# Collaborative filtering

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## Latest revision as of 15:39, 20 April 2008

This page was added to discuss different versions of the code for collaborative filtering at Bryan's blog.

## [edit] 1 Chris' version

I renamed the variables and then reorganized the code a bit.

The(update,SlopeOne,predict) definitions can be compared with the newer (update2,SlopeOne',predict') definitions.

module WeightedSlopeOne (Rating, SlopeOne, empty, predict, update) where import Data.List (foldl',foldl1') import qualified Data.Map as M -- The item type is a polymorphic parameter. Since it goes into a Map -- it must be able to be compared, so item must be an instance of Ord. type Count = Int type RatingValue = Double -- The Rating is the known (item,Rating) information for a particular "user" type Rating item = M.Map item RatingValue -- The SlopeOne matrix is indexed by pairs of items and is implemented -- as a sparse map of maps. newtype SlopeOne item = SlopeOne (M.Map item (M.Map item (Count,RatingValue))) deriving (Show) -- The SlopeOne' matrix is an un-normalized version of SlopeOne newtype SlopeOne' item = SlopeOne' (M.Map item (M.Map item (Count,RatingValue))) deriving (Show) empty = SlopeOne M.empty empty' = SlopeOne' M.empty -- This performs a strict addition on pairs made of two nuumeric types addT (a,b) (c,d) = let (l,r) = (a+c, b+d) in l `seq` r `seq` (l, r) -- There is never an entry for the "diagonal" elements with equal -- items in the pair: (foo,foo) is never in the SlopeOne. update :: Ord item => SlopeOne item -> [Rating item] -> SlopeOne item update (SlopeOne matrixInNormed) usersRatings = SlopeOne . M.map (M.map norm) . foldl' update' matrixIn $ usersRatings where update' oldMatrix userRatings = foldl' (\oldMatrix (itemPair, rating) -> insert oldMatrix itemPair rating) oldMatrix itemCombos where itemCombos = [ ((item1, item2), (1, rating1 - rating2)) | (item1, rating1) <- ratings , (item2, rating2) <- ratings , item1 /= item2] ratings = M.toList userRatings insert outerMap (item1, item2) newRating = M.insertWith' outer item1 newOuterEntry outerMap where newOuterEntry = M.singleton item2 newRating outer _ innerMap = M.insertWith' addT item2 newRating innerMap norm (count,total_rating) = (count, total_rating / fromIntegral count) un_norm (count,rating) = (count, rating * fromIntegral count) matrixIn = M.map (M.map un_norm) matrixInNormed -- This version of update2 makes an unnormalize slopeOne' from each -- Rating and combines them using Map.union* operations and addT. update2 :: Ord item => SlopeOne' item -> [Rating item] -> SlopeOne' item update2 s@(SlopeOne' matrixIn) usersRatingsIn | null usersRatings = s | otherwise = SlopeOne' . M.unionsWith (M.unionWith addT) . (matrixIn:) . map fromRating $ usersRatings where usersRatings = filter ((1<) . M.size) usersRatingsIn fromRating userRating = M.mapWithKey expand1 userRating where expand1 item1 rating1 = M.mapMaybeWithKey expand2 userRating where expand2 item2 rating2 | item1 == item2 = Nothing | otherwise = Just (1,rating1 - rating2) predict :: Ord a => SlopeOne a -> Rating a -> Rating a predict (SlopeOne matrixIn) userRatings = let freqM = foldl' insert M.empty [ (item1,found_rating,user_rating) | (item1,innerMap) <- M.assocs matrixIn , M.notMember item1 userRatings , (user_item, user_rating) <- M.toList userRatings , item1 /= user_item , found_rating <- M.lookup user_item innerMap ] insert oldM (item1,found_rating,user_rating) = let (count,norm_rating) = found_rating total_rating = fromIntegral count * (norm_rating + user_rating) in M.insertWith' addT item1 (count,total_rating) oldM normM = M.map (\(count, total_rating) -> total_rating / fromIntegral count) freqM in M.filter (\norm_rating -> norm_rating > 0) normM -- This is a modified version of predict. It also expect the -- unnormalized SlopeOne' but this is a small detail predict' :: Ord a => SlopeOne' a -> Rating a -> Rating a predict' (SlopeOne' matrixIn) userRatings = M.mapMaybe calcItem (M.difference matrixIn userRatings) where calcItem innerMap | M.null combined = Nothing | norm_rating <= 0 = Nothing | otherwise = Just norm_rating where combined = M.intersectionWith weight innerMap userRatings (total_count,total_rating) = foldl1' addT (M.elems combined) norm_rating = total_rating / fromIntegral total_count weight (count,rating) user_rating = (count,rating + fromIntegral count * user_rating) userData :: [Rating String] userData = map M.fromList [ [("squid", 1.0), ("cuttlefish", 0.5), ("octopus", 0.2)], [("squid", 1.0), ("octopus", 0.5), ("nautilus", 0.2)], [("squid", 0.2), ("octopus", 1.0), ("cuttlefish", 0.4), ("nautilus", 0.4)], [("cuttlefish", 0.9), ("octopus", 0.4), ("nautilus", 0.5)] ] userInfo = M.fromList [("squid", 0.4),("cuttlefish",0.9),("dolphin",1.0)] predictions = predict (update empty userData) userInfo predictions' = predict' (update2 empty' userData) userInfo

