1 Exam 2 Spring 2016 STAT 305B (Take-Home) Due 3/29 (T In class) Name_____________(Updated 3/17) NOTE: All work (except code) associated with a given part MUST be placed DIRECTLY beneath that part. Problem 1 (25+5pts) In this problem you will analyze the uncertainty claims made by physics Professor Walter Lewin in https://www.youtube.com/watch?v=4a0FbQdH3dY in relation to the measured period of a swinging pendulum. [This topic begins @ ~ 2 minutes into the video.] By the term 'uncertainty' we will assume it is a 2 uncertainty. (a)(4pts) Let L=the act of measuring the length of the pendulum. Assume L ~ N ( L , L ) . (i)Give the numerical values for L and L . Then (ii) verify Professor Lewin's claim that the percent uncertainty is ~0.1%. Solution: (b)(4pts) To mathematically compute Solution: [See my code @ 1(b)] L and L is very difficult. Use 106 simulations to obtain their values. (c)(6pts) Professor Lewin claims that because the percent uncertainty in L is ~1%, the resulting uncertainty in calculating the pendulum period T (2 / g ) L is only ~0.5%. Use your answers in (a) to arrive at (i) T , (ii) T , and (iii) to verify his claim re: percent uncertainty. [Do not use simulations of T. You should be able to compute these directly.] Solution: (d)(6pts) Professor Lewin points out that the uncertainty in the calculation of T must be followed by a recognition that one must then measure T as the pendulum swings. Let Q denote the measurment error associated with the stop time after any chosen number of pendulum swings.. Assume this error is independent of T. Let W mT Q denote the act of measuring m periods of the pendulum swing prior to stopping. The period estimator is then: T W / m T Q / m . Arrive at the expression for T as a function of only m. [Note: Assume Q ~ N (0, Q ) . You should be able to identify the numerical value for Q .] Solution: (e)(5pts) Professor Lewin claims that by counting m 10 periods, the resulting uncertainty in relation to T is 0.02 sec. (i) Use your expression in (d) to show that his claim is incorrect, and then (ii) explain why he 'messed up'. Solution: (f)(+5pts) Use your simulations in (a) to compute simulations of T, and then run 106 simulations of Q in order to arrive at simulations of T . From these, compute T , and note whether it validates your answer in (e). Solution: [See my code @ 1(e)] 2 PROBLEM 2 (25pts) The performance of a wind turbine is tied directly to the velocity of the wind impinging on the blades at any given time. Uitilization of wind speed requires that it be measured at discrete times. Let V (V1 , , Vn ) denote the act of measuring wind speed at n successive time intervals seconds apart from one another. n n (a)(4pts) Let V (1 / n) Vk . Use the linearity of E(*) to show that E ( V ) (1 / n) Vk . Recall that this linearity k 1 k 1 includes the following two properties: (i) E (cX ) cE ( X ) , and (ii) E ( X Y ) E ( X ) E ( X ) E (Y ) . In each step of your solution, identify which of these properties is being applied. Solution: (b)(3pts) Assume that is sufficiently large, so that the elements of V can be assumed to be mutually uncorrelated. On n 2 2 Var ( V ) (1 / n) 2 Vk . p.185 of the book, give the equation that will allow you to then show that V k 1 Solution: NOTE: From this point on, we will assume that the elements of V are identically distributed. Hence, from (a) it should be clear that E ( V ) V . From (b) it should also be clear that, if the elements of V can be assumed to be mutually 2 independent, then Var ( V ) V / n . (c)(3pts) Suppose