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QBUS6840 Predictive Analytics

创建日期：2024-04-17 15:40:59

所属分类：数学mathematics

标签： QBUS6840

Predictive Analytics

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QBUS6840 Predictive Analytics

Mid-term exam sample question solutions

You might try to refine and improve the given sample solutions. Question 1 () Solution: The simple exponential smoothing for forecasting in the weighted average form can be shown as: Ŷt+1 = αYt + (1− α)Ŷt The forecast Ŷt+1 at time t + 1 is equal to a weighted average between the most recent observation Yt and its forecast Yˆt. The naive forecast method uses the most recent value Yt as the forecast Ŷt+1 at time t+ 1 and follows the form: Ŷt+1 = Yt = 1× Yt. Therefore, the simple exponential smoothing method includes the naive method as a special case. When α = 1, the simple exponential smoothing method has 100% weight on the most recent observation Yt and becomes the naive method. Question 2 () Solution: (1) There is an obvious up-trend in linear way. There is no obvious cycle compo- nent. Although the graph is cluttered, we can still see a potential seasonal component presented in the data. The seasonal magnitude (or the gap between each rise and fall) does not proportionally change along with the time. Also there is sharp drop around time range from 155 to 165 (these numbers are just some estimates), then it resumes the main trend. (In the exam, you will not lose marks if you miss this point). (2) Given the patterns identified in (1), the following methods can be tried: a. Additive decomposition model. This method is capable of picking up all the components identified in (1). In this method, the trend can be estimated by linear regression model. b. Additive Holt-Winters (Seasonal) Smoothing Model. This model could be used as its level, trend component and seasonal components can be all time-dependent. Question 3 () Solution: 1. Smooth the data to remove seasonality, leading to the initial trend-cycle esti- mate. Compute the de-trend series Yt T̂t 2. For each season (M in total), compute the average of the de-trended values for that season. 3. Compute the seasonal indices by normalising the previous step M values to make sure that they add up to M 4. The estimated seasonal component Ŝt is obtained by concatenating these sea- sonal indices, then compute seasonally adjusted series 5. If forecasting, fit a trend model such as linear regression to the seasonally adjusted time series, otherwise smooth the seasonally adjusted time series 6. Check the estimated remainder and see whether it shows any pattern. If no pattern displayed, stop; otherwise, estimate the cycle-reminder and go to the next step 7. Smooth the cycle-reminder to estimate the cycle 8. Finally estimate the remainder by removing the cycle, seasonal and trend components from the original series Yt Page 2 Question 4 () Solution: (1) Note: as mentioned in the question: “You only need to present your final solution.” Therefore, you only need to type your answers as below: ̂̂ Y t = 1 9(Yt−2 + Yt+2) + 2 9(Yt−1 + Yt+1) + 1 3Yt (2) Note: as mentioned in the question: “You only need to present your final solution.” Therefore, you only need to type your answers as below: ̂̂ Y t = 1 21(Yt−4 + Yt+4) + 2 21(Yt−3 + Yt+3) + 1 7(Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2) (3) For part (1) of this question, we will lose 2 data points at the start and 2 data points at the end. For part (2) of this question, we will lose 4 data points at the start and 4 data points at the end. (4) The smoother in part (1) of this question is a WMA-5, so we have k = 5. The smoother in part (2) of this question is a WMA-9, so we have k = 9. (5) Both smoothers in part (1) and (2) of this question are symmetric. Both smoothers are symmetric since the coefficients of Yt−i and Yt+i are the same in the average formula. Solution: The details of the solutions for part (1) and (2) of this questions are shown below, while you do NOT have to type such solving details into the answer box. (1) In step 1, we do three MA-3 on the given time series for time t− 1, t, t+ 1: Page 3 Ŷt−1 = Yt−2 + Yt−1 + Yt 3 . Ŷt = Yt−1 + Yt + Yt+1 3 . Ŷt+1 = Yt + Yt+1 + Yt+2 3 . In step 2, we do another MA-3 over the smoothed series of step 1: ̂̂ Y t = Ŷt−1 + Ŷt + Ŷt+1 3 = 1 9(Yt−2 + Yt+2) + 2 9(Yt−1 + Yt+1) + 1 3Yt (2) In step 1, we do three MA-7 on the given time series for time t− 1, t, t+ 1: Ŷt−1 = Yt−4 + Yt−3 + Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2 7 . Ŷt = Yt−3 + Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2 + Yt+3 7 . Ŷt+1 = Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2 + Yt+3 + Yt+4 7 . In step 2, we do another MA-3 over the smoothed series of step 1: ̂̂ Y t = Ŷt−1 + Ŷt + Ŷt+1 3 = 121(Yt−4 + Yt+4) + 2 21(Yt−3 + Yt+3) + 1 7(Yt−2 + Yt−1 + Yt + Yt+1 + Yt+2) Question 5 () Solution: Note: as mentioned in the question: “For all questions, you only need to present your final solution.” Therefore, you only need to type your answers as below: (1) Page 4 The error correction form of lt equation is: lt = lt−1 + αεt. (2) The 1, 2 and 3-step-ahead point forecasts of the model are all equal to lt: ŶT+1 = ŶT+2 = ŶT+3 = lt (3) The 1-step-ahead forecast variance of the model is: Var(Yt+1|Y1:t) = σ2. The 2-step-ahead forecast variance of the model is: Var(Yt+2|Y1:t) = σ2(1 + α2) The 3-step-ahead forecast variance of the model is: Var(Yt+3|Y1:t) = σ2(1 + 2α2) Solution: The details of the solutions are shown below, while you do NOT have to type such solving details into the answer box. (1) The error correction form of lt equation can be derived as: lt = αYt + (1− α)lt−1 = lt−1 + α(Yt − lt−1) = lt−1 + αεt. (2) For the 1-step-ahead point forecast, we have: Ŷt+1 =E(lt + εt+1|Y1:t) =E(lt) + E(εt+1|Y1:t) = lt + E(εt+1) = lt + 0 = lt where we have used the assumption εt+1 ∼ N(0, σ2), so E(εt+1) = 0. For the 2-step-ahead point forecast, we have: Ŷt+2 = E(lt+1 + εt+2|Y1:t) = E(lt + αεt+1|Y1:t) = lt Page 5 For the 3-step-ahead point forecast, we have: Ŷt+3 = E(lt+2 + εt+3|Y1:t) = E(lt+1 + αεt+2|Y1:t) = E(lt + αεt+1|Y1:t) = lt (3) For the 1-step-ahead forecast variance, we have: Var(Yt+1|Y1:t) = Var(lt + εt+1|Y1:t) = Var(lt) + Var(εt+1|Y1:t) = 0 + σ2 = σ2. For the 2-step-ahead forecast variance, we have: Var(Yt+2|Y1:t) = Var(lt+1 + εt+2|Y1:t) = Var(lt + αεt+1 + εt+2|Y1:t) = Var(lt) + Var(αεt+1) + Var(εt+2|Y1:t) = 0 + α2Var(εt+1) + Var(εt+2|Y1:t) = α2σ2 + σ2 = σ2(1 + α2) For the 3-step-ahead forecast variance, we have: Var(Yt+3|Y1:t) = Var(lt+2 + εt+3|Y1:t) = Var(lt+1 + αεt+2 + εt+3|Y1:t) = Var(lt + αεt+1 + αεt+2 + εt+3|Y1:t) = σ2(1 + 2α2) Question 6 () Solution: (1) Page 6 Note: as mentioned in the question: “You only need to present your final solution.” Therefore, you only need to type your answers as below: The lt equation error correction form is: lt = lt−1 + bt−1 + αεt. The bt equation error correction form is: bt = bt−1 + βαεt. (2) Note: as mentioned in the question: “You only need to present your final solution.” Therefore, you only need to type your answers as below: The 1-step-ahead point forecast is: Ŷt+1 = lt + bt. The 2-step-ahead point forecast is: Ŷt+2 = lt + 2bt. The 3-step-ahead point forecast is: Ŷt+3 = lt + 3bt. (3) When we simply applying a linear trend (regression), we assume the trend inter- cept and slope keep as constants in the entire time span. In Holt’s linear method (or trend corrected exponential smoothing), the intercept (the level lt) and slope (the trend/direction bt) both changes as the time changes. This demonstrates the Holts linear method is potentially more flexible and accurate than simply apply- ing a linear trend (regression). However in the forecast stage, as we do not have any future true data available, the forecast of Holt’s linear method is calculated according to the most recent level lt and the most recent trend bt (times h). Solution: The details of the solutions for part (1) and (2) of this question are shown below, while you do NOT have to type such solving details into the answer box. (1) The error correction form for the lt equation can be derived as: lt = αYt + (1− α)(lt−1 + bt−1) = lt−1 + bt−1 + α(Yt − lt−1 − bt−1) = lt−1 + bt−1 + αεt Page 7 The error correction form for the bt equation can be derived as: bt = β(lt − lt−1) + (1− β)bt−1 = bt−1 + β(lt − lt−1 − bt−1) = bt−1 + βαεt (2) The 1-step-ahead point forecast can be derived as: Ŷt+1 = E(Yt+1|Y1:t) = E(lt + bt + εt+1|Y1:t) = lt + bt The 2-step-ahead point forecast can be derived as: Ŷt+2 = E(lt+1 + bt+1 + εt+2|Y1:t) = E(lt + 2bt + α(1 + β)εt+1 + εt+2|Y1:t) = lt + 2bt The 3-step-ahead point forecast can be derived as: Ŷt+3 = E(lt+2 + bt+2 + εt+3|Y1:t) = E((lt+1 + bt+1 + αεt+2) + (bt+1 + βαεt+2) + εt+3) = E(lt+1 + 2bt+1 + α(1 + β)εt+2 + εt+3) = E((lt + bt + αεt+1) + 2(bt + βαεt+1) + α(1 + β)εt+2 + εt+3) = E(lt + 3bt + α(1 + 2β)εt+1 + α(1 + β)εt+2 + εt+3) = lt + 3bt END OF EXAMINATION.

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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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