%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

"""
Explanation: Applications of Linear Alebra: PCA
We will explore 3 applications of linear algebra in data analysis - change of basis (for dimension reduction), projections (for solving linear systems) and the quadratic form (for optimization). The first application is the change of basis to the eigenvector basis that underlies Principal Components Analysis s(PCA).
We will review the following in class:

The standard basis
Orthonormal basis and orthgonal matrices
Change of basis
Similar matrices
Eigendecomposition
Sample covariance
Covariance as a linear transform
PCA and dimension reduction
PCA and "explained variance"
SVD
End of explanation
"""


def cov(x, y):
    """Returns covariance of vectors x and y)."""
    xbar = x.mean()
    ybar = y.mean()
    return np.sum((x - xbar)*(y - ybar))/(len(x) - 1)

X = np.random.random(10)
Y = np.random.random(10)

np.array([[cov(X, X), cov(X, Y)], [cov(Y, X), cov(Y,Y)]])

# This can of course be calculated using numpy's built in cov() function
np.cov(X, Y)

# Extension to more variables is done in a pair-wise way
Z = np.random.random(10)
np.cov([X, Y, Z])

"""
Explanation: Variance and covariance
Remember the formula for covariance
$$
\text{Cov}(X, Y) = \frac{\sum_{i=1}^n(X_i - \bar{X})(Y_i - \bar{Y})}{n-1}
$$
where $\text{Cov}(X, X)$ is the sample variance of $X$.
End of explanation
"""


mu = [0,0]
sigma = [[0.6,0.2],[0.2,0.2]]
n = 1000
x = np.random.multivariate_normal(mu, sigma, n).T

A = np.cov(x)

m = np.array([[1,2,3],[6,5,4]])
ms = m - m.mean(1).reshape(2,1)
np.dot(ms, ms.T)/2

e, v = np.linalg.eig(A)

plt.scatter(x[0,:], x[1,:], alpha=0.2)
for e_, v_ in zip(e, v.T):
    plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)
plt.axis([-3,3,-3,3])
plt.title('Eigenvectors of covariance matrix scaled by eigenvalue.');

"""
Explanation: Eigendecomposition of the covariance matrix
End of explanation
"""


covx = np.array([[1,0.6],[0.6,1]])

"""
Explanation: Covariance matrix as a linear transformation
The covariance matrix is a linear transformation that maps $\mathbb{R}^n$ in the direction of its eigenvectors with scaling factor given by the eigenvalues. Here we see it applied to a collection of random vectors in the box bounded by [-1, 1].
We will assume we have a covariance matrix
End of explanation
"""


u = np.random.uniform(-1, 1, (100, 2)).T

"""
Explanation: Create random vectors in a box
End of explanation
"""


y = covx @ u

e1, v1 = np.linalg.eig(covx)

"""
Explanation: Apply covariance matrix as linear transformation
End of explanation
"""


plt.scatter(u[0], u[1], c='blue')
plt.scatter(y[0], y[1], c='orange')
for e_, v_ in zip(e1, v1.T):
    plt.plot([0, e_*v_[0]], [0, e_*v_[1]], 'r-', lw=2)
plt.xticks([])
plt.yticks([])
pass

"""
Explanation: The linear transform maps the random vectors as described.
End of explanation
"""


np.set_printoptions(precision=3)

X = np.random.random((5,4))
X

### Subtract the row mean from each row

Y = X - X.mean(1)[:, None]

Y.mean(1)

Y

### Calculate the covariance

np.cov(X)

np.cov(Y)

"""
Explanation: PCA
Principal Components Analysis (PCA) basically means to find and rank all the eigenvalues and eigenvectors of a covariance matrix. This is useful because high-dimensional data (with $p$ features) may have nearly all their variation in a small number of dimensions $k$, i.e. in the subspace spanned by the eigenvectors of the covariance matrix that have the $k$ largest eigenvalues. If we project the original data into this subspace, we can have a dimension reduction (from $p$ to $k$) with hopefully little loss of information.
Numerically, PCA is typically done using SVD on the data matrix rather than eigendecomposition on the covariance matrix. The next section explains why this works.
Data matrices that have zero mean for all feature vectors
\begin{align}
\text{Cov}(X, Y) &= \frac{\sum_{i=1}^n(X_i - \bar{X})(Y_i - \bar{Y})}{n-1} \
  &= \frac{\sum_{i=1}^nX_iY_i}{n-1} \
  &= \frac{XY^T}{n-1}
\end{align}
and so the covariance matrix for a data set X that has zero mean in each feature vector is just $XX^T/(n-1)$. 
In other words, we can also get the eigendecomposition of the covariance matrix from the positive semi-definite matrix $XX^T$.
Note that zeroing the feature vector does not affect the covariance matrix
End of explanation
"""


e1, v1 = np.linalg.eig(np.dot(x, x.T)/(n-1))

plt.scatter(x[0,:], x[1,:], alpha=0.2)
for e_, v_ in zip(e1, v1.T):
    plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)
plt.axis([-3,3,-3,3]);

