Linear prediction (LPC) of Sine wave samples around maximas
Disclaimer: This is explanation is based on observation of my MatLab plots and my note be 100% textbook correct! 

Well after much looking around and experimenting I read something about the 'yule-walker' method for estimating the coefficients assuming the signal to be 'zero' outside of the observation point. This is not the case for my co-/sine-wave, so the estimated coefficients and the resulting predicted samples $\hat{x}(n)$ will trend towards zero. The further away from the $x$-Axis the sharper the slope towards 'zero' when using yule-walker.

For my problem an approach using Burg's method (coeffs = arburg(x_kn, P);) seems to yield the result I'm looking for, since burg does 'not' assume the Signal trending towards 'zero'?? At least this is how I can explain the result of my two plots. The 'star' marks the point where the prediction of future values start.

Yule-Walker:



Burg-Method: