[45,46] In the optimization procedure, the linear parameters of the consequent models were estimated Gamma-Secretase Inhibitors using the LS approach[47] while the parameters of the input membership functions were tuned using Levenberg–Marquardt nonlinear optimization algorithm.[38] For each subject and each MVC percentage, the best complexity (number of rules) was determined based on a 10-fold cross-validation procedure and complexity
analysis on the training data.[48] The range of rule numbers was specified between 4 and 11 rules, a-prior. It was observed that more than 11 rules caused over-fitting in many cases while using <4 rules, it was possible to capture the dynamics of the system. Then the model was produced with a given complexity, and finally it was evaluated using the test data. Finally, a two-sided 10 point
moving average filter was applied to the estimated torque signal to remove possible fluctuations. Validation For each trial, the difference between measured (y) and estimated (ỹ) torque signals was calculated using % Variance Accounted For (VAF) criteria.[49] The VAF formula is represented in the EQ.3. Moreover, a nonlinear dynamic model proposed by Clancy et al.[12] (3rd-degree polynomial, 28th-order dynamic model, whose model parameters were determined using the pseudo-inverse method), was implemented and applied on the same data sets for comparison. In each 100s trial, an epoch of 17 s of the torque signal (selected arbitrarily as to contain a flexion peak and
an extension peak and environs) was used for training and the rest of the samples were used as test data. RESULTS Here, the procedure used for selecting optimal number of fuzzy rules is discussed in details [Figure 1]: Displays the root mean square error (RMSE) central tendency and dispersion when changing the number of rules from 4 to 11 for the subject no. 4 at 70% MVC. Figure 1 10-fold cross-validation of the root mean square error versus the number of fuzzy rules for the 4th subject at 70% maximal voluntary contractions Based on the mean and standard deviation of the 10-fold cross-validation analysis, five and ten rules are possible candidates. However, when changing the number of fuzzy rules from 5 to 10, the number of unknown parameters in the FIS Brefeldin_A increases from 65 to 130 [Table 1]; thus increasing the probability of over-fitting. Table 1 The number of unknown parameters of the proposed fuzzy system for tuning as a function of number of fuzzy rules* The over-fitting problem could be assessed based on the model selection criteria. One of which is the Akaike information criterion (AIC)[50] whose cost function could be defined as: Where, VAIC is the AIC RMSE, VN is the RMSE in the training set, θ is the vector of the unknown parameters and N is the number of data samples used for training. Thus, there will be a penalty for increasing the number of unknown parameters.
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