levmar-1.0: Numeric/LevMar.hs
{-# LANGUAGE CPP
, NoImplicitPrelude
, UnicodeSyntax
, ScopedTypeVariables
, DeriveDataTypeable
#-}
--------------------------------------------------------------------------------
-- |
-- Module: Numeric.LevMar
-- Copyright: (c) 2009 - 2011 Roel van Dijk & Bas van Dijk
-- License: BSD-style (see the file LICENSE)
-- Maintainer: Roel van Dijk <vandijk.roel@gmail.com>
-- Bas van Dijk <v.dijk.bas@gmail.com>
-- Stability: Experimental
--
-- For additional documentation see the documentation of the levmar C
-- library which this library is based on:
-- <http://www.ics.forth.gr/~lourakis/levmar/>
--
--------------------------------------------------------------------------------
module Numeric.LevMar
( -- * Model & Jacobian.
Model
, Jacobian
-- * Levenberg-Marquardt algorithm.
, LevMarable(levmar)
-- * Minimization options.
, Options(..)
, defaultOpts
-- * Constraints
, Constraints(..)
, LinearConstraints
-- * Output
, Info(..)
, StopReason(..)
, LevMarError(..)
) where
-------------------------------------------------------------------------------
-- Imports
-------------------------------------------------------------------------------
-- from base:
import Control.Monad ( return, mplus )
import Control.Exception ( Exception )
import Data.Typeable ( Typeable )
import Data.Either ( Either(Left, Right) )
import Data.Function ( ($) )
import Data.Functor ( (<$>) )
import Data.Int ( Int )
import Data.List ( lookup, (++) )
import Data.Maybe ( Maybe(Nothing, Just), isJust, fromJust, fromMaybe )
import Data.Monoid ( Monoid(mempty, mappend) )
import Data.Ord ( (<) )
import Foreign.Marshal.Array ( allocaArray, withArray, peekArray, copyArray )
import Foreign.Ptr ( Ptr, nullPtr )
import Foreign.ForeignPtr ( ForeignPtr, newForeignPtr_, withForeignPtr )
import Foreign.Storable ( Storable )
import Prelude ( Enum, Fractional, RealFrac, Float, Double
, toEnum, (-), (*), error, floor
)
import System.IO ( IO )
import System.IO.Unsafe ( unsafePerformIO )
import Text.Read ( Read )
import Text.Show ( Show, show )
#if __GLASGOW_HASKELL__ >= 605
import GHC.ForeignPtr ( mallocPlainForeignPtrBytes )
import Prelude ( undefined )
import Foreign.Storable ( sizeOf )
#else
import Foreign.ForeignPtr ( mallocForeignPtrArray )
#endif
#if __GLASGOW_HASKELL__ < 700
import Prelude ( fromInteger )
#endif
-- from base-unicode-symbols:
import Data.Bool.Unicode ( (∧), (∨) )
import Data.Eq.Unicode ( (≢) )
import Data.Function.Unicode ( (∘) )
-- from hmatrix:
import Data.Packed.Vector ( Vector )
import Data.Packed.Matrix ( Matrix, Element, flatten, rows, reshape )
-- from vector:
import qualified Data.Vector.Storable as VS ( unsafeWith, length
, unsafeFromForeignPtr
, length
)
-- from bindings-levmar:
import Bindings.LevMar ( c'LM_INFO_SZ
, withModel
, withJacobian
, c'LM_ERROR
, c'LM_ERROR_LAPACK_ERROR
, c'LM_ERROR_FAILED_BOX_CHECK
, c'LM_ERROR_MEMORY_ALLOCATION_FAILURE
, c'LM_ERROR_CONSTRAINT_MATRIX_ROWS_GT_COLS
, c'LM_ERROR_CONSTRAINT_MATRIX_NOT_FULL_ROW_RANK
, c'LM_ERROR_TOO_FEW_MEASUREMENTS
, c'LM_ERROR_SINGULAR_MATRIX
, c'LM_ERROR_SUM_OF_SQUARES_NOT_FINITE
, c'LM_INIT_MU
, c'LM_STOP_THRESH
, c'LM_DIFF_DELTA
)
import qualified Bindings.LevMar ( Model, Jacobian )
-- from levmar (this package):
import Bindings.LevMar.CurryFriendly ( LevMarDer
, LevMarDif
, LevMarBCDer
, LevMarBCDif
, LevMarLecDer
, LevMarLecDif
, LevMarBLecDer
, LevMarBLecDif
, dlevmar_der, slevmar_der
, dlevmar_dif, slevmar_dif
, dlevmar_bc_der, slevmar_bc_der
, dlevmar_bc_dif, slevmar_bc_dif
, dlevmar_lec_der, slevmar_lec_der
, dlevmar_lec_dif, slevmar_lec_dif
, dlevmar_blec_der, slevmar_blec_der
, dlevmar_blec_dif, slevmar_blec_dif
)
--------------------------------------------------------------------------------
-- Model & Jacobian.
