULATED Value Just Above x act y int  Unknown INTERPOLATED value y lo  TABULATED Value Corresponding to x lo y hi  TABULATED Value Corresponding to x hi

ENGR-25_Lec-20_Statistics-2.ppt 31 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Linear InTerp PorPortionality ii.e.; y int −y lo is to y hi −y lo AS x act −x lo is to x hi −x lo

ENGR-25_Lec-20_Statistics-2.ppt 32 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 The LOCAL slope evaluated about x act

ENGR-25_Lec-20_Statistics-2.ppt 33 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Linear Interpolation Example  From the Rockwell Tester Manual 67.3 x lo x hi y hi y lo TThe Interp Eqn

ENGR-25_Lec-20_Statistics-2.ppt 34 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Linear Interp With MATLAB  Use the interp1 Command to find y int >> Xtab = [60, 70]; % = [xlo, xhi] >> Ytab = [3.5, 3.0]; % = [ylo, yhi] >> yint = interp1(Xtab, Ytab, 67.3) 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. zint = interp2(x,y,z,xint,yint)  interp2 Does Linear Interp in 2D

ENGR-25_Lec-20_Statistics-2.ppt 35 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

ENGR-25_Lec-20_Statistics-2.ppt 36 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods INterp vs. Extrap Graphically Interpolation Extrapolation Known Data ENDS

ENGR-25_Lec-20_Statistics-2.ppt 37 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 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

ENGR-25_Lec-20_Statistics-2.ppt 38 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today Consider the Source  Most Engineering Data is NOT Sufficiently ACCURATE nand/nor PRECISE to Justify Anything But LINEAR Interpolation

ENGR-25_Lec-20_Statistics-2.ppt 39 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engr/Math/Physics 25 Appendix

ENGR-25_Lec-20_Statistics-2.ppt 40 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Random No. Table