<｜begin▁of▁sentence｜>### Instruction:
As a pilot, we're used to relying on our instruments to navigate through the skies professionally and safely. Imagine we're charting a new course, and we've gathered data from 300 flight paths that apparently form three distinct flight patterns or 'blobs'. We need to organize these flight paths into distinct airways effectively, yet we're not quite sure how many separate routes we should create for optimal navigation.

Now, consider how we might employ the AutoGMMCluster tool from graspologic to serve as our automatic navigation system. This instrument is designed to analyse the patterns within our flight route data and determine the ideal number of airways – akin to establishing the number of clusters – and apply a Gaussian Mixture Model to better define these paths. How could we use AutoGMMCluster to perform this analysis and assist us in structuring our airspace efficiently, based on the data we've collected?
### Response:
import numpy as np
from graspologic.cluster import AutoGMMCluster
from sklearn.datasets import make_blobs

# Generate synthetic data
n_samples = 300
centers = [(-5, -5), (0, 0), (5, 5)]
X, y_true = make_blobs(n_samples=n_samples, centers=centers, cluster_std=1.0, random_state=42)

# Fit the AutoGMMCluster
gmm_cluster = AutoGMMCluster(min_components=2, max_components=4, affinity='euclidean', linkage='ward')
gmm_results = gmm_cluster.fit(X)

# Predict the cluster labels
labels = gmm_cluster.predict(X)

# Print best model based on BIC
print(f'Best model - number of clusters: {gmm_cluster.n_components_}')
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