January 8, 2014 5:40 AM Subscribe

I have a series of physical data with a fairly straightforward property: it follows a path in space. If I plot a bunch of successive (x, y) values from the data, they follow a curving, looping line rather than randomly jumping all over the plot. More formally, if I have a point (x[t], y[t]) at time t and (x[t+1], y[t+1]) at time t+1, there is an upper limit on the distance between these two points (and this limit is small relative to the overall variation in the data). Is there a well-defined technical term for this property?

If the variation were random I'd call it a random walk. But it's governed by a bunch of complex physical processes that aren't relevant to the analysis I'm trying to do: the only thing I'm currently interested in is the "walk" property. Of course it's easy to give an ad-hoc formalization as I did above, but I'm sure there must be some standard terminology for this, and maybe some general results on series with these properties. I've skimmed several works on random walks and time-series analysis without finding a suitable term. I'm sure that this wheel has been invented before; what have I missed? Links and references very welcome.

If the variation were random I'd call it a random walk. But it's governed by a bunch of complex physical processes that aren't relevant to the analysis I'm trying to do: the only thing I'm currently interested in is the "walk" property. Of course it's easy to give an ad-hoc formalization as I did above, but I'm sure there must be some standard terminology for this, and maybe some general results on series with these properties. I've skimmed several works on random walks and time-series analysis without finding a suitable term. I'm sure that this wheel has been invented before; what have I missed? Links and references very welcome.

A stronger property than continuity is differentiability, which basically means that your function has a well-defined "direction" at every point. For example, if you look at the function abs(x), it's continuous because the plot has no jumps in value, and it's differentiable at every point except x=0.

The set of continuous functions includes some very strange functions, like the "Weierstrass function" which is continuous yet not differentiable*anywhere*.

posted by mbrock at 6:39 AM on January 8 [1 favorite]

The set of continuous functions includes some very strange functions, like the "Weierstrass function" which is continuous yet not differentiable

posted by mbrock at 6:39 AM on January 8 [1 favorite]

If this is a discrete time series, you might also look into the correlation between the *changes* between successive data points. You could define (∆x[t], ∆y[t]) = (x[t+1],y[t+1]) - (x[t], y[t]). If the process was truly random, there wouldn't be any correlation between (∆x[t], ∆y[t]) and (∆x[t+1], ∆y[t+1]). However, you seem to be describing a process in which a given "step" is more likely to be in the direction of the previous "step" (i.e., the "steps" don't abruptly change direction very much.)

posted by Johnny Assay at 6:44 AM on January 8

posted by Johnny Assay at 6:44 AM on January 8

So, my interest is actually in a discretely sampled time-series of an underlying continuous process. If anyone happens to know good resources along those lines, I'd be glad of them, but many thanks in any case for the hefty shove in the right direction.

posted by pont at 6:51 AM on January 8

Any sequence of points is a discrete sample from a continuous (or even differentiable) process, so that's not going to help you much. If it's true that the distance between (x[t], y[t]) and (x[t+1], y[t+1]) is bounded independently of t, then what you might want to think about is a continuous process with bounded derivative -- i.e. it has a certain maximum speed.

posted by escabeche at 7:23 AM on January 8 [1 favorite]

posted by escabeche at 7:23 AM on January 8 [1 favorite]

This is not a description of continuity, though continuous functions can have this property. You simply cannot infer whether a signal is continuous from discrete samples of that signal. I don't know of a

posted by Jpfed at 7:31 AM on January 8 [1 favorite]

You simply cannot infer whether a signal is continuous from discrete samples of that signal.Thank you; filed under "in retrospect, that should have been obvious" :). OK, I'm clearly not going to get what I wanted purely from the property of continuity. Back to the drawing board, but now at least with a slightly clarified notion of what I'm trying to draw.

posted by pont at 8:01 AM on January 8

I think you are also concerned with "smooth" data. Take a look at splines for example, they have some constraints on the value of second derivatives which can be used to interpolate a set of discrete data with a set of curves which look "smooth" and natural to an observer - apparently we are really good at detecting changes in the second derivative of plotted curves...

posted by NoDef at 8:50 AM on January 8

posted by NoDef at 8:50 AM on January 8

I think escabeche's suggestion of a speed limit--a limited first derivative-- is the way to go; if either or both x and y are simple exponentials in t, for example, you could have continuity *and* unbounded distance between (x[t], y[t]) and (x[t+1], y[t+1]) for very large t:

if x[t] = e^{t}, (x[t+1] - x[t]) = (e^{t+1} - e^{t}) = e^{t}(e-1), which grows without limit as t increases.

posted by jamjam at 9:44 AM on January 8

if x[t] = e

posted by jamjam at 9:44 AM on January 8

I agree with escabeche that "bounded first derivative" is what you described. I'd also think that the time series of x(t) and y(t) should be autocorrelated -- you're more likely to be near where you were a short time ago.

As for further reading, try keywords "analysis of animal movement". This turns up

Turchin, P. 1998. Quantitative Analysis of Movement: measuring and modeling population redistribution in plants and animals. Sinauer Associates, Sunderland, MA.

and a lot of more recent references.

posted by Ian Scuffling at 7:04 PM on January 8

As for further reading, try keywords "analysis of animal movement". This turns up

Turchin, P. 1998. Quantitative Analysis of Movement: measuring and modeling population redistribution in plants and animals. Sinauer Associates, Sunderland, MA.

and a lot of more recent references.

posted by Ian Scuffling at 7:04 PM on January 8

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If this is the property you want to name it is called "continuity."

posted by Obscure Reference at 6:16 AM on January 8