balanced differencing hides the jolt
That is what I believe, and I did explain that to prove it isn't one has to answer the question of how the average about the data point in question can be higher than the previous average.
It's simple arithmetic. The discrepancy between the raw and the smoothed data comes about because of the smoothing procedure you're applying. It should be obvious that a smoothing algorithm removes spurious peaks and troughs in data; that is, after all, exactly what a smoothing algorithm is supposed to do. Unfortunately, it will just as easily remove significant peaks and troughs in data. Since you're looking for a significant trough, you shouldn't be smoothing the data. If your data is noisy enough that you need to smooth it, then it's too noisy to detect the presence or absence of the feature you're looking for.
This is first year undergraduate stuff, if that.
Because this is first year undergraduate stuff, it's easy to explain using a simple example. Suppose we have sampled the vertical position s of a particle or other object at times t=0, 1, 2, and so on. Suppose further that there is some controversy as to whether its velocity ds/dt ever goes negative. (For the purposes of this thread, we might think of a negative instantaneous velocity as a "jolt".)
Suppose further that, unbeknownst to us, the true position of the particle is given by s=1+t+cos(pi*t), which means the velocity really does go negative (there really are jolts, and they occur at regular intervals). As it happens, however, the resolution of our data is just barely good enough to reveal those jolts because we sampled the position at its Nyquist rate
WP. The true value of the position is as follows, with our sampled values shown by the plus marks:
The true velocity of this object ranges from a little under -2 to a little more than +4 (from 1-pi to 1+pi). If we calculate the velocity using simple backward differencing, we'll see the velocity alternating between -1 and +3. If we calculate the velocity using balanced (symmetric) differencing as was done by MacQueen and Szamboti, we'll see only the general trend, which is a constant velocity of +1:
As can be seen from this example, simple differencing tends to underestimate the extremal values of the instantaneous velocity, but it provides a far more accurate picture of the instantaneous velocity than we'd get from balanced differencing.
If we were to rely on balanced differencing only, as advocated by Tony Szamboti, then we would conclude that the object's velocity never goes negative, and there is no jolt. That conclusion would be false; the smoothing performed by balanced differencing would have led us astray.
