This post expresses the key ideas of a talk I gave at FP-SYD this week.

Monoids are a pretty simple concept in haskell. Some years ago I learnt of them through the excellent Typeclassopedia, looked at the examples, and understood them quickly (which is more than can be said for many of the new ideas that one learns in haskell). However that was it. Having learnt the idea, I realised that monoids are everywhere in programming, but I’d not found much use for the Monoid typeclass abstraction itself. Recently, I’ve found they can be a useful tool for data analysis…

## Monoids

First a quick recap. A monoid is a type with a binary operation, and an identity element:

```
class Monoid a where
mempty :: a
mappend :: a -> a -> a
```

It must satisfy a simple set of laws, specifically that the binary operation much be associative, and the identity element must actually be the identity for the given operation:

```
mappend a (mappend b c) = mappend (mappend a b) c
mappend mempty x = x
mappend x mempty = x
```

As is hinted by the names of the typeclass functions, lists are an obvious Monoid instance:

```
instance Monoid [a] where
mempty = []
mappend = (++)
```

However, many types can be Monoids. In fact, often a type can be a monoid in multiple ways. Numbers are monoids under both addition and multiplication, with 0 and 1 as their respective identity elements. In the haskell standard libraries, rather than choose one kind of monoid for numbers, newtype declarations are used to given instances for both:

```
newtype Sum a = Sum { getSum :: a }
deriving (Eq, Ord, Read, Show, Bounded)
instance Num a => Monoid (Sum a) where
mempty = Sum 0
Sum x `mappend` Sum y = Sum (x + y)
newtype Product a = Product { getProduct :: a }
deriving (Eq, Ord, Read, Show, Bounded)
instance Num a => Monoid (Product a) where
mempty = Product 1
Product x `mappend` Product y = Product (x * y)
```

We’ve now established and codified the common structure for a few monoids, but it’s not yet clear what it has gained us. The Sum and Product instances are unwieldly – you are unlikely to want to use Sum directly to add two numbers:

```
Prelude> :m Data.Monoid
Prelude Data.Monoid> 5+4
9
Prelude Data.Monoid> getSum (mappend (Sum 5) (Sum 4))
9
```

Before we progress, however, let’s define a few more monoid instances, potentially useful for data analysis.

```
data Min a = Min a | MinEmpty deriving (Show)
data Max a = Max a | MaxEmpty deriving (Show)
newtype Count = Count Int deriving (Show)
instance (Ord a) => Monoid (Min a) where
mempty = MinEmpty
mappend MinEmpty m = m
mappend m MinEmpty = m
mappend (Min a) (Min b) = (Min (P.min a b))
instance (Ord a) => Monoid (Max a) where
mempty = MaxEmpty
mappend MaxEmpty m = m
mappend m MaxEmpty = m
mappend (Max a) (Max b) = (Max (P.max a b))
instance Monoid Count where
mempty = Count 0
mappend (Count n1) (Count n2) = Count (n1+n2)
```

Also some helper functions to construct values of all these monoid types:

```
sum :: (Num a) => a -> Sum a
sum = Sum
product :: (Num a) => a -> Product a
product = Product
min :: (Ord a) => a -> Min a
min = Min
max :: (Ord a) => a -> Max a
max = Max
count :: a -> Count
count _ = Count 1
```

These functions are trivial, but they put a consistent interface on creating monoid values. They all have a signature (a -> m) where m is some monoid. For lack of a better name, I’ll call functions with such signatures "monoid functions".

## Foldable

It’s time to introduce another typeclass, Foldable. This class abstracts the classic foldr and foldl functions away from lists, making them applicable to arbitrary structures. (There’s a robust debate going on right now about the merits of replacing the list specific fold functions in the standard prelude with the more general versions from Foldable.) Foldable is a large typeclass – here’s the key function of interest to us:

```
class Foldable t where
...
foldMap :: Monoid m => (a -> m) -> t a -> m
...
```

foldMap takes a monoid function and a Foldable structure, and reduces the structure down to a single value of the monoid. Lists are, of course, instances of foldable, so we can demo our helper functions:

```
*Examples> let as = [45,23,78,10,11,1]
*Examples> foldMap count as
Count 6
*Examples> foldMap sum as
Sum {getSum = 168}
*Examples> foldMap max as
Max 78
```

Notice how the results are all still wrapped with the newtype constructors. We’ll deal with this later.

## Composition

As it turns out, tuples are already instances of Monoids:

```
instance (Monoid a,Monoid b) => Monoid (a,b) where
mempty = (mempty,mempty)
mappend (a1,b1) (a2,b2) = (mappend a1 a2,mappend b1 b2)
```

A pair is a monoid if it’s elements are monoids. There are similar instances for longer tuples. We need some helper monoid functions for tuples also:

```
a2 :: (a -> b) -> (a -> c) -> a -> (b,c)
a2 b c = (,) <$> b <*> c
a3 :: (a -> b) -> (a -> c) -> (a -> d) -> a -> (b,c,d)
a3 b c d = (,,) <$> b <*> c <*> d
```

These are implemented above using Applicative operators, though I’ve given them more restrictive types to make their intended use here clearer. Now I can compose monoid functions:

```
*Examples> let as = [45,23,78,10,11,1]
*Examples> :t (a2 min max)
(a2 min max) :: Ord a => a -> (Min a, Max a)
*Examples> foldMap (a2 min max) as
(Min 1,Max 78)
*Examples> :t (a3 count (a2 min max) (a2 sum product))
(a3 count (a2 min max) (a2 sum product))
:: (Num a, Ord a) =>
a -> (Count, (Min a, Max a), (Sum a, Product a))
*Examples> foldMap (a3 count (a2 min max) (a2 sum product)) as
(Count 6,(Min 1,Max 78),(Sum {getSum = 168},Product {getProduct = 8880300}))
```

It’s worth noting here that the composite computations are done in a single traversal of the input list.

## More complex calculations

Happy with this, I decide to extend my set of basic computations with the arithmetic mean. There is a problem, however. The arithmetic mean doesn’t "fit" as a monoid – there’s no binary operation such that a mean for a combined set of data can be calculated from the mean of two subsets.

What to do? Well, the mean is the sum divided by the count, both of which are monoids:

```
newtype Mean a = Mean (Sum a,Count) deriving (Show)
instance (Num a) => Monoid (Mean a) where
mempty = Mean mempty
mappend (Mean m1) (Mean m2) = Mean (mappend m1 m2)
mean v = Mean (Sum v,Count 1)
```

So I can calculate the mean if I am prepared to do a calculation after the foldMap:

```
*Examples> let as = [45,23,78,10,11,1.5]
*Examples> foldMap mean as
Mean (Sum {getSum = 168.5},Count 6)
*Examples> let (Mean (Sum t,Count n)) = foldMap mean as in t / fromIntegral n
28.083333333333332
```

## The Aggregation type class

For calculations like `mean`

, I need something more than a monoid. I need a monoid for accumulating the values, and then, once the accumulation is complete, a postprocessing function to compute the final result. Hence a new typeclass to extend Monoid:

```
{-# LANGUAGE TypeFamilies #-}
class (Monoid a) => Aggregation a where
type AggResult a :: *
aggResult :: a -> AggResult a
```

This makes use of the type families ghc extension. We need this to express the fact that our postprocessing function aggResult has a different return type to the type of the monoid. In the above definition:

- aggResult is a function that gives you the
*value*of the final result from the*value*of the monoid - AggResult is a
*type*function that gives you the*type*of the final result from the*type*of the monoid

We can write an instance of Aggregation for Mean:

```
instance (Fractional a) => Aggregation (Mean a) where
type AggResult (Mean a) = a
aggResult (Mean (Sum t,Count n)) = t/fromIntegral n
```

and test it out:

```
*Examples> let as = [45,23,78,10,11,1.5]
*Examples> aggResult (foldMap mean as)
28.083333333333332
*Examples>
```

Nice. Given that `aggResult (foldMap ...)`

will be a common pattern, lets write a helper:

```
afoldMap :: (Foldable t, Aggregation a) => (v -> a) -> t v -> AggResult a
afoldMap f vs = aggResult (foldMap f vs)
```

