비선형 최소 제곱법.rst

single: nonlinear least squares single: least squares, nonlinear

비선형 최소 제곱법

Note

번역중

This chapter describes functions for multidimensional nonlinear least-squares fitting. There are generally two classes of algorithms for solving nonlinear least squares problems, which fall under line search methods and trust region methods. GSL currently implements only trust region methods and provides the user with full access to intermediate steps of the iteration. The user also has the ability to tune a number of parameters which affect low-level aspects of the algorithm which can help to accelerate convergence for the specific problem at hand. GSL provides two separate interfaces for nonlinear least squares fitting. The first is designed for small to moderate sized problems, and the second is designed for very large problems, which may or may not have significant sparse structure.

The header file gsl_multifit_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions and related declarations relating to the small to moderate sized systems.

The header file gsl_multilarge_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions and related declarations relating to large systems.

single: nonlinear least squares, overview

Overview

The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_i, in p parameters, x_i,

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$$\begin{aligned} \Phi(x) &= {1 \over 2} || f(x) ||^2 \\\ &= {1 \over 2} \sum_{i=1}^{n} f_i (x_1, \dots, x_p)^2 \end{aligned}$$

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\Phi(x) = (1/2) || f(x) ||^2
        = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2 

In trust region methods, the objective (or cost) function Φ(x) is approximated by a model function m_k(δ) in the vicinity of some point x_k. The model function is often simply a second order Taylor series expansion around the point x_k, ie:

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$$\Phi(x_k + \delta) \approx m_k(\delta) = \Phi(x_k) + g_k^T \delta + {1 \over 2} \delta^T B_k \delta$$

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\Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta

where g_k = ∇Φ(x_k) = J^Tf is the gradient vector at the point x_k, B_k = ∇²Φ(x_k) is the Hessian matrix at x_k, or some approximation to it, and J is the n-by-p Jacobian matrix

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J_ij = ∂f_i/∂x_j

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J_{ij} = d f_i / d x_j

In order to find the next step δ, we minimize the model function m_k(δ), but search for solutions only within a region where we trust that m_k(δ) is a good approximation to the objective function Φ(x_k + δ). In other words, we seek a solution of the trust region subproblem (TRS)

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$$\min_{\delta \in R^p} m_k(\delta) = \Phi(x_k) + g_k^T \delta + {1 \over 2} \delta^T B_k \delta, \qquad\hbox{s.t.}\quad || D_k \delta || \le \Delta_k$$

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\min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k

where Δ_k > 0 is the trust region radius and D_k is a scaling matrix. If D_k = I, then the trust region is a ball of radius Δ_k centered at x_k. In some applications, the parameter vector x may have widely different scales. For example, one parameter might be a temperature on the order of 10³ K, while another might be a length on the order of 10^− 6 m. In such cases, a spherical trust region may not be the best choice, since if Φ changes rapidly along directions with one scale, and more slowly along directions with a different scale, the model function m_k may be a poor approximation to Φ along the rapidly changing directions. In such problems, it may be best to use an elliptical trust region, by setting D_k to a diagonal matrix whose entries are designed so that the scaled step D_kδ has entries of approximately the same order of magnitude.

The trust region subproblem above normally amounts to solving a linear least squares system (or multiple systems) for the step δ. Once δ is computed, it is checked whether or not it reduces the objective function Φ(x). A useful statistic for this is to look at the ratio

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$$\rho_k = { \Phi(x_k) - \Phi(x_k + \delta_k) \over m_k(0) - m_k(\delta_k) }$$

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\rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )

where the numerator is the actual reduction of the objective function due to the step δ_k, and the denominator is the predicted reduction due to the model m_k. If ρ_k is negative, it means that the step δ_k increased the objective function and so it is rejected. If ρ_k is positive, then we have found a step which reduced the objective function and it is accepted. Furthermore, if ρ_k is close to 1, then this indicates that the model function is a good approximation to the objective function in the trust region, and so on the next iteration the trust region is enlarged in order to take more ambitious steps. When a step is rejected, the trust region is made smaller and the TRS is solved again. An outline for the general trust region method used by GSL can now be given.

Trust Region Algorithm

1. Initialize: given x₀, construct m₀(δ), D₀ and Δ₀ > 0
2. For k = 0, 1, 2, ...
   1. If converged, then stop
   2. Solve TRS for trial step δ_k
   3. Evaluate trial step by computing ρ_k

      1). if step is accepted, set x_k + 1 = x_k + δ_k and increase radius,

      Δ_k + 1 = αΔ_k

      2). if step is rejected, set x_k + 1 = x_k and decrease radius,

      $\Delta_{k+1} = {\Delta_k \over \beta}$; goto 2(b)

   4. Construct m_k + 1(δ) and D_k + 1

GSL offers the user a number of different algorithms for solving the trust region subproblem in 2(b), as well as different choices of scaling matrices D_k and different methods of updating the trust region radius Δ_k. Therefore, while reasonable default methods are provided, the user has a lot of control to fine-tune the various steps of the algorithm for their specific problem.

Solving the Trust Region Subproblem (TRS)

Below we describe the methods available for solving the trust region subproblem. The methods available provide either exact or approximate solutions to the trust region subproblem. In all algorithms below, the Hessian matrix B_k is approximated as B_k ≈ J_k^TJ_k, where J_k = J(x_k). In all methods, the solution of the TRS involves solving a linear least squares system involving the Jacobian matrix. For small to moderate sized problems (gsl_multifit_nlinear interface), this is accomplished by factoring the full Jacobian matrix, which is provided by the user, with the Cholesky, QR, or SVD decompositions. For large systems (gsl_multilarge_nlinear interface), the user has two choices. One is to solve the system iteratively, without needing to store the full Jacobian matrix in memory. With this method, the user must provide a routine to calculate the matrix-vector products Ju or J^Tu for a given vector u. This iterative method is particularly useful for systems where the Jacobian has sparse structure, since forming matrix-vector products can be done cheaply. The second option for large systems involves forming the normal equations matrix J^TJ and then factoring it using a Cholesky decomposition. The normal equations matrix is p-by-p, typically much smaller than the full n-by-p Jacobian, and can usually be stored in memory even if the full Jacobian matrix cannot. This option is useful for large, dense systems, or if the iterative method has difficulty converging.

single: Levenberg-Marquardt algorithm single: nonlinear least squares, levenberg-marquardt

Levenberg-Marquardt

There is a theorem which states that if δ_k is a solution to the trust region subproblem given above, then there exists μ_k ≥ 0 such that

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(B_k+μ_kD_k^TD_k)δ_k =  − g_k

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( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k

with μ_k(Δ_k − ||D_kδ_k||) = 0. This forms the basis of the Levenberg-Marquardt algorithm, which controls the trust region size by adjusting the parameter μ_k rather than the radius Δ_k directly. For each radius Δ_k, there is a unique parameter μ_k which solves the TRS, and they have an inverse relationship, so that large values of μ_k correspond to smaller trust regions, while small values of μ_k correspond to larger trust regions.

