How to fit an equation to a curve with disturbances

Question

For example, I have the following data:

$Y = 366$ measured values

$X = 366$ measured values

$t = [ 1 : 366 ]$, representing the days of the year (index)

So at each $t$ (day), we have value of $Y$ and corresponding value of $X$. When drawing $Y$

and $X$ vs. $t$, it shows a continued curve for $Y$ with disturbances. These disturbances

are caused by the change of $X$ and it is clear that $Y$ is mainly affected by $X$, meaning

that: $Y = f(X)$

This figure is shown here:

This figure shows $Y$ vs. $X$:

My aim is to find this relation between $Y$ and $X$ or in another words: $Y = f(X)$.

What I have tried and think so far is first to smooth the curve $Y$ and from the

smoothed points and smoothed curve, some function may be established.

Then, including the effect of disturbances (up and down) by some function,

may be exponential.

Could you please guide me how can I manipulate this problem to get

a final model $Y = f(X)$.

regards

Answer 1

There are applications around that will fit curves to data, using a number of functions such as polynomial, exponential, and trig functions. For example, Matlab has built-in curve fitting and there are some add-on toolboxes for it as well.

Answer 2

Usually people do regression, ananlysis of variance, and perhaps also time series for problems like this. Regression means least-squares fitting of the data to a curve of predefined form but unknown parameters. The analysis of variance is a breakdown of the residual errors to search for systematic components, to see how well the model fits the data. Time series can help you find further relationships. It might also be interesing to see your second plot with data points on either side of $t=200$ (or near the ends versus near the middle, or into $7$ partitions by day of week, etc.) having alternate colors.