Data.TDigest.Tree

A new data structure for accurate on-line accumulation of rank-based statistics such as quantiles and trimmed means. . See original paper: "Computing extremely accurate quantiles using t-digest" by Ted Dunning and Otmar Ertl for more details https://github.com/tdunning/t-digest/blob/master/docs/t-digest-paper/histo.pdf.

Examples

>>> quantile 0.99 (tdigest [1..1000] :: TDigest 25)
Just 990.5

>>> quantile 0.99 (tdigest [1..1000] :: TDigest 3)
Just 989.0...

t-Digest is more precise in tails, especially median is imprecise:

>>> median (forceCompress $ tdigest [1..1000] :: TDigest 25)
Just 497.6...

Semigroup

This operation isn't strictly associative, but statistical variables shouldn't be affected.

>>> let td xs = tdigest xs :: TDigest 10

>>> median (td [1..500] <> (td [501..1000] <> td [1001..1500]))
Just 802...

>>> median ((td [1..500] <> td [501..1000]) <> td [1001..1500])
Just 726...

The linear is worst-case scenario:

>>> let td' xs = tdigest (fairshuffle xs) :: TDigest 10

>>> median (td' [1..500] <> (td' [501..1000] <> td' [1001..1500]))
Just 750.3789...

>>> median ((td' [1..500] <> td' [501..1000]) <> td' [1001..1500])
Just 750.3789...

Construction

data TDigest (compression :: Nat)

TDigest is a tree of centroids.

compression is a 1/δ. The greater the value of compression the less likely value merging will happen.

Instances

KnownNat comp => Reducer Double (TDigest comp)

Both cons and snoc are insert

Defined in Data.TDigest.Tree.Internal

Methods

unit :: Double -> TDigest comp

snoc :: TDigest comp -> Double -> TDigest comp

cons :: Double -> TDigest comp -> TDigest comp

Show (TDigest compression)

Defined in Data.TDigest.Tree.Internal

Methods

showsPrec :: Int -> TDigest compression -> ShowS

show :: TDigest compression -> String

showList :: [TDigest compression] -> ShowS

KnownNat comp => Semigroup (TDigest comp)

Defined in Data.TDigest.Tree.Internal

Methods

(<>) :: TDigest comp -> TDigest comp -> TDigest comp

sconcat :: NonEmpty (TDigest comp) -> TDigest comp

stimes :: Integral b => b -> TDigest comp -> TDigest comp

KnownNat comp => Monoid (TDigest comp)

Defined in Data.TDigest.Tree.Internal

Methods

mempty :: TDigest comp

mappend :: TDigest comp -> TDigest comp -> TDigest comp

mconcat :: [TDigest comp] -> TDigest comp

KnownNat comp => Binary (TDigest comp)

TDigest isn't compressed after de-serialisation, but it can be still smaller.

Defined in Data.TDigest.Tree.Internal

Methods

put :: TDigest comp -> Put

get :: Get (TDigest comp)

putList :: [TDigest comp] -> Put

NFData (TDigest comp)

TDigest has only strict fields.

Defined in Data.TDigest.Tree.Internal

Methods

rnf :: TDigest comp -> ()

HasHistogram (TDigest comp) Maybe

Defined in Data.TDigest.Tree.Internal

Methods

histogram :: TDigest comp -> Maybe (NonEmpty HistBin)

totalWeight :: TDigest comp -> Weight

tdigest :: (Foldable f, KnownNat comp) => f Double -> TDigest comp

Strict foldl' over Foldable structure.

Population

singleton :: KnownNat comp => Double -> TDigest comp

Make a TDigest of a single data point.

insert

Arguments

:: KnownNat comp

=> Double

element

-> TDigest comp

-> TDigest comp

Insert single value into TDigest.

insert'

Arguments

:: KnownNat comp

=> Double

element

-> TDigest comp

-> TDigest comp

Insert single value, don't compress TDigest even if needed.

For sensibly bounded input, it makes sense to let TDigest grow (it might grow linearly in size), and after that compress it once.

Compression

>>> let digest = foldl' (flip insert') mempty [0..1000] :: TDigest 10
>>> (size digest, size $ compress digest)
(1001,52)

>>> (quantile 0.1 digest, quantile 0.1 $ compress digest)
(Just 99.6...,Just 89.7...)

Note: when values are inserted in more random order, t-Digest self-compresses on the fly:

>>> let digest = foldl' (flip insert') mempty (fairshuffle [0..1000]) :: TDigest 10
>>> (size digest, size $ compress digest, size $ forceCompress digest)
(78,78,48)

>>> quantile 0.1 digest
Just 98.9...

compress :: forall comp. KnownNat comp => TDigest comp -> TDigest comp

Compress TDigest.

Reinsert the centroids in "better" order (in original paper: in random) so they have opportunity to merge.

Compression will happen only if size is both: bigger than relMaxSize * comp and bigger than absMaxSize.

forceCompress :: forall comp. KnownNat comp => TDigest comp -> TDigest comp

Perform compression, even if current size says it's not necessary.

Statistics

minimumValue :: TDigest comp -> Mean

Center of left-most centroid. Note: may be different than min element inserted.

>>> minimumValue (tdigest [1..100] :: TDigest 3)
1.0

maximumValue :: TDigest comp -> Mean

Center of right-most centroid. Note: may be different than max element inserted.

>>> maximumValue (tdigest [1..100] :: TDigest 3)
99.0

Percentile

median :: TDigest comp -> Maybe Double

Median, i.e. quantile 0.5.

quantile :: Double -> TDigest comp -> Maybe Double

Calculate quantile of a specific value.

Mean & Variance

• - >>> stddev (tdigest $ fairshuffle [0..100] :: TDigest 10) Just 29.1...

mean :: TDigest comp -> Maybe Double

Mean.

>>> mean (tdigest [1..100] :: TDigest 10)
Just 50.5

Note: if you only need the mean, calculate it directly.

variance :: TDigest comp -> Maybe Double

Variance.

stddev :: TDigest comp -> Maybe Double

Standard deviation, square root of variance.

CDF

cdf :: Double -> TDigest comp -> Double

Cumulative distribution function.

Note: if this is the only thing you need, it's more efficient to count this directly.

icdf :: Double -> TDigest comp -> Maybe Double

An alias for quantile

Debug

size :: TDigest comp -> Int

valid :: TDigest comp -> Bool

isRight . validate

validate :: TDigest comp -> Either String (TDigest comp)

Check various invariants in the TDigest tree.

debugPrint :: TDigest comp -> IO ()

Output the TDigest tree.