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LevMar.Fitting | Stability | Experimental | Maintainer | vandijk.roel@gmail.com, v.dijk.bas@gmail.com |
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Description |
This module provides the Levenberg-Marquardt algorithm specialised
for curve-fitting.
For additional documentation see the documentation of the levmar C
library which this library is based on:
http://www.ics.forth.gr/~lourakis/levmar/
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Synopsis |
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Model & Jacobian.
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A functional relation describing measurements represented as a function
from m parameters and an x-value to an expected measurement.
For example, the quadratic function f(x) = a*x^2 + b*x + c can be
written as:
type N3 = S (S (S Z))
quad :: Num r => Model N3 r r
quad a b c x = a*x^2 + b*x + c
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This type synonym expresses that usually the a in Model m r a
equals the type of the parameters.
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The jacobian of the Model function. Expressed as a function from n
parameters and an x-value to the m partial derivatives of the parameters.
See: http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
For example, the jacobian of the quadratic function f(x) = a*x^2 +
b*x + c can be written as:
type N3 = S (S (S Z))
quadJacob :: Num r => Jacobian N3 r r
quadJacob _ _ _ x = x^2 -- with respect to a
::: x -- with respect to b
::: 1 -- with respect to c
::: Nil
Notice you don't have to differentiate for x.
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This type synonym expresses that usually the a in Jacobian m r a
equals the type of the parameters.
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Levenberg-Marquardt algorithm.
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The Levenberg-Marquardt algorithm is overloaded to work on Double and Float.
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Linear constraints consisting of a constraints matrix, kxn and
a right hand constraints vector, kx1 where n is the number of
parameters and k is the number of constraints.
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Value to denote the absense of any linear constraints over the
parameters of the model function. Use this instead of Nothing
because the type parameter which contains the number of constraints
can't be inferred.
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A nxm matrix is a sized list of n sized lists of length m.
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Minimization options.
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Minimization options
| Constructors | Opts | | optScaleInitMu :: r | Scale factor for initial mu.
| optStopNormInfJacTe :: r | Stopping thresholds for ||J^T e||_inf.
| optStopNorm2Dp :: r | Stopping thresholds for ||Dp||_2.
| optStopNorm2E :: r | Stopping thresholds for ||e||_2.
| optDelta :: r | Step used in the difference approximation to the Jacobian.
If optDelta<0, the Jacobian is approximated
with central differences which are more accurate
(but slower!) compared to the forward differences
employed by default.
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Default minimization options
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Output
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Information regarding the minimization.
| Constructors | Info | | infNorm2initE :: r | ||e||_2 at initial parameters.
| infNorm2E :: r | ||e||_2 at estimated parameters.
| infNormInfJacTe :: r | ||J^T e||_inf at estimated parameters.
| infNorm2Dp :: r | ||Dp||_2 at estimated parameters.
| infMuDivMax :: r | mu/max[J^T J]_ii ] at estimated parameters.
| infNumIter :: Integer | Number of iterations.
| infStopReason :: StopReason | Reason for terminating.
| infNumFuncEvals :: Integer | Number of function evaluations.
| infNumJacobEvals :: Integer | Number of jacobian evaluations.
| infNumLinSysSolved :: Integer | Number of linear systems solved, i.e. attempts for reducing error.
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Reason for terminating.
| Constructors | SmallGradient | Stopped because of small gradient J^T e.
| SmallDp | Stopped because of small Dp.
| MaxIterations | Stopped because maximum iterations was reached.
| SingularMatrix | Stopped because of singular matrix. Restart from current estimated parameters with increased optScaleInitMu.
| SmallestError | Stopped because no further error reduction is possible. Restart with increased optScaleInitMu.
| SmallNorm2E | Stopped because of small ||e||_2.
| InvalidValues | Stopped because model function returned invalid values (i.e. NaN or Inf). This is a user error.
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Covariance matrix corresponding to LS solution.
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Constructors | LevMarError | Generic error (not one of the others)
| LapackError | A call to a lapack subroutine failed in the underlying C levmar library.
| FailedBoxCheck | At least one lower bound exceeds the upper one.
| MemoryAllocationFailure | A call to malloc failed in the underlying C levmar library.
| ConstraintMatrixRowsGtCols | The matrix of constraints cannot have more rows than columns.
| ConstraintMatrixNotFullRowRank | Constraints matrix is not of full row rank.
| TooFewMeasurements | Cannot solve a problem with fewer measurements than unknowns.
In case linear constraints are provided, this error is also returned
when the number of measurements is smaller than the number of unknowns
minus the number of equality constraints.
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Type-level machinery
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Type-level natural denoting zero
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Type-level natural denoting the Successor of another type-level natural.
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Class of all type-level naturals.
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data SizedList n a where | Source |
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A list which is indexed with a type-level natural that denotes the size of
the list.
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type family NFunction n a b :: * | Source |
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A NFunction n a b is a function which takes n arguments of
type a and returns a b.
For example: NFunction (S (S (S Z))) a b ~ (a -> a -> a -> b)
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