Non-linear Least Squares - Theory

Theory

Consider a set of data points, and a curve (model function) that in addition to the variable also depends on parameters, with It is desired to find the vector of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares

is minimized, where the residuals (errors) r_i are given by

for

The minimum value of S occurs when the gradient is zero. Since the model contains n parameters there are n gradient equations:

In a non-linear system, the derivatives are functions of both the independent variable and the parameters, so these gradient equations do not have a closed solution. Instead, initial values must be chosen for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation,

Here, k is an iteration number and the vector of increments, is known as the shift vector. At each iteration the model is linearized by approximation to a first-order Taylor series expansion about

The Jacobian, J, is a function of constants, the independent variable and the parameters, so it changes from one iteration to the next. Thus, in terms of the linearized model, and the residuals are given by

Substituting these expressions into the gradient equations, they become

which, on rearrangement, become n simultaneous linear equations, the normal equations

The normal equations are written in matrix notation as

When the observations are not equally reliable, a weighted sum of squares may be minimized,

Each element of the diagonal weight matrix W should, ideally, be equal to the reciprocal of the error variance of the measurement. The normal equations are then

These equations form the basis for the Gauss–Newton algorithm for a non-linear least squares problem.