非线性最小二乘法Levenberg-Marquardt method

Levenberg-Marquardt Method(麦夸尔特法)Levenberg-Marquardt is a popular alternative to the Gauss-Newton method of finding the minimum of afunction that is a sum of squares of nonlinear functions,Let the Jacobian of be denoted , then the Levenberg-Marquardt method searches in thedirection given by the solution to the equationswhere are nonnegative scalars and is the identity matrix. The method has the nice property that, forsome scalar related to , the vector is the solution of the constrained subproblem of minimizingsubject to (Gill et al. 1981, p. 136).The method is used by the command FindMinimum[f, x, x0] when given the Method -> Levenberg Marquardt option.SEE A LSO:Minimum, OptimizationREFERENCES:Bates, D. M. and Watts, D. G. N onlinear Regr ession and Its Applications. New York: Wiley, 1988.Gill, P. R.; Murray, W.; and Wright, M. H. "The Levenberg-Marquardt Method." §4.7.3 in Practical Optim ization. London: Academic Press, pp. 136-137, 1981.Levenberg, K. "A Method for the Solution of Certain Problems in Least Squares." Quart. Appl. Math.2, 164-168, 1944. Marquardt, D. "An Algor ithm for Least-Squares Estimation of Nonlinear Parameters." SIAM J. Appl. Math.11, 431-441, 1963.Levenberg–Marquardt algorithmFrom Wikipedia, the free encyclopediaJump to: navigation, searchIn mathematics and computing, the Levenberg–Marquardt algorithm (LMA)[1] provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function. These minimization problems arise especially in least squares curve fitting and nonlinear programming.The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be a bit slower than the GNA. LMA can also be viewed as Gauss–Newton using a trust region approach.The LMA is a very popular curve-fitting algorithm used in many software applications for solving generic curve-fitting problems. However, the LMA finds only a local minimum, not a global minimum.Contents[hide]∙ 1 Caveat Emptor∙ 2 The problem∙ 3 The solutiono 3.1 Choice of damping parameter∙ 4 Example∙ 5 Notes∙ 6 See also∙7 References∙8 External linkso8.1 Descriptionso8.2 Implementations[edit] Caveat EmptorOne important limitation that is very often over-looked is that it only optimises for residual errors in the dependant variable (y). It thereby implicitly assumes that any errors in the independent variable are zero or at least ratio of the two is so small as to be negligible. This is not a defect, it is intentional, but it must be taken into account when deciding whether to use this technique to do a fit. While this may be suitable in context of a controlled experiment there are many situations where this assumption cannot be made. In such situations either non-least squares methods should be used or the least-squares fit should be done in proportion to the relative errors in the two variables, not simply the vertical "y" error. Failing to recognise this can lead to a fit which is significantly incorrect and fundamentally wrong. It will usually underestimate the slope. This may or may not be obvious to the eye.MicroSoft Excel's chart offers a trend fit that has this limitation that is undocumented. Users often fall into this trap assuming the fit is correctly calculated for all situations. OpenOffice spreadsheet copied this feature and presents the same problem.[edit] The problemThe primary application of the Levenberg–Marquardt algorithm is in the least squares curve fitting problem: given a set of m empirical datum pairs of independent and dependent variables, (x i, y i), optimize the parameters β of the model curve f(x,β) so that the sum of the squares of the deviationsbecomes minimal.[edit] The solutionLike other numeric minimization algorithms, the Levenberg–Marquardt algorithm is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, β. In many cases, an uninformed standard guess like βT=(1,1,...,1) will work fine;in other cases, the algorithm converges only if the initial guess is already somewhat close to the final solution.In each iteration step, the parameter vector, β, is replaced by a new estimate, β + δ. To determine δ, the functions are approximated by their linearizationswhereis the gradient(row-vector in this case) of f with respect to β.At its minimum, the sum of squares, S(β), the gradient of S with respect to δwill be zero. The above first-order approximation of gives.Or in vector notation,.Taking the derivative with respect to δand setting theresult to zero gives:where is the Jacobian matrix whose i th row equals J i,and where and are vectors with i th componentand y i, respectively. This is a set of linear equations which can be solved for δ.Levenberg's contribution is to replace this equation by a "damped version",where I is the identity matrix, giving as the increment, δ, to the estimated parameter vector, β.The (non-negative) damping factor, λ, isadjusted at each iteration. If reduction of S is rapid, a smaller value can be used, bringing the algorithm closer to the Gauss–Newton algorithm, whereas if an iteration gives insufficientreduction in the residual, λ can be increased, giving a step closer to the gradient descentdirection. Note that the gradient of S withrespect to β equals .Therefore, for large values of λ, the step will be taken approximately in the direction of the gradient. If either the length of the calculated step, δ, or the reduction of sum of squares from the latest parameter vector, β + δ, fall below predefined limits, iteration stops and the last parameter vector, β, is considered to be the solution.Levenberg's algorithm has the disadvantage that if the value of damping factor, λ, is large, inverting J T J + λI is not used at all. Marquardt provided the insight that we can scale eachcomponent of the gradient according to thecurvature so that there is larger movement along the directions where the gradient is smaller. This avoids slow convergence in the direction of small gradient. Therefore, Marquardt replaced theidentity matrix, I, with the diagonal matrixconsisting of the diagonal elements of J T J,resulting in the Levenberg–Marquardt algorithm:.A similar damping factor appears in Tikhonov regularization, which is used to solve linear ill-posed problems, as well as in ridge regression, an estimation technique in statistics.