Optimization Toolbox™

lsqcurvefit

Solve nonlinear curve-fitting (data-fitting) problems in least-squares sense

Equation

Find coefficients x that best fit the equation

given input data xdata, and the observed output ydata, where xdata and ydata are matrices or vectors of length m, and F (x, xdata) is a matrix-valued or vector-valued function.

The function lsqcurvefit uses the same algorithm as lsqnonlin. Its purpose is to provide an interface designed specifically for data-fitting problems.

Syntax

x = lsqcurvefit(fun,x0,xdata,ydata)
x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)
x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)
x = lsqcurvefit(problem)
[x,resnorm] = lsqcurvefit(...)
[x,resnorm,residual] = lsqcurvefit(...)
[x,resnorm,residual,exitflag] = lsqcurvefit(...)
[x,resnorm,residual,exitflag,output] = lsqcurvefit(...)
[x,resnorm,residual,exitflag,output,lambda] = lsqcurvefit(...)
[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(...)

Description

lsqcurvefit solves nonlinear data-fitting problems. lsqcurvefit requires a user-defined function to compute the vector-valued function F (x, xdata). The size of the vector returned by the user-defined function must be the same as the size of the vectors ydata and xdata.

x = lsqcurvefit(fun,x0,xdata,ydata) starts at x0 and finds coefficients x to best fit the nonlinear function fun(x,xdata) to the data ydata (in the least-squares sense). ydata must be the same size as the vector (or matrix) F returned by fun.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub) defines a set of lower and upper bounds on the design variables in x so that the solution is always in the range lb ≤ x ≤ ub.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options. Pass empty matrices for lb and ub if no bounds exist.

x = lsqcurvefit(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB^® Workspace.

[x,resnorm] = lsqcurvefit(...) returns the value of the squared 2-norm of the residual at x: sum((fun(x,xdata)-ydata).^2).

[x,resnorm,residual] = lsqcurvefit(...) returns the value of the residual fun(x,xdata)-ydata at the solution x.

[x,resnorm,residual,exitflag] = lsqcurvefit(...) returns a value exitflag that describes the exit condition.

[x,resnorm,residual,exitflag,output] = lsqcurvefit(...) returns a structure output that contains information about the optimization.

[x,resnorm,residual,exitflag,output,lambda] = lsqcurvefit(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(...) returns the Jacobian of fun at the solution x.

Note   If the specified input bounds for a problem are inconsistent, the output x is x0 and the outputs resnorm and residual are []. Components of x0 that violate the bounds lb ≤ x ≤ ub are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed.

Input Arguments

Function Arguments contains general descriptions of arguments passed into lsqcurvefit. This section provides function-specific details for fun, options, and problem:

x = lsqcurvefit(@myfun,x0,xdata,ydata)

function F = myfun(x,xdata)
F = ...            % Compute function values at x

f = @(x,xdata)x(1)*xdata.^2+x(2)*sin(xdata),...
           'x','xdata';
x = lsqcurvefit(f,x0,xdata,ydata);

options = optimset('Jacobian','on')

function [F,J] = myfun(x,xdata)
F = ...          % objective function values at x
if nargout > 1   % two output arguments
   J = ...   % Jacobian of the function evaluated at x
end

Output Arguments

Function Arguments contains general descriptions of arguments returned by lsqcurvefit. This section provides function-specific details for exitflag, lambda, and output:

Note   The sum of squares should not be formed explicitly. Instead, your function should return a vector of function values. See the examples below.

Options

Optimization options used by lsqcurvefit. Some options apply to all algorithms, some are only relevant when using the large-scale algorithm, and others are only relevant when you are using the medium-scale algorithm. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

The LargeScale option specifies a preference for which algorithm to use. It is only a preference, because certain conditions must be met to use the large-scale or medium-scale algorithm. For the large-scale algorithm, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of F returned by fun) must be at least as many as the length of x. Furthermore, only the large-scale algorithm handles bound constraints:

The large-scale algorithm is a more modern algorithm than the medium-scale algorithms. The large-scale algorithm handles both large-scale and medium-scale problems effectively.

Medium-Scale and Large-Scale Algorithms

These options are used by both the medium-scale and large-scale algorithms:

Large-Scale Algorithm Only

These options are used only by the large-scale algorithm:

W = jmfun(Jinfo,Y,flag,p1,p2,...) 

[F,Jinfo] = fun(x)

Medium-Scale Algorithm Only

These options are used only by the medium-scale algorithm:

Examples

Given vectors of data xdata and ydata, suppose you want to find coefficients x to find the best fit to the exponential decay equation

That is, you want to minimize

where m is the length of xdata and ydata, the function F is defined by

F(x,xdata) = x(1)*exp(x(2)*xdata);

and the starting point is x0 = [100; -1];.

