{-# LANGUAGE CPP
, NoImplicitPrelude
, UnicodeSyntax
, ScopedTypeVariables
, DeriveDataTypeable
, StandaloneDeriving
, FlexibleContexts
, UndecidableInstances
#-}

--------------------------------------------------------------------------------
-- |
-- Module:     Numeric.LevMar
-- Copyright:  (c) 2009 - 2011 Roel van Dijk & Bas van Dijk
-- 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.Data             ( Data )
import Data.Typeable         ( Typeable )
import Data.Either           ( Either(Left, Right) )
import Data.Eq               ( Eq )
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              ( 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.Show             ( Show, show )

import GHC.ForeignPtr        ( mallocPlainForeignPtrBytes )
import Prelude               ( undefined )
import Foreign.Storable      ( sizeOf )
#else
import Foreign.ForeignPtr    ( mallocForeignPtrArray )
#endif

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.Matrix    ( Matrix, Element, flatten, rows, reshape )
import Numeric.Container     ( Container )
import Numeric.LinearAlgebra ( {- Instances for Matrix -} )

-- from vector:
import           Data.Vector.Storable       ( 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 (Eq, Ord, Read, Show, Data, Typeable) -- | 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 } deriving (Read, Show, Typeable) deriving instance Container Vector r ⇒ Eq (Constraints r) -- | 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 (Eq, Ord, Read, Show, Data, Typeable) 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
| 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
| 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 (Eq, Ord, Read, Show, Data, Typeable, 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 (Eq, Ord, Read, Show, Data, 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)