How to calculate signatures numerically in rough path theory. The question is related to the 
link:https://en.wikipedia.org/wiki/Rough_path#Signature I search the internet looking for tutorial on how to calculate the signature sequence, but could not find a clear document. The first term is one the second term seems to be $X_t−X_s$ but after the 2nd term how can I calculate the other integral? Consider a simplest curve, I have a set of numbers which form a line and sorted properly. This is a function also a very simple curve. For such curve can we have a signature? How to calculate the terms?  Any hint or reference would be helpful.  
 A: A rough path postulates the iterated integrals. So given a regular path, you must postulate or define (part of) the signature to make it into a rough path. 
These are not unique! This is the point of rough path theory. Given a second order rough path, $(X_{st},\Bbb X_{st})$ with $X\in C^\alpha$ for $\alpha\in (1/3,1/2]$ you can define another rough path $(X_{st},\Bbb X_{st}+F_t-F_s)$ where $F$ is a $2\alpha$ Hoelder function.
These both have the same underlying path. However in the first case we have 
$$\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}=\Bbb X_{st}$$
And in the second case:
$$\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}=\Bbb X_{st}+F_t-F_s$$
Both of these have the same underlying path but you have different iterated integrals.

The entire point of rough path theory is that you CANNOT CALCULATE ITERATED INTEGRALS FROM $X_t$ ALONE.

In rough paths, you postulate those values, and there are infinitely many ways to do this. 
However from Lyons Victoir extension theorem, for $C^\alpha$ paths with $\alpha\in(1/3,1/2)$ is is always possible to construct a rough path lift (i.e. $\int_{s<s_1<s_2<t}dX_{s_1}\otimes dX_{s_2}$) and there are other cases where we can define that iterated integral.