## [edit] 2 More optimized storage

The changes to SlopeOne/update/predict below use a different internal data structure for storing the sparse matrix of SlopeOne. Instead of a Map of Map design it uses a Map of List design and keeps the List in distinct ascending form. The list values are a strict (data Tup) type which should help save space compared to the previous inner Map design and yet efficiently provide all the operations needed by update and predict.

Much of the logic of prediction is in the computeRating helper function.

-- The SlopeOne matrix is indexed by pairs of items and is implemented -- as a sparse map of distinct ascending lists. The 'update' and -- 'predict' functions do not need the inner type to actually be a -- map, so the list saves space and complexity. newtype SlopeOne item = SlopeOne (M.Map item [Tup item]) deriving (Show) -- Strict triple tuple type for SlopeOne internals data Tup item = Tup { itemT :: !item, countT :: !Count, ratingT :: !RatingValue } deriving (Show) empty :: SlopeOne item empty = SlopeOne M.empty update :: Ord item => SlopeOne item -> [Rating item] -> SlopeOne item update s@(SlopeOne matrixIn) usersRatingsIn | null usersRatings = s | otherwise = SlopeOne . M.unionsWith mergeAdd . (matrixIn:) . map fromRating $ usersRatings where usersRatings = filter ((1<) . M.size) usersRatingsIn -- fromRating converts a Rating into a Map of Lists, a singleton SlopeOne. fromRating userRatings = M.mapWithKey expand userRatings where expand item1 rating1 = map makeTup . M.toAscList . M.delete item1 $ userRatings where makeTup (item2,rating2) = Tup item2 1 (rating1-rating2) -- 'mergeAdd' is a helper for 'update'. -- Optimized traversal of distinct ascending lists to perform additive merge. mergeAdd :: Ord item => [Tup item] -> [Tup item] -> [Tup item] mergeAdd !xa@(x:xs) !ya@(y:ys) = case compare (itemT x) (itemT y) of LT -> x : mergeAdd xs ya GT -> y : mergeAdd xa ys EQ -> Tup (itemT x) (countT x + countT y) (ratingT x + ratingT y) : mergeAdd xs ys mergeAdd xs [] = xs mergeAdd [] ys = ys -- The output Rating has no items in common with the input Rating and -- only includes positively weighted ratings. predict :: Ord item => SlopeOne item -> Rating item -> Rating item predict (SlopeOne matrixIn) userRatings = M.mapMaybe (computeRating ratingList) (M.difference matrixIn userRatings) where ratingList = M.toAscList userRatings -- 'computeRating' is a helper for 'predict'. -- Optimized traversal of distinct ascending lists to compute positive weighted rating. computeRating :: (Ord item) => [(item,RatingValue)] -> [Tup item] -> Maybe RatingValue computeRating !xa@(x:xs) !ya@(y:ys) = case compare (fst x) (itemT y) of LT -> computeRating xs ya GT -> computeRating xa ys EQ -> helper (countT y) (ratingT y + fromIntegral (countT y) * snd x) xs ys where helper :: (Ord item) => Count -> RatingValue -> [(item,RatingValue)] -> [Tup item] -> Maybe RatingValue helper !count !rating !xa@(x:xs) !ya@(y:ys) = case compare (fst x) (itemT y) of LT -> helper count rating xs ya GT -> helper count rating xa ys EQ -> helper (count + countT y) (rating + ratingT y + fromIntegral (countT y) * (snd x)) xs ys helper !count !rating _ _ | rating > 0 = Just (rating / fromIntegral count) | otherwise = Nothing computeRating _ _ = Nothing