{-# 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 -}