It's been a couple years since I studied calculus but I wondered if it was possible to calculate the double differential of a set of discrete data points, and if so, how do I go about doing this. What I mean is:
If I have a sequence of six data points: [1,2,3,4,5,6], can I process these in some way so that I find out what the second order differential is in the middle of this sequence (i.e. between 3 and 4). Obviously for this data it's 0 but I wondered if there was a standard method that could be applied to any sequence?
Something in my head says it's to do with the Taylor series or finite difference, but I am struggling to follow them and I might be way off.
Thanks so much.
Sorry, I might be completely missunderstanding but I thought that a step function was only for single dimensional data? My understanding was that as my data is a sequence it could be almost be considered as a set of six two dimensional points.
i.e. if I take the sequence [a,b,c,d,e] then the points would be: [(1,a),(2,b),(3,c),(4,d),(5,e)].
Does that make any sense?
There are a number of different methods you might try. Derivatives are not defined at a single point, so you'd have to "connect the dots" in some way in order to get a continuous function (which is what you were talking about anyway: you seem to want the second derivative in-between data points). You have to decide how you want to do that. If you need the second derivative to be continuous, then you might try cubic splines to interpolate your data. If you go that route, pay attention to how you want the splines to look at the endpoints of your intervals.