82. The sample data sometimes represents a time series, where 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 is chosen . Then with at time t, we set , and for

, .

a. Consider the following time series in which 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 using . Repeat using .

Which value of gives a smoother x series? a t x a 5 .1 a 5 .5 t ’s xt 5 temperature (8F)

xt 5 axt 1 (1 2 a)xt 5 2, 3, t21

c, n x x1 5 x1 t 5 smoothed value a (0 , a , 1)

xt 5 the observed x1, x2,

c, xn 90th percentile 5 640 95th percentile 5 720 5th percentile 5 400 10th percentile 5 430 sd 5 96 minimum 5 220 maximum 5 925 mean 5 535 median 5 500 mode 5 500

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 does depend? What happens to the coefficient on 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.]