PROJECT 2 K-MEANS CLUSTERING & THE IRIS PLANT DATASET Monique Kirkman-Bey ELEN-857: Advanced Pattern Recognition Methods
OCTOBER 7, 2015
Kirkman-Bey Project 2
Abstract In this work the K-means clustering algorithm is applied to Fisher’s Iris Plant Dataset. The dataset is known to include 3 classes of Iris plant data – Setosa, Virginica, and Versicolor - one of which is linearly separable form the other two. To assess the capabilities of the clustering algorithm, it is applied to the dataset with varied number of initial centers, and stopping thresholds. It will be shown that the K-Means Clustering algorithm is capable of perfectly separating the Setosa dataset from the other two, as expected, and able to achieve acceptable recognition of the other two plant species.
Methodology The K-means algorithm is an unsupervised algorithm that attempts to cluster data into groups based on a chosen similarity measure. In this work, the similarity measure of choice is Euclidean distance. To create the clusters, the K-Means algorithm iteratively implements the following steps: 1.
Initialize - Initialize the center of each cluster.
2.
Distribute data points - Assign each data point to the cluster whose center is the smallest
3.
distance from the data point. Compute new cluster centers – Set the position of the cluster center to the mean of all data points belonging to that cluster.
4.
Compare new centers to old center – If the new centers are the same as the old centers, then
the algorithm converges. The clusters and ce nters computed in step 3 are the final clusters and centers. If they are not the same, return to step 2. In reality, it may not always be possible to find centers that do not change from iteration to iteration. In other words, this algorithm may not always lead to a perfect solution. Some datasets may lead to centers that oscillate between two values, for example. So, to avoid an infinite loop when iterating through the algorithm, the threshold is used as another stopping condition. When the new cluster centers are identified at the end of each iteration, the amount of change in the clusters is also t aken into consideration. This is done by measuring the distance between the new centers and old centers. If this distance is less than the threshold distance, the algorithm converges.
Experimental Setup This project was broken up into two tasks. First, the K-means algorithm was coded into a general function so that the number of cente rs and threshold value could be easily varied. Next, a shell to call the function iteratively for each of the three k values and two threshold values was created. The results of each run was saved to a cell array. The K-means code from homework 2 was modified and made to be a more general K-means function. The function can be reviewed in Appendix A. In addition to making the function more general, the convergence test step was modified to include the threshold as a stopping condition for the algorithm. So, the function takes in the following inputs: data, number of centers, and threshold value. Given these parameters, the Kmeans.m function will return the following two cells: centers per iteration, assigned classes per iteration. More in-depth consideration of pertinent steps are presented below. 1
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Initializing the centers An important consideration in the K-means algorithm is the choice in initial centers. Since this is an unsupervised algorithm, I chose to use sample values as the initial centers. However, instead of using the first k sample values, I decided to choose the centers at random for each iteration. This was achieved using the code snippet below. Z = x(randi([1,numSamples],centers,1),:);
%k initial centers
Implementing the threshold as a stopping condition To avoid an infinite loop when the Kmeans function is called, step 4 of the algorithm was modified to include the threshold as a stopping condition. First, the new centers are compared to the old centers. If the new centers do not match the old centers, then the distance between them is computed using the norm function as shown in the code snippet below. This difference is compared to the threshold value. If the distance is less than the threshold, the algorithm converges. This is summarized in the code snippet below. case 4
%Step 4: compare new centers to old centers if Z_iter{m+1} == Z_iter{m} %if new center = current center NotEq = 0; %algorithm converges break; elseif m == 1 %if 1st iter, no prev distance, just proceed m = m+1; step = 2; else %if not 1st iter and Z \= Znew %check stopping conditions if abs(Dist_iter{m} - Dist_iter{m-1}) < threshold NotEq = 0; break; else m = m+1; %new iteration step = 2; %go back to step 2 end
end
Confusion Matrix After calling the Kmeans function, the confusion matrix was generated for each simulation using the Matlab confusionmat function. The results can be seen in the subsequent section.
Results As previously stated, K-means clustering was applied six times to the dataset. The results are broken up into two groups and presented based on the chosen threshold value. 2
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Threshold = 0.01 Below, the confusion matrix for the t hree different choices of number of initial centers is shown. In each simulation, the stopping threshold was set to 0.01. Table 1. K = 2 Confusion Matrix
Setosa
1 50
2 0
Versicolor
3
Virginica
0
Table 2. K = 3 Confusion Matrix
1
2
3
Setosa
0
50
0
47
Versicolor
2
0
50
Virginica
36
0
Table 3
1
2
3
4
Setosa
50
0
0
0
48
Versicolor
0
0
21
29
14
Virginica
0
23
26
1
It can be seen that in each case, the Setosa plant species was easily separated from the others. However, the Versicolor and Virginica datasets we re not as easily distinguished from each other as they were from the Setosa. However, it is interesting to see that when there were just 2 centers, the Virginica dataset was able to be perfectly separated from the others. The Versicolor. However is still straddling between the two clusters. It is mostly clustered with the Virginica dataset, but there are several pieces that were clustered with the Setosa plants.
