Vector Field Reconstruction - Formulation

Formulation

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-∑_(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 nth 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.