79. 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 dif cult 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 a is chosen (0  a  1). Then with 

smoothed value at time t, we set  x1, and for t 

2, 3, . . . , n, .

a. Consider the following time series in which xt 

temperature (F) of ef uent 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 twodimensional coordinate system (a time-series plot). Does there appear to be any pattern?

b. Calculate the s using a  .1. Repeat using a  .5. Which value of a gives a smoother series?

c. Substitute on the right-hand side of the expression for , then substitute xt2 in terms of xt2 and xt3, and so on.On how many of the values xt, xt1, . . . , x1 does depend?
What happens to the coef cient on xtk as k increases?

d. Refer to part (c). If t is large, how sensitive is to the initialization  x1? Explain.
(Note: A relevant reference is the article Simple Statistics for Interpreting Environmental Data, Water Pollution Control Fed. J., 1981: 167—175.)