The sample data sometimes represents a timeseries, where value of a response variablex

Chapter 1, Problem 82

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The sample data sometimes represents a timeseries, where value of a response variablex at time t. Often the observed series shows a great deal ofrandom variation, which makes it difficult to study longertermbehavior. In such situations, it is desirable to producea smoothed version of the series. One technique for doing soinvolves exponential smoothing. The value of a smoothingconstant is chosen . Then withat time t, we set , and for, .a. Consider the following time series in whichof 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 a two-dimensionalcoordinate system (a time-series plot). Does there appearto be any pattern?b. Calculate the using . Repeat using .Which value of gives a smoother series?c. Substitute on the right-handside of the expression for , then substitute in termsof and , and so on. On how many of the valuesdoes depend? What happens to thecoefficient 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 Statisticsfor Interpreting Environmental Data, Water PollutionControl Fed. J., 1981: 167175.]

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