K-Means Clustering: Finding Manhattan Distance in Second Iteration

This problem involves understanding the K-means clustering algorithm, specifically how centroids are updated and how distance is calculated. We are given the clusters formed after the first iteration and asked to find the Manhattan distance of a specific observation from a cluster's centroid in the second iteration.

Understanding K-Means Iterations

In the K-means algorithm, after the initial cluster assignments (either randomly or based on initial centroids), the algorithm proceeds iteratively:

1. Update Centroids: Calculate the mean of all observations assigned to each cluster. These means become the new centroids for the next iteration.
2. Assign Observations: Re-assign each observation to the cluster whose new centroid is closest (based on a chosen distance metric, like Euclidean or Manhattan distance).

These steps are repeated until cluster assignments no longer change or a maximum number of iterations is reached.

Calculating the Centroid for Cluster C1 for the Second Iteration

The clusters given are the result *after* the first iteration. To find the centroids for the *second* iteration, we must calculate the mean of the points within each cluster given. The observations in Cluster C1 after the first iteration are: {(3,3), (5,5), (7,7)}.

The centroid of C1 for the second iteration is the mean of these points. Let the points be $(x_i, y_i)$. The centroid $(\bar{x}, \bar{y})$ is calculated as:

$$\bar{x} = \frac{\sum x_i}{n}$$

$$\bar{y} = \frac{\sum y_i}{n}$$

where $n$ is the number of points in the cluster.

For C1:

• Sum of x-coordinates: $3 + 5 + 7 = 15$
• Sum of y-coordinates: $3 + 5 + 7 = 15$
• Number of points: $n = 3$

The new centroid for C1 is:

$$\bar{x}_{C1} = \frac{15}{3} = 5$$

$$\bar{y}_{C1} = \frac{15}{3} = 5$$

So, the centroid of cluster C1 for the second iteration is (5, 5).

Calculating Manhattan Distance

The question asks for the Manhattan distance of observation (4,4) from the cluster centroid C1 in the second iteration. The observation is (4,4) and the calculated new centroid for C1 is (5,5).

The Manhattan distance (also known as Taxicab or L1 distance) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$$D_{Manhattan} = |x_1 - x_2| + |y_1 - y_2|$$

Using this formula for observation (4,4) and C1 centroid (5,5):

$$D = |4 - 5| + |4 - 5|$$

$$D = |-1| + |-1|$$

$$D = 1 + 1$$

$$D = 2$$

The Manhattan distance of observation (4,4) from cluster centroid C1 in the second iteration is 2.

Conclusion

By first calculating the updated centroid for cluster C1 based on the points assigned to it after the first iteration, and then applying the Manhattan distance formula between the observation (4,4) and the new C1 centroid, we find the distance to be 2.

Revision Table: K-Means Clustering Key Steps

Additional Information: K-Means Distance Metrics

The choice of distance metric in K-means clustering can significantly impact the resulting clusters. While Euclidean distance is the most common, Manhattan distance is also used, particularly when dealing with grid-like data or when differences along dimensions are considered equally important regardless of their interaction.

Euclidean Distance

This is the "straight-line" distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $$D_{Euclidean} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ It is sensitive to the magnitude of differences in each dimension.

Manhattan Distance

This is the sum of the absolute differences of their coordinates: $$D_{Manhattan} = |x_1 - x_2| + |y_1 - y_2|$$ It represents the distance if you can only travel horizontally or vertically (like walking in a city grid). It is less sensitive to outliers than Euclidean distance in some cases.

In this problem, the Manhattan distance was specifically requested for the final calculation involving the observation (4,4) and the updated C1 centroid (5,5).