Co-occurrence Approach to an Item Based Recommender

Article
06/23/2012

For a while I thought I would tackle the problem of creating an item-based recommender. Firstly I will start with a local variant before moving onto a MapReduce version. The current version of the code can be found at:

https://code.msdn.microsoft.com/Co-occurrence-Approach-to-57027db7

The approach taken for the item-based recommender will be to define a co-occurrence matrix based on purchased items; products purchased on an order. As an example I will use the Sales data from the Adventure Works SQL Server sample database.. The algorithm/approach is best explained, and is implemented, using a matrix.

For the matrix implementation I will be using the Math.Net Numerics libraries. To install the libraries from NuGet one can use the Manage NuGet Packages browser, or run these commands in the Package Manager Console:

Install-Package MathNet.Numerics
Install-Package MathNet.Numerics.FSharp

Co-occurrence Sample

Lets start with a simple sample. Consider the following order summary:

Order ID	Product ID1	Product ID2	Product ID3
500001	1001	1003	1005
500002	1001	1002	1003
500003	1002	1003
500004	1003	1004	1005
500005	1003	1005
500006	1003	1004
500007	1002	1003
500008	1001	1003	1004
500009	1003	1004	1005

If you look at the orders containing products 1002, you will see that there is 1 occurrence of product 1001 and 3 occurrences of product 1003. From this, one can deduce that if someone purchases product 1002 then it is likely that they will want to purchase 1003, and to a lesser extent product 1001. This concept forms the crux of the item-based recommender.

So if we computed the co-occurrence for every pair of products and placed them into a square matrix we would get the following:

Product ID	1001	1002	1003	1004	1005
1001		1	3	1	1
1002	1		3
1003	3	3		4	4
1004	1		4		2
1005	1		4	2

There are a few things we can say about such a matrix, other than it will be square. More than likely, for a large product set the matrix will be sparse, where the number of rows and columns is equal to the number of items. Also, each row (and column as the matrix is symmetric) expresses similarities between an item and all others.

Due to the diagonal symmetric nature of the matrix (a_xy = a_yx) one can think of the rows and columns as vectors, where similarity between items X and Y is the same as the similarity between items Y and X.

The algorithm/approach I will be outlining will essentially be a means to build and query this co-occurrence matrix. Co-occurrence is like similarity, the more two items occur together (in this case in a shopping basket), the more they are probably related.

Working Data

Before talking about the code a word is warranted on how I have created the sample data.

As mentioned before I am using the Adventure Works database sales information. The approach I have taken is to export the sales detail lines, ordered by the sales order identifier, into flat files. The rationale for this being that the processing of the matrix can then occur with a low impact to the OLTP system. Also, the code will support the parallel processing of multiple files. Thus one can take the approach of just exporting more recent data and using the previously exported archived data to generate the matrix.

To support this process I have created a simple view for exporting the data:

CREATE VIEW [Sales].[vSalesSummary]
AS
    SELECT SOH.[SalesOrderID], CAST(SOH.[OrderDate] AS date) [OrderDate], SOD.[ProductID], SOD.[OrderQty]
    FROM [Sales].[SalesOrderHeader] SOH
    INNER JOIN [Sales].[SalesOrderDetail] SOD ON SOH.[SalesOrderID] = SOD.[SalesOrderID]
GO

I have then used the BCP command to export the data into a Tab delimited file:

bcp
  "SELECT [SalesOrderID], [OrderDate], [ProductID], [OrderQty] FROM [AdventureWorks2012].[Sales].[vSalesSummary] ORDER BY [SalesOrderID], [ProductID]"
  queryout SalesOrders.dat
  -T -S (local) -c

One could easily define a similar process where a file is generated for each month, with the latest month being updated on a nightly basis. One could then easily ensure only orders for the past X months are including in the metric. The only important aspect in how the data is generated is that the output must be ordered on the sales order identifier. As you will see later, this grouping is necessary to allow the co-occurrence data to be derived.

A sample output from the Adventure Works sales data is as follows, with the fields being tab delimited:

43659    2005-07-01    776    1
43659    2005-07-01    777    3
43659    2005-07-01    778    1
43660    2005-07-01    758    1
43660    2005-07-01    762    1
43661    2005-07-01    708    5
43661    2005-07-01    711    2
43661    2005-07-01    712    4
43661    2005-07-01    715    4
43661    2005-07-01    716    2
43661    2005-07-01    741    2
43661    2005-07-01    742    2
43661    2005-07-01    743    1
43661    2005-07-01    745    1
43662    2005-07-01    730    2
43662    2005-07-01    732    1
43662    2005-07-01    733    1
43662    2005-07-01    738    1

Once the data has been exported building the matrix becomes a matter of counting the co-occurrence of products associated with each sales order.