In order to use the monoids we defined before (sum,product etc) we need to define Aggregation instances for them also. Even though they are trivial, it turns out to be useful, as we can make the aggResult function strip off the newtype constructors that were put there to enable the Monoid typeclass:

```
instance (Num a) => Aggregation (Sum a) where
type AggResult (Sum a) = a
aggResult (Sum a) = a
instance (Num a) => Aggregation (Product a) where
type AggResult (Product a) = a
aggResult (Product a) = a
instance (Ord a) => Aggregation (Min a) where
type AggResult (Min a) = a
aggResult (Min a) = a
instance (Ord a) => Aggregation (Max a) where
type AggResult (Max a) = a
aggResult (Max a) = a
instance Aggregation Count where
type AggResult Count = Int
aggResult (Count n) = n
instance (Aggregation a, Aggregation b) => Aggregation (a,b) where
type AggResult (a,b) = (AggResult a, AggResult b)
aggResult (a,b) = (aggResult a, aggResult b)
instance (Aggregation a, Aggregation b, Aggregation c) => Aggregation (a,b,c) where
type AggResult (a,b,c) = (AggResult a, AggResult b, AggResult c)
aggResult (a,b,c) = (aggResult a, aggResult b, aggResult c)
```

This is mostly boilerplate, though notice how the tuple instances delve into their components in order to postprocess the results. Now everything fits together cleanly:

```
*Examples> let as = [45,23,78,10,11,1.5]
*Examples> :t (a3 count (a2 min max) mean)
(a3 count (a2 min max) mean)
:: Ord a => a -> (Count, (Min a, Max a), Mean a)
*Examples> afoldMap (a3 count (a2 min max) mean) as
(6,(1.5,78.0),28.083333333333332)
*Examples>
```

The 4 computations have been calculated all in a single pass over the input list, and the results are free of the type constructors that are no longer required once the aggregation is complete.

Another example of an Aggregation where we need to postprocess the result is counting the number of unique items. For this we will keep a set of the items seen, and then return the size of this set at the end:

```
newtype CountUnique a = CountUnique (Set.Set a)
instance Ord a => Monoid (CountUnique a) where
mempty = CountUnique Set.empty
mappend (CountUnique s1) (CountUnique s2) = CountUnique (Set.union s1 s2)
instance Ord a => Aggregation (CountUnique a) where
type AggResult (CountUnique a) = Int
aggResult (CountUnique s1) = Set.size s1
countUnique :: Ord a => a -> CountUnique a
countUnique a = CountUnique (Set.singleton a)
```

.. in use:

```
*Examples> let as = [5,7,8,7,11,10,11]
*Examples> afoldMap (a2 countUnique count) as
(5,7)
```

## Higher order aggregation functions

All of the calculations seen so far have worked consistently across all values in the source data structure. We can make use of the `mempty`

monoid value in order to filter our data set, and or aggregate in groups. Here’s a couple of higher order monoid functions for this:

```
afilter :: Aggregation m => (a -> Bool) -> (a -> m) -> (a -> m)
afilter match mf = \a -> if match a then mf a else mempty
newtype MMap k v = MMap (Map.Map k v)
deriving Show
instance (Ord k, Monoid v) => Monoid (MMap k v) where
mempty = MMap (Map.empty)
mappend (MMap m1) (MMap m2) = MMap (Map.unionWith mappend m1 m2)
instance (Ord k, Aggregation v) => Aggregation (MMap k v) where
type AggResult (MMap k v) = Map.Map k (AggResult v)
aggResult (MMap m) = Map.map aggResult m
groupBy :: (Ord k, Aggregation m) => (a -> k) -> (a -> m) -> (a -> MMap k m)
groupBy keyf valuef = \a -> MMap (Map.singleton (keyf a) (valuef a))
```

`afilter`

restricts the application of a monoid function to a subset of the input data. eg to calculate the sum of all the values, and the sum of values less than 20:

```
*Examples> let as = [5,10,20,45.4,35,1,3.4]
*Examples> afoldMap (a2 sum (afilter (<=20) sum)) as
(119.8,39.4)
```

