# A Neural Network Model for Estimating the Heart Rate Response to Constant Intensity Exercises
## 3 Results  
The evaluation of the performance of the NN model proposed herein is the last step in the flow chart of Figure 4, as indicated by the block at the lower right-hand corner of the chart. The testing procedure is performed with an independent dataset, which is kept unseen from the NNs during training. The testing dataset comprises samples of the form \(\big{(}tb_{k}^{te},hr_{k}^{te}\big{)}\), \(k=1\), \(2\), \(\ldots\), \(K\) with \(tb_{k}^{te}\) and \(hr_{k}^{te}\) being the input and the output, respectively, of the NN during testing; \(K\) stands for the number of samples used for testing. The testing mean absolute error is defined as follows:  
\[MAE_{te}=\frac{1}{K}\sum_{k=1}^{K}\left|hr_{k}^{pr}-hr_{k}^{te}\right| \tag{4}\]whereas the testing relative mean square error is given by:  
\[RMSE_{te}=\frac{1}{K}\sum_{k=1}^{K}\left(\frac{hr_{k}^{pr}-hr_{k}^{te}}{hr_{k}^{te }}\right)^{2} \tag{5}\]  
Both \(MAE_{te}\) and \(RMSE_{te}\) have been calculated for all exercise sets; their values are listed in Table 3. The latter indicates that the \(RMSE_{te}\) may vary between \(10^{-3}\) and \(10^{-5}\), while the \(MAE_{te}\) for all sets is 1.19 bpm on average; this is a satisfactory outcome, since results reported by other researchers, that employ techniques based on PPG signals to estimate HR, refer to average absolute errors of the order of 1.06 [4], 1.37 [5], 1.17 [6], 1 [7], and 1.03 [3] bpm. The main remark about Table 3 is that both the \(MAE_{te}\) and the \(RMSE_{te}\) are smaller during the exercise than the recovery period for all sets presented herein; a similar remark has been reported for the \(MSE_{tr}\). On average, the \(MAE_{te}\) is 0.84 bpm during exercise, whereas it increases to 1.54 bpm for the recovery periods. As regards the \(RMSE_{te}\), it deteriorates by roughly one order of magnitude for the recovery compared to the exercise periods. This may be attributed to the high non-linearity and the steep decrease of the HR during recovery, as well as to the inherent physiological noise, which is much more pronounced during the recovery than the exercise periods. Future work may focus on improving the accuracy of the model by applying the appropriate noise suppression techniques. A comparison of the HR predicted from the NN models with measured HR time series data is presented in Figure 5, for the exercise and recovery periods of sets (I) and (II). It is readily apparent from Figure 5 that the proposed NN models are able to predict the HR accurately. Moreover, the familiar remark that the performance of the NNs is slightly better during the exercise than the recovery periods may be verified by the diagrams of Figure 5. Still, the overall performance of all NNs is of sufficient accuracy, since the \(RMSE_{te}\) does not exceed \(10^{-3}\) for all cases examined, as indicated by Table 3.  
Figure 6 offers a comparison of results predicted by the proposed NN model (denoted as predicted hereafter) with the corresponding results produced by the mathematical model presented by [PERSON] [9, 10] (denoted as simulated hereafter). The exercise and recovery periods of sets (III) and (IV) are considered, while the measured HR time series data (henceforth denoted as measured) are also included for reference. A careful examination of the results in Figure 6 leads to the conclusion that the NN model provides more accurate predictions than the mathematical model. This remark is more apparent from the plots for the recovery period, i.e., for \(k>100\); where the NN model is more efficient in capturing the frequent variations of the \(hr\). It is worth mentioning that the dashed and dash-dotted lines in Figure 6 are representative examples of the HR values simulated by the dynamical systems model of Equation (3).