Methods ISYE 3039
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This 0 page Class Notes was uploaded by Maryse Thiel on Monday November 2, 2015. The Class Notes belongs to ISYE 3039 at Georgia Institute of Technology - Main Campus taught by Jye-Chyi Lu in Fall. Since its upload, it has received 53 views. For similar materials see /class/234205/isye-3039-georgia-institute-of-technology-main-campus in Industrial Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
My ISyE 3039 Final Exam Review W 6ur r n MI 1 The leastsquares estimation idea 39 6a 2 Estimation formulas of intercept and slope 17 C54 74 5 2 Multiple Linear Regression 767m 04477 lm 1 Matrix representation of linear regression model with several input variables and the least squares 547 1 Simple Linear Regression estimation formula bvectorr f39I ZI l X Y Enquot 0 2 Inference procedures of mean response and individual new observation 4 g 3 ANOVA of regression MSE and multipleRSquare skipped A 3 NonLinear Regression skipped 3 of model parameters by using transformations 1 Nonlinear model 2 Taylorseries expansi nd its matrix format representation 39 GAnt x y 3 GaussNewt ethod of e imating model parameters 3 4 U and momtormg conve ence of model parameters 1 re Ca 7 4 Time eries Regression l AR1 model and test of autocorrelation H0 rho 0 against Ha rho gt 0 2 Estimating model parameters rho beta0 and betal A C 9 a 2 3 Forecasting future observation Ynlhat 4 Con dence interval of Ynl gagh f Note Exam grades will be announced on 39 r 39 noon on my door with the last four digits of your 8811 W Collection of ISyE 3039 Sample Final Exams Show details of your work for possibly receiving partial credits 1 Consider the following regression model Yi beta xi epsiloni for i l 2 n Verify that the least squares estimate of the parameter beta is equal to J betahat sumxi Yisumxiquot2 Page L 2 The Zarthan Company sells a special skin cream through fashion stores exclusively It operates in 15 marketing districts and is interested in predicting district sales The rst three rows of the sales data are provided below District 139 Yi X1i X2i 1 162 274 2450 2 120 180 3254 3 223 375 3802 where Yi is sales in gross of jars a gross 12 dozen X1i Target POpulation in thousands of persons and X2i Per Capita Discretionary Income in dollars a What will be the design matrix X for the following multiple regression model Yi betaO betal X1i beta2 X2i betal l X1iquot2 beta22 X2iquot2 beta 12 X IiX2i epsiloni fori 1 2 n 15 b What is the dimension of the X X matrix Give a numerical value c Give the formula of computing the variance covariance matrix of the regression coef cient estimates d How do you obtain the MSE mean squared error needed to estimate sigmaquot2 e For given values of the input variables X1h 236 X2h 2660 construct a 95 con dence interval of the mean response You need to specify the values in the vector of Xh in computing the variance of the mean response estimation Give the degree of freedom of the ttable value 0 How to obtain the coef cient of multiple determination Rquot2 show details with the given vectormatrix Y X and b betahat 3 Consider the following nonlinear regression model Yi gamma0 03expgammalXi epsiloni i 1 2 n SKIPPED 4 Consider the following timeseries regression model Yt betaO betal xt epsilont and epsilont rho epsilontl ut where ut s are random samples from the normal distribution with mean 0 and variance sigma 2 rhollt land t 12 Given the following data My 3 Time point xt yt that 1 353 1098 1081 2 297 1113 1125 3 308 1251 1116 0 o o n sample size 30 Explain how to forecast the response at time point 31 and how to construct its con dence interval Provide all the needed details for computations without numerical illustrations That is answer the following questions i How to obtain the residual et t l 2 n ii How to use the residual et to estimate the autocorrelation parameter rho iii What are the transformation formulas Yt and xt for constructing the model Yt beta039 betal xt ut such that the simple linear regression procedure can be used to estimate beta0 and beta1 and thus nd beta0 and betal estimates Note that Yt and Yt are different What are the formulas for beta0 and beta1 iv Given the estimates of beta0 betal rho and en how to obtain the forecasting point v How to use the simple linear regression prediction interval ideas to construct the 1001 a prediction interval of the forecasting point What is the degree of freedom in the t distribution Consider the following new timeseries regression model Yt beta0 beta1 xlt beta2 x2t epsilont and epsilont rho epsilont1 ut where ut s are random samples from the normal distribution with mean 0 and variance sigmaquot2 rho I lt 1 and t 1 2 Given the following data Time point x1t x2t yt et 1 1273 232 2096 0026 2 1300 243 2140 0062 3 1327 251 2196 0022 0 o o n sample size 20 vi Use the de nition oflthe residual et to nd ythat Show one example with t 2 vii How to use the residual et to estimate the autocorrelation parameter rho viii Use the DurbinWatson test to see if there is enough evidence to support the hypothesis rho gt 0 Use 005 as the level of signi cance alpha Be sure to write down the test hypotheses test statistic and the decision rule No computation is needed Use the attached table to nd the needed table values dL and dU ie what are the values of p and n ix Extend the following transformation method given in the class notes to 39 our new time series regression model by providing detailed de nitions of X xi Yt x1t x2t39 betaO betal and beta2 Note that Yt and Yt are i different 39 Note If you are familiar with the following derivation from the class notes you can skip reading these details It is just a copy from the notes For the model Yt betaO betal xt epsilont and epsilont rho epsilontl ut considered in the class we de ne Yt Yt rho Ytl and xt xt rho xtl Replacing Yt in the equation Yt Yt rho Ytl with the de nition of Yt betaO betal xt epsilont at the time points t and H we obtain the following equation Yt39 betaO betal xt epsilont rho betaO betal xt l epsilont1 Then reorganize them into the regression format with the error term ut which is iid normal with the time series dependence structure That is Yt beta01 rho betal xt rhoxtl epsilont rho epsilont1 betaO betal xt ut as presented in the class notes where some of the symbols were cut in the very right Given the estimates of betaO betal how to obtain the estimates of betaO and betal Moreover if rhohat and en are also provided how to obtain the forecasting point How to use the prediction interval ideas taught in the regression class for the new observation to construct the 1001 0t prediction interval of the forecasting point What is the degree of freedom in the tdistribution Note I know that it is dif cult for you to construct the formula with two input variables x1t and x2t You can answer this question by using the results presented in the class with only one input variable xt However if you are so clever to combine the multiple regression and the time series regression 39 materials together for extending the con dence interval of the forecasting point presented in the class to the two input variables case successfully you will get 5 points extra credits One hint for this extension is that you need to use the matrix notation somewhere J H w 4 501737 I J m mam TL SE w 74 7 hfh ija I A quot A 2 m ElKW xi dang I I 1 h A quot A quot W 2 2f X39Xc 23 2lt7 z8 1 I r H 392 O u 2 1 x l 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