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 t = smoothed value at time t, we set 1 = x1, and for t = 2, 3,..., n, t = αxt + (1 - α) t-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 (a time-series plot). Does there appear to be any pattern?
b. Calculate the t s using α = .1. Repeat using α = .5. Which value of α gives a smoother t series?
c. Substitute t-1 = αxt-1 + (1 - α) t-2 on the right-hand side of the expression for t, then substitute t-2 in terms of xt-2 and t-3, and so on. On how many of the values xt, xt-1,..., x1 does t depend? What happens to the coefficient on xt-k as k increases?
d. Refer to part (c). If t is large, how sensitive is t to the initialization 1 = x1? Explain.