Tham khảo tài liệu 'handbook of empirical economics and finance _10', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 260 Handbook of Empirical Economics and Finance which means that the recursive form yields tighter intervals than the error correction form. Due to this fact the error correction form should not be considered in ITS forecasting. In addition the error correction representation is not equivalent to the ITS moving average with exponentially decreasing weights while the recursive form is. By backward substitution in Equation and for t large the simple exponential smoothing becomes t . t i - Z2 1 - -1 x t- j-1 which is a moving average with exponentially decreasing weights. Since the interval arithmetic subsumes the classical arithmetic the smoothing methods for ITS subsume those for classic time series so that if the intervals in the ITS are degenerated then the smoothing results will be identical to those obtained with the classical smoothing methods. When using Equation all the components of the interval center radius minimum and maximum are equally smoothed . xr t 1 axr t 1 - xr t where r e L U C R which means that in a smoothed ITS both the position and the width of the intervals will show less variability than in the original ITS and that the smoothing factor will be the same for all components of the interval. Additional smoothing procedures like exponential smoothing with trend or damped trend or seasonality can be adapted to ITS following the same principles presented in this section. k-NN Method The k-Nearest Neighbors k-NN method is a classic pattern recognition procedure that can be used for time series forecasting Yakowitz 1987 . The k-NN forecasting method in classic time series consists of two steps identification of the k sequences in the time series that are more similar to the current one and computation of the forecast as the weighted or unweighted average of the k-closest sequences determined in the previous step. The adaptation of the k-NN method to forecast ITS consists of the following steps 1. ThelTS x t with t
