{-# LANGUAGE NoImplicitPrelude #-}
{-
Reciprocal of variance of a Gaussian bell curve.
We describe the curve only in terms of its variance
thus we represent a bell curve at the coordinate origin
neglecting its amplitude.
We could also define the amplitude as @root 4 c@,
but then @dilate@ and @shrink@ also include an amplification.
We could do some projective geometry in the exponent
in order to also have zero variance,
which corresponds to the dirac impulse.
-}
module MathObj.Gaussian.Variance where
import qualified MathObj.Polynomial as Poly
import qualified Algebra.Transcendental as Trans
import qualified Algebra.Algebraic as Algebraic
import qualified Algebra.Field as Field
import qualified Algebra.Real as Real
import qualified Algebra.Ring as Ring
import qualified Algebra.Additive as Additive
import Algebra.Transcendental (pi, )
import Algebra.Ring ((*), (^), )
import Algebra.Additive ((+))
import Test.QuickCheck (Arbitrary, arbitrary, coarbitrary, )
-- import Prelude (($))
import NumericPrelude
import PreludeBase
data T a = Cons {c :: a}
deriving (Eq, Show)
instance (Real.C a, Arbitrary a) => Arbitrary (T a) where
arbitrary = fmap (Cons . (1+) . abs) arbitrary
coarbitrary = undefined
constant :: Additive.C a => T a
constant = Cons zero
{-# INLINE evaluate #-}
evaluate :: (Trans.C a) =>
T a -> a -> a
evaluate f x =
exp $ (-pi * c f * x^2)
exponentPolynomial :: (Additive.C a) =>
T a -> Poly.T a
exponentPolynomial f =
Poly.fromCoeffs [zero, zero, c f]
norm1 :: (Algebraic.C a) => T a -> a
norm1 f =
recip $ sqrt $ c f
norm2 :: (Algebraic.C a) => T a -> a
norm2 f =
recip $ sqrt $ sqrt $ 2 * c f
normP :: (Trans.C a) => a -> T a -> a
normP p f =
(p * c f) ^? (- recip (2*p))
variance :: (Trans.C a) =>
T a -> a
variance f =
recip $ c f * 2*pi
multiply :: (Additive.C a) =>
T a -> T a -> T a
multiply f g =
Cons $ c f + c g
{- |
> convolve x y t =
> integrate $ \s -> x s * y(t-s)
-}
convolve :: (Field.C a) =>
T a -> T a -> T a
convolve f g =
Cons $ recip $ recip (c f) + recip (c g)
{- |
> fourier x f =
> integrate $ \t -> x t * cis (-2*pi*t*f)
-}
fourier :: (Field.C a) =>
T a -> T a
fourier f =
Cons $ recip $ c f
{-
fourier (t -> exp(-(a*t)^2))
-}
dilate :: (Field.C a) => a -> T a -> T a
dilate k f =
Cons $ c f / k^2
shrink :: (Ring.C a) => a -> T a -> T a
shrink k f =
Cons $ c f * k^2
{- laws
fourier (convolve f g) = multiply (fourier f) (fourier g)
dilate k (dilate m f) = dilate (k*m) f
dilate k (shrink k f) = f
variance (dilate k f) = k^2 * variance f
variance (convolve f g) = variance f + variance g
-}