Safe Haskell | None |
---|---|
Language | Haskell2010 |
Online statistics for ordered data (such as time-series data), modelled as mealy machines
Synopsis
- newtype Mealy a b = Mealy {}
- pattern M :: (c -> b) -> (c -> a -> c) -> (a -> c) -> Mealy a b
- scan :: Mealy a b -> [a] -> [b]
- fold :: Mealy a b -> [a] -> b
- newtype Averager a b = Averager {
- sumCount :: (a, b)
- pattern A :: a -> b -> Averager a b
- av :: Divisive a => Averager a a -> a
- av_ :: (Eq a, Additive a, Divisive a) => Averager a a -> a -> a
- online :: (Divisive b, Additive b) => (a -> b) -> (b -> b) -> Mealy a b
- ma :: (Divisive a, Additive a) => a -> Mealy a a
- absma :: (Divisive a, Additive a, Signed a) => a -> Mealy a a
- sqma :: (Divisive a, Additive a) => a -> Mealy a a
- std :: (Divisive a, ExpField a) => a -> Mealy a a
- cov :: Field a => Mealy a a -> Mealy (a, a) a
- corrGauss :: ExpField a => a -> Mealy (a, a) a
- corr :: ExpField a => Mealy a a -> Mealy a a -> Mealy (a, a) a
- beta1 :: ExpField a => Mealy a a -> Mealy (a, a) a
- alpha1 :: ExpField a => Mealy a a -> Mealy (a, a) a
- reg1 :: ExpField a => Mealy a a -> Mealy (a, a) (a, a)
- beta :: (Field a, Field a, KnownNat n) => a -> Mealy (Array '[n] a, a) (Array '[n] a)
- alpha :: (Field a, ExpField a, KnownNat n) => a -> Mealy (Array '[n] a, a) a
- reg :: (Field a, ExpField a, KnownNat n) => a -> Mealy (Array '[n] a, a) (Array '[n] a, a)
- asum :: Additive a => Mealy a a
- aconst :: b -> Mealy a b
- delay1 :: a -> Mealy a a
- delay :: [a] -> Mealy a a
- depState :: (a -> b -> a) -> Mealy a b -> Mealy a a
- data Model1 = Model1 {}
- zeroModel1 :: Model1
- depModel1 :: Double -> Model1 -> Mealy Double Double
- fromFoldl :: Fold a b -> Mealy a b
- foldB :: Reifies s W => (BVar s Double -> BVar s Double) -> BVar s Double -> BVar s [Double] -> BVar s Double
- maB :: Reifies s W => BVar s Double -> BVar s [Double] -> BVar s Double
- data Medianer a b = Medianer {}
- onlineL1 :: (Ord b, Field b, Signed b) => b -> b -> (a -> b) -> (b -> b) -> Fold a b
- onlineL1' :: (Ord b, Field b, Signed b) => b -> b -> (a -> b) -> (b -> b) -> Fold a (b, b)
- maL1 :: (Ord a, Field a, Signed a) => a -> a -> a -> Fold a a
- absmaL1 :: (Ord a, Field a, Signed a) => a -> a -> a -> Fold a a
Types
A Mealy
is a triple of functions
- (c -> b) extract Convert state to the output type.
- (c -> a -> c) step Update state given prior state and (new) input.
- (a -> c) inject Convert an input into the state type.
The type is a newtype wrapper around L1
in Fold
.
inject is necessary to kick off state on a fold
or scan
, rather than a state existing prior to the fold or scan (this is a Moore machine or M1
in Fold
).
scan (M e s i) (x : xs) = e <$> scanl' s (i x) xs
pattern M :: (c -> b) -> (c -> a -> c) -> (a -> c) -> Mealy a b Source #
Pattern for a Mealy
.
M extract step inject
scan :: Mealy a b -> [a] -> [b] Source #
Run a list through a Mealy
and return a list of values for every step
length (scan _ xs) == length xs
Most common statistics are averages, which are some sort of aggregation of values (sum) and some sort of sample size (count).
av :: Divisive a => Averager a a -> a Source #
extract the average from an Averager
av gives NaN on zero divide
av_ :: (Eq a, Additive a, Divisive a) => Averager a a -> a -> a Source #
substitute a default value on zero-divide
av_ (Averager (0,0)) x == x
online :: (Divisive b, Additive b) => (a -> b) -> (b -> b) -> Mealy a b Source #
online f g
is a Mealy
where f is a transformation of the data and g is a decay function (convergent tozero) applied at each step.
online id id == av
Statistics
Generate some random variates for the examples.
xs0, xs1 & xs2 are samples from N(0,1)
xsp is a pair of N(0,1)s with a correlation of 0.8
>>>
:set -XDataKinds
>>>
import Control.Category ((>>>))
>>>
import Data.List
>>>
import Data.Simulate
>>>
g <- create
>>>
xs0 <- rvs g 10000
>>>
xs1 <- rvs g 10000
>>>
xs2 <- rvs g 10000
>>>
xsp <- rvsp g 10000 0.8
ma :: (Divisive a, Additive a) => a -> Mealy a a Source #
A moving average using a decay rate of r. r=1 represents the simple average, and r=0 represents the latest value.
>>>
fold (ma 0) (fromList [1..100])
100.0
>>>
fold (ma 1) (fromList [1..100])
50.5
>>>
fold (ma 0.99) xs0
-4.292501077490672e-2
A change in the underlying mean at n=10000 in the chart below highlights the trade-off between stability of the statistic and response to non-stationarity.
absma :: (Divisive a, Additive a, Signed a) => a -> Mealy a a Source #
absolute average
>>>
fold (absma 1) xs0
0.7894201075535578
sqma :: (Divisive a, Additive a) => a -> Mealy a a Source #
average square
fold (ma r) . fmap (**2) == fold (sqma r)
std :: (Divisive a, ExpField a) => a -> Mealy a a Source #
standard deviation
The construction of standard deviation, using the Applicative instance of a Mealy
:
(\s ss -> sqrt (ss - s ** (one+one))) <$> ma r <*> sqma r
The average deviation of the numbers 1..1000 is about 1 / sqrt 12 * 1000 https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)#Standard_uniform
>>>
fold (std 1) [0..1000]
288.9636655359978
The average deviation with a decay of 0.99
>>>
fold (std 0.99) [0..1000]
99.28328803163829
>>>
fold (std 1) xs0
0.9923523681261158
cov :: Field a => Mealy a a -> Mealy (a, a) a Source #
The covariance of a tuple given an underlying central tendency fold.
>>>
fold (cov (ma 1)) xsp
0.8011368250045314
corrGauss :: ExpField a => a -> Mealy (a, a) a Source #
correlation of a tuple, specialised to Guassian
>>>
fold (corrGauss 1) xsp
0.8020637696465039
corr :: ExpField a => Mealy a a -> Mealy a a -> Mealy (a, a) a Source #
a generalised version of correlation of a tuple
>>>
fold (corr (ma 1) (std 1)) xsp
0.8020637696465039
corr (ma r) (std r) == corrGauss r
beta1 :: ExpField a => Mealy a a -> Mealy (a, a) a Source #
The beta in a simple linear regression of an (independent variable, single dependent variable) tuple given an underlying central tendency fold.
