foldl-incremental- incremental folds

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This module provides incremental statistical folds based upon the foldl library

An incremental statsitical fold can be thought of as exponentially-weighting statistics designed to be efficient computations over a Foldable.

Some throat clearing is required, however.

The common usage term "exponential moving ..." refers to the cumulative effect of the fold referencing the original data. From the point of view of a single step, the algorithm could be better described as "constant proportion" or "geometric" decay. Many other methods are also possible and future versions of the library may introduce some more.

A main point of the library is that the traditional simple moving average uses a sliding window of past data and thus requires keeping track of the last n elements in State (in a LIFO queue most likey). It may be simple for the human brain but its a more complex and costly computational than this single-pass version.

For clarity, moving average (and moving whatever) below refers to geometric decay rather than the common usage. So with the throat clearing out of the way:

To avoid clashes, Control.Foldl should be qualified.

>>> import Control.Foldl.Incremental
>>> import qualified Control.Foldl as L

The folds represent incremental statistics such as moving averages`.

The stream of moving averages with a forgetting rate of 0.9 is:

>>> L.scan (ma 0.9) [1..10]

or if you just want the moving average at the end.

>>> L.fold (ma 0.9) [1..10]

The simple average is obtained via a decay rate of 1.0 (ie no decay)

>>> L.fold (ma 1.0) [1..10]

further reading:

Online Algorithms in High-frequency Trading recursive least squares with exponential forgetting



incrementalize :: (a -> Double) -> Double -> Fold a Double Source

Incrementalize takes a function and turns it into a Fold where the step is an Increment similar to the typical step in an exponential moving average calculation.

>>> incrementalize id

is a moving average of a foldable

>>> incrementalize (*2)

is a moving average of the square of a foldable

This lets you build an exponential standard deviation computation (using Foldl) as

>>> std r = (\s ss -> sqrt (ss - s**2)) <$> incrementalize id r <*> incrementalize (*2) r

incrementalize works with any function that produces a double. A correlation fold of a tuple is quite intuitive:

>>> cov r = (\xy xbar ybar -> xy - xbar * ybar) <$> incrementalize (uncurry (*)) r <*> incrementalize fst r <*> incrementalize snd r
>>> corr r = (\cov' stdx stdy -> cov' / (stdx * stdy)) <$> cov r <*> L.premap fst (std r) <*> L.premap snd (std r)

The rate is the parameter regulating the discount (or forgetting) of current state and the introduction of the current value.

>>> incrementalize id 1

tracks the sum/average of an entire Foldable. In other words, prior values are never forgotten.

>>> incrementalize id 0

produces the latest value (ie current state is discounted (or decays) to zero). In other words, prior values are immediately forgotten.

A exponential moving average with an exponetially-weighted length (duration if its a time series) of 10 (the average lag of the values effecting the calculation) is

>>> incrementalize id (1 - 1/10)
>>> L.fold (length 0.9) [1..100]

There is no particular reason for different parts to have the same rate. A standard deviation where mean is expected to be static (eg equal to the unconditional sample average) would be:

>>> std' r = (\s ss -> sqrt (ss - s**2)) <$> incrementalize id 1 <*> incrementalize (*2) r

and a standard deviation with a prior for the mean (eg ignoring sample averges) would be:

>>> std'' mean r = incrementalize (\x -> x*2 - mean**2) r

common incremental folds

ma :: Double -> Fold Double Double Source

incremental average

absma :: Double -> Fold Double Double Source

incremental absolute average

sqma :: Double -> Fold Double Double Source

incremental average square

std :: Double -> Fold Double Double Source

incremental standard deviation

cov :: Double -> Fold (Double, Double) Double Source

incremental covariance

corr :: Double -> Fold (Double, Double) Double Source

incremental corelation

length :: Double -> Fold a Double Source

the exponentially weighted length of a rate, which is 1/(1-rate) at infinity

beta :: Double -> Fold (Double, Double) Double Source

the beta in a simple linear regression of snd on fst

alpha :: Double -> Fold (Double, Double) Double Source

the alpha in a simple linear regression of snd on fst

autocorr :: Double -> Double -> Fold Double Double Source

autocorrelation is a slippery concept. This method starts with the concept that there is an underlying random error process (e), and autocorrelation is a process on top of that ie for a one-step correlation relationship.

valuet = et + k * e@t-1

where k is the autocorrelation.

There are thus two decay rates needed: one for the average being considered to be the dependent variable, and one for the decay of the correlation calculation between the most recent value and the moving average.

>>> L.fold (autoCorr 0 1)

Would estimate the one-step autocorrelation relationship of the previous value and the current value over the entire sample set.