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IEOR 165 – Homework 5 Due Thursday, May 7, 2015 Electronic copies will not be accepted except under extenuating circumstances. Each student must turn in their own homework solutions. Question 1. a. Assuming the weather on a given day only depends on the weather on the previous day, weather forecasting can be modeled as a Markov Chain. Weather can be sunny, rainy or cloudy. The data coming from the US Weather Service can be summarized as follows: today sunny today rainy today cloudy tomorrow sunny 286 128 245 tomorrow rainy 123 333 255 tomorrow cloudy 165 256 189 Find the maximum likelihood estimates of the transition probabilities. b. Each time a certain horse runs in a three-horse race, he can win, come in second, and come in third. This race is modeled as Markov chain and the following data is collected. first second third first 55 44 20 second 71 62 39 third 22 56 53 Find the maximum likelihood estimates of the transition probabilities. Question 2. A popular brand of tennis shoe has had the following demand history by quarters over a three-year period. 1990 1 2 3 4 Demand 12 25 76 52 1991 1 2 3 4 Demand 16 32 71 62 1 1992 1 2 3 4 Demand 14 45 84 47 a. Using the data from 1991 and 1992, determine initial values of the intercept, slope and seasonal factors for Winter’s method. b. Assume that the observed demand for the first quarter of 1993 was 18. Using α = 0.2, β = 0.15 and γ = 0.1, update the estimates of the series, the slope and the seasonal factors. c. What are the forecasts made at the end of the first quarter of 1993 for the remaining three quarters of 1993? Question 3. The energy consumption can be modeled as a the linear time invariance (LTI) dynamical system in discrete time. xt+1 = Axt + But The following data shows how energy consumption (thousand metric ton of oil equivalent per capita) is related to the Gross Domestic Product (GDP) per capita. Write a code to determine the values of A and B. 2 GDP per capita 16,176 16,472 15,779 16,076 16,836 17,582 17,862 18,348 18,993 19,108 18,872 18,255 18,139 18,303 18,936 19,227 19,319 19,953 20,613 21,563 22,488 22,599 23,156 23,409 23,943 24,470 24,971 25,809 25,267 Energy Consumption (thousand metric ton of oil equivalent per capita) 5468.54 5566.28 5665.80 5764.13 5864.23 5967.52 6073.97 6182.22 6292.91 6406.52 6521.51 6636.81 6751.31 6866.49 6983.20 7102.85 7224.46 7347.44 7473.42 7602.52 7735.64 7872.72 8011.37 8152.94 8296.55 8443.04 8592.44 8744.69 8900.81 3