# Relation between original points and 1st/2nd derivative points

I have points $$\{p_0, p_1, ... p_n\}$$, I create a discrete derivative consisting of $$\{d_0, d_1, ..., d_{n-1}\}$$ like this: $$d_k=p_{k+1}-p_k$$. I'd like to choose a point in the original signal by exploring the derivative. But after I find a specific point $$d_k$$, which point from the original set should I choose: $$p_k$$ or $$p_{k+1}$$? It looks like I can't really choose between them.

Similarly I explore 2nd derivative for inflection points. But in 2nd derivative I'm guessing that $$p_{k+1}$$ is a sensible choice because it's a midpoint out of 3 participants ($$p_k, p_{k+1}, p_{k+2}$$). Though this is still not perfect because of the 1st and last points in the set.

What are possible strategies to resolve this ambiguity?

• that depends on how you calculated the discrete derivative! How did you do that? – Marcus Müller May 10 '20 at 12:25
• Huh, I thought there was only 1 way, sorry. Added the algorithm. – Stanislav Bashkyrtsev May 10 '20 at 13:13

Numerical signals are discrete and finite. So you are likely to hit issues either on end-points or discrete locations. I won't treat end-points here.

If you use only one operator, many signal processing folks care relatively little about a shift, as long as it is known, and constant. Here, the two-point derivative yields a $$0.5$$-point shift. You can consider that the derivative is valid at some mid-point. And you can interpolate it an integer locations if needed. Smoothing first can help limit a too-high fluctuations of differences.

A classical option is to choose an odd-length derivative operator. The three-point derivative writes:

$$d_3[k] = (p[k+1]-p[k-1])/2$$

and gives you an information at the center of the schemes. Indeed, it coincides with the average of the left- and the right two-points skewed differences: $$d_2[k] = p[k]-p[k-1]$$ and $$d_2[k+1] = p[k+1]-p[k]$$, since:

$$d_3[k] = (d_2[k] +d_2[k+1])/2$$

Many domains uses such schemes, and name them differently. To better find resources, some uses other terms, like generic ones, like "Finite difference coefficients", or specific ones, like "five-point stencil" or $$n$$-point gradient/Laplacian. One example is:

$$d_5[k] = (p[k+2]- 8p[k+1] + 8p[k-1]-p[k-2])/12$$

The above thinking extends to higher order derivatives. One can further constraints methods using data properties, knowledge of noise, additional penalties, etc. To start with, some literature on numerical derivatives: