Concept explainers
a.
Construct a time series plot for temperature using the given data.
Comment on the pattern of obtained time series plot.
a.
Answer to Problem 82SE
Output obtained from MINITAB is given below:
The obtained time series plot represents a cyclic pattern of temperature.
Explanation of Solution
Given info:
The data represents the observed value of response variable x at time t. The observed values are
Calculation:
Software Procedure:
Step-by-step procedure to draw the time series plot for temperature using the MINITAB software:
- Choose Graph > Time Series Plot.
- Choose Simple, and then click OK.
- In Series, enter the column of Temperature.
- Click OK.
Observation:
Time series plot shows that the highest temperature is at time 3 and the lowest temperature is at time 6 and 7. From the graph it can be concluded that temperature oscillates with the change in time. Overall the plot represents a cyclic pattern of temperature.
b.
Find the smoothed value
b.
Answer to Problem 82SE
Smoothed values
Time t | ||
2 | 47 | 47 |
3 | 54 | 47.7 |
4 | 53 | 48.2 |
5 | 50 | 48.4 |
6 | 46 | 48.2 |
7 | 46 | 48 |
8 | 47 | 47.9 |
9 | 50 | 48.1 |
10 | 51 | 48.4 |
11 | 50 | 48.5 |
12 | 46 | 48.3 |
13 | 52 | 48.6 |
14 | 50 | 48.8 |
15 | 50 | 48.9 |
Smoothed values
Time t | ||
2 | 47 | 47 |
3 | 54 | 50.5 |
4 | 53 | 51.8 |
5 | 50 | 50.9 |
6 | 46 | 48.4 |
7 | 46 | 47.2 |
8 | 47 | 47.1 |
9 | 50 | 48.6 |
10 | 51 | 49.8 |
11 | 50 | 49.9 |
12 | 46 | 47.9 |
13 | 52 | 50 |
14 | 50 | 50 |
15 | 50 | 50 |
Explanation of Solution
Calculation:
The exponential smoothing equation is
Smoothed values
Here, to calculate the smoothed value
That is,
Moreover, it is given that
Hence, the smoothed value
Thus, the smoothed value at time 2 is
The smoothed value
Thus, the smoothed value at time 3 is
Similarly, smoothed values for the remaining times are given below:
Time t | ||
2 | 47 | 47 |
3 | 54 | 47.7 |
4 | 53 | 48.2 |
5 | 50 | 48.4 |
6 | 46 | 48.2 |
7 | 46 | 48 |
8 | 47 | 47.9 |
9 | 50 | 48.1 |
10 | 51 | 48.4 |
11 | 50 | 48.5 |
12 | 46 | 48.3 |
13 | 52 | 48.6 |
14 | 50 | 48.8 |
15 | 50 | 48.9 |
Smoothed values
Here, to calculate the smoothed value
That is,
Moreover, it is given that
Hence, the smoothed value
Thus, the smoothed value at time 2 is
The smoothed value
Thus, the smoothed value at time 3 is
Similarly, smoothed values for the remaining times are given below:
Time t | ||
2 | 47 | 47 |
3 | 54 | 50.5 |
4 | 53 | 51.8 |
5 | 50 | 50.9 |
6 | 46 | 48.4 |
7 | 46 | 47.2 |
8 | 47 | 47.1 |
9 | 50 | 48.6 |
10 | 51 | 49.8 |
11 | 50 | 49.9 |
12 | 46 | 47.9 |
13 | 52 | 50 |
14 | 50 | 50 |
15 | 50 | 50 |
c.
Find the number of values of
Find the change in the coefficient of
c.
Answer to Problem 82SE
The value of
The coefficient on
Explanation of Solution
Calculation:
The exponential smoothing equation is
Substituting the value,
Substituting the value,
Continuing the same computational procedure till the value of
Now, the equation reduces as follows:
The value of
Now, the exponential smoothing equation
Here, from the above obtained equation it is seen that the value of
Here, form the equation it can be said that the coefficient of
Moreover, the smoothing constant
That is,
Hence the value of
Therefore, the coefficient on
c.
Explain the sensitivity of the initialization of
c.
Answer to Problem 82SE
The smoothed series
Explanation of Solution
Calculation:
From part (c), the exponential smoothing equation is,
The substitution of of
Here, form the equation it can be said that the coefficient of
Moreover, the smoothing constant
That is,
Hence the value of
Therefore, the smoothed series
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