that at any index k, we have Vk ~ Weibull (a 31.7, b 9) . [A Google search will reveal that the Weibull pdf is the pdf of chice in relation to modeling wind.] These parameter values correspond to V 30 mph and V 4 mph . Suppose further, that the time between successive measurements is 60 sec . This sampling interval is sufficiently large that we can assume that the elements of V can be assumed to be mutually uncorrelated. Then for a 10 minute observation time we have a total of n 10 measurements. Use the above NOTE to compute the numerical values of the mean and standard deviation of the sample mean. Solution: 3 (d)(6pts) The pdf for any Vk is given shown below (left). Clearly, it does not have the symmetry of the normal pdf. The Central Limit Theorem (CLT) says that if the sample size, n, is 'sufficiently large', then no matter what the pdf might be for Vk , the pdf for V will have a pdf that is approximately normal. This issue here is that one might claim that n 10 is NOT AT ALL large enough for this assumption to hold. Use 105 simulations of V to arrive at (i) the simulation-based pdf overlaid against a normal pdf for V . Then (iii) use your results to determine whether you think the CLT is applicable. Figure 2(d) The given pdf for any Vk (LEFT), and your pdfs (normal and simulation-based) for V (RIGHT). Solution:[See code @2d used to arrive at the plot at RIGHT] Determination: (e)(4pts) Now, rather than using a 60-second sampling interval, let =5 seconds. Then, over a one minute observation time we will have a total of n 12 measurements. In this case we can no longer assume that Vk and Vk j will be j 2 uncorrelated for small values of j. Specifically, assume that Cov(Vk ,Vk j ) 0.6 V . Show that V1 and V12 are essentially uncorrelated by computing the correlation coefficient 1,12 . Solution: 1 0.6 0.6 2 0.63 0.611 1 0.6 (f)(5pts) In view of (e), the covariance matrix for V is given at right. 0.6 On p.185 of the book we have, as a special case of (5-26): 0.6 2 0.6 0.63 2 Cov(V, V ) V 12 12 12 3 0.62 0.6 Var Vk Cov (V j ,Vk ) . 1 1 1 0.6 k j k 11 Notice that this double sum is exactly the sum of all the elements 0.63 0.6 2 0.6 1 0.6 of the matrix Cov ( V, V ) . (i) Compute this double sum. Then (ii) use it to arrive at V . [HINT: Cov (V, V ) is a symmetric Toeplitz matrix, which can be easily formulated in Matlab using the 'toeplitz' command, if you want. To add up all its elements, simply use the command sum(sum( Cov (V, V ) )).] Solution: [See code @ 2(f)] 4 PROBLEM 3 (25pts) Future development of drones that are solar-powered will necessitate addressing the influence of Reynolds number ( Re ), which is a function of speed, on the drag Coefficient ( Cd ). In this problem you will investigate the basic problem of the influence of Re on Cd in relation to a sphere. Specifically, you will attempt to arrive at a linear model for the first 20 points of the data given in Figure 1 of the 2013 paper by Professor Faith Morrison: www.chem.mtu.edu/~fmorriso/DataCorrelationForSphereDrag2013.pdf. [For those with a keen interest, see also: http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html ] The code for this problem is given in Appendix 2; specifically, the code related to PROBLEM 3. I have written most of this code for you. The only code that you need to write is in relation to the linear model Y mX b , where, X log10 (Re) and Y log10 (Cd ) . As you will note, the data associated with ( X , Y ) is given as the 125x2 array X1Y. [NOTE: You must have the data file ReCd_FM.txt in your folder, so