Vector field reconstruction

In the best possible case, one has data streams of measurements of all the system variables, equally spaced in time, say

s₁(t), s₂(t), ... , s_k(t)

for

t = t₁, t₂,..., t_n,

beginning at several different initial conditions. Then the task of finding a vector field, and thus a differential equation model consists of fitting functions, for instance, a cubic spline, to the data to obtain a set of continuous time functions

x₁(t), x₂(t), ... , x_k(t),

computing time derivatives dx₁/dt, dx₂/dt,...,dx_k/dt of the functions, then making a least squares fit using some sort of orthogonal basis functions (orthogonal polynomials, radial basis functions, etc.) to each component of the tangent vectors to find a global vector field. A differential equation then can be read off the global vector field.

There are various methods of creating the basis functions for the least squares fit. The most common method is the Gram–Schmidt process. Which creates a set of orthogonal basis vectors, which can then easily be normalized. This method begins by first selecting any standard basis β={v₁, v₂,...,v_n}. Next, set the first vector v₁=u₁. Then, we set u₂=v₂-proj_u1v₂. This process is repeated to for k vectors, with the final vector being u_k= v_k-Σ_(j=1)^(k-1)proj_ukv_k. This then creates a set of orthogonal standard basis vectors.

The reason for using a standard orthogonal basis rather than a standard basis arises from the creation of the least squares fitting done next. Creating a least-squares fit begins by assuming some function, in the case of the reconstruction an n^th degree polynomial, and fitting the curve to the data using constants. The accuracy of the fit can be increased by increasing the degree of the polynomial being used to fit the data. If a set of non-orthogonal standard basis functions was used, it becomes necessary to recalculate the constant coefficients of the function describing the fit. However, by using the orthogonal set of basis functions, it is not necessary to recalculate the constant coefficients.

Vector field reconstruction has several applications, and many different approaches. Some mathematicians have not only used radial basis functions and polynomials to reconstruct a vector field, but they have used Lyapunov exponents and singular value decomposition.[2] Gouesbet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field. This method was applied to the Rössler system, and the Lorenz system, as well as thermal lens oscillations.

The Rossler system, Lorenz system and Thermal lens oscillation follows the differential equations in standard system as

X'=Y, Y'=Z and Z'=F(X,Y,Z)

where F(X,Y,Z) is known as the standard function.[3]

In some situation the model is not very efficient and difficulties can arise if the model has a large number of coefficients and demonstrates a divergent solution. For example, nonautonomous differential equations give the previously described results.[4] In this case the modification of the standard approach in application gives a better way of further development of global vector reconstruction.

Usually the system being modeled in this way is a chaotic dynamical system, because chaotic systems explore a large part of the phase space and the estimate of the global dynamics based on the local dynamics will be better than with a system exploring only a small part of the space.

Frequently, one has only a single scalar time series measurement from a system known to have more than one degree of freedom. The time series may not even be from a system variable, but may be instead of a function of all the variables, such as temperature in a stirred tank reactor using several chemical species. In this case, one must use the technique of delay coordinate embedding,[5] where a state vector consisting of the data at time t and several delayed versions of the data is constructed.

A comprehensive review of the topic is available from [6]