I'm sorry, but high order polynomials (as suggested by @akshatsood) are always a bad idea. You will run into numerical problems working in double precision. And as well, people tend to think that if a degree 9 polynomial gave an ok fit, then we can get a better fit from a degree 10, or 12 or 15 degree polynomial. In the end, they end up running into the realm of overfitting their data. And worst of all, the @Chaudhary P Patel talks about using that equation for prediction in a different interval. Trying to extrapolate data using a high degree polynomial is just insanity. In fact, trying to extrapolate any model that fits data like that, where you have no rational method for having chosen the model is as crazy. The same applies to the sum of exponentials model produced by @Alex Sha. Just because the model predicts well for the existing data is no reason it will predict anything meaningful for data outside of the support of the data. In the end, there is no magical way to know the true model that produced any set of data. What is needed is an understanding of the physics of what produced the data. If you do understand the physics, then you can SOMETIMES use that knowledge to suggest a model form, and then you can then use that model form to infer estimates of the parameters that might have existed.
This is why splines exist, and why people use splines as heavily as they do. For example:
spl1 = spline(xy1(:,1),xy1(:,2));
plot(xy1(:,1),xy1(:,2),'ro')
Even here though, note that at the right end of that curve, the spline predicts a quick turnaround. Is that true? Perhaps. There is clearly some curvature in the data at that end. But lacking any information to the right, it is IMPOSSIBLE to guess what will really happen there. And since it looks like there is also probably some noise in the data, we need to guess if that is just noise, or if it is real. Most of the time, I would guess some of the bumps in your curve are noise. But maybe they are real signal. We cannot know. (I recall one of my clients who positively knew that if there was a small jiggle in a curve, they ABSOLUTELY needed to know about it, and needed to see it reproduced. But that is not always the case.)
A problem of course is there is no simple function you can write down for the spline I just produced above. But the sum of exponentials model given by @Alex Sha, or the high degree polynomial from @akshatsood are nearly as bad. They will both be terribly sensitive to even small perturbations in the coefficients in those models. And extrapolative predictions of those curves will be just as sensitive. Again, I'm sorry. But there is no simple way to know the true equation of such a set of data. There is not even a sophisticated, complex way to know that.
