definition of Wikipedia
Advertizing ▼
In mathematics and computing, the Levenberg–Marquardt algorithm (LMA)^{[1]}, also known as the damped leastsquares (DLS) method, 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 wellbehaved 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 curvefitting algorithm used in many software applications for solving generic curvefitting problems. However, the LMA finds only a local minimum, not a global minimum.
Contents 
The 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 deviations
becomes minimal.
Like 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 cases with only one minimum, an uninformed standard guess like β^{T}=(1,1,...,1) will work fine; in cases with multiple minima, 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 linearizations
where
is the gradient (rowvector in this case) of f with respect to β.
At the minimum of the sum of squares, , the gradient of with respect to δ will be zero. The above firstorder approximation of gives
Or in vector notation,
Taking the derivative with respect to δ and setting the result to zero gives:
where is the Jacobian matrix whose i^{th} row equals , and where and are vectors with i^{th} component and , 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 (nonnegative) damping factor, λ, is adjusted 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 insufficient reduction in the residual, λ can be increased, giving a step closer to the gradient descent direction. Note that the gradient of S with respect 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 each component of the gradient according to the curvature 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 the identity matrix, I, with the diagonal matrix consisting 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 illposed problems, as well as in ridge regression, an estimation technique in statistics.
Various moreorless 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 steepestdescent, in particular very slow convergence close to the optimum.
The absolute values of any choice depends on how wellscaled the initial problem is. Marquardt recommended starting with a value λ_{0} and a factor ν>1. Initially setting λ=λ_{0} and computing the residual sum of squares S(β) after one step from the starting point with the damping factor of λ=λ_{0} and secondly with λ_{0}/ν. If both of these are worse than the initial point then the damping is increased by successive multiplication by ν until a better point is found with a new damping factor of λ_{0}ν^{k} 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 λ resulted in a better residual then λ is left unchanged and the new optimum is taken as the value obtained with λ as damping factor.
In this example we try to fit the function using the Levenberg–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 the curves fitting exactly. This equation is an example of very sensitive initial conditions for the Levenberg–Marquardt algorithm. One reason for this sensitivity is the existence of multiple minima — the function has minima at parameter value and
scipy.optimize.leastsq
,

sensagent's content
Dictionary and translator for handheld
New : sensagent is now available on your handheld
Advertising ▼
Webmaster Solution
Alexandria
A windows (popinto) of information (fullcontent of Sensagent) triggered by doubleclicking any word on your webpage. Give contextual explanation and translation from your sites !
SensagentBox
With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.
Business solution
Improve your site content
Add new content to your site from Sensagent by XML.
Crawl products or adds
Get XML access to reach the best products.
Index images and define metadata
Get XML access to fix the meaning of your metadata.
Please, email us to describe your idea.
Lettris
Lettris is a curious tetrisclone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
boggle
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
English dictionary
Main references
Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).
Copyrights
The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.
Translation
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
last searches on the dictionary :
computed in 0.031s