Time series analysis

Air pollution control specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time series data exhibit seasonality, with the levels of pollutants showing similar patterns over the hours in the day. On July 15,16, and 17, the observed levels of nitrogen dioxide in a city's downtown are for the 12 hours from 6:00 a.m. to 6:00 p.m. were as follows.

July 15 25 28 35 50 60 60 40 35 30 25 25 20
July 16 28 30 35 48 60 65 50 40 35 25 20 20
July 17 35 42 45 70 72 75 60 45 40 25 25 25

a. Identify the hourly seasonal indexes for the 12 hourly daily readings.

Centered Moving Average and Deseasonalization

Centered
Moving Ratio to Seasonal of nitrogen dioxide
t Year Month of nitrogen dioxide Average CMA Indexes Deseasonalized
1 1 1 25 0.771 32.4
2 1 2 28 0.864 32.4
3 1 3 35 0.954 36.7
4 1 4 50 1.392 35.9
5 1 5 60 1.571 38.2
6 1 6 60 1.667 36.0
7 1 7 40 36.208 1.105 1.207 33.1
8 1 8 35 36.417 0.961 0.994 35.2
9 1 9 30 36.500 0.822 0.850 35.3
10 1 10 25 36.417 0.686 0.647 38.7
11 1 11 25 36.333 0.688 0.579 43.2
12 1 12 20 36.542 0.547 0.504 39.7
13 2 1 28 37.167 0.753 0.771 36.3
14 2 2 30 37.792 0.794 0.864 34.7
15 2 3 35 38.208 0.916 0.954 36.7
16 2 4 48 38.417 1.249 1.392 34.5
17 2 5 60 38.208 1.570 1.571 38.2
18 2 6 65 38.000 1.711 1.667 39.0
19 2 7 50 38.292 1.306 1.207 41.4
20 2 8 40 39.083 1.023 0.994 40.2
21 2 9 35 40.000 0.875 0.850 41.2
22 2 10 25 41.333 0.605 0.647 38.7
23 2 11 20 42.750 0.468 0.579 34.5
24 2 12 20 43.667 0.458 0.504 39.7
25 3 1 35 44.500 0.787 0.771 45.4
26 3 2 42 45.125 0.931 0.864 48.6
27 3 3 45 45.542 0.988 0.954 47.2
28 3 4 70 45.750 1.530 1.392 50.3
29 3 5 72 45.958 1.567 1.571 45.8
30 3 6 75 46.375 1.617 1.667 45.0
31 3 7 60 1.207 49.7
32 3 8 45 0.994 45.3
33 3 9 40 0.850 47.1
34 3 10 25 0.647 38.7
35 3 11 25 0.579 43.2
36 3 12 25 0.504 49.6

1 2 3 mean: adjusted:
Calculation of Seasonal Indexes 1 0.753 0.787 0.770 0.771
2 0.794 0.931 0.862 0.864
3 0.916 0.988 0.952 0.954
4 1.249 1.530 1.390 1.392
5 1.570 1.567 1.568 1.571
6 1.711 1.617 1.664 1.667
7 1.105 1.306 1.205 1.207
8 0.961 1.023 0.992 0.994
9 0.822 0.875 0.848 0.850
10 0.686 0.605 0.646 0.647
11 0.688 0.468 0.578 0.579
12 0.547 0.458 0.503 0.504
11.979 12.000

b. Based on the seasonal indexes in part (a), the trend equation developed for the depersonalized data is = 32.983 + 0.3922t. Using only the trend equation, develop forecasts for the 12 hours for July 18.

The t for 12 hours for July 18 is 42

= 32.983 + 0.3922*12=37.6894

c. Use the seasonal indexes from part (a) to adjust the trend forecasts in part (b).

Adjusting with the seasonal index
Forecast for 12 hours for July 18 = 1.667*37.6894=62.8282