This is a generalisation of the classical regression formula, where averages are replaced by Mealy
statistics.
\[ \begin{align} \beta & = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \\ & = \frac{n^2 \overline{xy} - n^2 \bar{x} \bar{y}}{n^2 \overline{x^2} - n^2 \bar{x}^2} \\ & = \frac{\overline{xy} - \bar{x} \bar{y}}{\overline{x^2} - \bar{x}^2} \\ \end{align} \]
>>>
fold (beta1 (ma 1)) $ zipWith (\x y -> (y, x + y)) xs0 xs1
0.9953875263096014
alpha1 :: ExpField a => Mealy a a -> Mealy (a, a) a Source #
The alpha in a simple linear regression of an (independent variable, single dependent variable) tuple given an underlying central tendency fold.
\[ \begin{align} \alpha & = \frac{\sum y \sum x^2 - \sum x \sum xy}{n\sum x^2 - (\sum x)^2} \\ & = \frac{n^2 \bar{y} \overline{x^2} - n^2 \bar{x} \overline{xy}}{n^2 \overline{x^2} - n^2 \bar{x}^2} \\ & = \frac{\bar{y} \overline{x^2} - \bar{x} \overline{xy}}{\overline{x^2} - \bar{x}^2} \\ \end{align} \]
>>>
fold (alpha1 (ma 1)) $ zipWith (\x y -> ((3+y), x + 0.5 * (3 + y))) xs0 xs1
1.1880996822796197e-2
reg1 :: ExpField a => Mealy a a -> Mealy (a, a) (a, a) Source #
The (alpha, beta) tuple in a simple linear regression of an (independent variable, single dependent variable) tuple given an underlying central tendency fold.
>>>
fold (reg1 (ma 1)) $ zipWith (\x y -> ((3+y), x + 0.5 * (3 + y))) xs0 xs1
(1.1880996822796197e-2,0.49538752630956845)
beta :: (Field a, Field a, KnownNat n) => a -> Mealy (Array '[n] a, a) (Array '[n] a) Source #
multiple regression
\[ \begin{align} {\hat {{\mathbf {B}}}}=({\mathbf {X}}^{{{\rm {T}}}}{\mathbf {X}})^{{ -1}}{\mathbf {X}}^{{{\rm {T}}}}{\mathbf {Y}} \end{align} \]
\[ \begin{align} {\mathbf {X}}={\begin{bmatrix}{\mathbf {x}}_{1}^{{{\rm {T}}}}\\{\mathbf {x}}_{2}^{{{\rm {T}}}}\\\vdots \\{\mathbf {x}}_{n}^{{{\rm {T}}}}\end{bmatrix}}={\begin{bmatrix}x_{{1,1}}&\cdots &x_{{1,k}}\\x_{{2,1}}&\cdots &x_{{2,k}}\\\vdots &\ddots &\vdots \\x_{{n,1}}&\cdots &x_{{n,k}}\end{bmatrix}} \end{align} \]
let ys = zipWith3 (\x y z -> 0.1 * x + 0.5 * y + 1 * z) xs0 xs1 xs2 let zs = zip (zipWith (\x y -> fromList [x,y] :: F.Array '[2] Double) xs1 xs2) ys fold (beta 0.99) zs
- 0.4982692361226971, 1.038192474255091
alpha :: (Field a, ExpField a, KnownNat n) => a -> Mealy (Array '[n] a, a) a Source #
alpha in a multiple regression
reg :: (Field a, ExpField a, KnownNat n) => a -> Mealy (Array '[n] a, a) (Array '[n] a, a) Source #
multiple regression
let ys = zipWith3 (\x y z -> 0.1 * x + 0.5 * y + 1 * z) xs0 xs1 xs2 let zs = zip (zipWith (\x y -> fromList [x,y] :: F.Array '[2] Double) xs1 xs2) ys fold (reg 0.99) zs
([0.4982692361226971, 1.038192474255091],2.087160803386695e-3)
:: [a] | initial statistical values, delay equals length |
-> Mealy a a |
delays values by n steps
delay [0] == delay1 0
delay [] == id
delay [1,2] = delay1 2 . delay1 1
>>>
scan (delay [-2,-1]) [0..3]
[-2,-1,0,1]
Autocorrelation example:
scan (((,) <$> id <*> delay [0]) >>> beta (ma 0.99)) xs0
depState :: (a -> b -> a) -> Mealy a b -> Mealy a a Source #
Add a state dependency to a series.
