Question

The sample data x1,x2,...,xn sometimes represents a time series, where xt = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with ¯ xt = smoothed value at time t, we set ¯ x1 = x1, and for t = 2,3,...,n, ¯ xt = αxt + (1−α)¯ xt−1.

(a) Consider the following time series in which xt = temperature (◦F) of effluent at a sewage treatment plant on day t : 47,54,53,50,46,46,47,50,51,50,46,52,50,50. Plot each xt against t on a two-dimensional coordinate system. Does there appear to be any pattern?

(b) Calculate the ¯ xts using α = 0.1. Plot each ¯ xt against t on a twodimensional coordinate system. Repeat using α = 0.5. Which value of α gives a smoother ¯ xt series?

Answer 1

The time series temperature dataset has a range of values from 46 to 54. Plot the time series data with the y-axis having values between 40 and 60

Temperature Time Series Plot 60 58 56 54 C 50 48 ー46 42 40 0 4 10 12 14 lime

Looking at the time series plot, the data appears to have somewhat of a cyclical pattern similar to a sine wave.

Use the exponential smoothing technique with α-0.1 to determine the-s. Recall that 耳=f -24. Compute x2 Compute =47.7 Use the same method to compute X 48.23

Repeat this process to get a table for all values in the time series dataset. Exponential Smoothing with a- 0.1 147 2 47.7 3 48.23 4 48.41 5 48.17 6| 47.95 7 47.85 txt 848.07 948.36 10 48.53 1148.27 12 48.65 13 48.78 14 48.90

Repeat the exponential smoothing technique with a-0.5 to obtain another table of data. The process is exactly the same as before, only with a different value of α Exponential Smoothing with a-0.5 1 47 2 50.5 3 51.75 450.88 5 48.44 6 47.22 747.11 tXt 848.55 9 49.78 1049.89 1147.94 12 49.97 13|49.99 14 49.99

The first series with α=0.1 seems to be smoother than the second series with α=0.5 The first series is smoother because there is less variability between values in the table. Additionally, the first series switches between increasing and decreasing less frequently When the first series does switch between increasing and decreasing, the magnitude of the switch is much less than that of the switches in the second series Therefore, after examining both series, α-0.1 yields a smoother series.