With the approximation B_k ≈ J_k^TJ_k, on each iteration, in order to calculate the step δ_k, the following linear least squares problem is solved:

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$$\begin{aligned} \left[ \begin{array}{c} J_k \\\ \sqrt{\mu_k} D_k \end{array} \right] \delta_k = - \left[ \begin{array}{c} f_k \cr 0 \end{array} \right] \end{aligned}$$

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[J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]

If the step δ_k is accepted, then μ_k is decreased on the next iteration in order to take a larger step, otherwise it is increased to take a smaller step. The Levenberg-Marquardt algorithm provides an exact solution of the trust region subproblem, but typically has a higher computational cost per iteration than the approximate methods discussed below, since it may need to solve the least squares system above several times for different values of μ_k.

single: Levenberg-Marquardt algorithm, geodesic acceleration single: nonlinear least squares, levenberg-marquardt, geodesic acceleration

Levenberg-Marquardt with Geodesic Acceleration

This method applies a so-called geodesic acceleration correction to the standard Levenberg-Marquardt step δ_k (Transtrum et al, 2011). By interpreting δ_k as a first order step along a geodesic in the model parameter space (ie: a velocity δ_k = v_k), the geodesic acceleration a_k is a second order correction along the geodesic which is determined by solving the linear least squares system

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$$\begin{aligned} \left[ \begin{array}{c} J_k \\\ \sqrt{\mu_k} D_k \end{array} \right] a_k = - \left[ \begin{array}{c} f_{vv}(x_k) \\\ 0 \end{array} \right] \end{aligned}$$

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[J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]

where f_vv is the second directional derivative of the residual vector in the velocity direction v, f_vv(x) = D_v²f = ∑_αβv_αv_β∂_α∂_βf(x), where α and β are summed over the p parameters. The new total step is then $\delta_k' = v_k + {1 \over 2}a_k$. The second order correction a_k can be calculated with a modest additional cost, and has been shown to dramatically reduce the number of iterations (and expensive Jacobian evaluations) required to reach convergence on a variety of different problems. In order to utilize the geodesic acceleration, the user must supply a function which provides the second directional derivative vector f_vv(x), or alternatively the library can use a finite difference method to estimate this vector with one additional function evaluation of f(x + hv) where h is a tunable step size (see the h_fvv parameter description).

single: Dogleg algorithm single: nonlinear least squares, dogleg

Dogleg

This is Powell's dogleg method, which finds an approximate solution to the trust region subproblem, by restricting its search to a piecewise linear "dogleg" path, composed of the origin, the Cauchy point which represents the model minimizer along the steepest descent direction, and the Gauss-Newton point, which is the overall minimizer of the unconstrained model. The Gauss-Newton step is calculated by solving

J_kδ_gn =  − f_k

which is the main computational task for each iteration, but only needs to be performed once per iteration. If the Gauss-Newton point is inside the trust region, it is selected as the step. If it is outside, the method then calculates the Cauchy point, which is located along the gradient direction. If the Cauchy point is also outside the trust region, the method assumes that it is still far from the minimum and so proceeds along the gradient direction, truncating the step at the trust region boundary. If the Cauchy point is inside the trust region, with the Gauss-Newton point outside, the method uses a dogleg step, which is a linear combination of the gradient direction and the Gauss-Newton direction, stopping at the trust region boundary.

single: double Dogleg algorithm single: Dogleg algorithm, double single: nonlinear least squares, double dogleg

Double Dogleg

This method is an improvement over the classical dogleg algorithm, which attempts to include information about the Gauss-Newton step while the iteration is still far from the minimum. When the Cauchy point is inside the trust region and the Gauss-Newton point is outside, the method computes a scaled Gauss-Newton point and then takes a dogleg step between the Cauchy point and the scaled Gauss-Newton point. The scaling is calculated to ensure that the reduction in the model m_k is about the same as the reduction provided by the Cauchy point.

Two Dimensional Subspace

The dogleg methods restrict the search for the TRS solution to a 1D curve defined by the Cauchy and Gauss-Newton points. An improvement to this is to search for a solution using the full two dimensional subspace spanned by the Cauchy and Gauss-Newton directions. The dogleg path is of course inside this subspace, and so this method solves the TRS at least as accurately as the dogleg methods. Since this method searches a larger subspace for a solution, it can converge more quickly than dogleg on some problems. Because the subspace is only two dimensional, this method is very efficient and the main computation per iteration is to determine the Gauss-Newton point.

Steihaug-Toint Conjugate Gradient

One difficulty of the dogleg methods is calculating the Gauss-Newton step when the Jacobian matrix is singular. The Steihaug-Toint method also computes a generalized dogleg step, but avoids solving for the Gauss-Newton step directly, instead using an iterative conjugate gradient algorithm. This method performs well at points where the Jacobian is singular, and is also suitable for large-scale problems where factoring the Jacobian matrix could be prohibitively expensive.

Weighted Nonlinear Least-Squares

Weighted nonlinear least-squares fitting minimizes the function

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$$\begin{aligned} \Phi(x) &= {1 \over 2} || f ||_W^2 \\\ &= {1 \over 2} \sum_{i=1}^{n} w_i f_i (x_1, \dots, x_p)^2 \end{aligned}$$

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\Phi(x) = (1/2) || f(x) ||_W^2
        = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2 

where $W = \diag(w_1,w_2,...,w_n)$ is the weighting matrix, and ||f||_W² = f^TWf. The weights w_i are commonly defined as w_i = 1/σ_i², where σ_i is the error in the i-th measurement. A simple change of variables $\tilde{f} = W^{1 \over 2} f$ yields $\Phi(x) = {1 \over 2} ||\tilde{f}||^2$, which is in the same form as the unweighted case. The user can either perform this transform directly on their function residuals and Jacobian, or use the gsl_multifit_nlinear_winit interface which automatically performs the correct scaling. To manually perform this transformation, the residuals and Jacobian should be modified according to

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$$\begin{aligned} \tilde{f}_i & = \sqrt{w_i} f_i = {f_i \over \sigma_i} \\\ \tilde{J}_{ij} & = \sqrt{w_i} { \partial f_i \over \partial x_j } = { 1 \over \sigma_i} { \partial f_i \over \partial x_j } \end{aligned}$$

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f~_i = f_i / \sigma_i
J~_ij = 1 / \sigma_i df_i/dx_j

For large systems, the user must perform their own weighting.

Tunable Parameters

The user can tune nearly all aspects of the iteration at allocation time. For the gsl_multifit_nlinear interface, the user may modify the gsl_multifit_nlinear_parameters structure, which is defined as follows:

gsl_multifit_nlinear_parameters

typedef struct
{
  const gsl_multifit_nlinear_trs *trs;        /* trust region subproblem method */
  const gsl_multifit_nlinear_scale *scale;    /* scaling method */
  const gsl_multifit_nlinear_solver *solver;  /* solver method */
  gsl_multifit_nlinear_fdtype fdtype;         /* finite difference method */
  double factor_up;                           /* factor for increasing trust radius */
  double factor_down;                         /* factor for decreasing trust radius */
  double avmax;                               /* max allowed |a|/|v| */
  double h_df;                                /* step size for finite difference Jacobian */
  double h_fvv;                               /* step size for finite difference fvv */
} gsl_multifit_nlinear_parameters;

For the gsl_multilarge_nlinear interface, the user may modify the gsl_multilarge_nlinear_parameters structure, which is defined as follows:

gsl_multilarge_nlinear_parameters

typedef struct
{
  const gsl_multilarge_nlinear_trs *trs;       /* trust region subproblem method */
  const gsl_multilarge_nlinear_scale *scale;   /* scaling method */
  const gsl_multilarge_nlinear_solver *solver; /* solver method */
  gsl_multilarge_nlinear_fdtype fdtype;        /* finite difference method */
  double factor_up;                            /* factor for increasing trust radius */
  double factor_down;                          /* factor for decreasing trust radius */
  double avmax;                                /* max allowed |a|/|v| */
  double h_df;                                 /* step size for finite difference Jacobian */
  double h_fvv;                                /* step size for finite difference fvv */
  size_t max_iter;                             /* maximum iterations for trs method */
  double tol;                                  /* tolerance for solving trs */
} gsl_multilarge_nlinear_parameters;

Each of these parameters is discussed in further detail below.