--------------------------------------------------------------------------------
{-| A functional relation describing measurements represented as a function
from a vector of parameters to a vector of expected measurements.
* Ensure that the length of the parameters vector equals the length of the
initial parameters vector in 'levmar'.
* Ensure that the length of the ouput vector equals the length of the samples
vector in 'levmar'.
-}
type Model r = Vector r → Vector r
{-| The jacobian of the 'Model' function. Expressed as a function from a vector
of parameters to a matrix which for each expected measurement describes
the partial derivatives of the parameters.
See: <http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant>
* Ensure that the length of the parameter vector equals the length of the initial
parameter vector in 'levmar'.
* Ensure that the output matrix has the dimension @n><m@ where @n@ is the
number of samples and @m@ is the number of parameters.
-}
type Jacobian r = Vector r → Matrix r
--------------------------------------------------------------------------------
-- Levenberg-Marquardt algorithm.
--------------------------------------------------------------------------------
-- | The Levenberg-Marquardt algorithm is overloaded to work on 'Double' and 'Float'.
class LevMarable r where
-- | The Levenberg-Marquardt algorithm.
--
-- Returns a tuple of the found parameters, a structure containing
-- information about the minimization and the covariance matrix
-- corresponding to LS solution.
levmar ∷ Model r -- ^ Model
→ Maybe (Jacobian r) -- ^ Optional jacobian
→ Vector r -- ^ Initial parameters
→ Vector r -- ^ Samples
→ Int -- ^ Maximum iterations
→ Options r -- ^ Minimization options
→ Constraints r -- ^ Constraints
→ Either LevMarError (Vector r, Info r, Matrix r)
instance LevMarable Float where
levmar = gen_levmar slevmar_der
slevmar_dif
slevmar_bc_der
slevmar_bc_dif
slevmar_lec_der
slevmar_lec_dif
slevmar_blec_der
slevmar_blec_dif
instance LevMarable Double where
levmar = gen_levmar dlevmar_der
dlevmar_dif
dlevmar_bc_der
dlevmar_bc_dif
dlevmar_lec_der
dlevmar_lec_dif
dlevmar_blec_der
dlevmar_blec_dif
{-| @gen_levmar@ takes the low-level C functions as arguments and
executes one of them depending on the optional jacobian and constraints.
