Describe a curve by other than formula, fitting or interpolation

Question

I have a curve defined by a set of $(x,y)$ given points. After ploting these points I get :

My aim is to study the sensitivity of the operation that results these points. In order to do that, the reasonable idea was to have a tengible description of this curve. Thus my question here is about some ideas that may help me to describe a curve other than its formula( y=f(x)).

In fact what I have tried is:

1. As you see the shape of the curve starts to be straight then rounded,, then straight again. So my firts idea was to define the initial point, then find the curvature numerically at the centred points,and lastly approximate a slope for the last straight part of the curve(eventhough it hs some fluctuations we can assueme it smooth enough).
2. I try the idea of Bezier curves. Having the sample points I try to find the controls points of the bezier curve that best fit my curve. Thihs method is much helpful, however, to get a cubic BEzier curve we just need 4 sample points to get the two central control points. And as the number of the sample points increase(in my case I have about 200) this will be no more applicable, since either I have to define my curve by several patches and thus several cubic bezier curves, or I have to find just 4 control points from the huge data I have(In fact I wrote a matlab code to this but the fitting was worth!!).

So does any one help me by suggesting ideas that I can use to describe a curve other than interpolation and fitting ??

Edit: To be more precise: my aim is to do sensitivity analysis, so it would be great for me if I can tell that this graph is determined by X, Y ,... where X , Y and ... are parameters. For example: according to my second try my curve was determined by the four control points of the bezier curve that fit my curve. Or for example the slopes of the two lines discribing the begining and the neding part of my curve + the center and radius of the circle that best firt the central part of my curve.

Answer 1

You could perform Fourier analysis to express this curve as the sum of different harmonic trigonometric functions. This way you can convey information about the different transient behaviours (longer lasting transient superimposed over a perturbation with higher frequency). Of course, given that your function is non-periodic you will have to choose window function carefully so as to not introduce severe distortions.

Example of Fourier analysis perfromed on four signals below:

Answer 2

From your question, it seems that you want to examine the relations of certain independent variables (which you call parameters X, Y in your edit) to some characteristics of curves like the one you plotted.

I think it would be best to use your knowledge of the subject matter to devise a specific form of the relation $y=f(x,a,b,...)$, where $(x,y)$ are the points of your observed curve and $a,b,...$ are parameters describing the shape of such a curve in terms of the underlying physical processes. For each observed curve you would find the best-fitting values of $a,b,...$ and then examine their relations to the variables X, Y.

For example, the curve you plotted looks like it might be a combination of two exponential decays, characterized by time constants and magnitudes. If so, you would use non-linear curve fitting to find the time constants and magnitudes that best fit each specific observed curve, then examine how the variables X, Y are related to those characteristics of the fitted curves. If you have a better idea of the processes underlying these curves, then write a function that describes the relation in terms of a few interpretable parameters and proceed similarly.

There are ways to fit curves that do not depend on having a particular functional form, as discussed for example on this page. Some types of splines, for example, are related to the Bézier curves you have examined. These types of methods are used, for example, for transformations of independent (predictor) variables in regressions. But you want to relate characteristics of the $(x,y)$ curves to other variables like your X, Y. The values returned by analyses based on Bézier curves or splines are unlikely to give you anything that you can easily analyze in this way, and they would be difficult to generalize or to relate cleanly to the underlying physical processes.