Paul
Posts:
258
Registered:
7/12/10
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Re: A natural definition of smoothness?
Posted:
Feb 21, 2012 9:31 AM
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On Feb 21, 12:57 pm, David C. Ullrich <ullr...@math.okstate.edu>
wrote:
> On Mon, 20 Feb 2012 10:17:36 -0800 (PST), Paul <pepste...@gmail.com>
> wrote:
>
> >I would like to analyse how the price of something changes through
> >time. Hence I get a time series of prices P0, P1, P2 etc., consisting
> >of prices at time 0, time 1, time 2 etc.
>
> >The term "price" can be defined in many ways.
>
> Was "price" a typo for "smooth"? If you really meant "price" I
> don't follow this at all - I thought the prices P_j were _given_,
> not something to be defined.
>
> >I want to select the
> >definition where the above time series (which can be thought of as a
> >graph of points (t, Pt) ) is as smooth as possible. For example, a
> >perfect polynomial approximation would be ideal.
>
> >What is a standard definition of "smooth" to use in the above so that
> >the notion of finding the smoothest possible definition of P is a
> >meaningful notion?
>
> The way you state this is not so clear to me. I imagine that
> what you meant is that you have a bunch of observed data
> points and you want to find the "smoothest" curve that
> fits the observed data "as well as possible".
>
> The bad news is also good news: There are hundreds of
> standard definitions of "smooth" and hundreds of standard
> definitions of "best fit".
>
> One comment: If by "perfect polynomial approximation"
> you mean a polynomial that fits the observed data exactly,
> that's almost certainly _not_ what you want!
>
> An obvious reason why: Suppose the observed
> data points are all very close to a straight line.
> Then probably what you want is the straight line
> that best fits the data - least squares is a very common
> notion of "best fit" here, that has the advantage that
> it's easy to deal with computationally.
>
> On the other hand, if you take those n data points
> (which we're still assuming lie close to a straight
> line) and find a polynomial that passes through all of them
> exactly, that will be a polynomial of degree n-1. Assuming
> that the "real" function is in fact a straight line, and the
> deviations from that line in the observed data are due to
> various sorts of "error", that polynomial is going to be
> infinitely far from the truth for large t; no polynomial of
> degree n-1 gives a good approximation to a straight line.
>
> That's just a crude reason why the interpolating polynomial
> is probably not what you want, which has the advantage
> that it's easy to explain. In fact even if you're just
> considering t in some bounded interval, the polynomial
> that passes through all the observed data points exactly
> is typically going to have all sorts of wild fluctuations
> that you don't want - the "best fit" is going to be a
> polynomial of much lower degree that just comes close
> to the observed data.
>
>
>
Thanks a lot to all who replied. Actually, "price" was not a typo,
but was unclear. Apologies for my lack of clarity. More precisely,
there are many conventions for defining the price of a stock through
time -- and they all give different results.
Some examples to illustrate the point: price could mean "last price at
which the stock traded." Price could mean "(mean bid price in the last
second + mean ask price in the last second)/2". Price could also mean
"(median bid price in the last second + median ask price in the last
second)/2"
But there are many more possible definitions. I'm interested in the
definition of "price" which maximises the "smoothness" of the
resulting definition of price. However, my problem (one of them,
anyway) is that I'm unsure which definition of "smooth" to use.
Thank you,
Paul Epstein
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