samples/texts/1072043/page_11.md

Nonlinear behavior characteristics. For example, linear interval of saturation, zero interval of dead-zone, wavenet, sigmoid network requiring the simplest and less complex function to explain the system. Model properties, estimators, percentage of accuracy, final Prediction Error-FPE and Akaikae Information Criterion-AIC are as follows [58]:

Fig. 20. Data divided into Estimated and validated data

Criteria for Model selection

The percentage of the best fit accuracy in equation (13) is obtained from comparison between experimental waveform and simulation modeling waveform.

$$\text{Best fit}=100\ast \left(\frac{1-\mathrm{norm}(y^{\ast }-y)}{\mathrm{norm}(\bar{y}-{\bar{y}}^{\ast })}\right)(13)\backslash text\{Best\; fit\}\; =\; 100\; *\; \backslash left(\; \backslash frac\{1\; -\; \backslash operatorname\{norm\}(y^*\; -\; y)\}\{\backslash operatorname\{norm\}(\backslash bar\{y\}\; -\; \backslash bar\{y\}^*)\}\; \backslash right)\; \backslash quad\; (13)$$

where $y^*$ is the simulated output, $y$ is the measured output and $\bar{y}$ is the mean of output. FPE is the Akaike Final Prediction Error for the estimated model, of which the error calculation is defined as equation (14)

$$FPE=V\left(\frac{1+d\mathrm{/}N}{1-d\mathrm{/}N}\right)(14)FPE\; =\; V\; \backslash left(\; \backslash frac\{1\; +\; d/N\}\{1\; -\; d/N\}\; \backslash right)\; \backslash qquad\; (14)$$

where V is the loss function, d is the number of estimated parameters, N is the number of estimation data. The loss function V is defined in Equation (15) where θ_N represents the estimated parameters.

$$V=\det (\frac{1}{N}\sum _1^N\epsilon (t,{\theta }_N)(\epsilon (t,{\theta }_N){)}^T)(15)V\; =\; \backslash det\; \backslash left(\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{1\}^\{N\}\; \backslash varepsilon(t,\; \backslash theta\_\{N\})\; (\backslash varepsilon(t,\; \backslash theta\_\{N\}))^\{T\}\; \backslash right)\; \backslash quad\; (15)$$

The Final Prediction Error (FPE) provides a measure of a model quality by simulating situations where the model is tested on a different data set. The Akaike Information