## ChiMerge discretization algorithm

November 2, 2010

Chi merge is a simple algorithm that uses the chi-square statistic to discretize numeric attributes. It is a supervised, bottom-up data discretization method. It checks each pair of adjacent rows in order to determine if the class frequencies of the two intervals are significantly different. It tests the following hypothesis: that the two adjacent intervals are independent. If the hypothesis is confirmed the intervals are merged into a single interval, if not, they remain separated.

In this problem we select one of the following as an attribute: sepal length, sepal width, petal length, and petal width. Three Classes are available which are Iris-setosa, Iris-versicolor, Iris-virginica.

Chi merge algorithm is described below:

- Set the values for minintervals and maxintervals in order not to result in a large number or small number of intervals. Choose minintervals and maxintervals such that:

**2 ≤ minintervals < maxintervals**

** **

- Look up the threshold value to use based on the significance level and the number of degrees of freedom. The threshold value can be found in statistical tables depending on two factors:

- The significance level, the higher the significance level, the higher the threshold value and the more likely the adjacent intervals will be merged.
- The number of degrees of freedom=(2-1)*([number of classes]-1) = [number of classes] -1

Here we have 3 classes thus the number of degrees of freedom is 2.

- A) Sort the attributes into ascending numerical order.

B) Create a frequency table containing one row for each distinct attribute value and one column for each class. Enter the number of occurrences of each distinct value of the attribute for each possible class in the cells of the table. Then generate a set of distinct intervals. Set the interval lower bound equal to the attribute value (inclusive) that belongs to this interval, and set its upper bound to the attribute value (exclusive) belonging to the next interval.

C) If the number of rows = minintervals then stop otherwise go to next step.

D) For each pair of adjacent rows in the frequency table calculate e the expected frequency value for that combination from the product of row and column sums divided by the total number of occurrences in the two rows combined. The expected value is the frequency value that would be expected to occur by chance given the assumption of independence. Calculate the value of (O – E) ^{2}/E (if E<0.5, replace E by 0.5) where O is the observed frequency value for that combination, then add the values obtained to give chi ^{2} for that pair of adjacent rows. No chi^{2} is calculated for the final interval because there is not one below it.

E) Chi merge selects the smallest value of chi ^{2}.This value is compared with a threshold, the two intervals are merged based on the following condition if chi2 is less than threshold or if the number of rows > maxintervals, if the condition is met then the two intervals are merged and the attribute for the merged row to that of the first of the two consistent rows and the upper interval bound to the upper bound interval of the next attribute. Then go to step C, but if the condition is not met then the algorithm stops.

The chi^{2} values are calculated for the revised frequency table, chi merge proceeds iteratively in this way merging two intervals at each stage until the chi^{2} for the remaining pairs of intervals are greater than the threshold value and the number of intervals is less than the maximum number of intervals, hence no further merging of intervals is possible and the discretisation is complete.

Check these ChiMerge powerpoint slides that visualizes the above algorithm.

Now lets take the IRIS data set, obtained from http://www.ics.uci.edu/~mlearn/MLRepository.html (UC-Irvine Machine Learning Data Repository), as a data set to be discretized. We will perform data discretization for each of the four numerical attributes using the Chi-Merge method having the stopping criteria be: **max-interval = 6**.

5.1,3.5,1.4,0.2,Iris-setosa 4.9,3.0,1.4,0.2,Iris-setosa 4.7,3.2,1.3,0.2,Iris-setosa 4.6,3.1,1.5,0.2,Iris-setosa 5.0,3.6,1.4,0.2,Iris-setosa 5.4,3.9,1.7,0.4,Iris-setosa 4.6,3.4,1.4,0.3,Iris-setosa 5.0,3.4,1.5,0.2,Iris-setosa 4.4,2.9,1.4,0.2,Iris-setosa 4.9,3.1,1.5,0.1,Iris-setosa 5.4,3.7,1.5,0.2,Iris-setosa 4.8,3.4,1.6,0.2,Iris-setosa 4.8,3.0,1.4,0.1,Iris-setosa

These attributes refer to Sepal Length, Sepal Width, Petal Length,Petal Width & Class respectively. Lets create two classes that i’ll call “IRISDataItem” and “IRISWorkItem”. The former will be used to store the attribute values for each when the data file is read while the latter will be used for later processing and storing the ChiSquare for each distinct attribute.

