# What statistics to use when the data are polynomial functions (aka change over time)?

EDIT: OK, having spent the last couple of hours checking out some pubs on functional data analysis, I am actually starting to feel a little silly having asked the question in the first place! It's becoming obvious (well. Still a tad -ish but getting there.) - Thanks, @PeterFlom for pointing it out! I'll leave the question up for now, should anyone see something worth looking out for as I venture into FDA with my data...?

Here's what I have (this is for voice analysis):

Suppose I have three speakers; each speaker produces say 10 tokens of a particular vowel; I've measured each token and fitted time-normalised 3rd order polynomial curves to the vowel formants (I'm using three formants, which in layman's terms are the frequencies determining the vowel sound, which change over time; the formants are determined by vocal tract shape, so they are interrelated). I have also calculated the average polynomial function to each formant for each speaker, to give me a general idea of what this vowel looks like for each speaker on average, leaving me with a set of three related polynomial functions (one for each vowel formant) for each of the three speakers.

Pictures might help clarify! The first figure shows 10 individual tokens (three time-normalised polynomials (because three formants) per token) for three speakers; the second figure shows the average polynomials for each formant for each speaker. (I've indicated the frequency range of each formant, hope that makes a little clearer what I'm talking about!):

(You can see that the degree of variability is quite massive within a single speaker; but you can still guess from the graphs that speakers 1 & 2 are very similar, at least in terms of these particular vowel tokens (our guess is they're the same speaker), and speaker 3 is someone else (this we happen to know).)

To start with, I would be happy if I could find a way that would give me a basic distance measure between the average polynomials for each speaker. (Something to say, but in numerical terms, e.g. that speaker 1 and speaker 2's formants look very similar to each other (at least with regards to the first and second formants), but quite different to speaker 3's formants.)

Ideally, I want to be able to compare all the speakers' tokens, where each token is described by three polynomial functions, and get a probability to tell me how likely it is that the speakers are the same, based on their productions for this vowel (i.e. taking into account within-speaker variability). But if someone could just help me with my first question I just might be able to figure everything else out from there!

Thank you so much!

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I am not sure but have you looked at functional data analysis? e.g Ramsay and Silverman – Peter Flom Nov 3 '12 at 19:18
Oh hey, that sounds just like what I should be looking at, never even heard of functional data analysis before! Thank you, that should at least give me somewhere to start. – crs Nov 3 '12 at 19:37