## 3 Results and Discussion
### ANN Prediction  
In this article, we have used values of laser power 100 and 150 W, the number of annual rings 3 to 11 and cutting speed (3, 6, 9)\(\cdot\)10\({}^{-3}\) m\(\cdot\)s\({}^{-1}\) as input values for teaching and measured values of WKL, WKU, WHAZL and WHAZU and WKR, thus the training groups have 54 lines. By routine, we obtained the five best artificial neural networks for prediction of WKU, WHAZU, WKL, WHAZL and WKR.  
In the measured values teaching and generalization we have used all the basic neural networks used in the Statistica 12 software. According to a characteristic sum of squares, we obtained the five best neural networks which are shown in Table 4. All the studied multilayer perceptron networks used the Quasi-Newton training algorithm and the best neural network MLP 3-3-5 used the BFGS 73 training algorithm with error function sum of squares. The hidden layer was activated by logistic function and the output activation function was exponential. The number of the hidden neurons' possibilities were between 1 and 54 (length of dataset).  
In this article we will present the results of the \(P\) and \(AR\) effect on the abovementioned parameters of wood cutting for all the possible parameters of CO\({}_{2}\) lasers from which it is possible to predict Picea abies wood cutting properties at any laser power between 100 and 500 W and the number of annual rings between 3 and 11 with a goal to optimize the cutting process. The results are given for Picea abies wood, therefore they can be successfully used for CO\({}_{2}\) laser cutting of Picea abies.  
Artificial neural networks have been used for prediction of WKU, WKL, WKR, WHAZU, WHAZL parameters for input parameters \(P\) = 200, 300, 400, 500 W, cutting speed \(v\) = 1.2\(\cdot\)10\({}^{-2}\) m\(\cdot\)s\({}^{-1}\) and the number of annual rings AR = 3, 4, 5,..., 11. The error propagation of the networks are presented in Table 4. All the types of errors are decimal form of variance coefficient.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Net**} & **Training Error** & **Testing Error** & **Validation Error** \\  & **(-)** & **(-)** & **(-)** \\ \hline MLP 3-7-5 & 0.019 & 0.023 & 0.022 \\ \hline MLP 3-3-5 & 0.010 & 0.013 & 0.011 \\ \hline MLP 3-10-5 & 0.018 & 0.022 & 0.022 \\ \hline MLP 3-6-5 & 0.004 & 0.014 & 0.011 \\ \hline MLP 3-10-5 & 0.030 & 0.030 & 0.036 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Error propagation in the best neural networks.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Value at \(AR\)**} & **WKU (10\({}^{-3}\) m)** & **WKL (10\({}^{-3}\) m)** & **WKR (-)** & **WHAZU (10\({}^{-3}\) m)** & **WHAZL (10\({}^{-3}\) m)** \\ \hline
3 & \(0.90\pm 0.24\) & \(0.57\pm 0.15\) & \(0.67\pm 0.19\) & \(0.17\pm 0.03\) & \(0.21\pm 0.06\) \\ \hline
4 & \(0.91\pm 0.23\) & \(0.57\pm 0.14\) & \(  
From Table 4, it is good to see that the lowest error (the standard is the mean square error) is obtained for network MLP 3-3-5 which was also found to be the best according to the slope of the predicted vs. measured values, coefficient of determination R\({}^{2}\), root mean square error RMSE and relative root mean square error Rel_RMSE, which are shown in Table 5.  
From Table 5, we can conclude:  
* The coefficient of determination R\({}^{2}\) for all the output parameters dependent on the input parameters is at least 0.933.
* The slope which should be close to 1 is for all the studied parameters higher than 0.94, which is a very good result.
* The maximum value of Rel_RMSE which in percentage is equal to 3.5% is highly under the measuring accuracy from which we can conclude that ANN can predict with accuracy all the studied cutting results' parameters.
* ANN can teach and predict with very high accuracy all the studied parameters.
* In the next three chapters we will deal with prediction of (WKU, WHAZU), (WKL, WHAZL) and WKR vs. \(P\) and AR and discuss the effects of \(P\) and AR on all the predicted parameters.