Threshold = 0.1 Below, the confusion matrix for the t hree different choices of number of initial centers is shown. In each simulation, the stopping threshold was set to 0.1. Table 4. K = 2 Confusion Matrix
Setosa
1 0
2 50
Versicolor
47
Virginica
50
Table 5. K = 3 Confusion Matrix
1
2
3
Setosa
0
50
0
3
Versicolor
47
0
0
Virginica
14
0
Table 6
1
2
3
4
Setosa
50
0
0
0
3
Versicolor
0
25
25
0
36
Virginica
0
17
1
32
Increasing the threshold did not have much of an impact on the final confusion matrices. Although there is some shifting of the data points, as is e videnced by the values shown in the tables, the overall clustering results are quite similar. In all three runs, the Setosa species was perfectly separated from the other two species. The other two species, on average could not be perfectly separated. However, when there are two centers, the Setosa and Virginica sets are again easily separated from one another while the Versicolor is split (unevenly) between the two clusters.
Conclusions Using Matlab and a personal computer, the K-means algorithm was applied to the Iris plant dataset. It was shown that the Setosa dataset was able to be perfectly classified in each case. The other two species – Versicolor and Virginica - were not as easily separated from each other as they were from the Setosa plant. After randomly selecting the initial centers , varying the number of centers, and manipulating the stopping threshold, these results remained true. Since these results are typical of the Iris plant dataset and the recognition using the K -means clustering algorithm was able to reach these results, the K-means algorithm was shown to be a reliable method of clustering.
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Appendix A - Kmeans.m %Monique Kirkman-Bey %This function takes in a dataset and number of clusters and returns %the clustered data after applying the K-means algorithm %inputs: data, number of clusters, threshold %output: clustered data function [clusCenters,clusData] = Kmeans (x,cen,t) centers=cen; %number of centers threshold = t; numSamples = size(x,1); %number of samples sampleLength = size(x,2); %dimension of samples Dist = zeros(numSamples,centers); %array to hold distances Class = zeros(numSamples,1); %array to hold classes Znew = zeros(centers,sampleLength); %array to hold new centers %step 1 Z = x(randi([1,numSamples],centers,1),:); m=1; Z_iter{m} = Z; step = 2; NotEq = true;
%k initial centers
%save Z values
while NotEq switch step case 2 %distribute samples to clusters Z = Z_iter{m}; %grab current centers for k = 1:centers %for each center for N = 1:numSamples %for each sample %compute distance between sample and center Dist(N,k) = norm(x(N,:)-Z(k,:)); end end Dist_iter{m} = norm(Dist); for N = 1:numSamples %for each sample minDist = min(Dist(N,:)); %get min dist for sample [i,j] = find(Dist(N,:) == minDist); %index of min Class(N) = j(1); %index=class, save index/class end Class_Iter{m} = Class;
%save class assignments for m
step = 3; case 3 %compute new centers for k = 1:centers%for each center 4
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C = find(Class == k); %find all samples in class zt = [0,0]; for i = 1:size(C,1) %for every sample in class zt = zt + x(C(i),:); %add sample to sum end Znew(k,:) = zt/size(C,1);
%center = sample mean
end Z_iter{m+1} = Znew; step = 4;
%save next centers
case 4 %compare new centers to old centers if Z_iter{m+1} == Z_iter{m} %if new = current NotEq = 0; %algorithm converges break; elseif m == 1 %if 1st iter, no prev distance, proceed m = m+1; step = 2; else %if not 1st iter and Z \= Znew %check stopping conditions if abs(Dist_iter{m} - Dist_iter{m-1}) < threshold NotEq = 0; break; else m = m+1; %new iteration step = 2; %go back to step 2 end end end end clusCenters = Z_iter; %return cluster centers per iteration clusData = Class_Iter; %return clusters per iteration end
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Appendix B - Proj2Shell.m %Monique Kirkman-Bey %Pattern Recognition Project 2 Shell %October 6, 2015 %This program loads the iris data set, then iteratively calls the kmeans %function to implement the kmeans clustering algorithm for k values of 2, %3, and 4 cluster center using different threshold values. clear; iris = csvread('iris.csv'); %load dataset x = iris(:,1:end-1); %get all but the class value y = iris(:,end); %set simulation parameters k = [2 3 4]; %number of centers t = [0.01 0.1]; %stopping threshold for i = 1:size(k,2) for j = 1:size(t,2) [centers, clusters] = Kmeans(x,k(i),t(j)); CONF{i,j} = confusionmat(y,clusters{end}); end end
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