By R.L. Kashyap and A. Ramachandra Rao (Eds.)
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4 and they are not very good. The prediction error can be reduced to a fraction of these values by using the stochastic difference equation model discussed in Chapter XI. This example dramatically illustrates the limitations of pure deterministic modeling of any time series, even with the aid of complicated functions such as the logistic function. 3. 1810 1850 1890 1930 1970 2010 Year FIG. 3. Population of the United States. Logistic curve fitted so that observed points at 1840, 1900, and 1960 are exact.
Such models are usually very uneconomical in the sense that they may involve quite a large number of sinusoidal or polynomial terms. The more important criticism of such models is that their predictive ability is usually very small. For example, if we have a time series having 100 points and if we use the first 50 points to fit an nth-order polynomial to the data, and then use the resulting model to predict the values of the 51st observation, 52nd observation, and so on up to the 100th observation, then usually the prediction error so obtained will be extremely high.
L. Description The correlation function R(k) oflag k associated with a weak stationary ARMA equation is either an exponentially decaying function of k or a sum of terms every one of which decays exponentially with k. Jn modeling processes such as atmospheric turbulence, we need to consider models in which the correlation function R(k) decays at a rate slower than the exponential rate. , the rate is considerably slower than the exponential rate. Our interest in this class of processes arises because of the suggestion that many of the naturally occurring processes such as river flows are better handled by such models rather than ARMA models.