commit61b5bd98535154a88821ad37e70412d6747674f6parentdf1d0c89c01cb7435fceceb8c0df50f3f623c5d2Author:Jared Tobin <jared@jtobin.ca>Date:Thu, 17 Oct 2013 17:16:22 +1300 Add identity measure.Diffstat:

M | src/Measurable.hs | | | 51 | ++++++++++++++++++++++++++++----------------------- |

1 file changed, 28 insertions(+), 23 deletions(-)diff --git a/src/Measurable.hs b/src/Measurable.hs@@ -23,11 +23,7 @@ import Numeric.Integration.TanhSinh -- a set and returning a probability, we can take a function and return a -- probability. --- NOTE I probably want to generalize this to something like --- --- newtype Measure a = Measure { measure :: (a -> a) -> Double } --- -type Measure = (Double -> Double) -> Double +type Measure a = (a -> Double) -> Double -- | Once we have a measure, we use it by integrating against it. Take a -- real-valued random variable (i.e., measurable function) /X/. The mean of X @@ -37,7 +33,7 @@ type Measure = (Double -> Double) -> Double -- measures. Expectation can be defined by taking a measurable function and -- applying a measure to it - i.e., it's just function application. -- --- expectation :: Measure -> (Double -> Double) -> Double +-- expectation :: Measure Double -> (Double -> Double) -> Double -- expectation -- :: ((Double -> Double) -> Double) -- (a -> b) -- -> (Double -> Double) -> Double -- a -> b @@ -52,54 +48,54 @@ type Measure = (Double -> Double) -> Double -- NOTE want to add cumulants -- | The nth raw moment. -rawMoment :: Integral a => Measure -> a -> Double +rawMoment :: Integral a => Measure Double -> a -> Double rawMoment mu n = mu (^^ n) -- | All positive raw moments. -rawMoments :: Measure -> [Double] +rawMoments :: Measure Double -> [Double] rawMoments mu = map (rawMoment mu) [1..] -- | Alias for first raw moment. -mean :: Measure -> Double +mean :: Measure Double -> Double mean mu = rawMoment mu 1 -- | The nth central moment. -- -- NOTE slow-as in ghci. Need to memoize or something, or this might just -- disappear when compiling. -centralMoment :: Integral a => Measure -> a -> Double +centralMoment :: Integral a => Measure Double -> a -> Double centralMoment mu n = mu $ (^^ n) . \x -> x - rawMoment mu 1 -- | All positive central moments. -centralMoments :: Measure -> [Double] +centralMoments :: Measure Double -> [Double] centralMoments mu = map (centralMoment mu) [1..] -- | Alias for second central moment. -variance :: Measure -> Double +variance :: Measure Double -> Double variance mu = centralMoment mu 2 -- | The nth normalized moment. -normalizedMoment :: Integral a => Measure -> a -> Double +normalizedMoment :: Integral a => Measure Double -> a -> Double normalizedMoment mu n = (/ (sd ^ n)) $ centralMoment mu n where sd = sqrt $ centralMoment mu 2 -- | All normalized moments. -normalizedMoments :: Measure -> [Double] +normalizedMoments :: Measure Double -> [Double] normalizedMoments mu = map (normalizedMoment mu) [1..] -- | The moment generating function about a point. -momentGeneratingFunction :: Double -> Measure -> Double +momentGeneratingFunction :: Double -> Measure Double -> Double momentGeneratingFunction t mu = mu $ exp . (* t) . id -- | The cumulant generating function about a point. -cumulantGeneratingFunction :: Double -> Measure -> Double +cumulantGeneratingFunction :: Double -> Measure Double -> Double cumulantGeneratingFunction t mu = log $ momentGeneratingFunction t mu -- | We want two ways to create measures; empirically (i.e. from observations) -- or directly from some integrable function (i.e. a density). -- | Construct an empirical measure from observations. -fromObservations :: [Double] -> Measure +fromObservations :: [Double] -> Measure Double fromObservations xs f = normalize . sum . map f $ xs where normalize = (/ fromIntegral (length xs)) @@ -108,15 +104,24 @@ fromDensity :: (Double -> Double) -> (Double -> Double) -> Double fromDensity d f = quadratureTanhSinh $ liftA2 (*) f d where quadratureTanhSinh = result . last . everywhere trap +-- | For a random variable X, measurable function f, and measure P, we can +-- construct the image (pushforward) measure P_(f X). +push :: (Double -> Double) -> Measure Double -> Measure Double +push f mu g = mu $ g . f + -- | Measure composition is convolution. This allows us to compose measures, -- independent of how they were constructed. -- --- NOTE I think this is '.' on the category of measures. -convolute :: Measure -> Measure -> Measure +-- This is (.) on the category of measures. +convolute :: Measure Double -> Measure Double -> Measure Double convolute mu nu f = nu $ \y -> mu $ \x -> f $ x + y --- | For a random variable X, measurable function f, and measure P, we can --- construct the image (pushforward) measure P_(f X). -push :: (Double -> Double) -> Measure -> Measure -push f mu g = mu $ g . f +-- | The identity measure. +-- +-- This is 'id' on the category of measures. +identity :: Measure Double +identity = fromObservations [0] + +-- instance Functor Measure where +-- fmap f mu = push f mu