Typical regression analytics tend to assume that moments of a distributional assumption are unconditional with respect to prior instantiations of the stochastics being studied.
For time series analytics, a major preoccupation is estimation of the current moments given what has happened in the past.
IID:
\[ \begin{align} x_{t+1} & = alpha_t^x + s_{t+1}\\ s_{t+1} & = alpha_t^s * N(0,1) \end{align} \]
Example: including a linear dependency on moving average history:
\[ \begin{align} x_{t+1} & = (alpha_t^x + beta_t^{x->x} * ma_t^x) + s_{t+1}\\ s_{t+1} & = alpha_t^s * N(0,1) \end{align} \]
>>>
let xs' = scan (depState (\a m -> a + 0.1 * m) (ma 0.99)) xs0
>>>
let ma' = scan ((ma (1 - 0.01)) >>> delay [0]) xs'
>>>
let xsb = fold (beta1 (ma (1 - 0.001))) $ drop 1 $ zip ma' xs'
>>>
-- beta measurement if beta of ma was, in reality, zero.
>>>
let xsb0 = fold (beta1 (ma (1 - 0.001))) $ drop 1 $ zip ma' xs0
>>>
xsb - xsb0
9.999999999999976e-2
This simple model of relationship between a series and it's historical average shows how fragile the evidence can be.
In unravelling the drivers of this result, the standard deviation of a moving average scan seems well behaved for r > 0.01, but increases substantively for values less than this. This result seems to occur for wide beta values. For high r, the standard deviation of the moving average seems to be proprtional to r**0.5, and equal to around (0.5*r)**0.5.
fold (std 1) (scan (ma (1 - 0.01)) xs0)
a linear model of state dependencies for the first two moments
\[ \begin{align} x_{t+1} & = (alpha_t^x + beta_t^{x->x} * ma_t^x + beta_t^{s->x} * std_t^x) + s_{t+1}\\ s_{t+1} & = (alpha_t^s + beta_t^{x->s} * ma_t^x + beta_t^{s->s} * std_t^x) * N(0,1) \end{align} \]
Instances
Eq Model1 Source # | |
Show Model1 Source # | |
Generic Model1 Source # | |
type Rep Model1 Source # | |
Defined in Data.Mealy type Rep Model1 = D1 (MetaData "Model1" "Data.Mealy" "online-0.6.0-HLmMvMxp7IYLqRFxok4q8s" False) (C1 (MetaCons "Model1" PrefixI True) ((S1 (MetaSel (Just "alphaX") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double) :*: (S1 (MetaSel (Just "alphaS") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double) :*: S1 (MetaSel (Just "betaMa2X") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double))) :*: (S1 (MetaSel (Just "betaMa2S") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double) :*: (S1 (MetaSel (Just "betaStd2X") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double) :*: S1 (MetaSel (Just "betaStd2S") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double))))) |
zeroModel1 :: Model1 Source #
depModel1 :: Double -> Model1 -> Mealy Double Double Source #
Apply a model1 relationship using a single decay factor.
>>>
:set -XOverloadedLabels
>>>
fold (depModel1 0.01 (zeroModel1 & #betaMa2X .~ 0.1)) xs0
-0.47228537123218206
conversion
foldB :: Reifies s W => (BVar s Double -> BVar s Double) -> BVar s Double -> BVar s [Double] -> BVar s Double Source #
median
A rough Median. The average absolute value of the stat is used to callibrate estimate drift towards the median
onlineL1 :: (Ord b, Field b, Signed b) => b -> b -> (a -> b) -> (b -> b) -> Fold a b Source #
onlineL1 takes a function and turns it into a Fold
where the step is an incremental update of an (isomorphic) median statistic.
onlineL1' :: (Ord b, Field b, Signed b) => b -> b -> (a -> b) -> (b -> b) -> Fold a (b, b) Source #
onlineL1' takes a function and turns it into a Fold
where the step is an incremental update of an (isomorphic) median statistic.