that the program can load it.] (a)(10pts) (i)Use the data X1Y to arrive at estimates of the model parameters m and b. Then (ii) Insert the two plots generated after this code HERE Solution: [Highlight your code beneath the line: INSERT YOU CODE HERE] m _____ ; b _____ Figure 3(a) Linear model for the log data (LEFT), and associated nonlinear model for the data (RIGHT). (b)(5pts) Give a zoomed plot of your figure at RIGHT in (a), and discuss how well you think it matches Morrison's model in ths region. Discussion: Figure 3(b) Zoomed plot of the plot at RIGHT in (a). (c)(5pts) Prove that the model for predicting Cd from Re is: Cd 10b Re m . Proof: (d)(5pts) (i) Use your answers in (a) to compute Cd 10b Re m . Then (ii) identify the single term in Morrison's model that your model most closely approximates. Solution: 5 PROBLEM 4 (25pts) In this problem you will attempt to arrive at a quadratic prediction model for Cd using the first 73 points in of the data given in Figure 1 of the paper by Morrison. The code for this problem is given in Appendix 2; specifically, it is the code related to PROBLEM 4. Again, I have written most of this code for you. The only code that you need to write is in relation to the quadratic model Y m1 X m2 X 2 b , where, X log10 (Re) and Y log10 (Cd ) . Let X X and X X 2 . Then prediction model can be expressed as the linear model: 1 2 X Y m1 X 1 m2 X 2 b m1 m2 1 b M X b . X2 In Section 5.4 of my Chapter 5 lecture notes it is shown that the optimal predictor has: M 1 XY and b Y M tr X . XX As you will note, the data associated with ( X 1 , X 2 , Y ) is given as the 759x3 array XY2. (4.1a) (4.1b) (a)(10pts) (i)Use the data XY2 to arrive at estimates of the model parameters M and b. Then (ii) Insert the two plots generated after this code HERE Solution: [Highlight your code beneath the line: INSERT YOU CODE HERE] M ; b Figure 4(a) Quadratic model for the log data (LEFT), and associated nonlinear model for the data (RIGHT). (b)(5pts) Prove that the model for predicting Cd from Re is: Cd 10b Re[ m1 m2 log10 (Re)] . Proof: (c)(5pts) (i) Use your answers in (a) to compute Cd 10b Re[ m1 m2 log10 (Re)] Then (ii) identify the two terms in Morrison's model, whose sum most closely resembles your model. Overlay plots of Morrison's full model, the partial model you identified, and your quadratic model. Solution: Figure 4(d) Full model, T1+T3 model, & quad. model (e)(5pts) Your quadratic model includes 3 parameters. Give the number of parameters in Morrison's partial model you identified in (d). Then discuss which of these two models would be preferred for predicting Cd in the Re range of interest. Answer/Discussion: 6 APPENDIX 1 Your Matlab code for PROBLEMS 1 & 2: %PROGRAM NAME: exam2.m %PROBLEM 1(b): %1(f): %========================= %PROBLEM 2: 2(d): %2(f): 7 APPENDIX 2 Matlab code for PROBLEMS 3 & 4: %PROGRAM NAME: ReCdModels.m %PROBLEM 3: load ReCd_FM.txt Re=ReCd_FM(:,1); Cd_data=ReCd_FM(:,2); %---------------------------------------%Contruction of Morrison's Model fo Cd: T1=24*Re.^-1; T2n=2.6*(Re/5); T2d=1+(Re/5).^1.52; T3n=0.411*(Re/263000).^-7.94; T3d=1+(Re/263000).^-8; T4=(Re.