gsl_multifit_nlinear_trs gsl_multilarge_nlinear_trs

The parameter trs determines the method used to solve the trust region subproblem, and may be selected from the following choices,

gsl_multifit_nlinear_trs * gsl_multifit_nlinear_trs_lm gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_lm

This selects the Levenberg-Marquardt algorithm.

gsl_multifit_nlinear_trs * gsl_multifit_nlinear_trs_lmaccel gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_lmaccel

This selects the Levenberg-Marquardt algorithm with geodesic acceleration.

gsl_multifit_nlinear_trs * gsl_multifit_nlinear_trs_dogleg gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_dogleg

This selects the dogleg algorithm.

gsl_multifit_nlinear_trs * gsl_multifit_nlinear_trs_ddogleg gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_ddogleg

This selects the double dogleg algorithm.

gsl_multifit_nlinear_trs * gsl_multifit_nlinear_trs_subspace2D gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_subspace2D

This selects the 2D subspace algorithm.

gsl_multilarge_nlinear_trs * gsl_multilarge_nlinear_trs_cgst

This selects the Steihaug-Toint conjugate gradient algorithm. This method is available only for large systems.

gsl_multifit_nlinear_scale gsl_multilarge_nlinear_scale

The parameter scale determines the diagonal scaling matrix D and may be selected from the following choices,

gsl_multifit_nlinear_scale * gsl_multifit_nlinear_scale_more gsl_multilarge_nlinear_scale * gsl_multilarge_nlinear_scale_more

This damping strategy was suggested by , and corresponds to $D^T D = \max(\diag(J^T J))$, in other words the maximum elements of $\diag(J^T J)$ encountered thus far in the iteration. This choice of D makes the problem scale-invariant, so that if the model parameters x_i are each scaled by an arbitrary constant, x̃_i = a_ix_i, then the sequence of iterates produced by the algorithm would be unchanged. This method can work very well in cases where the model parameters have widely different scales (ie: if some parameters are measured in nanometers, while others are measured in degrees Kelvin). This strategy has been proven effective on a large class of problems and so it is the library default, but it may not be the best choice for all problems.

gsl_multifit_nlinear_scale * gsl_multifit_nlinear_scale_levenberg gsl_multilarge_nlinear_scale * gsl_multilarge_nlinear_scale_levenberg

This damping strategy was originally suggested by Levenberg, and corresponds to D^TD = I. This method has also proven effective on a large class of problems, but is not scale-invariant. However, some authors (e.g. Transtrum and Sethna 2012) argue that this choice is better for problems which are susceptible to parameter evaporation (ie: parameters go to infinity)

gsl_multifit_nlinear_scale * gsl_multifit_nlinear_scale_marquardt gsl_multilarge_nlinear_scale * gsl_multilarge_nlinear_scale_marquardt

This damping strategy was suggested by Marquardt, and corresponds to $D^T D = \diag(J^T J)$. This method is scale-invariant, but it is generally considered inferior to both the Levenberg and strategies, though may work well on certain classes of problems.

gsl_multifit_nlinear_solver gsl_multilarge_nlinear_solver

Solving the trust region subproblem on each iteration almost always requires the solution of the following linear least squares system

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$$\begin{aligned} \left[ \begin{array}{c} J \\\ \sqrt{\mu} D \end{array} \right] \delta = - \left[ \begin{array}{c} f \\\ 0 \end{array} \right] \end{aligned}$$

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[J; sqrt(mu) D] \delta = - [f; 0]

The solver parameter determines how the system is solved and can be selected from the following choices:

gsl_multifit_nlinear_solver * gsl_multifit_nlinear_solver_qr

This method solves the system using a rank revealing QR decomposition of the Jacobian J. This method will produce reliable solutions in cases where the Jacobian is rank deficient or near-singular but does require about twice as many operations as the Cholesky method discussed below.

gsl_multifit_nlinear_solver * gsl_multifit_nlinear_solver_cholesky gsl_multilarge_nlinear_solver * gsl_multilarge_nlinear_solver_cholesky

This method solves the alternate normal equations problem

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(J^TJ+μD^TD)δ =  − J^Tf

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( J^T J + \mu D^T D ) \delta = -J^T f

by using a Cholesky decomposition of the matrix J^TJ + μD^TD. This method is faster than the QR approach, however it is susceptible to numerical instabilities if the Jacobian matrix is rank deficient or near-singular. In these cases, an attempt is made to reduce the condition number of the matrix using Jacobi preconditioning, but for highly ill-conditioned problems the QR approach is better. If it is known that the Jacobian matrix is well conditioned, this method is accurate and will perform faster than the QR approach.

gsl_multifit_nlinear_solver * gsl_multifit_nlinear_solver_mcholesky gsl_multilarge_nlinear_solver * gsl_multilarge_nlinear_solver_mcholesky

This method solves the alternate normal equations problem

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(J^TJ+μD^TD)δ =  − J^Tf

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( J^T J + \mu D^T D ) \delta = -J^T f

by using a modified Cholesky decomposition of the matrix J^TJ + μD^TD. This is more suitable for the dogleg methods where the parameter μ = 0, and the matrix J^TJ may be ill-conditioned or indefinite causing the standard Cholesky decomposition to fail. This method is based on Level 2 BLAS and is thus slower than the standard Cholesky decomposition, which is based on Level 3 BLAS.

gsl_multifit_nlinear_solver * gsl_multifit_nlinear_solver_svd

This method solves the system using a singular value decomposition of the Jacobian J. This method will produce the most reliable solutions for ill-conditioned Jacobians but is also the slowest solver method.

gsl_multifit_nlinear_fdtype

The parameter fdtype specifies whether to use forward or centered differences when approximating the Jacobian. This is only used when an analytic Jacobian is not provided to the solver. This parameter may be set to one of the following choices.

GSL_MULTIFIT_NLINEAR_FWDIFF

This specifies a forward finite difference to approximate the Jacobian matrix. The Jacobian matrix will be calculated as

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$$J_{ij} = {1 \over \Delta_j} \left( f_i(x + \Delta_j e_j) - f_i(x) \right)$$

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J_ij = 1 / \Delta_j ( f_i(x + \Delta_j e_j) - f_i(x) )

where Δ_j = h|x_j| and e_j is the standard j-th Cartesian unit basis vector so that x + Δ_je_j represents a small (forward) perturbation of the j-th parameter by an amount Δ_j. The perturbation Δ_j is proportional to the current value |x_j| which helps to calculate an accurate Jacobian when the various parameters have different scale sizes. The value of h is specified by the h_df parameter. The accuracy of this method is O(h), and evaluating this matrix requires an additional p function evaluations.

GSL_MULTIFIT_NLINEAR_CTRDIFF

This specifies a centered finite difference to approximate the Jacobian matrix. The Jacobian matrix will be calculated as

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$$J_{ij} = {1 \over \Delta_j} \left( f_i(x + {1 \over 2} \Delta_j e_j) - f_i(x - {1 \over 2} \Delta_j e_j) \right)$$

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J_ij = 1 / \Delta_j ( f_i(x + 1/2 \Delta_j e_j) - f_i(x - 1/2 \Delta_j e_j) )

See above for a description of Δ_j. The accuracy of this method is O(h²), but evaluating this matrix requires an additional 2p function evaluations.

double factor_up

When a step is accepted, the trust region radius will be increased by this factor. The default value is 3.

double factor_down

When a step is rejected, the trust region radius will be decreased by this factor. The default value is 2.

double avmax

When using geodesic acceleration to solve a nonlinear least squares problem, an important parameter to monitor is the ratio of the acceleration term to the velocity term,

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$${ ||a|| \over ||v|| }$$

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|a| / |v|

If this ratio is small, it means the acceleration correction is contributing very little to the step. This could be because the problem is not "nonlinear" enough to benefit from the acceleration. If the ratio is large ( > 1) it means that the acceleration is larger than the velocity, which shouldn't happen since the step represents a truncated series and so the second order term a should be smaller than the first order term v to guarantee convergence. Therefore any steps with a ratio larger than the parameter avmax are rejected. avmax is set to 0.75 by default. For problems which experience difficulty converging, this threshold could be lowered.

double h_df

This parameter specifies the step size for approximating the Jacobian matrix with finite differences. It is set to $\sqrt{\epsilon}$ by default, where ϵ is GSL_DBL_EPSILON.

double h_fvv

When using geodesic acceleration, the user must either supply a function to calculate f_vv(x) or the library can estimate this second directional derivative using a finite difference method. When using finite differences, the library must calculate f(x + hv) where h represents a small step in the velocity direction. The parameter h_fvv defines this step size and is set to 0.02 by default.

