Forecasting noisy data with polynomials What's the best way to incorporate polynomials or sinusoidal equations into a linear model to predict a future amplitude in this sort of data? You can see that if those polynomials continued to go on, they wouldnt be anywhere close to fitting the final location. Im trying to use the greeen shaded area to predict the non shaded area.


Those light grey lines are polynomials like this: $$y_1 = a_1(x)^2 + b_1(x) + c_1\\$$ that are fitted to different sections of the data. Do I just use the a b and c variables from each polynomial, or do extrapolate out and use some other value?
Starting with your last question: to extrapolate you do just use your fitted coefficients and extend them to a greater $x$ range.

However, I think this approach is inherently flawed for data such as yours over any significant time period, especially when using a polynomial model. The problem is that any polynomial has a limited number of turning points. Once you pass these the data will tend to head off to $\pm \infty$ which is unlikely to model anything well.

Frequency domain analysis may be a possibility, but I expect you would run into similar problems. Another approach could be to use some sort of moving average or auto-regressive model.

Banks and similar financial institutions invest millions into modelling this sort of data so it definitely isn't easy.