New clustering metrics

### Describe the workflow you want to enable

Scikit-learn defines three popular metrics for evaluating clustering performance when there are no ground-truth cluster labels: [sklearn.metrics.silhouette_score](https://scikit-learn.org/stable/modules/generated/sklearn.metrics.silhouette_score.html), [sklearn.metrics.calinski_harabasz_score](https://scikit-learn.org/stable/modules/generated/sklearn.metrics.calinski_harabasz_score.html) and [sklearn.metrics.davies_bouldin_score](https://scikit-learn.org/stable/modules/generated/sklearn.metrics.davies_bouldin_score.html). But there are lots of others, and it's previously been [discussed](https://github.com/scikit-learn/scikit-learn/discussions/21164) whether to integrate more into scikit-learn.

### Describe your proposed solution

I've implemented four relatively popular ones, using the same interface and code style as in https://github.com/scikit-learn/scikit-learn/blob/main/sklearn/metrics/cluster/_unsupervised.py. Would there be interest in integrating these into scikit-learn?

```python
import numpy as np
from itertools import combinations
from sklearn.utils import check_X_y
from sklearn.metrics.cluster._unsupervised import check_number_of_labels
from sklearn.preprocessing import LabelEncoder


def log_ss_ratio(X, labels):
    X, labels = check_X_y(X, labels)
    le = LabelEncoder()
    labels = le.fit_transform(labels)

    n_samples, _ = X.shape
    n_labels = len(le.classes_)

    check_number_of_labels(n_labels, n_samples)

    extra_disp, intra_disp = 0.0, 0.0
    mean = X.mean(axis=0)
    for k in range(n_labels):
        cluster_k = X[labels == k]
        mean_k = cluster_k.mean(axis=0)
        extra_disp += len(cluster_k) * ((mean_k - mean) ** 2).sum()
        intra_disp += ((cluster_k - mean_k) ** 2).sum()

    return np.log(extra_disp / intra_disp)


def ball_hall(X, labels):
    X, labels = check_X_y(X, labels)
    le = LabelEncoder()
    labels = le.fit_transform(labels)

    n_samples, _ = X.shape
    n_labels = len(le.classes_)

    check_number_of_labels(n_labels, n_samples)

    sum_mean_dispersions = 0

    for k in range(n_labels):
        cluster_k = X[labels == k]
        mean_k = cluster_k.mean(axis=0)
        intra_disp_cluster = ((cluster_k - mean_k) ** 2).sum()
        sum_mean_dispersions += intra_disp_cluster / len(cluster_k)

    return sum_mean_dispersions / n_labels


def banfeld_raftery(X, labels):
    X, labels = check_X_y(X, labels)
    le = LabelEncoder()
    labels = le.fit_transform(labels)

    n_samples, _ = X.shape
    n_labels = len(le.classes_)

    check_number_of_labels(n_labels, n_samples)

    br_index = 0
    for k in range(n_labels):
        cluster_k = X[labels == k]
        mean_k = cluster_k.mean(axis=0)
        intra_disp_cluster = ((cluster_k - mean_k) ** 2).sum()
        br_index += len(cluster_k) * \
                    np.log(intra_disp_cluster / len(cluster_k))

    return br_index


def ray_turi(X, labels):
    X, labels = check_X_y(X, labels)
    le = LabelEncoder()
    labels = le.fit_transform(labels)

    n_samples, _ = X.shape
    n_labels = len(le.classes_)

    check_number_of_labels(n_labels, n_samples)

    intra_disp = 0.0

    cluster_means = []
    for k in range(n_labels):
        cluster_k = X[labels == k]
        mean_k = cluster_k.mean(axis=0)
        intra_disp += ((cluster_k - mean_k) ** 2).sum()
        cluster_means.append(mean_k)

    min_cluster_mean_diff = min(
        ((mean_i - mean_j) ** 2).sum()
        for mean_i, mean_j in combinations(cluster_means, 2))

    return intra_disp / (min_cluster_mean_diff * n_samples)
```

### Describe alternatives you've considered, if relevant

_No response_

### Additional context

_No response_

COMMENT:
> You can ping people who have shown interest in this thread every now and then and hopefully they can allocate some time to review.

Hello

as i'm not involved in the project but highly interested. Im curious.
Is this Issue for implementing DBCV ongoing or solved in another way?

*sending a small health check on that issue*  ;)

Labels: New Feature, Needs Decision - Include Feature