TITLE: Find formula to changing cyclic behavior
QUESTION [2 upvotes]: Could you please help me in this problem?
I have 3 independent variables, $(T,H,t)$, as inputs and
one output $P$  ( I have all data for these inputs and the output, done
experimentally measured every hour during one year).
I want to find a formula of this form:
$$P = f ( T , H , t )$$
where $t$ is the time in hours and it is always in the $x$-axis (index),
$T$ is temperature, $H$ is humidity, and $P$ is power.
I have all the data, and when I draw them in the same graph during one year, meaning that $P$, $T$,  and $H$ vs. hours. I found that the behavior of $P$ is oscillating, making a sinusoidal shape over the entire, year as you can see from the following ($P$ and time only):
So, if I make a zoom view to this figure, for example from
the $2000$th hour of the year to the $3000$th hour, it is clear that it almost has the same shape but it is oscillating.
So, it keeps oscillating and increasing up to the peak point and then
it starts decreasing till the end of the year.
But this is only for only one independent variable which is time in hours.
Now, how if we include the effect of temperature and humidity and draw 
these vectors along with the power vs. time, to see how power is changing
with respect to $T$, $H$, and $t$  rather than $t$ only:

So, how can I predict the structure of the formula that relates $P$ with $T$, $H$, and $t$?
Is there any approach that you advise me to follow?
Sorry for this long question and any help from you is highly appreciated. I read many papers but I could not know how to solve the problem.
thanks

REPLY [1 votes]: It seems you have one independent variable (t) and three dependent variables (P, H, and T) in that you do not have independent control of H and T.  This does not prevent finding a relation that would allow you to predict P based on H and T.
Looking at your data from year 2000, P and T overlay very closely.  A naive point of view would be to predict that P is a linear function of T and H doesn't matter.
First, you need to decide if you think the oscillations are data or noise.  If they are noise, you should filter them out.  The easiest, but far from the best, approach is just to average a number of data points.  There is discussion of filtering in Numerical Recipes as well as any numeric analysis book.  As suggested by J.M., doing an FFT makes this much easier, and there are routines to do that.
After any filtering, you can write P in some functional form involving T, H, and t and use a multidimensional minimizer to find the best fit.
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