# Approximating temperature data beyond Fourier series [closed]

I have some hourly/daily temperature data at:

and want to approximate it using a "simple" mathematical function. Notes:

If you look at the mean daily temperature graph for KABQ, it looks like a sine wave with a period of one year.

Similarly, the hourly temperature for a given day in KABQ also looks like a sine wave with a period of one day.

However, closer inspection (Fourier analysis) shows that they're not. There are fairly strong components of frequency 2/year and 3/year for the daily temperature, and the hourly temperature also has strong non-single-period terms.

Is there a function that describes the daily mean temperature and (a separate function) the hourly mean temperature somewhat accurately?

The function would have constants (parameters) that would change for non-KABQ locations.

I realize I can keep taking more Fourier terms to increase accuracy, but I was hoping for something more elegant. For example, maybe the graph is a parametrized version of sin^2(x) or some other "well-known" function.

EDIT: Below's a hopefully better example of what I mean.

You take the Fourier transform of some data and the first 15 terms are:

2.82831
3.36979 + 0.322402 I
3.79182 + 2.34689 I
0.73987 + 4.71806 I
-3.16205 + 0.709054 I
2.76656 - 1.86274 I
-0.762084 + 3.94561 I
1.98004 - 2.6786 I
-2.51665 + 2.21764 I
3.6131 + 2.29278 I
2.79673 - 2.55569 I
-0.455474 - 2.7534 I
-1.51359 - 1.18634 I
-1.31561 - 0.177807 I
-0.928778 + 0.218559 I


You immediately recognize this as the Fourier series for cos(x^2), and model your data appropriately.

In this case, using just the first few Fourier terms for approximation is a bad idea, since the data follows a much "better" pattern.

That's what I want. Fourier series is one of many methods to analyze data. I want my approximation to be method neutral in some sense.

So I guess I'm sort of asking: given a Fourier series, how do I find a nearby "well known" function.

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Did you remember to subtract out linear trends before FFT'ing your data? – J. M. Dec 17 '10 at 5:28
If we're looking at average daily temperature (or average hourly), there couldn't be any linear trends, right? – barrycarter Dec 17 '10 at 6:15
Personally I think Fourier analysis is quite an elegant approach and see nothing inelegant about using the first few Fourier terms, but I guess elegance is in the eye of the beholder.. – onestop Dec 17 '10 at 9:13
I'm not sure of the question. Are you essentially asking for domain-specific knowledge that "Oh yeah, temperatures are modeled as ___"? Or are you asking for help in defining "simple" or "elegant"? In the former case, this may not be the right place. In the latter, it's really your call about whether 3 Fourier terms is "elegant". (Or if the very simple sine wave with period 1 is "accurate enough", which is also your call, depending on your intended use.) – Wayne Dec 17 '10 at 17:02
"Physically sensible" and "simple" may not necessarily go hand in hand. Especially not when describing relatively complicated phenomena like weather. If there was a simple formula for predicting the weather, I'm pretty sure you could get quite rich by discovering it. – Jakob Borg Dec 18 '10 at 22:14