Initializing the Solver

gsl_multifit_nlinear_type

This structure specifies the type of algorithm which will be used to solve a nonlinear least squares problem. It may be selected from the following choices,

gsl_multifit_nlinear_type * gsl_multifit_nlinear_trust

This specifies a trust region method. It is currently the only implemented nonlinear least squares method.

gsl_multifit_nlinear_workspace * gsl_multifit_nlinear_alloc (const gsl_multifit_nlinear_type * T, const gsl_multifit_nlinear_parameters * params, const size_t n, const size_t p) gsl_multilarge_nlinear_workspace * gsl_multilarge_nlinear_alloc (const gsl_multilarge_nlinear_type * T, const gsl_multilarge_nlinear_parameters * params, const size_t n, const size_t p)

These functions return a pointer to a newly allocated instance of a derivative solver of type T for n observations and p parameters. The params input specifies a tunable set of parameters which will affect important details in each iteration of the trust region subproblem algorithm. It is recommended to start with the suggested default parameters (see gsl_multifit_nlinear_default_parameters and gsl_multilarge_nlinear_default_parameters) and then tune the parameters once the code is working correctly. See sec_tunable-parameters. for descriptions of the various parameters. For example, the following code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters, using suggested defaults:

const gsl_multifit_nlinear_type * T = gsl_multifit_nlinear_trust;
gsl_multifit_nlinear_parameters params = gsl_multifit_nlinear_default_parameters();
gsl_multifit_nlinear_workspace * w = gsl_multifit_nlinear_alloc (T, &params, 100, 3);

The number of observations n must be greater than or equal to parameters p.

If there is insufficient memory to create the solver then the function returns a null pointer and the error handler is invoked with an error code of GSL_ENOMEM.

gsl_multifit_nlinear_parameters gsl_multifit_nlinear_default_parameters (void) gsl_multilarge_nlinear_parameters gsl_multilarge_nlinear_default_parameters (void)

These functions return a set of recommended default parameters for use in solving nonlinear least squares problems. The user can tune each parameter to improve the performance on their particular problem, see sec_tunable-parameters.

int gsl_multifit_nlinear_init (const gsl_vector * x, gsl_multifit_nlinear_fdf * fdf, gsl_multifit_nlinear_workspace * w) int gsl_multifit_nlinear_winit (const gsl_vector * x, const gsl_vector * wts, gsl_multifit_nlinear_fdf * fdf, gsl_multifit_nlinear_workspace * w) int gsl_multilarge_nlinear_init (const gsl_vector * x, gsl_multilarge_nlinear_fdf * fdf, gsl_multilarge_nlinear_workspace * w)

These functions initialize, or reinitialize, an existing workspace w to use the system fdf and the initial guess x. See sec_providing-function-minimized for a description of the fdf structure.

Optionally, a weight vector wts can be given to perform a weighted nonlinear regression. Here, the weighting matrix is $W = \diag(w_1,w_2,...,w_n)$.

void gsl_multifit_nlinear_free (gsl_multifit_nlinear_workspace * w) void gsl_multilarge_nlinear_free (gsl_multilarge_nlinear_workspace * w)

These functions free all the memory associated with the workspace w.

const char * gsl_multifit_nlinear_name (const gsl_multifit_nlinear_workspace * w) const char * gsl_multilarge_nlinear_name (const gsl_multilarge_nlinear_workspace * w)

These functions return a pointer to the name of the solver. For example:

printf ("w is a '%s' solver\n", gsl_multifit_nlinear_name (w));

would print something like w is a 'trust-region' solver.

const char * gsl_multifit_nlinear_trs_name (const gsl_multifit_nlinear_workspace * w) const char * gsl_multilarge_nlinear_trs_name (const gsl_multilarge_nlinear_workspace * w)

These functions return a pointer to the name of the trust region subproblem method. For example:

printf ("w is a '%s' solver\n", gsl_multifit_nlinear_trs_name (w));

would print something like w is a 'levenberg-marquardt' solver.

Providing the Function to be Minimized

The user must provide n functions of p variables for the minimization algorithm to operate on. In order to allow for arbitrary parameters the functions are defined by the following data types:

gsl_multifit_nlinear_fdf

This data type defines a general system of functions with arbitrary parameters, the corresponding Jacobian matrix of derivatives, and optionally the second directional derivative of the functions for geodesic acceleration.

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

This function should store the n components of the vector f(x) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.

int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)

This function should store the n-by-p matrix result

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J_ij = ∂f_i(x)/∂x_j

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J_ij = d f_i(x) / d x_j

in J for argument x and arbitrary parameters params, returning an appropriate error code if the matrix cannot be computed. If an analytic Jacobian is unavailable, or too expensive to compute, this function pointer may be set to NULL, in which case the Jacobian will be internally computed using finite difference approximations of the function f.

int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector * fvv)

When geodesic acceleration is enabled, this function should store the n components of the vector $f_{vv}(x) = \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x)$, representing second directional derivatives of the function to be minimized, into the output fvv. The parameter vector is provided in x and the velocity vector is provided in v, both of which have p components. The arbitrary parameters are given in params. If analytic expressions for f_vv(x) are unavailable or too difficult to compute, this function pointer may be set to NULL, in which case f_vv(x) will be computed internally using a finite difference approximation.

size_t n

the number of functions, i.e. the number of components of the vector f.

size_t p

the number of independent variables, i.e. the number of components of the vector x.

void * params

a pointer to the arbitrary parameters of the function.

size_t nevalf

This does not need to be set by the user. It counts the number of function evaluations and is initialized by the _init function.

size_t nevaldf

This does not need to be set by the user. It counts the number of Jacobian evaluations and is initialized by the _init function.

size_t nevalfvv

This does not need to be set by the user. It counts the number of f_vv(x) evaluations and is initialized by the _init function.

gsl_multilarge_nlinear_fdf

This data type defines a general system of functions with arbitrary parameters, a function to compute Ju or J^Tu for a given vector u, the normal equations matrix J^TJ, and optionally the second directional derivative of the functions for geodesic acceleration.

int (* f) (const gsl_vector * x, void * params, gsl_vector * f)

This function should store the n components of the vector f(x) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.

int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const gsl_vector * u, void * params, gsl_vector * v, gsl_matrix * JTJ)

If TransJ is equal to CblasNoTrans, then this function should compute the matrix-vector product Ju and store the result in v. If TransJ is equal to CblasTrans, then this function should compute the matrix-vector product J^Tu and store the result in v. Additionally, the normal equations matrix J^TJ should be stored in the lower half of JTJ. The input matrix JTJ could be set to NULL, for example by iterative methods which do not require this matrix, so the user should check for this prior to constructing the matrix. The input params contains the arbitrary parameters.

int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector * fvv)

When geodesic acceleration is enabled, this function should store the n components of the vector $f_{vv}(x) = \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x)$, representing second directional derivatives of the function to be minimized, into the output fvv. The parameter vector is provided in x and the velocity vector is provided in v, both of which have p components. The arbitrary parameters are given in params. If analytic expressions for f_vv(x) are unavailable or too difficult to compute, this function pointer may be set to NULL, in which case f_vv(x) will be computed internally using a finite difference approximation.

size_t n

the number of functions, i.e. the number of components of the vector f.

size_t p

the number of independent variables, i.e. the number of components of the vector x.

void * params

a pointer to the arbitrary parameters of the function.

size_t nevalf

This does not need to be set by the user. It counts the number of function evaluations and is initialized by the _init function.

size_t nevaldfu

This does not need to be set by the user. It counts the number of Jacobian matrix-vector evaluations (Ju or J^Tu) and is initialized by the _init function.

size_t nevaldf2

This does not need to be set by the user. It counts the number of J^TJ evaluations and is initialized by the _init function.

size_t nevalfvv

This does not need to be set by the user. It counts the number of f_vv(x) evaluations and is initialized by the _init function.

Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using the params argument and the trial best-fit parameters through the x argument.

Iteration

The following functions drive the iteration of each algorithm. Each function performs one iteration of the trust region method and updates the state of the solver.

int gsl_multifit_nlinear_iterate (gsl_multifit_nlinear_workspace * w) int gsl_multilarge_nlinear_iterate (gsl_multilarge_nlinear_workspace * w)

These functions perform a single iteration of the solver w. If the iteration encounters an unexpected problem then an error code will be returned. The solver workspace maintains a current estimate of the best-fit parameters at all times.

The solver workspace w contains the following entries, which can be used to track the progress of the solution:

gsl_vector * x

The current position, length p.

gsl_vector * f

The function residual vector at the current position f(x), length n.

gsl_matrix * J

The Jacobian matrix at the current position J(x), size n-by-p (only for gsl_multifit_nlinear interface).

gsl_vector * dx

The difference between the current position and the previous position, i.e. the last step δ, taken as a vector, length p.

These quantities can be accessed with the following functions,

gsl_vector * gsl_multifit_nlinear_position (const gsl_multifit_nlinear_workspace * w) gsl_vector * gsl_multilarge_nlinear_position (const gsl_multilarge_nlinear_workspace * w)

These functions return the current position x (i.e. best-fit parameters) of the solver w.

gsl_vector * gsl_multifit_nlinear_residual (const gsl_multifit_nlinear_workspace * w) gsl_vector * gsl_multilarge_nlinear_residual (const gsl_multilarge_nlinear_workspace * w)

These functions return the current residual vector f(x) of the solver w. For weighted systems, the residual vector includes the weighting factor $\sqrt{W}$.

gsl_matrix * gsl_multifit_nlinear_jac (const gsl_multifit_nlinear_workspace * w)

This function returns a pointer to the n-by-p Jacobian matrix for the current iteration of the solver w. This function is available only for the gsl_multifit_nlinear interface.

size_t gsl_multifit_nlinear_niter (const gsl_multifit_nlinear_workspace * w) size_t gsl_multilarge_nlinear_niter (const gsl_multilarge_nlinear_workspace * w)

These functions return the number of iterations performed so far. The iteration counter is updated on each call to the _iterate functions above, and reset to 0 in the _init functions.

int gsl_multifit_nlinear_rcond (double * rcond, const gsl_multifit_nlinear_workspace * w) int gsl_multilarge_nlinear_rcond (double * rcond, const gsl_multilarge_nlinear_workspace * w)

This function estimates the reciprocal condition number of the Jacobian matrix at the current position x and stores it in rcond. The computed value is only an estimate to give the user a guideline as to the conditioning of their particular problem. Its calculation is based on which factorization method is used (Cholesky, QR, or SVD).

• For the Cholesky solver, the matrix J^TJ is factored at each iteration. Therefore this function will estimate the 1-norm condition number rcond² = 1/(||J^TJ||₁ ⋅ ||(J^TJ)^− 1||₁)
• For the QR solver, J is factored as J = QR at each iteration. For simplicity, this function calculates the 1-norm conditioning of only the R factor, rcond = 1/(||R||₁ ⋅ ||R^− 1||₁). This can be computed efficiently since R is upper triangular.
• For the SVD solver, in order to efficiently solve the trust region subproblem, the matrix which is factored is JD^− 1, instead of J itself. The resulting singular values are used to provide the 2-norm reciprocal condition number, as rcond = σ_min/σ_max. Note that when using scaling, D ≠ I and the resulting rcond estimate may be significantly different from the true rcond of J itself.

double gsl_multifit_nlinear_avratio (const gsl_multifit_nlinear_workspace * w) double gsl_multilarge_nlinear_avratio (const gsl_multilarge_nlinear_workspace * w)

This function returns the current ratio |a|/|v| of the acceleration correction term to the velocity step term. The acceleration term is computed only by the gsl_multifit_nlinear_trs_lmaccel and gsl_multilarge_nlinear_trs_lmaccel methods, so this ratio will be zero for other TRS methods.

single: nonlinear fitting, stopping parameters, convergence

Testing for Convergence

A minimization procedure should stop when one of the following conditions is true:

• A minimum has been found to within the user-specified precision.
• A user-specified maximum number of iterations has been reached.
• An error has occurred.

The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.

int gsl_multifit_nlinear_test (const double xtol, const double gtol, const double ftol, int * info, const gsl_multifit_nlinear_workspace * w) int gsl_multilarge_nlinear_test (const double xtol, const double gtol, const double ftol, int * info, const gsl_multilarge_nlinear_workspace * w)

These functions test for convergence of the minimization method using the following criteria:

• Testing for a small step size relative to the current parameter vector

  
  |δ_i| ≤ xtol(|x_i| + xtol)
  

  for each 0 <  = i < p. Each element of the step vector δ is tested individually in case the different parameters have widely different scales. Adding xtol to |x_i| helps the test avoid breaking down in situations where the true solution value x_i = 0. If this test succeeds, info is set to 1 and the function returns GSL_SUCCESS.

  A general guideline for selecting the step tolerance is to choose xtol = 10^− d where d is the number of accurate decimal digits desired in the solution x. See Dennis and Schnabel for more information.

• Testing for a small gradient (g = ∇Φ(x) = J^Tf) indicating a local function minimum:

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  max_i|g_i × max (x_i, 1)| ≤ gtol × max (Φ(x), 1)
  

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  ||g||_inf <= gtol

  This expression tests whether the ratio (∇Φ)_ix_i/Φ is small. Testing this scaled gradient is a better than ∇Φ alone since it is a dimensionless quantity and so independent of the scale of the problem. The max arguments help ensure the test doesn't break down in regions where x_i or Φ(x) are close to 0. If this test succeeds, info is set to 2 and the function returns GSL_SUCCESS.

  A general guideline for choosing the gradient tolerance is to set gtol = GSL_DBL_EPSILON^(1/3). See Dennis and Schnabel for more information.

If none of the tests succeed, info is set to 0 and the function returns GSL_CONTINUE, indicating further iterations are required.

High Level Driver

These routines provide a high level wrapper that combines the iteration and convergence testing for easy use.

int gsl_multifit_nlinear_driver (const size_t maxiter, const double xtol, const double gtol, const double ftol, void (* callback)(const size_t iter, void * params, const gsl_multifit_linear_workspace * w), void * callback_params, int * info, gsl_multifit_nlinear_workspace * w) int gsl_multilarge_nlinear_driver (const size_t maxiter, const double xtol, const double gtol, const double ftol, void (* callback)(const size_t iter, void * params, const gsl_multilarge_linear_workspace * w), void * callback_params, int * info, gsl_multilarge_nlinear_workspace * w)

These functions iterate the nonlinear least squares solver w for a maximum of maxiter iterations. After each iteration, the system is tested for convergence with the error tolerances xtol, gtol and ftol. Additionally, the user may supply a callback function callback which is called after each iteration, so that the user may save or print relevant quantities for each iteration. The parameter callback_params is passed to the callback function. The parameters callback and callback_params may be set to NULL to disable this feature. Upon successful convergence, the function returns GSL_SUCCESS and sets info to the reason for convergence (see gsl_multifit_nlinear_test). If the function has not converged after maxiter iterations, GSL_EMAXITER is returned. In rare cases, during an iteration the algorithm may be unable to find a new acceptable step δ to take. In this case, GSL_ENOPROG is returned indicating no further progress can be made. If your problem is having difficulty converging, see sec_nlinear-troubleshooting for further guidance.

single: best-fit parameters, covariance single: least squares, covariance of best-fit parameters single: covariance matrix, nonlinear fits

Covariance matrix of best fit parameters

int gsl_multifit_nlinear_covar (const gsl_matrix * J, const double epsrel, gsl_matrix * covar) int gsl_multilarge_nlinear_covar (gsl_matrix * covar, gsl_multilarge_nlinear_workspace * w)

This function computes the covariance matrix of best-fit parameters using the Jacobian matrix J and stores it in covar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient.

The covariance matrix is given by,

C = (J^TJ)^− 1

or in the weighted case,

C = (J^TWJ)^− 1

and is computed using the factored form of the Jacobian (Cholesky, QR, or SVD). Any columns of R which satisfy

|R_kk| ≤ epsrel|R₁₁|

are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero).

If the minimisation uses the weighted least-squares function f_i = (Y(x, t_i) − y_i)/σ_i then the covariance matrix above gives the statistical error on the best-fit parameters resulting from the Gaussian errors σ_i on the underlying data y_i. This can be verified from the relation δf = Jδc and the fact that the fluctuations in f from the data y_i are normalised by σ_i and so satisfy

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⟨δfδf^T⟩ = I

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<\delta f \delta f^T> = I

For an unweighted least-squares function f_i = (Y(x, t_i) − y_i) the covariance matrix above should be multiplied by the variance of the residuals about the best-fit σ² = ∑(y_i − Y(x, t_i))²/(n − p) to give the variance-covariance matrix σ²C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data.

For more information about covariance matrices see Linear Least-Squares Overview <sec_lls-overview>.

Troubleshooting

When developing a code to solve a nonlinear least squares problem, here are a few considerations to keep in mind.

1. The most common difficulty is the accurate implementation of the Jacobian matrix. If the analytic Jacobian is not properly provided to the solver, this can hinder and many times prevent convergence of the method. When developing a new nonlinear least squares code, it often helps to compare the program output with the internally computed finite difference Jacobian and the user supplied analytic Jacobian. If there is a large difference in coefficients, it is likely the analytic Jacobian is incorrectly implemented.
2. If your code is having difficulty converging, the next thing to check is the starting point provided to the solver. The methods of this chapter are local methods, meaning if you provide a starting point far away from the true minimum, the method may converge to a local minimum or not converge at all. Sometimes it is possible to solve a linearized approximation to the nonlinear problem, and use the linear solution as the starting point to the nonlinear problem.
3. If the various parameters of the coefficient vector x vary widely in magnitude, then the problem is said to be badly scaled. The methods of this chapter do attempt to automatically rescale the elements of x to have roughly the same order of magnitude, but in extreme cases this could still cause problems for convergence. In these cases it is recommended for the user to scale their parameter vector x so that each parameter spans roughly the same range, say [ − 1, 1]. The solution vector can be backscaled to recover the original units of the problem.

Examples

The following example programs demonstrate the nonlinear least squares fitting capabilities.

Exponential Fitting Example

The following example program fits a weighted exponential model with background to experimental data, Y = Aexp ( − λt) + b. The first part of the program sets up the functions expb_f and expb_df to calculate the model and its Jacobian. The appropriate fitting function is given by,

f_i = (Aexp ( − λt_i) + b) − y_i

where we have chosen t_i = iT/(N − 1), where N is the number of data points fitted, so that t_i ∈ [0, T]. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, λ, b). It is given by,

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$$J_{ij} = {\partial f_i \over \partial x_j}$$

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J_{ij} = d f_i / d x_j

where x₀ = A, x₁ = λ and x₂ = b. The i-th row of the Jacobian is therefore

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$$J_{i\cdot} = \left( \begin{array}{ccc} \exp(-\lambda t_i) & -t_i A \exp(-\lambda t_i) & 1 \end{array} \right)$$

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J(i,:) = [ \exp(-\lambda t_i) ; -t_i A \exp(-\lambda t_i) ; 1 ]

The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (5.0,1.5,1.0) combined with Gaussian noise (standard deviation = 0.1) with a maximum time T = 3 and N = 100 timesteps. The initial guess for the parameters is chosen as (1.0, 1.0, 0.0). The iteration terminates when the relative change in x is smaller than 10^− 8, or when the magnitude of the gradient falls below 10^− 8. Here are the results of running the program:

iter  0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) =      inf, |f(x)| = 88.4448
iter  1: A = 4.5109, lambda = 2.5258, b = 1.0704, cond(J) =  26.2686, |f(x)| = 24.0646
iter  2: A = 4.8565, lambda = 1.7442, b = 1.1669, cond(J) =  23.7470, |f(x)| = 11.9797
iter  3: A = 4.9356, lambda = 1.5713, b = 1.0767, cond(J) =  17.5849, |f(x)| = 10.7355
iter  4: A = 4.8678, lambda = 1.4838, b = 1.0252, cond(J) =  16.3428, |f(x)| = 10.5000
iter  5: A = 4.8118, lambda = 1.4481, b = 1.0076, cond(J) =  15.7925, |f(x)| = 10.4786
iter  6: A = 4.7983, lambda = 1.4404, b = 1.0041, cond(J) =  15.5840, |f(x)| = 10.4778
iter  7: A = 4.7967, lambda = 1.4395, b = 1.0037, cond(J) =  15.5396, |f(x)| = 10.4778
iter  8: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5344, |f(x)| = 10.4778
iter  9: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
iter 10: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
iter 11: A = 4.7965, lambda = 1.4394, b = 1.0037, cond(J) =  15.5339, |f(x)| = 10.4778
summary from method 'trust-region/levenberg-marquardt'
number of iterations: 11
function evaluations: 16
Jacobian evaluations: 12
reason for stopping: small gradient
initial |f(x)| = 88.444756
final   |f(x)| = 10.477801
chisq/dof = 1.1318
A      = 4.79653 +/- 0.18704
lambda = 1.43937 +/- 0.07390
b      = 1.00368 +/- 0.03473
status = success

The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chi-squared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated from the square roots of the diagonal elements of the covariance matrix. If the chi-squared value shows a poor fit (i.e. χ²/(n − p) ≫ 1 then the error estimates obtained from the covariance matrix will be too small. In the example program the error estimates are multiplied by $\sqrt{\chi^2/(n-p)}$ in this case, a common way of increasing the errors for a poor fit. Note that a poor fit will result from the use of an inappropriate model, and the scaled error estimates may then be outside the range of validity for Gaussian errors.

Additionally, we see that the condition number of J(x) stays reasonably small throughout the iteration. This indicates we could safely switch to the Cholesky solver for speed improvement, although this particular system is too small to really benefit.

fig_fit-exp shows the fitted curve with the original data.

