As will be shown next, the kernel volume defined by this surface is equal to the number of points (sequence units), N, of the kernel-training dataset x. This result, strictly considered, disqualifies K as a kernel density function as kernel density volumes are unitary by definition. There are a number of reasons why having a volume that is the number of sequence units is desirable, particularly when sequences of different lengths are being compared. A compliant alternative definition of K is in any case obtained by dividing the expression in Equation 2, by the total length of the training sequences, N. This alternative will not be explicitly explored here because the scale alteration is so straight forward that it can easily be applied to any of the results reported here. The 2D density plots are offered without a scale in the z-axis to highlight the inconsequence of the correction. On the other hand, when multiple sequences are plotted together, as in Figure 4, the effect is that that the same motif in two sequences is represented with the same density height, Equation 3, even if the two sequences have very different lengths.