Predicting the next vector given a known sequence I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a finite unit vector.
As the sequence is generated by a poorly known process, I am interested in modelling $\vec{v}_k$ given previous generated vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_{k-1}$.
What are the available mathematical tools which allows me to discover a vector function $\vec{f}$ such that $\vec{v}_k\approx \vec{f}(\vec{v}_{k-1},\vec{v}_{k-2},\ldots,\vec{v}_{k-n})$, for a given $n$, in the $L_p$-norm sense?
EDIT: I am looking along the lines of the Newton's Forward Difference Formula, which predicts interpolated values between tabulated points, except for two differences for my problem: 1) Newton' Forward Difference is applicable for a scalar sequence, and 2) I am doing extrapolation at one end of the sequence, not interpolation in between given values.
ADDITIONAL INFO: Below are plots of the individual components of an 8-tuple unit vector from a sequence of 200:


A lot of methods effectively work by fitting a polynomial to your data, and then using that polynomial to guess a new value. The main reason for polynomials is that they are easy to work with.
Given that you know your functions have asymptotes, you may get better success by choosing a  form that incorporates that fact, such as a rational function. If nothing else, you can always use a sledgehammer to derive a method -- e.g. use the method of least squares to select the coefficients of your rational functions.