Exponential fitted curve with data

Geodesic Acceleration Example 1

The following example program minimizes a modified Rosenbrock function, which is characterized by a narrow canyon with steep walls. The starting point is selected high on the canyon wall, so the solver must first find the canyon bottom and then navigate to the minimum. The problem is solved both with and without using geodesic acceleration for comparison. The cost function is given by

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$$\begin{aligned} \Phi(x) &= {1 \over 2} (f_1^2 + f_2^2) \\\ f_1 &= 100 \left( x_2 - x_1^2 \right) \\\ f_2 &= 1 - x_1 \end{aligned}$$

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Phi(x) = 1/2 (f1^2 + f2^2)
f1 = 100 ( x2 - x1^2 )
f2 = 1 - x1

The Jacobian matrix is

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$$\begin{aligned} J = \left( \begin{array}{cc} {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} \\\ {\partial f_2 \over \partial x_1} & {\partial f_2 \over \partial x_2} \end{array} \right) = \left( \begin{array}{cc} -200 x_1 & 100 \\\ -1 & 0 \end{array} \right) \end{aligned}$$

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J = [ -200*x1 100  ]
    [   -1     0   ]

In order to use geodesic acceleration, the user must provide the second directional derivative of each residual in the velocity direction, D_v²f_i = ∑_αβv_αv_β∂_α∂_βf_i. The velocity vector v is provided by the solver. For this example, these derivatives are

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$$\begin{aligned} f_{vv} = D_v^2 \left( \begin{array}{c} f_1 \\\ f_2 \end{array} \right) = \left( \begin{array}{c} -200 v_1^2 \\\ 0 \end{array} \right) \end{aligned}$$

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fvv = [ -200 v1^2 ]
      [     0     ]

The solution of this minimization problem is

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$$\begin{aligned} x^{*} &= \left( \begin{array}{c} 1 \\\ 1 \end{array} \right) \\\ \Phi(x^{*}) &= 0 \end{aligned}$$

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x* = [ 1 ; 1 ]
Phi(x*) = 0

The program output is shown below:

=== Solving system without acceleration ===
NITER         = 53
NFEV          = 56
NJEV          = 54
NAEV          = 0
initial cost  = 2.250225000000e+04
final cost    = 6.674986031430e-18
final x       = (9.999999974165e-01, 9.999999948328e-01)
final cond(J) = 6.000096055094e+02
=== Solving system with acceleration ===
NITER         = 15
NFEV          = 17
NJEV          = 16
NAEV          = 16
initial cost  = 2.250225000000e+04
final cost    = 7.518932873279e-19
final x       = (9.999999991329e-01, 9.999999982657e-01)
final cond(J) = 6.000097233278e+02

Paths taken by solver for Rosenbrock function

We can see that enabling geodesic acceleration requires less than a third of the number of Jacobian evaluations in order to locate the minimum. The path taken by both methods is shown in fig_nlfit2. The contours show the cost function Φ(x₁, x₂). We see that both methods quickly find the canyon bottom, but the geodesic acceleration method navigates along the bottom to the solution with significantly fewer iterations.

The program is given below.

Geodesic Acceleration Example 2

The following example fits a set of data to a Gaussian model using the Levenberg-Marquardt method with geodesic acceleration. The cost function is

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$$\begin{aligned} \Phi(x) &= {1 \over 2} \sum_i f_i^2 \\\ f_i &= y_i - Y(a,b,c;t_i) \end{aligned}$$

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Phi(x) = 1/2 \sum_i f_i^2
f_i = y_i - Y(a,b,c;t_i)

where y_i is the measured data point at time t_i, and the model is specified by

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$$Y(a,b,c;t) = a \exp{ \left[ -{1 \over 2} \left( { t - b \over c } \right)^2 \right] }$$

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Y(a,b,c;t) = a exp(-1/2 ((t-b)/c)^2)

The parameters a, b, c represent the amplitude, mean, and width of the Gaussian respectively. The program below generates the y_i data on [0, 1] using the values a = 5, b = 0.4, c = 0.15 and adding random noise. The i-th row of the Jacobian is

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$$J_{i,:} = \left( \begin{array}{ccc} {\partial f_i \over \partial a} & {\partial f_i \over \partial b} & {\partial f_i \over \partial c} \end{array} \right) = \left( \begin{array}{ccc} -e_i & -{a \over c} z_i e_i & -{a \over c} z_i^2 e_i \end{array} \right)$$

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J(i,:) = ( -e_i  -(a/c)*z_i*e_i  -(a/c)*z_i^2*e_i )

where

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$$\begin{aligned} z_i &= { t_i - b \over c} \\\ e_i &= \exp{\left( -{1 \over 2} z_i^2 \right)} \end{aligned}$$

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z_i = (t_i - b) / c
e_i = \exp(-1/2 z_i^2)

In order to use geodesic acceleration, we need the second directional derivative of the residuals in the velocity direction, D_v²f_i = ∑_αβv_αv_β∂_α∂_βf_i, where v is provided by the solver. To compute this, it is helpful to make a table of all second derivatives of the residuals f_i with respect to each combination of model parameters. This table is

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$$\begin{array}{cccc} & {\partial \over \partial a} & {\partial \over \partial b} & {\partial \over \partial c} \cr {\partial \over \partial a} & 0 & -{z_i \over c} e_i & -{z_i^2 \over c} e_i \cr {\partial \over \partial b} & & {a \over c^2} \left( 1 - z_i^2 \right) e_i & {a \over c^2} z_i \left( 2 - z_i^2 \right) e_i \cr {\partial \over \partial c} & & & {a \over c^2} z_i^2 \left( 3 - z_i^2 \right) e_i \end{array}$$

The lower half of the table is omitted since it is symmetric. Then, the second directional derivative of f_i is

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D_v²f_i = v_a²∂_a²f_i + 2v_av_b∂_a∂_bf_i + 2v_av_c∂_a∂_cf_i + v_b²∂_b²f_i + 2v_bv_c∂_b∂_cf_i + v_c²∂_c²f_i

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D_v^2 f_i = v_a^2 (d/da)^2 f_i + 2 v_a v_b (d/da) (d/db) f_i + 2 v_a v_c (d/da) (d/dc) f_i + v_b^2 (d/db)^2 f_i + 2 v_b v_c (d/db) (d/dc) f_i + v_c^2 (d/dc)^2 f_i

The factors of 2 come from the symmetry of the mixed second partial derivatives. The iteration is started using the initial guess a = 1, b = 0, c = 1. The program output is shown below:

iter  0: a = 1.0000, b = 0.0000, c = 1.0000, |a|/|v| = 0.0000 cond(J) =      inf, |f(x)| = 35.4785
iter  1: a = 1.5708, b = 0.5321, c = 0.5219, |a|/|v| = 0.3093 cond(J) =  29.0443, |f(x)| = 31.1042
iter  2: a = 1.7387, b = 0.4040, c = 0.4568, |a|/|v| = 0.1199 cond(J) =   3.5256, |f(x)| = 28.7217
iter  3: a = 2.2340, b = 0.3829, c = 0.3053, |a|/|v| = 0.3308 cond(J) =   4.5121, |f(x)| = 23.8074
iter  4: a = 3.2275, b = 0.3952, c = 0.2243, |a|/|v| = 0.2784 cond(J) =   8.6499, |f(x)| = 15.6003
iter  5: a = 4.3347, b = 0.3974, c = 0.1752, |a|/|v| = 0.2029 cond(J) =  15.1732, |f(x)| = 7.5908
iter  6: a = 4.9352, b = 0.3992, c = 0.1536, |a|/|v| = 0.1001 cond(J) =  26.6621, |f(x)| = 4.8402
iter  7: a = 5.0716, b = 0.3994, c = 0.1498, |a|/|v| = 0.0166 cond(J) =  34.6922, |f(x)| = 4.7103
iter  8: a = 5.0828, b = 0.3994, c = 0.1495, |a|/|v| = 0.0012 cond(J) =  36.5422, |f(x)| = 4.7095
iter  9: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6929, |f(x)| = 4.7095
iter 10: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6975, |f(x)| = 4.7095
iter 11: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6976, |f(x)| = 4.7095
NITER         = 11
NFEV          = 18
NJEV          = 12
NAEV          = 17
initial cost  = 1.258724737288e+03
final cost    = 2.217977560180e+01
final x       = (5.083101559156e+00, 3.994484109594e-01, 1.494898e-01)
final cond(J) = 3.669757713403e+01

We see the method converges after 11 iterations. For comparison the standard Levenberg-Marquardt method requires 26 iterations and so the Gaussian fitting problem benefits substantially from the geodesic acceleration correction. The column marked |a|/|v| above shows the ratio of the acceleration term to the velocity term as the iteration progresses. Larger values of this ratio indicate that the geodesic acceleration correction term is contributing substantial information to the solver relative to the standard LM velocity step.