Building the Matrix

The problem of building a co-occurrence matrix is simply one of counting. The basic process is as follows:

Read the file and group each sales order along with the corresponding list of product identifiers
For each sales order product listing, calculate all the corresponding pairs of products
Maintain/Update the running total of each pair of products found
At the end of the file, output a sparse matrix based on the running totals of product pairs

To support parallelism steps 1-3 are run in parallel where the output of these steps will be a collection of the product pairs with the co-occurrence count. These collections are then combined to create a single sparse matrix.

In performing this processing it is important that the reading of the file data is such that it can efficiently create a sequence of products for each sales order identifier; the grouping key. If you have been following my Hadoop Streaming blog entries one will see that this is the same process undertaken in processing data within a reducer step.

The matrix building code, in full, is as follows:

module MatrixBuilder =
// Configuration values
let qtyMaximum = 5.0 // Maximum rating contribution for an item
let entryThreshold = 20.0 // Minimum correlations for matrix inclusion
let recentFactor = 2.0 // Quantity increase factor for recent items
let baseDate = DateTime.Today.AddMonths(-3) // Date for a recent item
let matrixSize = 10000*500 // Products*Correlations for Hash Table init
// Gets the base data for building the sparse matrix
let private getMatrixData filename =
let minItem = ref Int32.MaxValue
let maxItem = ref 0
let rcTable = Dictionary<(int*int), float>(matrixSize)
// Define file reader properties
use reader = new StreamReader(Path.GetFullPath(filename))
let getInput() =
if reader.EndOfStream then
None
else
let input = reader.ReadLine()
if not (String.IsNullOrEmpty(input)) then
Some(Helpers.ParseInputData input)
else
None
// calculates the pairs for an order
let rec pairs items = seq {
match items with
| head::tail -> for item in tail do yield head, item
yield! pairs tail
| _ -> ()
}
// Add a factor based on order properties - example recent orders
let orderFactor (order:OrderHeader) =
if DateTime.Compare(order.OrderDate, baseDate) > 0 then
recentFactor
else
1.0
// Adds to the table
let addRowColumn idx1 idx2 qty =
minItem := min (min idx1 idx2) !minItem
maxItem := max (max idx1 idx2) !maxItem
if rcTable.ContainsKey (idx1, idx2) then
rcTable.[(idx1, idx2)] <- rcTable.[(idx1, idx2)] + qty
else
rcTable.[(idx1, idx2)] <- qty
()
// The main processor for the sequence for an order
let processOrder (header:OrderHeader) (orders:OrderDetail seq) =
let orderList = Seq.toList orders
// if order only has one product then no correlation can be determined
if (orderList.Length > 1) then
pairs orderList
|> Seq.iter (fun (order1, order2) ->
let qty = (min (float (max order1.OrderQty order2.OrderQty)) qtyMaximum) * (orderFactor header)
addRowColumn order1.ProductId order2.ProductId qty
addRowColumn order2.ProductId order1.ProductId qty)
// Creates a sequence of the input based on the provided orderId
let lastInput = ref None
let continueDo = ref false
let inputsByKey key firstValue = seq {
// Yield any value from previous read
yield firstValue
continueDo := true
while !continueDo do
match getInput() with
| Some(orderDetail) when orderDetail.OrderId = key ->
// Yield found value and remainder of sequence
yield orderDetail
| Some(orderDetail) ->
// Have a value but different key
lastInput := Some(orderDetail)
continueDo := false
| None ->
// Have no more entries
lastInput := None
continueDo := false
}
// Controls the calling of the matrix maker
let rec processInput (input:OrderDetail option) =
match input with
| Some(orderDetail) ->
let header = {OrderHeader.OrderId = orderDetail.OrderId; OrderDate = orderDetail.OrderDate}
processOrder header (inputsByKey orderDetail.OrderId orderDetail)
processInput lastInput.contents
| _ -> ()
// Build the matrix/table from the input data
processInput (getInput())
// return the table defintion along with the size
(!minItem, !maxItem, rcTable)
/// Build a Sparse matrix from the file data
let GetMatrixParallel (filenames:string array) =
// In Parallel gets the RC tables
let results =
filenames
|> Array.Parallel.map (fun filename -> getMatrixData filename)
// define the max sparse array size
let (minItem, maxItem) =
results
|> Array.fold (fun acc (idxMin, idxMax, _) ->
(min idxMin (fst acc), max idxMax (snd acc))) (0, 0)
let offset = minItem
let size = maxItem + 1 - minItem
// convert to a sparse matrix and return
let items = seq {
for (_, _, rcTable) in results do
for item in rcTable do
if item.Value > entryThreshold then
yield ((fst item.Key) - offset, (snd item.Key) - offset, item.Value)
}
(offset, SparseMatrix.ofSeq size size items)
/// Interface for a single file
let GetMatrix (filename:string) =
let (minItem, maxItem, rcTable) = getMatrixData filename
let offset = minItem
let size = maxItem + 1 - minItem
// convert to a sparse matrix and return
let items = seq {
for item in rcTable do
if item.Value > entryThreshold then
yield ((fst item.Key) - offset, (snd item.Key) - offset, item.Value)
}
(offset, SparseMatrix.ofSeq size size items)