Preconditions:
@
length ys >= length ps
isJust mLowBs && length (fromJust mLowBs) == length ps
&& isJust mUpBs && length (fromJust mUpBs) == length ps
boxConstrained && (all $ zipWith (<=) (fromJust mLowBs) (fromJust mUpBs))
@
-}
gen_levmar ∷ ∀ r. (Storable r, RealFrac r, Element r)
⇒ LevMarDer r
→ LevMarDif r
→ LevMarBCDer r
→ LevMarBCDif r
→ LevMarLecDer r
→ LevMarLecDif r
→ LevMarBLecDer r
→ LevMarBLecDif r
→ Model r -- ^ Model
→ Maybe (Jacobian r) -- ^ Optional jacobian
→ Vector r -- ^ Initial parameters
→ Vector r -- ^ Samples
→ Int -- ^ Maximum iterations
→ Options r -- ^ Options
→ Constraints r -- ^ Constraints
→ Either LevMarError (Vector r, Info r, Matrix r)
gen_levmar f_der
f_dif
f_bc_der
f_bc_dif
f_lec_der
f_lec_dif
f_blec_der
f_blec_dif
model mJac ps ys itMax opts (Constraints mLowBs mUpBs mWeights mLinC) =
-- All effects are contained, so we can safely perform:
unsafePerformIO $ do
-- We need to pass the initial parameters 'ps' to the C function.
-- However, we can't just pass a pointer to them because the C function
-- will modify the parameters during execution which will violate
-- referential transparanency. Instead we allocate new space
-- and copy the parameters to it.
--
-- Note that, in the end, the array is returned from this function.
-- This means that the only way to guarantee its finalisation
-- is to allocate it using a ForeignPtr:
psFP ← fastMallocForeignPtrArray lenPs
withForeignPtr psFP $ \psPtr → do
VS.unsafeWith ps $ \psPtrInp →
copyArray psPtr psPtrInp lenPs
-- Retrieve the (read-only) pointer 'ysPtr' to the samples vector 'ys'
-- so we can pass it to the C function:
VS.unsafeWith ys $ \ysPtr →
-- Convert the Options 'opts' to a list and then to an array
-- so we can pass the (read-only) pointer 'optsPtr' to the C function:
withArray (optsToList opts) $ \optsPtr →
-- Allocate space for the info array
-- so we can pass it to the C function.
-- Note that, in the end, this array is converted to an Info value
-- and returned from this function.
allocaArray c'LM_INFO_SZ $ \infoPtr → do
-- Allocate space for the covariance matrix
-- so we can pass it to the C function.
-- Like the parameters array the matrix
-- needs to be returned from this function.
-- So we also allocate it using a ForeignPtr:
covarFP ← fastMallocForeignPtrArray covarLen
withForeignPtr covarFP $ \covarPtr →
-- 'cmodel' is the low-level model function which is converted
-- to the FunPtr 'modelFunPtr' and passed to the C function.
-- 'cmodel' will first convert the parameters pointer 'parPtr'
-- into a Vector after converting it into a ForeignPtr
-- (without a finalizer).
-- Then it will apply the high-level 'model' function
-- to this parameter vector. The resulting vector is then copied
-- to the output buffer 'hxPtr':
let cmodel ∷ Bindings.LevMar.Model r
cmodel parPtr hxPtr _ _ _ = do
parFP ← newForeignPtr_ parPtr
let psV = VS.unsafeFromForeignPtr parFP 0 lenPs
vector = model psV
VS.unsafeWith vector $ \p → copyArray hxPtr p (VS.length vector)
in withModel cmodel $ \modelFunPtr → do
-- All the low-level C functions share a common set of arguments.