First, write an M-file to return the value of F (F has n components).

function F = myfun(x,xdata)
F = x(1)*exp(x(2)*xdata);

Next, invoke an optimization routine:

% Assume you determined xdata and ydata experimentally
xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];
x0 = [100; -1] % Starting guess
[x,resnorm] = lsqcurvefit(@myfun,x0,xdata,ydata)

Note that at the time that lsqcurvefit is called, xdata and ydata are assumed to exist and are vectors of the same size. They must be the same size because the value F returned by fun must be the same size as ydata.

After 27 function evaluations, this example gives the solution

x = 
498.8309 -0.1013 
resnorm = 
9.5049

There may be slight variation in the number of iterations and the value of the returned x that is dependent upon the platform and release.

Algorithm

Large-Scale Optimization

By default lsqcurvefit chooses the large-scale algorithm. This algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region Methods for Nonlinear Minimization and Preconditioned Conjugate Gradients.

Medium-Scale Optimization

If you set the LargeScale option to 'off' with optimset, lsqcurvefit uses the Levenberg-Marquardt method with line search [4], [5], and [6]. Alternatively, you can select a Gauss-Newton method [3] with line search by setting the LevenbergMarquardt option to 'off' (and LargeScale to 'off') with optimset. The Gauss-Newton method is generally faster when the residual sum((fun(x,xdata)-ydata).^2) is small.

The default line search algorithm, i.e., the LineSearchType option, is 'quadcubic'. This is a safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation method. You can select a safeguarded cubic polynomial method by setting the LineSearchType option to 'cubicpoly'. This method generally requires fewer function evaluations but more gradient evaluations. Thus, if gradients are being supplied and can be calculated inexpensively, the cubic polynomial line search method is preferable. The algorithms used are described fully in Standard Algorithms.

Diagnostics

Large-Scale Optimization

The large-scale method does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), lsqlin gives the error

Equal upper and lower bounds not permitted.

(lsqcurvefit does not handle equality constraints, which is another way to formulate equal bounds. If equality constraints are present, use fmincon, fminimax, or fgoalattain for alternative formulations where equality constraints can be included.)

Limitations

The function to be minimized must be continuous. lsqcurvefit might only give local solutions.

lsqcurvefit only handles real variables (the user-defined function must only return real values). When x has complex variables, the variables must be split into real and imaginary parts.

Large-Scale Optimization

The large-scale algorithm for lsqcurvefit does not solve underdetermined systems; it requires that the number of equations, i.e., the row dimension of F, be at least as great as the number of variables. In the underdetermined case, the medium-scale algorithm is used instead. See Large-Scale Problem Coverage and Requirements for more information on what problem formulations are covered and what information must be provided.

The preconditioner computation used in the preconditioned conjugate gradient part of the large-scale method forms J^TJ (where J is the Jacobian matrix) before computing the preconditioner; therefore, a row of J with many nonzeros, which results in a nearly dense product J^TJ, can lead to a costly solution process for large problems.

If components of x have no upper (or lower) bounds, then lsqcurvefit prefers that the corresponding components of ub (or lb) be set to inf (or -inf for lower bounds) as opposed to an arbitrary but very large positive (or negative for lower bounds) number.

Medium-Scale Optimization

The medium-scale algorithm does not handle bound constraints.

Since the large-scale algorithm does not handle underdetermined systems and the medium-scale does not handle bound constraints, problems with both these characteristics cannot be solved by lsqcurvefit.

References

[1] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418-445, 1996.

[2] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp. 189-224, 1994.

[3] Dennis, J. E. Jr., "Nonlinear Least-Squares," State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269-312, 1977.

[4] Levenberg, K., "A Method for the Solution of Certain Problems in Least-Squares," Quarterly Applied Math. 2, pp. 164-168, 1944.

[5] Marquardt, D., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," SIAM Journal Applied Math., Vol. 11, pp. 431-441, 1963.

[6] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977.

See Also

@ (function_handle), \ (matrix left division), lsqlin, lsqnonlin, lsqnonneg, optimset, optimtool, nlinfit

Note   The Statistics Toolbox™ function nlinfit has more statistics-oriented outputs that are useful, for example, in finding confidence intervals for the coefficients. It also comes with the nlintool GUI for visualizing the fitted function. The lsqnonlin function has more outputs related to how well the optimization performed. It can put bounds on the parameters, and it accepts many options to control the optimization algorithm.

linprog

lsqlin