82. 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 longerterm 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 , and for t
2, 3, . . . , n, .

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 (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 in terms of and , and so on. On how many of the values xt

, xt-1, . . . , x1 does depend? What happens to the coefficient on xt-k as k increases?

d. Refer to part (c). If t is large, how sensitive is to the initialization ? Explain.

[Note: A relevant reference is the article “Simple Statistics for Interpreting Environmental Data,” Water Pollution Control Fed. J., 1981: 167–175.]