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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Bock 2011 paper claims they cut 4 different layers, so we will do k-means w/ 4 clusters.
Step1:  Note pairs of clusters with similar mean densities and y-values.
Since we have seen numerous times that it appears layers change along y-axis, lets cluster our data based only on density and y value and then compare these to the previous clustering.
Step2:  This clustering result is more along the lines of what we expected, as it partitions by different y-values and densities. Let's make some graphs of the previous two clusters.
Step3:  Observe the similarity in clusters when projected onto the y-axis. Also note difference in x and z coordinate means for the first 2 and second 2 clusters. Let's try kmeans with k=2.
Step4:  Fit 2 gaussians...
Step5:  Clustering results for the most part, do not seem to be indicating a particular cut-off between layers, although they indicate some correlation between y coordinates and density.
Step6:  Some negative correlation between density and y coordinate, although not extreme. Let's compare the correlation for each coordinate
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Python Code:
from sklearn import cluster
kmeans1 = cluster.KMeans(4)
kmeans1.fit_predict(data)
print kmeans1.cluster_centers_
Explanation: Bock 2011 paper claims they cut 4 different layers, so we will do k-means w/ 4 clusters.
End of explanation
data_yd = data[:, (1, 3)]
kmeans2 = cluster.KMeans(4)
kmeans2.fit_predict(data_yd)
print kmeans2.cluster_centers_
Explanation: Note pairs of clusters with similar mean densities and y-values.
Since we have seen numerous times that it appears layers change along y-axis, lets cluster our data based only on density and y value and then compare these to the previous clustering.
End of explanation
colors = ['b', 'g', 'r', 'c', 'm']
for i, c in zip(range(4), colors):
    a = np.where(kmeans2.labels_ == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('KMeans clustering, k=4, only y-coord and density considered')
plt.show()
print kmeans2.cluster_centers_
for i, c in zip(range(4), colors):
    a = np.where(kmeans1.labels_ == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('Kmeans clusters, k=4, x,y,z coords and density considered')
plt.show()
for i, c in zip(range(4), colors):
    a = np.where(kmeans1.labels_ == i)
    plt.scatter(data[a, 0], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('x coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('Kmeans clusters, k=4, x,y,z coords and density considered')
plt.show()
print kmeans1.cluster_centers_
Explanation: This clustering result is more along the lines of what we expected, as it partitions by different y-values and densities. Let's make some graphs of the previous two clusters.
End of explanation
kmeans3 = cluster.KMeans(2)
kmeans3.fit_predict(data)
for i, c in zip(range(2), colors):
    a = np.where(kmeans3.labels_ == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('KMeans clustering, k=2, xyz and density considered')
plt.show()
print kmeans3.cluster_centers_
kmeans4 = cluster.KMeans(2)
kmeans4.fit_predict(data[:, (1, 3)])
for i, c in zip(range(2), colors):
    a = np.where(kmeans4.labels_ == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('KMeans clustering, k=2, y coord and density considered')
plt.show()
print kmeans4.cluster_centers_
Explanation: Observe the similarity in clusters when projected onto the y-axis. Also note difference in x and z coordinate means for the first 2 and second 2 clusters. Let's try kmeans with k=2.
End of explanation
from sklearn import mixture
gmm = mixture.GMM(2)
labels = gmm.fit_predict(data)
print gmm.means_
print "compare with kmeans, k=2, all coordinates"
print kmeans3.cluster_centers_
for i, c in zip(range(2), colors):
    a = np.where(labels == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('2 Component Gaussian Mixture Model Prediction')
plt.show()
# now do GMM w/ 4 clusters, since Bock 2011 suggests 4 layers
gmm = mixture.GMM(4)
labels = gmm.fit_predict(data)
for i, c in zip(range(4), colors):
    a = np.where(labels == i)
    plt.scatter(data[a, 1], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('y coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('4 Component Gaussian Mixture Model Prediction')
plt.show()
for i, c in zip(range(4), colors):
    a = np.where(labels == i)
    plt.scatter(data[a, 0], data[a, -1], 
                alpha = .2,
                color = c, 
                label='cluster #' + str(i))
plt.legend(bbox_to_anchor=(1.4, 1.00))
plt.xlabel('x coordinate (normalized)')
plt.ylabel('synaptic density (normalized)')
plt.title('4 Component Gaussian Mixture Model Prediction')
plt.show()
print gmm.means_
Explanation: Fit 2 gaussians...
End of explanation
print np.cov(data[:, 1], data[:, -1])
Explanation: Clustering results for the most part, do not seem to be indicating a particular cut-off between layers, although they indicate some correlation between y coordinates and density.
End of explanation
for i, coord in enumerate(['cx', 'cy', 'cz']):
    print "correlation between density and " + coord 
    print np.cov(data[:, i], data[:, -1])
Explanation: Some negative correlation between density and y coordinate, although not extreme. Let's compare the correlation for each coordinate
End of explanation
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