tut1

pdf

School

The University of Hong Kong *

*We aren’t endorsed by this school

Course

3613

Subject

Statistics

Date

Nov 24, 2024

Type

pdf

Pages

Uploaded by KidNeutron12719

1 STAT3613 Tutorial 1 Review Response models • Output variable 𝑦𝑦 • Input variables 𝑥𝑥 1 , 𝑥𝑥 2 , … , 𝑥𝑥 𝑝𝑝 • Model 𝑦𝑦 = 𝑓𝑓�𝑥𝑥 1 , 𝑥𝑥 2 , … , 𝑥𝑥 𝑝𝑝 � • Predicted value 𝑦𝑦 � 𝑖𝑖 = 𝑓𝑓�𝑥𝑥 𝑖𝑖1 , 𝑥𝑥 𝑖𝑖2 , … , 𝑥𝑥 𝑖𝑖𝑝𝑝 � • Residual 𝑒𝑒 𝑖𝑖 = 𝑦𝑦 𝑖𝑖 − 𝑦𝑦 � 𝑖𝑖 • SSE � 𝑒𝑒 𝑖𝑖 2 𝑛𝑛 𝑖𝑖=1 • SST � ( 𝑦𝑦 𝑖𝑖 − 𝑦𝑦 � ) 2 𝑛𝑛 𝑖𝑖=1 • Coefficient of determination 𝑅𝑅 2 = 1 − 𝑆𝑆𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆 Simple response models Model Formula Linear Y = a + b X Power series Y = a 0 + a 1 X + a 2 X 2 + … + a p X p Fractional root Y = a + b X c Semilog Y = a + b ln( X ) Exponential Y = a e bX Modified exponential Y = a (1 – e - bX ) + c Logistic ( ) d e a Y cX b + + = + − 1 Gompertz d ab Y X c + = ADBUG d X X b a Y c c + + =

2 Phenomena • Linear: Linear, power series and fractional root models • Concave (decreasing returns): Power series, fractional root, semilog, modified exponential and ADBUDG models • Saturation: Fractional root, modified exponential, logistic, Gompertz and ADBUDG models • Convex (increasing returns): Power series, fractional root, semilog and exponential model • S-shape: Power series, logistic, Gompertz and ADBUDG models • Threshold: Fractional root, semilog, modified exponential • Super saturation: Power series Diffusion models • Logistic model, ADBUG model and Gompertz model • Bass model 𝑛𝑛 𝑡𝑡 = 𝑎𝑎𝑎𝑎 + ( 𝑏𝑏 − 𝑎𝑎 ) 𝑋𝑋 𝑡𝑡 − 𝑏𝑏 𝑎𝑎 𝑋𝑋 𝑡𝑡 2 𝑋𝑋 𝑡𝑡 = 𝑎𝑎𝑎𝑎 exp � ( 𝑎𝑎 + 𝑏𝑏 ) 𝑡𝑡� − 1 𝑏𝑏 + 𝑎𝑎 exp � ( 𝑎𝑎 + 𝑏𝑏 ) 𝑡𝑡� Forecast models Simple exponential smoothing 𝐿𝐿 𝑡𝑡 = 𝛼𝛼𝑌𝑌 𝑡𝑡 + (1 − 𝛼𝛼 ) 𝐿𝐿 𝑡𝑡−1 𝑌𝑌 � 𝑡𝑡 ( 𝑘𝑘 ) = 𝐿𝐿 𝑡𝑡 Linear exponential smoothing 𝐿𝐿 𝑡𝑡 = 𝛼𝛼𝑌𝑌 𝑡𝑡 + (1 − 𝛼𝛼 )( 𝐿𝐿 𝑡𝑡−1 + 𝑆𝑆 𝑡𝑡−1 ) 𝑆𝑆 𝑡𝑡 = 𝛽𝛽 ( 𝐿𝐿 𝑡𝑡 − 𝐿𝐿 𝑡𝑡−1 ) + (1 − 𝛽𝛽 ) 𝑆𝑆 𝑡𝑡−1 𝑌𝑌 � 𝑡𝑡 ( 𝑘𝑘 ) = 𝐿𝐿 𝑡𝑡 + 𝑘𝑘𝑆𝑆 𝑡𝑡 HW additive seasonal model 𝐿𝐿 𝑡𝑡 = 𝛼𝛼 ( 𝑌𝑌 𝑡𝑡 − 𝑆𝑆 𝑡𝑡−𝑐𝑐 ) + (1 − 𝛼𝛼 )( 𝐿𝐿 𝑡𝑡−1 + 𝑆𝑆 𝑡𝑡−1 ) 𝑆𝑆 𝑡𝑡 = 𝛽𝛽 ( 𝐿𝐿 𝑡𝑡 − 𝐿𝐿 𝑡𝑡−1 ) + (1 − 𝛽𝛽 ) 𝑆𝑆 𝑡𝑡−1 𝑆𝑆 𝑡𝑡 = 𝛾𝛾 ( 𝑌𝑌 𝑡𝑡 − 𝐿𝐿 𝑡𝑡 ) + (1 − 𝛾𝛾 ) 𝑆𝑆 𝑡𝑡−𝑐𝑐 𝑌𝑌 � 𝑡𝑡 ( 𝑘𝑘 ) = 𝐿𝐿 𝑡𝑡 + 𝑘𝑘𝑆𝑆 𝑡𝑡 + 𝑆𝑆 𝑡𝑡−𝑐𝑐+𝑘𝑘 ∗ HW multiplicative seasonal model 𝐿𝐿 𝑡𝑡 = 𝛼𝛼 𝑌𝑌 𝑡𝑡 𝑆𝑆 𝑡𝑡−𝑐𝑐 + (1 − 𝛼𝛼 )( 𝐿𝐿 𝑡𝑡−1 + 𝑆𝑆 𝑡𝑡−1 ) 𝑆𝑆 𝑡𝑡 = 𝛽𝛽 ( 𝐿𝐿 𝑡𝑡 − 𝐿𝐿 𝑡𝑡−1 ) + (1 − 𝛽𝛽 ) 𝑆𝑆 𝑡𝑡−1 𝑆𝑆 𝑡𝑡 = 𝛾𝛾 𝑌𝑌 𝑡𝑡 𝐿𝐿 𝑡𝑡 + (1 − 𝛾𝛾 ) 𝑆𝑆 𝑡𝑡−𝑐𝑐 𝑌𝑌 � 𝑡𝑡 ( 𝑘𝑘 ) = ( 𝐿𝐿 𝑡𝑡 + 𝑘𝑘𝑆𝑆 𝑡𝑡 ) 𝑆𝑆 𝑡𝑡−𝑐𝑐+𝑘𝑘 ∗

