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Is there a name for this property of a data series?
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.
posted by pont to Science & Nature (11 answers total) 1 user marked this as a favorite
 
I'm assuming you don't mean t+1 literally, by which I mean, that time isn't discrete but can be any (possibly positive or confined to some interval) real number. Then what you are saying is that if two times, t1 and t2, are near each other, so is (x[t1],y[t1]) and (x[t2],y[t2]).

If this is the property you want to name it is called "continuity."
posted by Obscure Reference at 6:16 AM on January 8


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]


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


Obscure Reference: Duh. Continuity, of course. I learned all about that once upon a time, but have had about 15 years to forget it now :). And yes, time is continuous, but my sampling are discrete (and for now I'm pretending that they're evenly spaced), which is probably what caused me to frame this problem in a slightly bizarre ass-backwards way.

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.

mbrock, Johnny Assay: I think my underlying process is actually differentiable as well, and has some constraint on how sharply it changes direction... but I'm going to see if I can produce anything useful just by invoking continuity before I tangle myself up in any other properties.
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]


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?

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 mathematical term for what you're talking about, but an engineer would say that your process has a limited slew rate- in short, it can only change so fast.
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


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] = et, (x[t+1] - x[t]) = (et+1 - et) = et(e-1), which grows without limit as t increases.
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


what measurements generated the data?
posted by serif at 12:04 AM on January 9


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