In this instance the file data is processed such that order data is grouped and exposed as an OrderHeader type, along with a collection of OrderDetail types. From this the pairs of products are easily calculated; using the pairs function.

To maintain a co-occurrence count for each product pair a Dictionary is used. The key for the Dictionary is a tuple of the pair of products, with the value being the co-occurrence count. The rationale for using a Dictionary is that it supports O(1) lookups; whereas maintaining an Array will incur an O(n) lookup.

The final decision to be made is of what quantity should be added into the running total. One could use a value of 1 for all co-occurrences, but other options are available. The first is the order quantity. If one has a co-occurrence for products quantities x and y, I have defined the product quantity as (max x y). The rationale for this is that having product quantities is logically equivalent to having multiple product lines in the same order. If this was the case then the co-occurrence counting would arrive at the same value. However I have treated only the maximum amount as the contributing factor. I have also limited the maximum quantity amount such that small products ordered in large volumes do not skew the numbers; such as restricting a rating from 1-5.

One optional quantity factor I have included is one based on the order header. The sample code applies a quantity scaling factor for recent orders. Again the rationale for this is such that recent orders have a greater affect on the co-occurrence values over older ones. All these scaling factors are optional and should be configured to give your desired results.

As mentioned, to achieve parallelism in the matrix building code the collection of input files can be processed in parallel with each parallel step independently outing its collection of product pairs and co-occurrence counts. This is achieved using an Array.Parallel.map function call. This maps each input file into the Dictionary for the specified file. Once all the Dictionary elements have been created they are then used to create a sequence of elements to create the sparse matrix.

One other critical element returned from defining the Dictionary is the maximum product identifier. It is assumed that the product identifier is an integer which is used as an index into the matrix; for both row and column values. Thus the maximum value is needed to define row and columns dimensions for the sparse matrix.

Whereas this approach works well for products ranges from 1 onwards, what if product identifiers are defined from a large integer base, such as 100,000, or are Guid’s. In the former case one has the option of calculating an offset such that the index for the matrix is the product identifier minus the starting offset; this being the approach I have taken. The other option is that the exported data is mapped such that the product numbers are defined from 1 onwards; again this can be a simple offset calculation. In the latter case for Guid’s, one would need to do a mapping from the Guid keys to a sequential integer.

Querying for Item Similarity

So once the sparse matrix has been built how does one make recommendations. There are basically two options, either recommendations based on a single product selection, or recommendations for multiple products, say based on the contents of a shopping basket.

The basic process for performing the recommendations, in either case, is as follows:

Select the sparse row vectors that correspond to the products for which a recommendation is required
Place the row values into a PriorityQueue where the key is the co-occurrence count and the value the product identifier
Dequeue and return, as a sequence, the required number of recommendations from the PriorityQueue

For the case of a single product the recommendations are just a case of selecting the correct single vector and returning the products with the highest co-occurrence count. The process for a selection of products is almost the same, except that multiple vectors are added into the PriorityQueue. One just needs to ensure that products that are already in the selection on which the recommendation is being made are excluded from the returned values. This is easily achieved with a HashSet lookup.

