More smoothness metrics

Recently I've been looking at whether subtracting an appliance's known signature from the aggregate demand increases the smoothness of the curve. The success of such an approach is clearly dependant on what smoothness metric is used. I've already looked at the sum of the changes in power:

This is equivalent to taking the first derivative if the power values, because the denominator, dt, is always 1.

However, recently I've been troubled by the way this metric indicates an optimal match for the five graphs shown below. They are all possible combinations of the fridge's signature (red) with other appliances (blue). The vertical corresponds to power (W) while the horizontal corresponds to time (minutes).

However, we'd prefer the metric to return an optimal match for graphs A-D, and a sub-optimal match for graph E. This is because it is far less likely that another appliance of the same duration as the fridge will coincide with both the fridge's 'on' and 'off' transitions.

In an effort to quantify this, I realised that subtracting the fridge's signature reduces the number corners in graphs A-D, but not in E. Since a corner corresponds to a change in gradient, the second derivative of P might be a useful metric for smoothness:

The absolute value of the second derivative of P with respect to t will produce a positive value corresponding to each corner, and 0 otherwise. Therefore, the number non-zero values produced by this function is a measure of the noisiness of a curve. Taking the reciprocal provides us with the desired smoothness metric.