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This paper evaluates the forecasting performance of two nonlinear models, k-nearest neighbor (kNN) and feed-forward neural networks (FFNN), using stream flow data of the Kizilirmak River, the longest river in Turkey. For the kNN model, the required parameters are delay time, number of nearest neighbors and embedding dimension. The optimal delay time was obtained with the mutual information function; the number of nearest neighbors was obtained with the optimization process that minimizes RMSE as a function of the neighbor number and the embedding dimension was obtained with the correlation dimension method. The correlation dimension of the Kizilirmak River was d = 2.702, which was used in forming the input structure of the FFNN. The nearest integer above the correlation dimension (i.e., 3) provided the minimal number of required variables to characterize the system, and the maximum number of required variables was obtained with the nearest integer above the value 2d+1 (Takens, 1981) (i.e., 7). Two FFNN models were developed that incorporate 3 and 7 lagged discharge values and the predicted performance compared to that of the kNN model. The results showed that the kNN model was superior to the FFNN model in stream flow forecasting. However, as a result from the kNN model structure, the model failed in the prediction of peak values. Additionally, it was found that the correlation dimension (if it existed) could successfully be used in time series where the determination of the input structure is difficult because of high inter-dependency, as in stream flow time series. |
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