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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:
 scikit-learn is a machine learning library for python, with a very easy to use API and great documentation.
Step1:  Lets load up our trajectory. This is the trajectory that we generated in
the "Running a simulation in OpenMM and analyzing the results with mdtraj"
example.
Step2:  Create a two component PCA model, and project our data down into this
reduced dimensional space. Using just the cartesian coordinates as
input to PCA, it's important to start with some kind of alignment.
Step3:  Now we can plot the data on this projection.
Step4:  Lets try cross-checking our result by using a different feature space that isn't sensitive to alignment, and instead to "featurize" our trajectory by computing the pairwise distance between every atom  in each frame, and using that as our high dimensional input space for PCA.
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Python Code:
%matplotlib inline
from __future__ import print_function
import mdtraj as md
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
Explanation: scikit-learn is a machine learning library for python, with a very easy to use API and great documentation.
End of explanation
traj = md.load('ala2.h5')
traj
Explanation: Lets load up our trajectory. This is the trajectory that we generated in
the "Running a simulation in OpenMM and analyzing the results with mdtraj"
example.
End of explanation
pca1 = PCA(n_components=2)
traj.superpose(traj, 0)
reduced_cartesian = pca1.fit_transform(traj.xyz.reshape(traj.n_frames, traj.n_atoms * 3))
print(reduced_cartesian.shape)
Explanation: Create a two component PCA model, and project our data down into this
reduced dimensional space. Using just the cartesian coordinates as
input to PCA, it's important to start with some kind of alignment.
End of explanation
plt.figure()
plt.scatter(reduced_cartesian[:, 0], reduced_cartesian[:,1], marker='x', c=traj.time)
plt.xlabel('PC1')
plt.ylabel('PC2')
plt.title('Cartesian coordinate PCA: alanine dipeptide')
cbar = plt.colorbar()
cbar.set_label('Time [ps]')
Explanation: Now we can plot the data on this projection.
End of explanation
pca2 = PCA(n_components=2)
from itertools import combinations
# this python function gives you all unique pairs of elements from a list
atom_pairs = list(combinations(range(traj.n_atoms), 2))
pairwise_distances = md.geometry.compute_distances(traj, atom_pairs)
print(pairwise_distances.shape)
reduced_distances = pca2.fit_transform(pairwise_distances)
plt.figure()
plt.scatter(reduced_distances[:, 0], reduced_distances[:,1], marker='x', c=traj.time)
plt.xlabel('PC1')
plt.ylabel('PC2')
plt.title('Pairwise distance PCA: alanine dipeptide')
cbar = plt.colorbar()
cbar.set_label('Time [ps]')
Explanation: Lets try cross-checking our result by using a different feature space that isn't sensitive to alignment, and instead to "featurize" our trajectory by computing the pairwise distance between every atom  in each frame, and using that as our high dimensional input space for PCA.
End of explanation
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