Algorithms for Data Fitting in Error-in-Variables Models with Linear and Quadratic Constraints

Fitting a surface to a given set of measurements is an essential function for engineers and geodesists, also known as trend analysis [7],[1],[4]. This technique uses Least-Squares (LS) adjustment to estimate the parameters (ξ) of a polynomial surface within a linear model (y = Aξ +e) that includes the vector of observed attribute values (y), a vector of normally distributed errors e, and a matrix of variables A, constructed from the geographical locations. However, in this linear model the matrix of variables A is considered as fixed or error-free. This is not the case in many physical situations where errors exist both in the vector of attributes (y) and in the geographical locations matrix (A). The Total Least-Squares (TLS) approach as applied to the Error-in-Variables model is the proper method to treat problems where all the data are affected by random errors. The traditional approach to solve the TLS problem [2],[6] utilizes the Eckart-Young-Mirsky theorem to derive the best estimate [Â, ŷ] for the augmented matrix by setting the minimum eigenvalue of the Singular Value Decomposition of [A; y] to zero. A different approach to solve the TLS problem is based on the Euler-Lagrange theorem [4],[3] and follows an optimization process that minimizes a target function. In this contribution, the Euler-Lagrange approach is first used to solve an enhanced TLS problem that includes linear constraints as follows: y − e = (A − E) · ξ, rk[A] = m < n, κ0 = K · ξ, rk[K] = l ≤ m, E{[E, e]} = 0, D{vec[E, e]} = Σ0 ⊗ In, C{E, e} = 0