3 Exercises 1. Consider an ADBUDG model given as d X X b a Y c c + + = Given that b , c, d > 0 and X ≥ 0 , show that (a) a = minimum value of Y (b) b = maximum value of Y – minimum value of Y Suppose Y = 1 when X = 1 and y ’ = (slope of Y when X = 1). Show that (c) 1 1 − − = a b d (d) ( ) d b d y c × + = 2 1 ' 2. A company develops promotional response model tools to help it decide the level and allocation of promotional spending using response modeling and optimization. The managers constructed a response model, relating promotional spending with sales. They explored the promotional spending response analysis using the following information: Promotional Spending X ($’000,000) Sales Y (’000,000 units) 0.00 6.3 0.44 6.7 0.87 8.0 1.31 9.3 2.50 10.9 5.00 11.8 a. Plot the sales Y against the promotional spending X. Describe the relationship between Y and X. Which response models seem appropriate? b. Find the starting values for each model. c. Estimate the parameters of the response models and choose the best model with largest R 2 . 3. Mobile phone user diffusion rates in China are given in mobile . mobile is the diffusion rate and m1 is the monthly change of the diffusion rate. Forecast the diffusion rate in 2015 to 2019. a. Plot the diffusion rate and change of the diffusion rate. b. Apply an ADBUG model. Evaluate the fitness. Plot the predicted values. Predict the diffusion rate in 2015 to 2019. c. Apply a Bass model. Evaluate the fitness. Plot the predicted values. Predict the diffusion rate in 2015 to 2019.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

4 4. The palm oil monthly price and the price change are given in palmoil . t is time index, Month is the month of the year, Price is the spot price of palm oil and Change is the rate of change of the price. Forecast the change in the next 6 months. a. Plot the series plot of the price change. b. Forecast by a simple exponential smoothing. c. Forecast by a linear exponential smoothing. d. Forecast by an Additive Holt-Winter’s seasonal model. 5. For electronic company B, salesforce is to be allocated to 3 different products, the digital camera (DC), the mobile phone (MP) and the laptop (LP). The current sales for DC, MP and laptop are 5000, 10000 and 4000 respectively. The current salespeople allocated to DC are 25 and that for MP are 35, while that for laptop are 20. The average cost for each salesperson is $40. The margin of DC is $0.8 and that of MP and laptop are $0.7 and $0.9. Historical data of the sales (in %) for various numbers of salespeople (in %) are shown in the table. % current sales % current salesforce DC MP LP 0.0 0.5 0.18 0.15 0.5 0.62 0.44 0.4 1.0 1 1 1 1.5 1.25 1.22 1.18 2.0 1.35 1.28 1.22 2.5 1.4 1.29 1.24 3.0 1.48 1.36 1.25 a. Calibrate ADBUDG models to predict the sales for DC, MP and laptop respectively. b. Set up functions to calculate the total net profit. c. Find the optimal salespeople allocation to maximize the total net profit. d. If the total number of salespeople is restricted by 110, find the optimal allocation.