[R] newbie: fourier series for time series data

[Mike Marchywka]

( hotmail won't mark text so I'm top posting... )
 
Can you post the data? Personally I'd just plot abs(fft(x))
and see what you see, as well as looking at Im(fft(x))/Re(fft(x))
or phase spectrum. Now, presumably you are nominally looking
for something with a period of 1 year, that part could
be expressed in harmonic specrum as suggested below, but you'd
also be looking for trends and noises of various types- additive
gausssian, amplitude modulation, maybe even frequncy modulation, etc.
I guess you could remove a few power terms ( average, linear, etc)
just to simplify ( often you get this spectrum with huge uniformative
DC or zero frequency component just gets in the way). You can probably
find book online dealing with signal processing and RF ( this is where
I'm used to seeing this things). It would of course be helpful to 
then examine known simple cases and see if you can tell them apart.
Create fft( sin(t)*(1+a*sin(epsilon*t)+b*t ) for example. 


I guess if you want to look at writing a model, you could look at
phase portrait ( plot derviative versus value ) to again get some idea what 
you may have that makes sense as model to fit.



----------------------------------------
Date: Tue, 31 May 2011 10:35:16 -0700
From: spencer.graves at structuremonitoring.com
To: eddieatr at gmail.com
CC: r-help at r-project.org
Subject: Re: [R] newbie: fourier series for time series data


On 5/31/2011 5:12 AM, eddie smith wrote:
> Hi Guys,
>
> I had a monthly time series's data of land temperature from 1980 to 2008.
> After plotting a scatter diagram, it seems that annually, there is a semi
> sinusoidal cycle. How do I run Fourier's series to the data so that I can
> fit model on it?

There are several methods.


1. The simplest would be to select the number of terms you
want, put the data into a data.frame, and use lm(y ~ sin(t/period) +
cos(t/period) + sin(2*t/period) + cos(2*t/period) + ..., data),
including as many terms as you want in the series. This is not
recommended, because it ignores the time series effects and does not
apply a smoothness penalty to the Fourier approximation.


2. A second is to use the 'fda' package. Examples are
provided (even indexed) in Ramsay, Hooker and Graves (2009) Functional
Data Analysis with R and Matlab (Springer). This is probably what
Ramsay and Hooker would do, but I wouldn't, because it doesn't treat the
time series as a time series. It also requires more work on your part.


3. A third general class of approaches uses Kalman
filtering, also called dynamic linear models or state space models.
This would allow you to estimate a differential equation model, whose
solution could be a damped sinusoid. It would also allow you to
estimate regression coefficients of a finite Fourier series but without
the smoothness penalty you would get with 'fda'. For this, I recommend
the 'dlm' package with its vignette and companion book, Petris, Petrone
and Campagnoli (2009) Dynamic Linear Models with R (Springer).


If you want something quick and dirty, you might want option 1.
For that, I might use option 2, because I know and understand it
moderately well (being third author on the book). However, if you
really want to understand time series, I recommend option 3. That has
the additional advantage that I think it would have the greatest chances
of acceptance in a refereed academic journal of the three approaches.


> I am really sorry for my question sound stupid, but I just don't know where
> to start.

There also are specialized email lists that you might consider
for a future post. Go to www.r-project.org 
-> "Mailing Lists". In particular, you might be most interested in
R-sig-ecology.


Hope this helps.
Spencer Graves

> I am desperately looking for help from you guys.
>
> Thanks in advance.
>
> Eddie
>
> [[alternative HTML version deleted]]
>
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--
Spencer Graves, PE, PhD
President and Chief Operating Officer
Structure Inspection and Monitoring, Inc.
751 Emerson Ct.
San José, CA 95126
ph: 408-655-4567


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