So the full code that wraps building the recommendation is as follows:

type MatrixQuery (filenames:string array) =
let defaultRecommendations = 12
let (offset, coMatrix) =
match filenames with
| [||] -> invalidArg "filename" "Filename cannot be an empty Array"
| [| filename |] -> MatrixBuilder.GetMatrix(filename)
| _ -> MatrixBuilder.GetMatrixParallel(filenames)
let getQueueSM (products:int array) =
// Define the priority queue and lookup table
let queue = PriorityQueue(coMatrix.ColumnCount)
let lookup = HashSet(products)
// Add the items into a priority queue
products
|> Array.iter (fun item ->
let itemIdx = item - offset
if itemIdx >= 0 && itemIdx < coMatrix.ColumnCount then
seq {
for idx = 0 to (coMatrix.ColumnCount - 1) do
let productIdx = idx + offset
if (not (lookup.Contains(productIdx))) && (coMatrix.[itemIdx, idx] > 0.0) then
yield KeyValuePair(coMatrix.[itemIdx, idx], productIdx)
}
|> queue.Merge)
// Return the queue
queue
let getItems (queue:PriorityQueue<float, int>) (items:int) =
let toDequeue = min items queue.Count
seq { for i in 1 .. toDequeue do yield queue.Dequeue().Value }
new(filename:string) =
MatrixQuery([| filename |])
/// Get the requested number of Recommendations for a Product
member self.GetRecommendations(product:int, items:int) =
let queue = getQueueSM([| product |])
getItems queue items
/// Get the requested number of Recommendations for a Product Array
member self.GetRecommendations(products:int array, items:int) =
let queue = getQueueSM(products)
getItems queue items
/// Get the default number of Recommendations for a Product
member self.GetRecommendations(product:int) =
self.GetRecommendations(product, defaultRecommendations)
/// Get the default number of Recommendations for a Product Array
member self.GetRecommendations(products:int array) =
self.GetRecommendations(products, defaultRecommendations)

Using the code is a simply a matter of creating the MatrixQuery type, with the files to load, and then calling the GetRecommendations() operator for the required products (shopping basket):

let filenames = [|
    @"C:\DataExport\SalesOrders201203.dat";
    @"C:\DataExport\SalesOrders201204.dat";
    @"C:\DataExport\SalesOrders201205.dat";
    @"C:\DataExport\SalesOrders201206.dat" |]

let itemQuery = MatrixQuery(filenames)
let products = itemQuery.GetRecommendations([| 860; 870; 873 |], 25)

In extracting the row vector associated with the required product one could just have used coMatrix.Row(item); but this creates a copy of the vector. To avoid this the code just does an enumeration of the required matrix row. Internally the sparse vector maintains three separate arrays for the column and row offsets and sparse value. Using these internal arrays and the associated element based operations would speed up obtaining recommendations; but currently these properties are not exposed. If one uses the sparse matrix from the F# power Pack then one can operate in this fashion using the following code:

let getQueueSM (products:int array) =
// Define the priority queue and lookup table
let queue = PriorityQueue(coMatrix.NumCols)
let lookup = HashSet(products)
// Add the items into a priority queue
products
|> Array.iter (fun item ->
let last = coMatrix.InternalSparseRowOffsets.Length - 1
if item >= 0 && item <= last then
let (startI, endI) =
if item = last then (coMatrix.InternalSparseRowOffsets.[item], coMatrix.InternalSparseRowOffsets.[item])
else (coMatrix.InternalSparseRowOffsets.[item], coMatrix.InternalSparseRowOffsets.[item + 1] - 1)
seq {
for idx = startI to endI do
if not (lookup.Contains(idx)) then
let product = coMatrix.InternalSparseColumnValues.[idx]
yield KeyValuePair(-coMatrix.InternalSparseValues.[idx], product)
}
|> queue.Merge
)
// Return the queue
queue

In considering a shopping basket I have assumed that each product has been selected only once. You could have the situation where one wanted to take into consideration the quantity of each product selected. In this instance one would take the approach of performing a scalar multiplication of the corresponding product vectors. What this would achieve is, for recommendations, prioritizing products for which a user has purchased multiple items.

Although this code does a lot of element-wise operations, as mentioned earlier, one can think of the operations in pure matrix terms. In this case one would just multiply the sparse matrix by a sparse column vector representing the items from which to make the recommendations.

Consider the previous sparse matrix example, and say one had a basket consisting of 2 products 1002 and 1 product 1004:

Product ID	1001	1002	1003	1004	1005		Product ID	Qty		Product ID	Rec
1001		1	3	1	1		1001			1001	3
1002	1		3				1002	2		1002
1003	3	3		4	4	×	1003		=	1003	10
1004	1		4		2		1004	1		1004
1005	1		4	2			1005			1005	2

In this case it should be easy to see the recommendation for products should be 1003, 1001, and lastly 1005.

Conclusion

The approach mentioned here will probably work well for the cases where one has 100,000’s products with similarities between 1000’s of items, with 10’s millions of orders to be considered. For most situations this should suffice. However, if you are not in this bracket, in my next post, I will show how this approach can be mapped over to Hadoop and MapReduce, allowing for even greater scale.

Also, in a future post I will port the code to use the Cloud Numerics implementation of matrices.

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