-- 'runDif' applies these arguments to the given C function 'f':
let runDif ∷ LevMarDif r → IO Int
runDif f = f modelFunPtr
psPtr
ysPtr
lenPs
lenYs
itMax
optsPtr
infoPtr
nullPtr
covarPtr
nullPtr
err ← case mJac of
Nothing → if boxConstrained
then if linConstrained
then withBoxConstraints
(withLinConstraints $ withWeights runDif)
f_blec_dif
else withBoxConstraints runDif f_bc_dif
else if linConstrained
then withLinConstraints runDif f_lec_dif
else runDif f_dif
Just jac →
let cjacobian ∷ Bindings.LevMar.Jacobian r
cjacobian parPtr jPtr _ _ _ = do
parFP ← newForeignPtr_ parPtr
let psV = VS.unsafeFromForeignPtr parFP 0 lenPs
matrix = jac psV
vector = flatten matrix
VS.unsafeWith vector $ \p →
copyArray jPtr p (VS.length vector)
in withJacobian cjacobian $ \jacobPtr →
let runDer ∷ LevMarDer r → IO Int
runDer f = runDif $ f jacobPtr
in if boxConstrained
then if linConstrained
then withBoxConstraints
(withLinConstraints $ withWeights runDer)
f_blec_der
else withBoxConstraints runDer f_bc_der
else if linConstrained
then withLinConstraints runDer f_lec_der
else runDer f_der
-- Handling errors:
if err < 0
-- we don't treat the following two as an error:
∧ err ≢ c'LM_ERROR_SINGULAR_MATRIX
∧ err ≢ c'LM_ERROR_SUM_OF_SQUARES_NOT_FINITE
then return $ Left $ convertLevMarError err
else do -- Converting results:
info ← listToInfo <$> peekArray c'LM_INFO_SZ infoPtr
let psV = VS.unsafeFromForeignPtr psFP 0 lenPs
let covarM = reshape lenPs $
VS.unsafeFromForeignPtr covarFP 0 covarLen
return $ Right (psV, info, covarM)
where
lenPs = VS.length ps
lenYs = VS.length ys
covarLen = lenPs*lenPs
(cMat, rhcVec) = fromJust mLinC
-- Whether the parameters are constrained by a linear equation.
linConstrained = isJust mLinC
-- Whether the parameters are constrained by a bounding box.
boxConstrained = isJust mLowBs ∨ isJust mUpBs
withBoxConstraints f g =
maybeWithArray mLowBs $ \lBsPtr →
maybeWithArray mUpBs $ \uBsPtr →
f $ g lBsPtr uBsPtr
withLinConstraints f g =
VS.unsafeWith (flatten cMat) $ \cMatPtr →
VS.unsafeWith rhcVec $ \rhcVecPtr →
f ∘ g cMatPtr rhcVecPtr $ rows cMat
withWeights f g = maybeWithArray mWeights $ f ∘ g
maybeWithArray ∷ (Storable α) ⇒ Maybe (Vector α) → (Ptr α → IO β) → IO β
maybeWithArray Nothing f = f nullPtr
maybeWithArray (Just v) f = VS.unsafeWith v f
#if __GLASGOW_HASKELL__ >= 605
{-# INLINE fastMallocForeignPtrArray #-}
fastMallocForeignPtrArray ∷ ∀ α. Storable α ⇒ Int → IO (ForeignPtr α)
fastMallocForeignPtrArray n = mallocPlainForeignPtrBytes
(n * sizeOf (undefined ∷ α))
#else
fastMallocForeignPtrArray ∷ ∀ α. Storable α ⇒ Int → IO (ForeignPtr α)
fastMallocForeignPtrArray = mallocForeignPtrArray
#endif
--------------------------------------------------------------------------------
-- Minimization options.
--------------------------------------------------------------------------------
-- | Minimization options
data Options r =
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.
} deriving (Read, Show)
-- | Default minimization options
defaultOpts ∷ Fractional r ⇒ Options r
defaultOpts = Opts { optScaleInitMu = c'LM_INIT_MU
, optStopNormInfJacTe = c'LM_STOP_THRESH
, optStopNorm2Dp = c'LM_STOP_THRESH
, optStopNorm2E = c'LM_STOP_THRESH
, optDelta = c'LM_DIFF_DELTA
}
optsToList ∷ Options r → [r]
optsToList (Opts mu eps1 eps2 eps3 delta) =
[mu, eps1, eps2, eps3, delta]
--------------------------------------------------------------------------------
-- Constraints
--------------------------------------------------------------------------------
-- | Ensure that these vectors have the same length as the number of parameters.
data Constraints r = Constraints
{ lowerBounds ∷ !(Maybe (Vector r)) -- ^ Optional lower bounds
, upperBounds ∷ !(Maybe (Vector r)) -- ^ Optional upper bounds
, weights ∷ !(Maybe (Vector r)) -- ^ Optional weights
, linearConstraints ∷ !(Maybe (LinearConstraints r)) -- ^ Optional linear constraints
}
-- | Linear constraints consisting of a constraints matrix, @k><m@ and
-- a right hand constraints vector, of length @k@ where @m@ is the number of
-- parameters and @k@ is the number of constraints.
type LinearConstraints r = (Matrix r, Vector r)
-- | * 'mempty' is defined as a 'Constraints' where all fields are 'Nothing'.