Public Class IrisDataItem Public Enum IRISClass As Byte IrisSetosa = 1 IrisVersicolour = 2 IrisVirginica = 3 End Enum Public Property SepalLength As Single Public Property SepalWidth As Single Public Property PetalLength As Single Public Property PetalWidth As Single Public Property [Class] As IRISClass End Class Public Class IrisWorkItem Public Property Attribute As Decimal Public Property Frequency As Integer Public Property SetosaCount As Integer Public Property VersicolorCount As Integer Public Property VirginicaCount As Integer Public Property Start As Decimal Public Property [End] As Decimal Public Property ChiSquare As Decimal 'Operator overloading for merging of two adjacent items Public Shared Operator ^(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As IrisWorkItem Item1.SetosaCount += Item2.SetosaCount Item1.VirginicaCount += Item2.VirginicaCount Item1.VersicolorCount += Item2.VersicolorCount Item1.End = Item2.End Item2.Frequency = -1 Return Item1 End Operator 'Operator overloading for calculating the ChiSqaure for two items Public Shared Operator &(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As Decimal Return ChiMerge.ChiSquare(Item1, Item2) End Operator End Class

Our next task is to actually read the data file and fill create a List(Of IRISDataItem) object. This can be easily done by reading the file from disc, or you can embed the file in the application resources and read from there. Here is a couple of functions that handle the two ways of reading the file:

Private Function ReadDataFile(ByVal Path As String) As StringCollection If Not String.IsNullOrWhiteSpace(Path) Then Dim reader As New IO.StreamReader(Path) Dim data As New StringCollection While Not reader.EndOfStream data.Add(reader.ReadLine) End While reader.Close() Return data Else Throw New Exception("Invalid IRIS File Path!") End If End Function Private Function ReadDataFileFromResource() As StringCollection Dim strFile As String = System.Text.Encoding.ASCII.GetString(My.Resources.iris) Dim reader As New IO.StringReader(strFile) Dim data As New StringCollection While reader.Peek > 0 data.Add(reader.ReadLine) End While reader.Close() Return data End Function Public Shared Function CreateIrisDataItems(ByRef StringItems As StringCollection) As List(Of IrisDataItem) If StringItems IsNot Nothing AndAlso StringItems.Count > 0 Then Dim items As New List(Of IrisDataItem) For Each i In StringItems If Not String.IsNullOrWhiteSpace(i) Then Dim item As New IrisDataItem Dim strItem() As String = i.Split(",") item.SepalLength = strItem(0) item.SepalWidth = strItem(1) item.PetalLength = strItem(2) item.PetalWidth = strItem(3) Select Case strItem(4) Case "Iris-setosa" item.Class = IrisDataItem.IRISClass.IrisSetosa Case "Iris-versicolor" item.Class = IrisDataItem.IRISClass.IrisVersicolour Case "Iris-virginica" item.Class = IrisDataItem.IRISClass.IrisVirginica End Select items.Add(item) End If Next Return items End If Return Nothing End Function Public Shared Function CreateIrisWorkItems(ByVal AttributeIndex As Integer, ByVal DataItems As List(Of IrisDataItem)) As List(Of IrisWorkItem) Dim workItems As New List(Of IrisWorkItem) DataItems.ForEach(Sub(dataItem) Dim workItem As IrisWorkItem = Nothing Dim value As Decimal Select Case AttributeIndex Case 0 workItem = workItems.Find(Function(w) w.Attribute = dataItem.SepalLength) value = dataItem.SepalLength Case 1 workItem = workItems.Find(Function(w) w.Attribute = dataItem.SepalWidth) value = dataItem.SepalWidth Case 2 workItem = workItems.Find(Function(w) w.Attribute = dataItem.PetalLength) value = dataItem.PetalLength Case 3 workItem = workItems.Find(Function(w) w.Attribute = dataItem.PetalWidth) value = dataItem.PetalWidth End Select If workItem IsNot Nothing Then workItem.Frequency += 1 Else workItem = New IrisWorkItem workItem.Frequency = 1 workItems.Add(workItem) End If workItem.Attribute = value Select Case dataItem.Class Case IrisDataItem.IRISClass.IrisSetosa workItem.SetosaCount += 1 Case IrisDataItem.IRISClass.IrisVersicolour workItem.VersicolorCount += 1 Case IrisDataItem.IRISClass.IrisVirginica workItem.VirginicaCount += 1 End Select End Sub) Return workItems.OrderBy(Function(w) w.Attribute).ToList End Function

