The sample data x1, X2..., Xn sometimes represents a time series, where 1 = 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 lite series. One technigue for doing so involves exponential smoothing.The value of a smoothing constant a is chosen (0 < a < 1). Then with 1 smoothed value at time t, we set 1 = x1, and for t = 2, 3..., n, X1 = ax1 + (1 - a) xt-1.
a. Consider the following time series in which x1 = 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 x1 against t on a two dimensional coordinate system (a time-series plot) Does there appear to be any pattern?
b. Calculate the 1s using a = 1. Repeal using a = 5, which value of a gives a smoother series?
c. Substitute X1-1 = ax1-1 + (1 - a)1-2 on the right-hand side of the expression for X1 then substitute 1-2 in terms of 1-2 und 1-3 and so on. On how many of the values 1, 1-1..., 1 does x depend? What happens to the coefficient on X.-k as k increases?
d. Refer to part (c). If t is large, how sensitive is x, to the initialization 1 = X,? Explain.
