“Nonlinear Least Squares Fitting”的概念、定义、习题解题方法及翻译-数学辞典

单词

Nonlinear Least Squares Fitting

释义

Nonlinear Least Squares Fitting

Given a function of a variable tabulated at values , ..., , assume the function isof known analytic form depending on parameters , and consider the overdetermined setof equations

			(1)
			(2)

We desire to solve these equations to obtain the values

, ...,

which best satisfy this systemof equations. Pick an initial guess for the

and then define

(3)

Now obtain a linearized estimate for the changes

needed to reduce

to 0,

(4)

for

, ...,

. This can be written in component form as

(5)

where

is the

Matrix

(6)

In more concise Matrix form,

(7)

where

and

are

-Vectors. Applying the Matrix Transpose of

to bothsides gives

(8)

Defining

			(9)
			(10)

in terms of the known quantities

and

then gives the Matrix Equation

(11)

which can be solved for

using standard matrix techniques such as Gaussian Elimination. This offset isthen applied to

and a new

is calculated. By iteratively applying this procedure until the elements of

become smaller than some prescribed limit, a solution is obtained. Note that the procedure may not convergevery well for some functions and also that convergence is often greatly improved by picking initial values close to thebest-fit value. The sum of square residuals is given by

after the final iteration.

An example of a nonlinear least squares fit to a noisy Gaussian Function

(12)

is shown above, where the thin solid curve is the initial guess, the dotted curves are intermediate iterations, and the heavysolid curve is the fit to which the solution converges. The actual parameters are

, the initialguess was (0.8, 15, 4), and the converged values are (1.03105, 20.1369, 4.86022), with

. The PartialDerivatives used to construct the matrix

are

			(13)
			(14)
			(15)

The technique could obviously be generalized to multiple Gaussians, to include slopes, etc., although theconvergence properties generally worsen as the number of free parameters is increased.

An analogous technique can be used to solve an overdetermined set of equations. This problem might, for example, arisewhen solving for the best-fit Euler Angles corresponding to a noisy Rotation Matrix, in which case there arethree unknown angles, but nine correlated matrix elements. In such a case, write the different functions as for , ..., , call their actual values , and define

(16)

and

(17)

where

are the numerical values obtained after the

th iteration. Again, set up the equations as

(18)

and proceed exactly as before.

See also Least Squares Fitting, Linear Regression, Moore-Penrose Generalized Matrix Inverse

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