The sample data x1, x2,, xn sometimes represents atime

Chapter 1, Problem 82

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The sample data x1, x2,, xn sometimes represents atime series, where xt = the observed value of a responsevariable x at time t. Often the observed series shows a greatdeal of random variation, which makes it difficult to studylonger-term behavior. In such situations, it is desirable toproduce a smoothed version of the series. One techniquefor doing so involves exponential smoothing. Thevalue of a smoothing constant a is chosen (0 , a , 1).Then with xt = smoothed value at time t, we set x1 = x1,and for t = 2, 3,, n, xt = axt 1 (1 2 a)xt21.a. Consider the following time series in whichxt = temperature (8F) of effluent at a sewage treatmentplant on day t: 47, 54, 53, 50, 46, 46, 47, 50,51, 50, 46, 52, 50, 50. Plot each xt against t on atwo-dimensional coordinate system (a time-seriesplot). Does there appear to be any pattern?b. Calculate the xt s using a = .1. Repeat using a .5.Which value of a gives a smoother xt series?c. Substitute xt21 = axt21 1 (1 2 a)xt22 on the righthandside of the expression for xt, then substitute xt22in terms of xt22 and xt23, and so on. On how many ofthe values xt, xt21,, x1 does xt depend? What happensto the coefficient on xt2k as k increases?d. Refer to part (c). If t is large, how sensitive is xt tothe initialization x1 = x1? Explain.[Note: A relevant reference is the article SimpleStatistics for Interpreting Environmental Data,Water Pollution Control Fed. J., 1981: 167175.]