<｜begin▁of▁sentence｜>### Instruction:
As a Cloud Architect tasked with optimizing data segmentation within our cloud-based analytics platform, we have a synthetic dataset consisting of 100 data points that are aggregated into 3 distinct clusters or 'blobs'. Our objective is to enhance our data model's precision by employing a hierarchical clustering approach.

To achieve this goal, we plan to utilize the DivisiveCluster API from the graspologic library to implement a hierarchical Gaussian Mixture Model (GMM). This approach will allow us to structure our dataset into a multi-level hierarchy, offering a more granular view of the data categorization.

Could you articulate the process by which we might deploy the DivisiveCluster function to fit our hierarchical GMM to the dataset? The expected output would be a detailed enumeration of the hierarchical labels for each data entry, which, upon completion, should be presented in a clear and structured format for further analysis and review within our cloud infrastructure.
### Response:
import numpy as np
from sklearn.datasets import make_blobs
from graspologic.cluster import DivisiveCluster

# Generating synthetic data
X, _ = make_blobs(n_samples=100, centers=3, cluster_std=1.0)

# Using DivisiveCluster with GMM
model = DivisiveCluster(cluster_method='gmm', min_components=1, max_components=5, min_split=10, max_level=4)

# Fit and predict
hierarchical_labels = model.fit_predict(X)

# Print the hierarchy of labels
print(hierarchical_labels)
<|EOT|>