The data and fitted model are shown in fig_nlfit2b.

Gaussian model fitted to data

The program is given below.

Comparing TRS Methods Example

The following program compares all available nonlinear least squares trust-region subproblem (TRS) methods on the Branin function, a common optimization test problem. The cost function is

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$$\begin{aligned} \Phi(x) &= {1 \over 2} (f_1^2 + f_2^2) \\\ f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3 \\\ f_2 &= \sqrt{a_4} \sqrt{1 + (1 - a_5) \cos{x_1}} \end{aligned}$$

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\Phi(x) &= 1/2 (f_1^2 + f_2^2)
f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3
f_2 &= sqrt(a_4) sqrt(1 + (1 - a_5) cos(x_1))

with $a_1 = -{5.1 \over 4 \pi^2}, a_2 = {5 \over \pi}, a_3 = -6, a_4 = 10, a_5 = {1 \over 8\pi}$. There are three minima of this function in the range (x₁, x₂) ∈ [ − 5, 15] × [ − 5, 15]. The program below uses the starting point (x₁, x₂) = (6, 14.5) and calculates the solution with all available nonlinear least squares TRS methods. The program output is shown below:

Method                    NITER  NFEV  NJEV  Initial Cost  Final cost   Final cond(J) Final x        
levenberg-marquardt       20     27    21    1.9874e+02    3.9789e-01   6.1399e+07    (-3.14e+00, 1.23e+01)
levenberg-marquardt+accel 27     36    28    1.9874e+02    3.9789e-01   1.4465e+07    (3.14e+00, 2.27e+00)
dogleg                    23     64    23    1.9874e+02    3.9789e-01   5.0692e+08    (3.14e+00, 2.28e+00)
double-dogleg             24     69    24    1.9874e+02    3.9789e-01   3.4879e+07    (3.14e+00, 2.27e+00)
2D-subspace               23     54    24    1.9874e+02    3.9789e-01   2.5142e+07    (3.14e+00, 2.27e+00)

The first row of output above corresponds to standard Levenberg-Marquardt, while the second row includes geodesic acceleration. We see that the standard LM method converges to the minimum at ( − π, 12.275) and also uses the least number of iterations and Jacobian evaluations. All other methods converge to the minimum (π, 2.275) and perform similarly in terms of number of Jacobian evaluations. We see that J is fairly ill-conditioned at both minima, indicating that the QR (or SVD) solver is the best choice for this problem. Since there are only two parameters in this optimization problem, we can easily visualize the paths taken by each method, which are shown in fig_nlfit3. The figure shows contours of the cost function Φ(x₁, x₂) which exhibits three global minima in the range [ − 5, 15] × [ − 5, 15]. The paths taken by each solver are shown as colored lines.

Paths taken for different TRS methods for the Branin function

The program is given below.

Large Nonlinear Least Squares Example

The following program illustrates the large nonlinear least squares solvers on a system with significant sparse structure in the Jacobian. The cost function is

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$$\begin{aligned} \Phi(x) &= {1 \over 2} \sum_{i=1}^{p+1} f_i^2 \\\ f_i &= \sqrt{\alpha} (x_i - 1), \quad 1 \le i \le p \\\ f_{p+1} &= ||x||^2 - {1 \over 4} \end{aligned}$$

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\Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
f_{p+1} &= ||x||^2 - 1/4

with α = 10^− 5. The residual f_p + 1 imposes a constraint on the p parameters x, to ensure that $||x||^2 \approx {1 \over 4}$. The (p + 1)-by-p Jacobian for this system is

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$$\begin{aligned} J(x) = \left( \begin{array}{c} \sqrt{\alpha} I_p \\\ 2 x^T \end{array} \right) \end{aligned}$$

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J(x) = [ \sqrt{alpha} I_p; 2 x^T ]

and the normal equations matrix is

J^TJ = αI_p + 4xx^T

Finally, the second directional derivative of f for the geodesic acceleration method is

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$$\begin{aligned} f_{vv} = D_v^2 f = \left( \begin{array}{c} 0 \\\ 2 ||v||^2 \end{array} \right) \end{aligned}$$

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fvv = [     0     ]
      [ 2 ||v||^2 ]

Since the upper p-by-p block of J is diagonal, this sparse structure should be exploited in the nonlinear solver. For comparison, the following program solves the system for p = 2000 using the dense direct Cholesky solver based on the normal equations matrix J^TJ, as well as the iterative Steihaug-Toint solver, based on sparse matrix-vector products Ju and J^Tu. The program output is shown below:

Method                    NITER NFEV NJUEV NJTJEV NAEV Init Cost  Final cost cond(J) Final |x|^2 Time (s)  
levenberg-marquardt       25    31   26    26     0    7.1218e+18 1.9555e-02 447.50  2.5044e-01  46.28
levenberg-marquardt+accel 22    23   45    23     22   7.1218e+18 1.9555e-02 447.64  2.5044e-01  33.92
dogleg                    37    87   36    36     0    7.1218e+18 1.9555e-02 447.59  2.5044e-01  56.05
double-dogleg             35    88   34    34     0    7.1218e+18 1.9555e-02 447.62  2.5044e-01  52.65
2D-subspace               37    88   36    36     0    7.1218e+18 1.9555e-02 447.71  2.5044e-01  59.75
steihaug-toint            35    88   345   0      0    7.1218e+18 1.9555e-02 inf     2.5044e-01  0.09

The first five rows use methods based on factoring the dense J^TJ matrix while the last row uses the iterative Steihaug-Toint method. While the number of Jacobian matrix-vector products (NJUEV) is less for the dense methods, the added time to construct and factor the J^TJ matrix (NJTJEV) results in a much larger runtime than the iterative method (see last column).

The program is given below.

References and Further Reading

The following publications are relevant to the algorithms described in this section,

• J.J. , The Levenberg-Marquardt Algorithm: Implementation and Theory, Lecture Notes in Mathematics, v630 (1978), ed G. Watson.
• H. B. Nielsen, "Damping Parameter in Marquardt's Method", IMM Department of Mathematical Modeling, DTU, Tech. Report IMM-REP-1999-05 (1999).
• K. Madsen and H. B. Nielsen, "Introduction to Optimization and Data Fitting", IMM Department of Mathematical Modeling, DTU, 2010.
• J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.
• M. K. Transtrum, B. B. Machta, and J. P. Sethna, Geometry of nonlinear least squares with applications to sloppy models and optimization, Phys. Rev. E 83, 036701, 2011.
• M. K. Transtrum and J. P. Sethna, Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization, arXiv:1201.5885, 2012.
• J.J. , B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained Optimization Software", ACM Transactions on Mathematical Software, Vol 7, No 1 (1981), p 17--41.
• H. B. Nielsen, "UCTP Test Problems for Unconstrained Optimization", IMM Department of Mathematical Modeling, DTU, Tech. Report IMM-REP-2000-17 (2000).