Having the data ready in our hands, we can now proceed to implement the ChiSquare function which is basically an implementation of the formula:

*Where : *

*k* = number of classes

*A _{ij}* = number of samples in

*i*th interval, jth class

*E _{ij}* = expected frequency of

*A*= (

_{ij}*R**

_{i}*C*) /

_{j}*N*

*R _{i}* = number of samples in

*i*th interval

*C _{j}* = number of samples in

*j*th class

*N* = total number of samples on the two intervals

If Eij=<0.5 then set Eij to 0.5

Public Shared Function ChiSquare(ByVal Item1 As IrisWorkItem, ByVal Item2 As IrisWorkItem) As Decimal ChiSquare = 0 Dim table(,) As Integer = {{Item1.SetosaCount, Item1.VersicolorCount, Item1.VirginicaCount}, {Item2.SetosaCount, Item2.VersicolorCount, Item2.VirginicaCount}} Dim rowSums() As Decimal = {Item1.SetosaCount + Item1.VersicolorCount + Item1.VirginicaCount, Item2.SetosaCount + Item2.VersicolorCount + Item2.VirginicaCount} Dim columnSums() As Decimal = {Item1.SetosaCount + Item2.SetosaCount, Item1.VersicolorCount + Item2.VersicolorCount, Item1.VirginicaCount + Item2.VirginicaCount} Dim totalSum As Decimal = rowSums.Sum Dim E As Decimal = 0 For i As Integer = 0 To table.GetLength(0) - 1 For j As Integer = 0 To table.GetLength(1) - 1 E = rowSums(i) * columnSums(j) / totalSum If E > 0 Then ChiSquare += ((table(i, j) - E) ^ 2) / IIf(E < 0.5, 0.5, E) Next Next End Function

One last thing to do before we implement Chi-Merge algorithm is to setup the initial interval bounds and prepare them. This is easily achieved by taking the attribute value itself to be the lower bound of the interval and the next attribute value to be the upper bound of the interval. Below is an easy way of doing this:

Public Shared Sub CreateIntervalBounds(ByVal Items As List(Of IrisWorkItem)) For i As Integer = 0 To Items.Count - 2 Items(i).Start = Items(i).Attribute Items(i).End = Items(i + 1).Attribute Next Items(Items.Count - 1).Start = Items(Items.Count - 1).Attribute Items(Items.Count - 1).End = Items(Items.Count - 1).Attribute + 1 End Sub

Finally, we are ready to implement the Chi-Merge algorithm. One thing to note about the below implementation is the use of LINQ to Objects and Lambda Expressions to filter out tuples that need to be dropped and merged. So you could probably that the code below will compile only using **Visual Studio 2010 and .Net Framework 4.0**. I haven’t tested it on Visual Studio 2008 Sp1 but if you do let me know if it compiles well.

Public Sub Compute(ByRef Items As List(Of IrisWorkItem)) For i As Integer = 0 To Items.Count - 2 Items(i).ChiSquare = Items(i) & Items(i + 1) 'calculate chiSquare Next Dim minChiSquare As Decimal = Items.Take(Items.Count - 1).Min(Function(x) x.ChiSquare) If minChiSquare < THRESHOLD Then Dim i As Integer = 0 While i < Items.Count If Items(i).ChiSquare = minChiSquare Then Items(i) ^= Items(i + 1) 'merge Dim j As Integer = i i += 1 While i < Items.Count - 1 AndAlso Items(i).ChiSquare = minChiSquare Items(j) ^= Items(i + 1) 'merge i += 1 End While End If i += 1 End While Items.RemoveAll(Function(x) x.Frequency = -1) If Items.Count >= MIN_INTERVAL Then Compute(Items) Else If Items.Count > MAX_INTERVAL Then THRESHOLD += 0.5 Compute(Items) End If End If End Sub

Once you run the algorithm, you can compare the obtained intervals and split points with the ones found on the internet. Below are my final results compared to the results on http://sci2s.ugr.es/keel/pdf/algorithm/congreso/1992-Kerber-ChimErge-AAAI92.pdf