`groupBy`

takes a key function and a monoid function. It partitions the data set using the key function, and applies a monoid function to each subset, returning all of the results in a map. Non-numeric data works better as an example here. Let’s take a set of words as input, and for each starting letter, calculate the number of words with that letter, the length of the shortest word, and and the length of longest word:

```
*Examples> let as = words "monoids are a pretty simple concept in haskell some years ago i learnt of them through the excellent typeclassopedia looked at the examples and understood them straight away which is more than can be said for many of the new ideas that one learns in haskell"
*Examples> :t groupBy head (a3 count (min.length) (max.length))
groupBy head (a3 count (min.length) (max.length))
:: Ord k => [k] -> MMap k (Count, Min Int, Max Int)
*Examples> afoldMap (groupBy head (a3 count (min.length) (max.length))) as
fromList [('a',(6,1,4)),('b',(1,2,2)),('c',(2,3,7)),('e',(2,8,9)),('f',(1,3,3)),('h',(2,7,7)),('i',(5,1,5)),('l',(3,6,6)),('m',(3,4,7)),('n',(1,3,3)),('o',(3,2,3)),('p',(1,6,6)),('s',(4,4,8)),('t',(9,3,15)),('u',(1,10,10)),('w',(1,5,5)),('y',(1,5,5))]
```

Many useful data analysis functions can be written through simple function application and composition using these primitive monoid functions, the product combinators a2 and a3 and these new filtering and grouping combinators.

## Disk-based data

As pointed out before, regardless of the complexity of the computation, it’s done with a single traversal of the input data. This means that we don’t need to limit ourselves to lists and other in memory Foldable data structures. Here’s a function similar to foldMap, but that works over the lines in a file:

```
foldFile :: Monoid m => FilePath -> (BS.ByteString -> Maybe a) -> (a -> m) -> IO m
foldFile fpath pf mf = do
h <- openFile fpath ReadMode
m <- loop h mempty
return m
where
loop h m = do
eof <- hIsEOF h
if eof
then (return m)
else do
l <- BS.hGetLine h
case pf l of
Nothing -> loop h m
(Just a) -> let m' = mappend m (mf a)
in loop h m'
afoldFile :: Aggregation m => FilePath -> (BS.ByteString -> Maybe a) -> (a -> m) -> IO (AggResult m)
afoldFile fpath pf mf = fmap aggResult (foldFile fpath pf mf)
```

foldFile take two parameters – a function to parse each line of the file, the other is the monoid function to do the aggregation. Lines that fail to parse are skipped. (I can here questions in the background "What about strictness and space leaks?? – I’ll come back to that). As an example usage of aFoldFile, I’ll analyse some stock data. Assume that I have it in a CSV file, and I’ve got a function to parse one CSV line into a sensible data value:

```
import qualified Data.ByteString.Char8 as BS
import Data.Time.Calendar
data Prices = Prices {
pName :: String, -- The stock code
pDate :: Day, -- The historic date
pOpen :: Double, -- The price at market open
pHigh :: Double, -- The highest price on the date
pLow :: Double, -- The lowest price on the date
pClose :: Double, -- The price at market close
pVolume :: Double -- How many shares were traded
} deriving (Show)
parsePrices :: BS.ByteString -> Maybe Prices
parsePrices = ...
```

Now I can use my monoid functions to analyse the file based data. How many google prices do I have, over what date range:

```
*Examples> let stats = afilter (("GOOG"==).pName) (a3 count (min.pDate) (max.pDate))
*Examples> :t stats
stats
:: Prices
-> (Count,
Min time-1.4:Data.Time.Calendar.Days.Day,
Max time-1.4:Data.Time.Calendar.Days.Day)
*Examples> afoldFile "prices.csv" parsePrices stats
(1257,2008-05-29,2013-05-24)
*Examples>
```

Perhaps I want to aggregate my data per month, getting traded price range and total volume. We need a helper function to work out the month of each date:

```
startOfMonth :: Day -> Day
startOfMonth t = let (y,m,d) = toGregorian t
in fromGregorian y m 1
```

And then we can use groupBy to collect data monthly:

```
:*Examples> let stats = afilter (("GOOG"==).pName) (groupBy (startOfMonth.pDate) (a3 (min.pLow) (max.pHigh) (sum.pVolume)))
*Examples> :t stats
stats
:: Prices
-> MMap
time-1.4:Data.Time.Calendar.Days.Day
(Min Double, Max Double, Sum Double)
*Examples> results <- afoldFile "prices.csv" parsePrices stats
*Examples> mapM_ print (Map.toList results)
(2008-05-01,(573.2,589.92,8073107.0))
(2008-06-01,(515.09,588.04,9.3842716e7))
(2008-07-01,(465.6,555.68,1.04137619e8))
...
(2013-03-01,(793.3,844.0,4.2559856e7))
(2013-04-01,(761.26,827.64,5.3574633e7))
(2013-05-01,(816.36,920.6,4.1080028e7))
```

## Conclusion

So, I hope I’ve shown that monoids are useful indeed. They can form the core of a framework for cleanly specifing quite complex data analysis tasks.

