Numeric.Measure.Discrete

Type

data Discrete a

A discrete, finite signed measure on the number line.

Instances

Show a => Show (Discrete a)

Defined in Numeric.Measure.Discrete

Methods

showsPrec :: Int -> Discrete a -> ShowS

show :: Discrete a -> String

showList :: [Discrete a] -> ShowS

(Ord a, Num a) => Eq (Discrete a)

Two measures are equal if they yield the same measures on every set.

mx == my
  implies
  forall t. eval (distribution mx) t = eval (distribution my) t

Defined in Numeric.Measure.Discrete

Methods

(==) :: Discrete a -> Discrete a -> Bool

(/=) :: Discrete a -> Discrete a -> Bool

fromMap :: (Ord a, Num a) => Map a a -> Discrete a

Construct a discrete measure from a collection of points and their measures.

toMap :: Num a => Discrete a -> Map a a

Decompose the discrete measure into a collection of points and their measures.

zero :: Num a => Discrete a

The measure that assigns 0 to every set.

dirac :: (Ord a, Num a) => a -> Discrete a

A Dirac measure at the given point x.

total (dirac x) = 1

distribution :: (Ord a, Num a) => Discrete a -> Piecewise (Poly a)

eval (distribution m) x is the measure of the interval $(-∞, x]$.

This is known as the distribution function.

Observations

total :: Num a => Discrete a -> a

The total of the measure applied to the set of real numbers.

integrate :: (Ord a, Num a) => (a -> a) -> Discrete a -> a

Integrate a function f with respect to the given measure m, $\int f(x) dm(x)$.

Operations, numerical

add :: (Ord a, Num a) => Discrete a -> Discrete a -> Discrete a

Add two measures.

total (add mx my) = total mx + total my

scale :: (Ord a, Num a) => a -> Discrete a -> Discrete a

Scale a measure by a constant.

total (scale a mx) = a * total mx

translate :: (Ord a, Num a) => a -> Discrete a -> Discrete a

Translate a measure along the number line.

eval (distribution (translate y m)) x
   = eval (distribution m) (x - y)

convolve :: (Ord a, Num a) => Discrete a -> Discrete a -> Discrete a

Additive convolution of two measures.

Properties:

convolve (dirac x) (dirac y) = dirac (x + y)