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.