Hi,

I'm trying to study a list of vectors. The vectors represent particular states of an organic system over time. The vectors grow in size (relatively fast) and (absolute) value of entries (relatively slowly) over time.

A sequence of these vectors displays the characteristic that adjacent vectors are quite similar. So the sequence of 'first derivatives' of these vectors display mostly 0s and small numbers.

However, once I 'differentiate' this sequence this 'similarity property' (my one piece of evidence for a persistent underlying pattern) disappears.

For e.g.

The list of vectors may be

[5, -2, 2, -3, 4, -1, 3, -2 ]

[4, -3, 3, -2, 3, -1, 3, -4 ]

[6, -3, 2, -1, 3, -1, 3, -3, 2, -2 ]

So the sequence of first derivatives is

[4-5, -3--2, 3-2, -2--3, 3-4, -1--1, 3-3, -4--2 ] =

[ -1, -1, 1, 1, -1, 0, 0, -2 ]

[2, 0, -1, 1, 0, 0, 0, 1, 2, -2 ]

The sequence of second derivatives (only 1 ) is

[3, 1, -2, 0, 1, 0, 0, 3 2, -2 ]

The thing is I am hoping to find a pattern where ( like polynomials ) when I keep 'differentiating' the vectors, I find smaller and smaller and more homogenous vectors that result. This however does not happen.

A typical sequence of 1000 vectors differentiated to the 100th degree, displays a sequence of (1ooth differentials) vectors with entries of order 10^29.

What I am wondering is -- What am I dealing with?

The systems states are very closely related at the first level. My attempts to use this 'differentiation' to discover an underlying structure (or 'function' as one can do with polynomials) is not working.

So I wonder

-- What does it mean about the system that these 'derivatives' although very similar at first, from then on keep growing?

-- What other options do I have for studying, searching for, finding patterns that could explain / model the way the vectors change over time?

Any fresh ideas are welcome in this case.

Best,

Cris