Fig. 1. Self-organizing feature map used for making predictions.

Fig. 2. Illustration of the modified U-matrix method. All six possible differences of weights of four neighboring neurons are used. Their root mean square is assigned to the center of the neurons.

Fig. 3. Validation errors (upper solid line) and training errors (lower solid line) were calculated after each learning epoch. Some of the training errors were calculated according to the quadratic exponential law of decrease for , while the other training errors (lower dashed line) were determined according to a linear exponential law of decrease for . The square in the figure surrounds the temporal global minimum for the validation errors, and the long vertical line indicates the time point from which the cut-off condition is fulfilled. A univariate temporal pattern is used. Therefore, validation and prediction errors are identical.

Fig. 4. Sequence of the U-matrices corresponding to the errors in Fig. 3. Network states are shown after selected numbers of the learning epochs. The state of initialization is omitted. The effects of the break-off criterion are: the minimum is after 47, and the moment the cut-off occurs is after 49 epochs.

Fig. 5. Weights vectors of a Kohonen network which has finished learning according to Fig. 3 and Fig. 4. The upper figure explains the lower figure. Weight vectors of the outermost corners and the innermost kernel of the Kohonen network have been merged together to illustrate the principle how the network has self-organized in this case. The temporal pattern is univariate. Therefore, the weights can be represented as time series.

Fig. 6. Distributions of residues for predicting anomalies during high tide at Cuxhaven using four models: (1) the optimal MR model; (2) the Sea Level Forecast Service; (3) the hydrodynamic sea level model; and (4) the statistical model of the “Gesamtansatz”. The bin size is 10 cm.

Table 1. General description of the neural models

Table 2.

Full-size image (<1K)

Table 3. Specified neural models

Table 4. General description of the models of comparison

Table 5. Hourly and derived hindcasts

Table 6. Direct hindcasts at times of high/low tides