^0.8)/461000; muCd=T1+(T2n./T2d)+(T3n./T3d)+T4; figure(31) loglog(Re,muCd,'k','LineWidth',2) grid on xlabel('Reynolds Number') ylabel('C_d') title('Various Plots Related to C_d as a Function of Re') hold on pause %---------------------------------------%Convert Morrison's Model to log10 values LRe=log10(Re); LmuCd=log10(muCd); figure(32) plot(LRe,LmuCd,'k','LineWidth',2) grid 'minor' xlabel('Log_1_0(Reynolds Number)') ylabel('Log_1_0(C_d)') title('Various Plots Related to Log_1_0(C_d) as a Function of Log_1_0(Re)') hold on pause figure(31) loglog(Re,Cd_data,'bo') pause LCd_data=log10(Cd_data); figure(32) plot(LRe,LCd_data,'bo') pause %****************************************************** %*** Compute LINEAR Model for 1ST Portion of Data *** n1=125; %max index of data for regression X1Y=[LRe(1:n1),LCd_data(1:n1)]; %*****INSERT YOUR CODE HERE***** %*****END OF YOUR CODE***** YLinhat=mhat*X1Y(:,1)+bhat; plot(X1Y(:,1),YLinhat,'r--','LineWidth',2) pause figure(31) chat=10^bhat; Cdhat=chat*Re(1:n1).^(mhat*ones(n1,1)); loglog(Re(1:n1),Cdhat,'r--','LineWidth',2) pause %=================================================== %PROBLEM 4: %Compute QUADRATIC model over the entire region (prior to the strong dip): n2max=759; X21=LRe(1:n2max); X22=X21.^2; Y2=LCd_data(1:n2max); XY2=[X21,X22,Y2]; %*****INSERT YOUR CODE HERE***** %*****END OF YOUR CODE***** figure(32) plot(X21,Y2hat,'g--','LineWidth',2) pause figure(31) ghat=10^Bhat; R2=Re(1:n2max); p=Mhat(1)+Mhat(2)*log10(R2); Cd2hat=ghat*R2.^p; loglog(Re(1:n2max),Cd2hat,'g--','LineWidth',2) 8 1.0276850e-01 2.2911520e+02 1.0466160e-01 2.2490330e+02 1.0658960e-01 2.2076870e+02 1.0855310e-01 2.1671020e+02 1.1055280e-01 2.1272630e+02 1.1258940e-01 2.0881560e+02 1.1466340e-01 2.0497700e+02 1.1677570e-01 2.0120880e+02 1.1892680e-01 1.9750990e+02 1.2111770e-01 1.9387890e+02 1.2334880e-01 1.9031470e+02 1.2562100e-01 1.8681610e+02 1.2793520e-01 1.8338170e+02 1.3029190e-01 1.8001050e+02 1.3269200e-01 1.7670120e+02 1.3513640e-01 1.7345280e+02 1.3762580e-01 1.7026410e+02 1.4016110e-01 1.6713400e+02 1.4274300e-01 1.6406150e+02 1.4537250e-01 1.6104540e+02 1.4805050e-01 1.5808480e+02 1.5077780e-01 1.5517870e+02 1.5355530e-01 1.5232600e+02 1.5638400e-01 1.4952570e+02 1.5926480e-01 1.4677690e+02 1.6219870e-01 1.4407860e+02 1.6518660e-01 1.4142990e+02 1.6822960e-01 1.3882990e+02 1.7132860e-01 1.3627770e+02 1.7448470e-01 1.3377240e+02 1.7769890e-01 1.3131320e+02 1.8097240e-01 1.2889920e+02 1.8430620e-01 1.2652960e+02 1.8770130e-01 1.2420350e+02 1.9115900e-01 1.2192020e+02 1.9468050e-01 1.1967890e+02 1.9826680e-01 1.1747880e+02 2.0191910e-01 1.1531910e+02 2.0563870e-01 1.1319910e+02 2.0942690e-01 1.1111810e+02 2.1328480e-01 1.0907540e+02 2.1720930e-01 1.0907760e+02 2.2121510e-01 1.0510180e+02 2.2529020e-01 1.0316970e+02 2.2943570e-01 1.0317180e+02 2.3366220e-01 1.0127520e+02 2.3796650e-01 9.9413370e+01 2.4235020e-01 9.7585780e+01 2.4681970e-01 9.4028840e+01 2.5136650e-01 9.2300290e+01 2.5599170e-01 9.2302270e+01 2.6070740e-01 9.0605420e+01 2.6551000e-01 8.8939770e+01 2.7040110e-01 8.7304730e+01 2.7538220e-01 8.5699760e+01 2.8045510e-01 8.4124290e+01 2.8562150e-01 8.2577780e+01 2.9088310e-01 8.1059700e+01 2.9624150e-01 7.9569530e+01 3.0169870e-01 7.8106760e+01 3.0725640e-01 7.6670880e+01 3.1291640e-01 7.5261380e+01 3.1868090e-01 7.3877810e+01 3.2455130e-01 7.2519670e+01 3.3053000e-01 7.1186490e+01 3.3661880e-01 6.9877860e+01 3.4281980e-01 6.8593250e+01 3.4913500e-01 6.7332270e+01 3.5555920e-01 6.7333680e+01 3.6210910e-01 6.6095840e+01 3.6877970e-01 6.4880760e+01 3.7557300e-01 6.3688020e+01 3.8249160e-01 6.2517200e+01 3.8953760e-01 6.1367910e+01 3.9671350e-01 6.0239750e+01 4.0402140e-01 5.9132320e+01 4.1146420e-01 5.8045260e+01 4.1904380e-01 5.6978180e+01 4.2676320e-01 5.5930710e+01 4.3462470e-01 5.4902500e+01 4.4262200e-01 5.4903680e+01 4.5078500e-01 5.2902470e+01 4.5907960e-01 5.2903580e+01 4.6753640e-01 5.1931020e+01 4.7614910e-01 5.0976350e+01 4.8492040e-01 5.0039220e+01 4.9385330e-01 