What are the right metrics to validate the performance of a custom clustering model with three possible outcomes?

Question

I have developed a custom clustering model on top of MiniBatchKmeans, that has three possible outcomes for each data point:

1. Assign the point to the correct cluster.
2. Assign the point to the wrong cluster.
3. Create a new cluster.

I'm looking for appropriate metrics to evaluate the performance of this model. I am aware of common clustering evaluation metrics like Adjusted Rand Index (ARI), Silhouette Score, and Davies-Bouldin Index, but I am not sure how they apply to the unique outcomes of my model.

Specifically, I want to understand how to:

• Measure the accuracy of assigning points to the correct clusters.
• Penalize the creation of unnecessary new clusters.
• Account for points that are assigned to the wrong clusters.

Can anyone recommend metrics or a combination of metrics that would best validate the performance of my custom clustering model? Additionally, if there are any best practices or examples of handling such scenarios, I would appreciate the insights.

Thank you!

While these metrics (Adjusted Rand Index (ARI), Silhouette Score, Davies-Bouldin Index) provide some insights, they don't fully capture the three specific outcomes of my model. I expected to find a metric or a combination of metrics that would:

• Clearly indicate when points are correctly clustered.
• Penalize unnecessary creation of new clusters.
• Account for misassigned points.