An additional typeclass which I called "Aggregation" extends Monoid and provides for a broader range of computations and also cleaner result types (thanks to type families). There was some discussion when I presented this talk as to whether a single method typeclass like Aggregation was a "true" abstraction, given it has no associated laws. This is a valid point, however using it simplifies the syntax and usage of monoidal calculations significantly, and for me, this makes it worth having.

There remains an elephant in the room, however, and this is space leakage. Lazy evalulation means that, as written, most of the calculations shown run in space proportional to the input data set. Appropriate strictness annotations and related modifications will fix this, but it turns out to be slightly irritating. This blog post is already long enough, so I’ll address space leaks in in a subsequent post…

I don’t understand Haskell well enough (former SML-NJ hacker, though), but it seems like the identity for the ‘min’ operator should be the MaxBound. That is, minimum of sensible number and +inf is the sensible number. Am I missing something?

Perhaps the name for the bounding values on my min and max monoids are less than ideal. I called it MinBound as it was the bounding value for the min operation, however you are right in that it has to behave as if it is bigger than all other values. It’s just a label though – I believe the code is correct.

ahh, I see; I had confused it with Haskell’s ‘minBound’ from the ‘Bounded’ class. My bad.

This was confusing to me too. I think it would throw off most haskell newbies. Maybe you should put a note about it?

The names “MinBound” and “MaxBound” on the Min and Max monoids have caused confusion, given the minBound and maxBound methods of the standard Bounded typeclass. Hence I’ve updated the post and renamed them to be “MinEmpty” and “MaxEmpty”.

I’m not sure that Aggregation type class is right choice. It’s possible to extract several statistics at once from the single monoid. For example from Mean it’s possible to extract both count and mean. With single pass variance estimator it’s possible to get count, mean, unbiased (N-1 in denominator) and biased (N in denominatro) variances and corresponding stadard deviations. It calls for multiparameter type class but MPTC without dependencies between types are difficult to work with.

Also fact that estimators form monoids reflects monoidal structure of samples. Empty sample is identity and joining of two samples is composition. Also if order of elements doesn’t matter estimator will form commutative monoid.

I understand that it would be possible to have a monoid that would calculate multiple statistics at the same time, and do so more efficiently. However, you say:

> from Mean it’s possible to extract both count and mean

It’s the Aggregation type class that is doing the extraction. How would you do this extraction without this type class?

There are several options. First is to write bunch of monomorphic functions which does the extraction. It doesn’t scale well so don’t really consider it. Another option is to make Aggregation multiparameter and create bunch of newtypes. (I hope WP won’t treat code snippets too badly

class Aggregation m a where

extract :: m -> a

newtype Mean a = Mean a

newtype Count = Count Int

instance Aggregation (MeanEst a) (Mean a) where

extract (MeanEst _ x) = Mean x

instance Aggregation (MeanEst a) Count where

extract (MeanEst n _ ) = Count n

It does have problem common to MPTC without fundeps or type families, in (extract . foldMap f) type of monoid couldn’t be inferred in many cases. This approach could be used to extract statistics from districutions as well. Currently Statistics.Distributions uses bunch of ad hoc type classes. So far I don’t know satisfactory solution.

I’ve played with this idea although in slightly different way but never finished it

https://github.com/Shimuuar/statistics/blob/77a6ae446ddfb8d48c186352b30b96312ae44aff/Statistics/Sample/Classes.hs

https://github.com/Shimuuar/statistics/blob/77a6ae446ddfb8d48c186352b30b96312ae44aff/Statistics/Sample/Estimators.hs

https://github.com/Shimuuar/statistics/blob/77a6ae446ddfb8d48c186352b30b96312ae44aff/Statistics/Sample/New.hs

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