4.9119310e+01 5.0295070e-01 4.8216320e+01 5.1221580e-01 4.7329930e+01 5.2165160e-01 4.6459840e+01 5.3126110e-01 4.5605740e+01 5.4104760e-01 4.4767340e+01 5.5101450e-01 4.3944360e+01 5.6116490e-01 4.3136500e+01 5.7149050e-01 4.3137420e+01 5.8201810e-01 4.2344400e+01 5.9273960e-01 4.1565960e+01 6.0365870e-01 4.0801830e+01 6.1477900e-01 4.0051740e+01 6.2610410e-01 3.9315450e+01 6.3763770e-01 3.8592700e+01 6.4938400e-01 3.7883230e+01 6.6134650e-01 3.7186800e+01 6.7352930e-01 3.6503170e+01 6.8593660e-01 3.5832120e+01 6.9857250e-01 3.5173390e+01 7.1142650e-01 3.5174130e+01 7.2453190e-01 3.4527500e+01 7.3787880e-01 3.3892760e+01 7.5147160e-01 3.3269710e+01 7.6531460e-01 3.2658090e+01 7.7941290e-01 3.2057720e+01 7.9377060e-01 3.1468380e+01 8.0837640e-01 3.1469040e+01 8.2328470e-01 3.0322010e+01 8.3843340e-01 3.0322650e+01 8.5387850e-01 2.9765210e+01 8.6960820e-01 2.9218010e+01 8.8562760e-01 2.8680880e+01 9.0194190e-01 2.8153620e+01 9.1855690e-01 2.7636050e+01 9.3547800e-01 2.7127990e+01 9.5271090e-01 2.6629280e+01 9.7026090e-01 2.6139740e+01 9.8811430e-01 2.6140310e+01 1.0063170e+00 2.5659760e+01 1.0248540e+00 2.5188040e+01 1.0437340e+00 2.4724990e+01 1.0629610e+00 2.4270450e+01 1.0825420e+00 2.3824270e+01 1.1024840e+00 2.3386300e+01 1.1227700e+00 2.3386790e+01 1.1434530e+00 2.2956850e+01 1.1645160e+00 2.2534820e+01 1.1859680e+00 2.2120550e+01 1.2078160e+00 2.1713900e+01 1.2300660e+00 2.1314720e+01 1.2527250e+00 2.0922870e+01 1.2758020e+00 2.0538250e+01 1.2992770e+00 2.0538680e+01 1.3232110e+00 2.0161100e+01 1.3475870e+00 1.9790470e+01 1.3724110e+00 1.9426650e+01 1.3976930e+00 1.9069520e+01 1.4234400e+00 1.8718950e+01 1.4496620e+00 1.8374830e+01 1.4763670e+00 1.8037030e+01 1.5035320e+00 1.8037410e+01 1.5312290e+00 1.7705820e+01 1.5594360e+00 1.7380320e+01 1.5881640e+00 1.7060810e+01 1.6173870e+00 1.7061160e+01 1.6471810e+00 1.6747520e+01 1.6775240e+00 1.6439660e+01 1.7084260e+00 1.6137440e+01 1.7398980e+00 1.5840770e+01 1.7719490e+00 1.5549560e+01 1.8045910e+00 1.5263700e+01 1.8377960e+00 1.5264020e+01 1.8716510e+00 1.4983420e+01 1.9061290e+00 1.4707970e+01 1.9412420e+00 1.4437580e+01 1.9770030e+00 1.4172170e+01 2.0133800e+00 1.4172460e+01 2.0504700e+00 1.3911920e+01 2.0882420e+00 1.3656170e+01 2.1267100e+00 1.3405120e+01 2.1658870e+00 1.3158690e+01 2.2057860e+00 1.2916790e+01 2.2463720e+00 1.2917060e+01 2.2877540e+00 1.2679600e+01 2.3298980e+00 1.2446500e+01 2.3728180e+00 1.2217690e+01 2.4165290e+00 1.1993080e+01 2.4610450e+00 1.1772610e+01 2.5063290e+00 1.1772850e+01 2.5524990e+00 1.1556430e+01 2.5995190e+00 1.1343980e+01 2.6474060e+00 1.1135440e+01 2.6961190e+00 1.1135670e+01 2.7457850e+00 1.0930950e+01 2.7963660e+00 1.0730010e+01 2.8478780e+00 1.0532750e+01 2.9003410e+00 1.0339120e+01 2.9537690e+00 1.0149050e+01 3.0081190e+00 1.0149270e+01 3.0635330e+00 9.9626870e+00 3.1199670e+00 9.7795360e+00 3.1774410e+00 9.5997520e+00 3.2359730e+00 9.4232750e+00 3.2955170e+00 9.4234720e+00 3.3562240e+00 9.2502350e+00 3.4180510e+00 9.0801820e+00 3.4810160e+00 8.9132560e+00 3.5450680e+00 8.9134430e+00 3.6103730e+00 8.7495810e+00 3.6768810e+00 8.5887330e+00 3.7446150e+00 8.4308440e+00 3.8135960e+00 8.2758550e+00 3.8837670e+00 8.2760290e+00 3.9553120e+00 8.1238860e+00 4.0281730e+00 7.9745390e+00 4.1022930e+00 7.9747070e+00 4.1778630e+00 7.8281030e+00 4.2548250e+00 7.6841940e+00 4.3332040e+00 7.5429310e+00 4.4130270e+00 7.4042640e+00 4.4942280e+00 7.4044200e+00 4.5770190e+00 7.2683000e+00 4.6613340e+00 7.1346820e+00 4.7471040e+00 7.1348320e+00 4.8345510e+00 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