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