Question 3: In a soft K-means clustering process, a dataset is cluste…

In a soft K-means clustering process, a dataset is clustered into two groups. A data point has distances $d_1 = 3$ and $d_2 = 6$ from the two cluster centroids. Using the soft K-means formula with $\beta = 1$, compute the probabilities of the data point belonging to each cluster.

After assigning probabilities, update the centroid $\mu_1$ given that the current centroid is $\mu_1 = 4$ and this data point's value is $x = 5$. Assume no other points are assigned to cluster 1. Use the centroid update formula:

$\mu_k^{\text{new}} = \frac{\sum_i P(C_k \mid x_i) x_i}{\sum_i P(C_k \mid x_i)}$

where $P(C_k \mid x)$ is computed using the softmax-like probability formula:

$P(C_k \mid x) = \frac{e^{-\beta d_k}}{\sum_{j=1}^{K} e^{-\beta d_j}}$