 Nonlinear least squares

Nonlinear least squares is the form of least squares analysis which is used to fit a set of m observations with a model that is nonlinear in n unknown parameters (m > n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences.
Contents
Theory
Consider a set of m data points, and a curve (model function) that in addition to the variable x also depends on n 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 nonlinear 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 firstorder 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 variance of the measurement.^{[1]} The normal equations are then
These equations form the basis for the GaussNewton algorithm for a nonlinear least squares problem.
Geometrical interpretation
In linear least squares the objective function, S, is a quadratic function of the parameters.
When there is only one parameter the graph of S with respect to that parameter will be a parabola. With two or more parameters the contours of S with respect to any pair of parameters will be concentric ellipses (assuming that the normal equations matrix is positive definite). The minimum parameter values are to be found at the centre of the ellipses. The geometry of the general objective function can be described as paraboloid elliptical. In NLLSQ the objective function is quadratic with respect to the parameters only in a region close to its minimum value, where the truncated Taylor series is a good approximation to the model.
The more the parameter values differ from their optimal values, the more the contours deviate from elliptical shape. A consequence of this is that initial parameter estimates should be as close as practicable to their (unknown!) optimal values. It also explains how divergence can come about as the GaussNewton algorithm is convergent only when the objective function is approximately quadratic in the parameters.
Initial parameter estimates
Problems of illconditioning and divergence can be ameliorated by finding initial parameter estimates that are near to the optimal values. A good way to do this is by computer simulation. Both the observed and calculated data are displayed on a screen. The parameters of the model are adjusted by hand until the agreement between observed and calculated data is reasonably good. Although this will be a subjective judgment, it is sufficient to find a good starting point for the nonlinear refinement.
Computation
Gauss–Newton method
The normal equations
may be solved for by Cholesky decomposition, as described in linear least squares. The parameters are updated iteratively
where k is an iteration number. While this method may be adequate for simple models, it will fail if divergence occurs. Therefore protection against divergence is essential.
Shiftcutting
If divergence occurs, a simple expedient is to reduce the length of the shift vector, , by a fraction, f
For example the length of the shift vector may be successively halved until the new value of the objective function is less than its value at the last iteration. The fraction, f could be optimized by a line search.^{[2]} As each trial value of f requires the objective function to be recalculated it is not worth optimizing its value too stringently.
When using shiftcutting, the direction of the shift vector remains unchanged. This limits the applicability of the method to situations where the direction of the shift vector is not very different from what it would be if the objective function were approximately quadratic in the parameters,
Marquardt parameter
Main article: LevenbergMarquardt algorithmIf divergence occurs and the direction of the shift vector is so far from its "ideal" direction that shiftcutting is not very effective, that is, the fraction, f required to avoid divergence is very small, the direction must be changed. This can achieved by using the Marquardt parameter.^{[3]} In this method the normal equations are modified
where λ is the Marquardt parameter and I is an identity matrix. Increasing the value of λ has the effect of changing both the direction and the length of the shift vector. The shift vector is rotated towards the direction of steepest descent
 when
is the steepest descent vector. So, when λ becomes very large, the shift vector becomes a small fraction of the steepest descent vector.
Various strategies have been proposed for the determination of the Marquardt parameter. As with shiftcutting, it is wasteful to optimize this parameter too stringently. Rather, once a value has been found that brings about a reduction in the value of the objective function, that value of the parameter is carried to the next iteration, reduced if possible, or increased if need be. When reducing the value of the Marquardt parameter, there is a cutoff value below which it is safe to set it to zero, that is, to continue with the unmodified GaussNewton method. The cutoff value may be set equal to the smallest singular value of the Jacobian.^{[4]} A bound for this value is given by .^{[5]}
QR decomposition
The minimum in the sum of squares can be found by a method that does not involve forming the normal equations. The residuals with the linearized model can be written as
The Jacobian is subjected to an orthogonal decomposition; the QR decomposition will serve to illustrate the process.
where Q is an orthogonal matrix and R is an matrix which is partitioned into a block, , and a zero block. is upper triangular.
The residual vector is leftmultiplied by .
This has no effect on the sum of squares since because Q is orthogonal The minimum value of S is attained when the upper block is zero. Therefore the shift vector is found by solving
These equations are easily solved as R is upper triangular.
Singular value decomposition
A variant of the method of orthogonal decomposition involves singular value decomposition, in which R is diagonalized by further orthogonal transformations.
where is orthogonal, is a diagonal matrix of singular values and is the orthogonal matrix of the eigenvectors of or equivalently the right singular vectors of . In this case the shift vector is given by
The relative simplicity of this expression is very useful in theoretical analysis of nonlinear least squares. The application of singular value decomposition is discussed in detail in Lawson and Hanson.^{[4]}
Convergence criteria
The common sense criterion for convergence is that the sum of squares does not decrease from one iteration to the next. However this criterion is often difficult to implement in practice, for various reasons. A useful convergence criterion is
The value 0.0001 is somewhat arbitrary and may need to changed. In particular it may need to be increased when experimental errors are large. An alternative criterion is
Again, the numerical value is somewhat arbitrary; 0.001 is equivalent to specifying that each parameter should be refined to 0.1% precision. This is reasonable when it is less than the largest relative standard deviation on the parameters.
Calculation of the Jacobian by numerical approximation
There are models for which it is either very difficult or even impossible to derive analytical expressions for the elements of the Jacobian. Then, the numerical approximation
is obtained by calculation of for and . The increment,, size should be chosen so the numerical derivative is not subject to approximation error by being too large, or roundoff error by being too small.
Parameter errors, confidence limits, residuals etc.
