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Can u answer my question, can u show me step by step , i dont understanding Con...
Jun 14, 2024
Can u answer my question, can u show me step by step , i dont understanding
Solution by Steps
step 1
Identify the initial cluster means: m1=(1,1)m_{1}=(1,1) and m2=(3,3)m_{2}=(3,3)
step 2
Calculate the Euclidean distance from each point to the cluster means. The Euclidean distance formula is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
step 3
Compute distances for each point to m1m_{1} and m2m_{2}: - Point 1: d((1,1),(1,1))=0d((1,1), (1,1)) = 0, d((1,1),(3,3))=(3−1)2+(3−1)2=8d((1,1), (3,3)) = \sqrt{(3-1)^2 + (3-1)^2} = \sqrt{8} - Point 2: d((1,2),(1,1))=(1−1)2+(2−1)2=1d((1,2), (1,1)) = \sqrt{(1-1)^2 + (2-1)^2} = 1, d((1,2),(3,3))=(3−1)2+(3−2)2=5d((1,2), (3,3)) = \sqrt{(3-1)^2 + (3-2)^2} = \sqrt{5} - Point 3: d((1,3),(1,1))=(1−1)2+(3−1)2=2d((1,3), (1,1)) = \sqrt{(1-1)^2 + (3-1)^2} = 2, d((1,3),(3,3))=(3−1)2+(3−3)2=2d((1,3), (3,3)) = \sqrt{(3-1)^2 + (3-3)^2} = 2 - Point 4: d((2,1),(1,1))=(2−1)2+(1−1)2=1d((2,1), (1,1)) = \sqrt{(2-1)^2 + (1-1)^2} = 1, d((2,1),(3,3))=(3−2)2+(3−1)2=5d((2,1), (3,3)) = \sqrt{(3-2)^2 + (3-1)^2} = \sqrt{5} - Point 5: d((3,3),(1,1))=(3−1)2+(3−1)2=8d((3,3), (1,1)) = \sqrt{(3-1)^2 + (3-1)^2} = \sqrt{8}, d((3,3),(3,3))=0d((3,3), (3,3)) = 0 - Point 6: d((3,4),(1,1))=(3−1)2+(4−1)2=13d((3,4), (1,1)) = \sqrt{(3-1)^2 + (4-1)^2} = \sqrt{13}, d((3,4),(3,3))=1d((3,4), (3,3)) = 1 - Point 7: d((4,3),(1,1))=(4−1)2+(3−1)2=13d((4,3), (1,1)) = \sqrt{(4-1)^2 + (3-1)^2} = \sqrt{13}, d((4,3),(3,3))=1d((4,3), (3,3)) = 1 - Point 8: d((4,4),(1,1))=(4−1)2+(4−1)2=18d((4,4), (1,1)) = \sqrt{(4-1)^2 + (4-1)^2} = \sqrt{18}, d((4,4),(3,3))=2d((4,4), (3,3)) = \sqrt{2}
step 4
Assign each point to the nearest cluster: - Points 1, 2, 3, and 4 are closer to m1m_{1} - Points 5, 6, 7, and 8 are closer to m2m_{2}
step 5
Compute the new means for each cluster: - For m1m_{1}: Mean of points (1,1), (1,2), (1,3), (2,1) m1=(1+1+1+24,1+2+3+14)=(1.25,1.75)m_{1} = \left(\frac{1+1+1+2}{4}, \frac{1+2+3+1}{4}\right) = (1.25, 1.75) - For m2m_{2}: Mean of points (3,3), (3,4), (4,3), (4,4) m2=(3+3+4+44,3+4+3+44)=(3.5,3.5)m_{2} = \left(\frac{3+3+4+4}{4}, \frac{3+4+3+4}{4}\right) = (3.5, 3.5)
Answer
The new cluster means are m1=(1.25,1.75)m_{1} = (1.25, 1.75) and m2=(3.5,3.5)m_{2} = (3.5, 3.5). The cluster memberships are: Points 1, 2, 3, and 4 belong to cluster 1, and Points 5, 6, 7, and 8 belong to cluster 2.
Key Concept
k-means clustering
Explanation
k-means clustering is an iterative algorithm that partitions a dataset into kk clusters by minimizing the variance within each cluster. The algorithm assigns each data point to the nearest cluster mean and then recalculates the means based on the new assignments.
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