## 4 Discussion
It may now be instructive to take a focused look at our data via an example of \(k\)-fold division and evaluation. Each item in the dataset is a sample of about a dozen (the number varies between 10 and 15) 4-dimensional vectors each containing four coefficients of a Daubechies wavelet representation of the original 300-dimensional knee rotation measurement vectors. There are 21 samples from each of two disease classes, and each sample obtained from a distinct patient. This is a small dataset. Let us use \(k=5\) folds, a sensible size which would give about 4-5 elements in each 5-fold. Following our discussion, the recognition rates should be lower than with leave-one-out validation if the folds are non redundant, i.e., if, in general, the samples left out from training to be in testing are \"different\" from the training fold data. Figure 8 shows the results of the 21 5-fold cross validation experiments. Each 5-fold division was produced independently of the others. The horizontal axis lists the experiments from 1 to 21. The vertical axis unit is percentage correct classification, the star indicating the average performance in that experiment. The width of the vertical interval, centered about this mean, is twice the standard deviation of the cross validation recognition rate in the experiment. The overall average rate, computed over all the experiments, is 79%, with a standard deviation of 4.6 units of correct classification. These numbers are consistent with the expectations outlined in the discussion of \(k\)-fold validation above. We also ran a PCA plot of the original data to gain some insight into the dataset layout (Matlab _pca_ and _scatter_ routines). The first five coefficients of PCA account for 92% of the variance. Figure 9 shows the scatter plot for PCA coefficient pairs (1,2), (1,3), (1,4), and (1,5). These plots are sufficient to indicate that the data of the two classes (FR and FT) are neither redundant nor do they cluster away from each other, i.e., the classification problem in this application is not trivial. In addition, the spread of the data of each class that the plots show is consistent with the variations in the classification results of the 21 5-fold cross validation experiments shown in Figure 8, confirming that, in general, the test data used in an experiment contains information not present in the training data. The spatial ordering of the maps is evident in Figure 6. In this display, neighboring node values are assigned neighboring colors. In each sub-figure, neighboring node values describe the same pathology class. Figure 8: Five-fold cross validation experimentation. Horizontal axis: the experiments from 1 to 21. Vertical axis: the unit is percentage correct classification, the star indicating the average performance in that experiment. The width of the vertical interval about the mean is twice the standard deviation of the cross validation recognition rate in the experiment. The overall average rate, computed over all the experiments, is 79%, with a standard deviation of 4.6 units of correct classification. Figure 9: Scatter plots for PCA coefficient pairs (1,2), (1,3), (1,4), and (1,5) indicate that the data of the two classes (FR and FT) are neither redundant nor do they cluster away from each other. We note that FR class labeled 2 is characterized by high values of the weight of the abduction/adduction angle (yellow color in Figure 6a,b) and low values of the internal/external rotation angle (blue color in Figure 6c,d). In contrast, that FT class labeled 1 is characterized by low values of the weight of the abduction/adduction angle (yellow color) and high value of the internal/external rotation angle (blue color). This is an interesting result because it correlates the kinematic data signals and their time-frequency transforms via wavelets to better understand the pathological knee behavior. The classification time is marginal (0.01 s/sample). This can be important in computer-aided medical analysis.