--
-- * 'mappend' merges two 'Constraints' by taking the first non-'Nothing' value
-- for each field.
instance Monoid (Constraints r) where
mempty = Constraints Nothing Nothing Nothing Nothing
mappend (Constraints lb1 ub1 w1 l1)
(Constraints lb2 ub2 w2 l2) = Constraints (lb1 `mplus` lb2)
(ub1 `mplus` ub2)
(w1 `mplus` w2)
(l1 `mplus` l2)
--------------------------------------------------------------------------------
-- Output
--------------------------------------------------------------------------------
-- | Information regarding the minimization.
data Info r = 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 ∷ !Int -- ^ Number of iterations.
, infStopReason ∷ !StopReason -- ^ Reason for terminating.
, infNumFuncEvals ∷ !Int -- ^ Number of function evaluations.
, infNumJacobEvals ∷ !Int -- ^ Number of jacobian evaluations.
, infNumLinSysSolved ∷ !Int -- ^ Number of linear systems solved,
-- i.e. attempts for reducing error.
} deriving (Read, Show)
listToInfo ∷ (RealFrac r) ⇒ [r] → Info r
listToInfo [a,b,c,d,e,f,g,h,i,j] =
Info { infNorm2initE = a
, infNorm2E = b
, infNormInfJacTe = c
, infNorm2Dp = d
, infMuDivMax = e
, infNumIter = floor f
, infStopReason = toEnum $ floor g - 1
, infNumFuncEvals = floor h
, infNumJacobEvals = floor i
, infNumLinSysSolved = floor j
}
listToInfo _ = error "liftToInfo: wrong list length"
-- | Reason for terminating.
data StopReason
= 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.
deriving (Read, Show, Enum)
--------------------------------------------------------------------------------
-- Error
--------------------------------------------------------------------------------
data LevMarError
= 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.
deriving (Show, Typeable)
-- Handy in case you want to thow a LevMarError as an exception:
instance Exception LevMarError
levmarCErrorToLevMarError ∷ [(Int, LevMarError)]
levmarCErrorToLevMarError =
[ (c'LM_ERROR, LevMarError)
, (c'LM_ERROR_LAPACK_ERROR, LapackError)
--, (c'LM_ERROR_NO_JACOBIAN, can never happen)
--, (c'LM_ERROR_NO_BOX_CONSTRAINTS, can never happen)
, (c'LM_ERROR_FAILED_BOX_CHECK, FailedBoxCheck)
, (c'LM_ERROR_MEMORY_ALLOCATION_FAILURE, MemoryAllocationFailure)
, (c'LM_ERROR_CONSTRAINT_MATRIX_ROWS_GT_COLS, ConstraintMatrixRowsGtCols)
, (c'LM_ERROR_CONSTRAINT_MATRIX_NOT_FULL_ROW_RANK, ConstraintMatrixNotFullRowRank)
, (c'LM_ERROR_TOO_FEW_MEASUREMENTS, TooFewMeasurements)
--, (c'LM_ERROR_SINGULAR_MATRIX, we don't treat this as an error)
--, (c'LM_ERROR_SUM_OF_SQUARES_NOT_FINITE, we don't treat this as an error)
]
convertLevMarError ∷ Int → LevMarError
convertLevMarError err = fromMaybe (error $ "Unknown levmar error: " ++ show err)
(lookup err levmarCErrorToLevMarError)
-- The End ---------------------------------------------------------------------