[edit] Choice of damping parameterVarious more-or-less heuristic arguments have been put forward for the best choice for the damping parameter λ. Theoretical arguments exist showing why some of these choices guaranteed local convergence of the algorithm; however these choices can make the global convergence of the algorithm suffer from the undesirable properties of steepest-descent, in particular very slow convergence close to the optimum.The absolute values of any choice depends on how well-scaled the initial problem is. Marquardt recommended starting with a value λ0 and a factor ν>1. Initially setting λ=λ0and computing the residual sum of squares S(β) after one step from the starting point with the damping factor of λ=λ0 and secondly withλ0/ν. If both of these are worse than the initial point then the damping is increased by successive multiplication by νuntil a better point is found with a new damping factor of λ0νk for some k.If use of the damping factor λ/ν results in a reduction in squared residual then this is taken as the new value of λ (and the new optimum location is taken as that obtained with this damping factor) and the process continues; if using λ/ν resulted in a worse residual, but using λresulted in a better residual then λ is left unchanged and the new optimum is taken as the value obtained with λas damping factor.[edit] ExamplePoor FitBetter FitBest FitIn this example we try to fit the function y = a cos(bX) + b sin(aX) using theLevenberg–Marquardt algorithm implemented in GNU Octave as the leasqr function. The 3 graphs Fig 1,2,3 show progressively better fitting for the parameters a=100, b=102 used in the initial curve. Only when the parameters in Fig 3 are chosen closest to the original, are thecurves fitting exactly. This equation is an example of very sensitive initial conditions for the Levenberg–Marquardt algorithm. One reason for this sensitivity is the existenceof multiple minima —the function cos(βx)has minima at parameter value and[edit] Notes1.^ The algorithm was first published byKenneth Levenberg, while working at theFrankford Army Arsenal. It was rediscoveredby Donald Marquardt who worked as astatistician at DuPont and independently byGirard, Wynn and Morrison.[edit] See also∙Trust region[edit] References∙Kenneth Levenberg(1944). "A Method for the Solution of Certain Non-Linear Problems in Least Squares". The Quarterly of Applied Mathematics2: 164–168.∙ A. Girard (1958). Rev. Opt37: 225, 397. ∙ C.G. Wynne (1959). "Lens Designing by Electronic Digital Computer: I". Proc.Phys. Soc. London73 (5): 777.doi:10.1088/0370-1328/73/5/310.∙Jorje J. Moré and Daniel C. Sorensen (1983)."Computing a Trust-Region Step". SIAM J.Sci. Stat. Comput. (4): 553–572.∙ D.D. Morrison (1960). Jet Propulsion Laboratory Seminar proceedings.∙Donald Marquardt (1963). "An Algorithm for Least-Squares Estimation of NonlinearParameters". SIAM Journal on AppliedMathematics11 (2): 431–441.doi:10.1137/0111030.∙Philip E. Gill and Walter Murray (1978)."Algorithms for the solution of thenonlinear least-squares problem". SIAMJournal on Numerical Analysis15 (5):977–992. doi:10.1137/0715063.∙Nocedal, Jorge; Wright, Stephen J. (2006).Numerical Optimization, 2nd Edition.Springer. ISBN0-387-30303-0.[edit] External links[edit] Descriptions∙Detailed description of the algorithm can be found in Numerical Recipes in C, Chapter15.5: Nonlinear models∙ C. T. Kelley, Iterative Methods for Optimization, SIAM Frontiers in AppliedMathematics, no 18, 1999, ISBN0-89871-433-8. Online copy∙History of the algorithm in SIAM news∙ A tutorial by Ananth Ranganathan∙Methods for Non-Linear Least Squares Problems by K. Madsen, H.B. Nielsen, O.Tingleff is a tutorial discussingnon-linear least-squares in general andthe Levenberg-Marquardt method inparticular∙T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner, ISBN 978-3-8348-1022-9.[edit] Implementations∙Levenberg-Marquardt is a built-in algorithm with Mathematica∙Levenberg-Marquardt is a built-in algorithm with Matlab∙The oldest implementation still in use is lmdif, from MINPACK, in Fortran, in thepublic domain. See also:o lmfit, a translation of lmdif into C/C++ with an easy-to-use wrapper for curvefitting, public domain.o The GNU Scientific Library library hasa C interface to MINPACK.o C/C++ Minpack includes theLevenberg–Marquardt algorithm.o Several high-level languages andmathematical packages have wrappers forthe MINPACK routines, among them:▪Python library scipy, modulescipy.optimize.leastsq,▪IDL, add-on MPFIT.▪R (programming language) has theminpack.lm package.∙levmar is an implementation in C/C++ with support for constraints, distributed under the GNU General Public License.o levmar includes a MEX file interface for MATLABo Perl (PDL), python and Haskellinterfaces to levmar are available: seePDL::Fit::Levmar, PyLevmar andHackageDB levmar.∙sparseLM is a C implementation aimed at minimizing functions with large,arbitrarily sparse Jacobians. Includes a MATLAB MEX interface.∙ALGLIB has implementations of improved LMA in C# / C++ / Delphi / Visual Basic.Improved algorithm takes less time toconverge and can use either Jacobian orexact Hessian.∙NMath has an implementation for the .NET Framework.∙gnuplot uses its own implementation .∙Java programming language implementations:1) Javanumerics, 2) LMA-package (a small,user friendly and well documentedimplementation with examples and support),3) Apache Commons Math∙OOoConv implements the L-M algorithm as an Calc spreadsheet.∙SAS, there are multiple ways to access SAS's implementation of the Levenberg-Marquardt algorithm: it can be accessed via NLPLMCall in PROC IML and it can also be accessed through the LSQ statement in PROC NLP, and the METHOD=MARQUARDT option in PROC NLIN.。