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Abstract:
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Nonlinear prediction of time series can offer potential accuracy gains over linear methods when the process is non-linear. As there are numerous examples of non-linearity in time series data (e.g., finance, macroeconomics, image and speech processing), there seems to be merit in developing a general theory and methodology. We explore the class of quadratic predictors, which directly generalize linear predictors, and show that they can be computed in terms of the second, third, and fourth auto-cumulant functions when the time series is stationary. The new formulas for quadratic predictors generalize the normal equations for linear prediction of stationary time series, and hence we obtain quadratic generalizations of the Yule-Walker equations; we explicitly quantify the prediction gains in quadratic over linear methods, and discuss the class of processes for which quadratic prediction is efficacious.
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