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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:
 Generate data to simulate a dataset of samples $x$ in which all features/columns (3) could be collected. x has size $n$ by $features$.
Step1:  Transform the generated data $x$ to a new basis. 
The basis is given by the eigenvectors of $cov(x)$. In this new basis, the eigenvectors are the same as the $x$, $y$, $z$ axis vectors $(1,0,0)$, $(0,1,0)$, etc.
Step2:  Generate another dataset $y$ with the same distribution as $x$ (this is a very strong assumption!)
Step3:  Lets simulate the fact that for $y$ we can't measure all values. In this case, we will create y_missing, which only has 2 features
Step4:  Now, lets assume that we can recover the last feature of y_missing  using information from the eigendecomposition of $cov(x)$. 
We perform PCA on $x$, but use only the two most important eigenvectors (those with bigger eigenvalues). pca_reconstruction allows us to perform a forward transform from $R^3$ to $R^2$ (compression) and an inverse transform from $R^2$ to $R^3$ (reconstruction). This can also be interpreted as going from the canonical basis to the eigenbasis and viceversa.
Step5:  We still can't reconstruct y from y_missing with pca_reconstruction because it is encoded with the canonical basis. We first need to encode it in the eigenbasis. To do that we will augment y_missing with a new feature with value equal to the mean of that feature (a reasonable assumption), and encode it.
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Python Code:
#mean and std of multivariate normal dist to generate samples
mu=np.array([5,0,-2])
σ=np.array([[9,1, -1],
            [1, 3, -2],
            [-1, -2,2],])
if not is_covariance(σ):
    print("Warning: σ is not a valid covariance matrix (not symmetric or positive definite)")
n=1000 # number of samples
x=np.random.multivariate_normal(mu,σ,size=(n,))
#plot generated data
plt.close("all")
plot_data(x,"data in original space",eigen=True)
plt.show()
Explanation: Generate data to simulate a dataset of samples $x$ in which all features/columns (3) could be collected. x has size $n$ by $features$.
End of explanation
pca_exact = PCA(n_components=3) # since x has 3 features, this PCA model does not do compression
pca_exact.fit(x) # calculate eigen decomposition of cov(x)
x_transformed=pca_exact.transform(x) #encode x with the eigenvectors as basis
plot_data(x_transformed,"x in natural (eigenvectors) space",eigen=True)
#save the eigenvectors and eigenvalues
eigenvectors=pca_exact.components_
eigenvalues=pca_exact.explained_variance_
Explanation: Transform the generated data $x$ to a new basis. 
The basis is given by the eigenvectors of $cov(x)$. In this new basis, the eigenvectors are the same as the $x$, $y$, $z$ axis vectors $(1,0,0)$, $(0,1,0)$, etc.
End of explanation
y=np.random.multivariate_normal(mu,σ,size=(n,))
plot_data(y,"y in original space",eigen=True)
Explanation: Generate another dataset $y$ with the same distribution as $x$ (this is a very strong assumption!)
End of explanation
y_missing=y[:,0:2]
plt.figure()
plt.scatter(y_missing[:,0],y_missing[:,1])
plt.title("y_missing in original space (2d)")
Explanation: Lets simulate the fact that for $y$ we can't measure all values. In this case, we will create y_missing, which only has 2 features
End of explanation
pca_reconstruction=PCA(n_components=2)
pca_reconstruction.fit(x)
print(pca_reconstruction.components_)
print(eigenvectors)
Explanation: Now, lets assume that we can recover the last feature of y_missing  using information from the eigendecomposition of $cov(x)$. 
We perform PCA on $x$, but use only the two most important eigenvectors (those with bigger eigenvalues). pca_reconstruction allows us to perform a forward transform from $R^3$ to $R^2$ (compression) and an inverse transform from $R^2$ to $R^3$ (reconstruction). This can also be interpreted as going from the canonical basis to the eigenbasis and viceversa.
End of explanation
y_augmented=np.copy(y_missing)
y3=np.zeros((n,1))+mu[2]
y_augmented=np.hstack([y_missing,y3])
y_eigen=pca_exact.transform(y_augmented)
least_eigenvalue_index=np.argmin(eigenvalues)
y_eigen_2d=y_eigen[:,np.arange(3)!=least_eigenvalue_index]
y_reconstructed=pca_reconstruction.inverse_transform(y_eigen_2d)
plot_data(y_reconstructed, "y_reconstructed",eigen=True)
mean_reconstruction_error=((y_reconstructed-y)**2).sum()/n
print(mean_reconstruction_error)
Explanation: We still can't reconstruct y from y_missing with pca_reconstruction because it is encoded with the canonical basis. We first need to encode it in the eigenbasis. To do that we will augment y_missing with a new feature with value equal to the mean of that feature (a reasonable assumption), and encode it.
End of explanation
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