For details concerning these topics see linear least squares#Parameter errors, correlation and confidence limits
Multiple minima
Multiple minima can occur in a variety of circumstances some of which are:
 A parameter is raised to a power of two or more. For example, when fitting data to a Lorentzian curve
where α is the height, γ is the position and β is the halfwidth at half height, there are two solutions for the halfwidth, and which give the same optimal value for the objective function.
 Two parameters can be interchanged without changing the value of the model. A simple example is when the model contains the product of two parameters, since αβ will give the same value as βα.
 A parameter is in a trigonometric function, such as , which has identical values at . See Levenberg–Marquardt algorithm for an example.
Not all multiple minima have equal values of the objective function. False minima, also known as local minima, occur when the objective function value is greater than its value at the socalled global minimum. To be certain that the minimum found is the global minimum, the refinement should be started with widely differing initial values of the parameters. When the same minimum is found regardless of starting point, it is likely to be the global minimum.
When multiple minima exist there is an important consequence: the objective function will have a maximum value somewhere between two minima. The normal equations matrix is not positive definite at a maximum in the objective function, as the gradient is zero and no unique direction of descent exists. Refinement from a point (a set of parameter values) close to a maximum will be illconditioned and should be avoided as a starting point. For example, when fitting a Lorentzian the normal equations matrix is not positive definite when the halfwidth of the band is zero.^{[6]}
Other methods
Transformation to a linear model
A nonlinear model can sometimes be transformed into a linear one. For example, when the model is a simple exponential function,
it can be transformed into a linear model by taking logarithms.
The sum of squares becomes
This procedure should be avoided unless the errors are multiplicative and lognormally distributed because it can give misleading results. This comes from the fact that whatever the experimental errors on y might be, the errors on log y are different. Therefore, when the transformed sum of squares is minimized different results will be obtained both for the parameter values and their calculated standard deviations. However, with multiplicative errors that are lognormally distributed, this procedure gives unbiased and consistent parameter estimates.
Another example is furnished by Michaelis–Menten kinetics, used to determine two parameters V_{max } and K_{m}.
of 1/v against [S] is very sensitive to data error and it is strongly biased toward fitting the data in a particular range of the independent variable, [S].
Gradient methods
There are many examples in the scientific literature where different methods have been used for nonlinear datafitting problems.
 Inclusion of second derivatives in The Taylor series expansion of the model function. This is Newton's method in optimization.
The matrix H is known as the Hessian matrix. Although this model has better convergence properties near to the minimum, it is much worse when the parameters are far from their optimal values. Calculation of the Hessian adds to the complexity of the algorithm. This method is not in general use.
 DavidonFletcherPowell method. This method, a form of pseudoNewton method, is similar to the one above but calculates the Hessian by successive approximation, to avoid having to use analytical expressions for the second derivatives.
 Steepest descent. Although a reduction in the sum of squares is guaranteed when the shift vector points in the direction of steepest descent, this method often performs poorly. When the parameter values are far from optimal the direction of the steepest descent vector, which is normal (perpendicular) to the contours of the objective function, is very different from the direction of the GaussNewton vector. This makes divergence much more likely, especially as the minimum along the direction of steepest descent may correspond to a small fraction of the length of the steepest descent vector. When the contours of the objective function are very eccentric, due to there being high correlation between parameters. the steepest descent iterations, with shiftcutting, follow a slow, zigzag trajectory towards the minimum.
 Conjugate gradient search. This is an improved steepest descent based method with good theoretical convergence properties, although it can fail on finiteprecision digital computers even when used on quadratic problems.^{[7]}
Direct search methods
Direct search methods depend on evaluations of the objective function at a variety of parameter values and do not use derivatives at all. They offer alternatives to the use of numerical derivatives in the GaussNewton method and gradient methods.
 Alternating variable search.^{[2]} Each parameter is varied in turn by adding a fixed or variable increment to it and retaining the value that brings about a reduction in the sum of squares. The method is simple and effective when the parameters are not highly correlated. It has very poor convergence properties, but may be useful for finding initial parameter estimates.
 NelderMead (Simplex) search A simplex in this context is a polytope of n + 1 vertices in n dimensions; a triangle on a plane, a tetrahedron in threedimensional space and so forth. Each vertex corresponds to a value of the objective function for a particular set of parameters. The shape and size of the simplex is adjusted by varying the parameters in such a way that the value of the objective function at the highest vertex always decreases. Although the sum of squares may initially decrease rapidly, it can converge to a nonstationary point on quasiconvex problems, by an example of M. J. D. Powell.
More detailed descriptions of these, and other, methods are available, in Numerical Recipes, together with computer code in various languages.
See also
 Least squares support vector machine
 Curve fitting
 Nonlinear programming
 Optimization (mathematics)
 Levenberg–Marquardt algorithm
Notes
 ^ This implies that the observations are uncorrelated. If the observations are correlated, the expression
 ^ ^{a} ^{b} M.J. Box, D. Davies and W.H. Swann, NonLinear optimisation Techniques, Oliver & Boyd, 1969
 ^ This technique was proposed independently by Levenberg (1944), Girard (1958), Wynne (1959), Morrison (1960) and Marquardt (1963). Marquardt's name alone is used for it in much of the scientific literature.
 ^ ^{a} ^{b} C.L. Lawson and R.J. Hanson, Solving Least Squares Problems, PrenticeHall,1974
 ^ R. Fletcher, UKAEA Report AERER 6799, H.M. Stationery Office, 1971
 ^ In the absence of roundoff error and of experimental error in the independent variable the normal equations matrix would be singular
 ^ M.J.D. Powell, Computer Journal, (1964), 7, 155.
References
 C. T. Kelley, Iterative Methods for Optimization, SIAM Frontiers in Applied Mathematics, no 18, 1999, ISBN 0898714338. Online copy
 T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner, ISBN 9783834810229.
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