"""
Explanation: Eigendecomposition of the covariance matrix
End of explanation
"""


e, v = np.linalg.eig(np.cov(x))
v.dot(v.T)

"""
Explanation: Change of basis via PCA
We can transform the original data set so that the eigenvectors are the basis vectors and find the new coordinates of the data points with respect to this new basis
This is the change of basis transformation covered in the Linear Alegebra module. First, note that the covariance matrix is a real symmetric matrix, and so the eigenvector matrix is an orthogonal matrix.
End of explanation
"""


ys = np.dot(v1.T, x)

"""
Explanation: Linear algebra review for change of basis
Graphical illustration of change of basis

Suppose we have a vector $u$ in the standard basis $B$ , and a matrix $A$ that maps $u$ to $v$, also in $B$. We can use the eigenvalues of $A$ to form a new basis $B'$. As explained above, to bring a vector $u$ from $B$-space to a vector $u'$ in $B'$-space, we multiply it by $Q^{-1}$, the inverse of the matrix having the eigenvctors as column vectors. Now, in the eigenvector basis, the equivalent operation to $A$ is the diagonal matrix $\Lambda$ - this takes $u'$ to $v'$. Finally, we convert $v'$ back to a vector $v$ in the standard basis by multiplying with $Q$.
End of explanation
"""


plt.scatter(ys[0,:], ys[1,:], alpha=0.2)
for e_, v_ in zip(e1, np.eye(2)):
    plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)
plt.axis([-3,3,-3,3]);

"""
Explanation: Principal components
Principal components are simply the eigenvectors of the covariance matrix used as basis vectors. Each of the original data points is expressed as a linear combination of the principal components, giving rise to a new set of coordinates.
End of explanation
"""


zs = np.dot(v1, ys)

plt.scatter(zs[0,:], zs[1,:], alpha=0.2)
for e_, v_ in zip(e1, v1.T):
    plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)
plt.axis([-3,3,-3,3]);

u, s, v = np.linalg.svd(x)
u.dot(u.T)

"""
Explanation: For example, if we only use the first column of ys, we will have the projection of the data onto the first principal component, capturing the majority of the variance in the data with a single feature that is a linear combination of the original features.
Transform back to original coordinates
We may need to transform the (reduced) data set to the original feature coordinates for interpretation. This is simply another linear transform (matrix multiplication).
End of explanation
"""


A

A.trace()

e, v = np.linalg.eig(A)
D = np.diag(e)
D

D.trace()

D[0,0]/D.trace()

"""
Explanation: Dimension reduction via PCA
We have the sepctral decomposition of the covariance matrix
$$
A = Q^{-1}\Lambda Q
$$
Suppose $\Lambda$ is a rank $p$ matrix. To reduce the dimensionality to $k \le p$, we simply set all but the first $k$ values of the diagonal of $\Lambda$ to zero. This is equivalent to ignoring all except the first $k$ principal components.
What does this achieve? Recall that $A$ is a covariance matrix, and the trace of the matrix is the overall variability, since it is the sum of the variances.
End of explanation
"""


u, s, v = np.linalg.svd(x)

e2 = s**2/(n-1)
v2 = u
plt.scatter(x[0,:], x[1,:], alpha=0.2)
for e_, v_ in zip(e2, v2):
    plt.plot([0, 3*e_*v_[0]], [0, 3*e_*v_[1]], 'r-', lw=2)
plt.axis([-3,3,-3,3]);

v1 # from eigenvectors of covariance matrix

v2 # from SVD

e1 # from eigenvalues of covariance matrix

e2 # from SVD

a0 = np.random.normal(0,1,100)
a1 = a0 + np.random.normal(0,.5,100)
a2 = 2*a0 + a1 + np.random.normal(5,0.01,100)
xs = np.vstack([a0, a1, a2])
xs.shape

U, s, V = np.linalg.svd(xs)

(s**2)/(99)

U

"""
Explanation: Since the trace is invariant under change of basis, the total variability is also unchanged by PCA. By keeping only the first $k$ principal components, we can still "explain" $\sum_{i=1}^k e[i]/\sum{e}$ of the total variability. Sometimes, the degree of dimension reduction is specified as keeping enough principal components so that (say) $90\%$ of the total variability is explained.
Using Singular Value Decomposition (SVD) for PCA
SVD is a decomposition of the data matrix $X = U S V^T$ where $U$ and $V$ are orthogonal matrices and $S$ is a diagnonal matrix. 
Recall that the transpose of an orthogonal matrix is also its inverse, so if we multiply on the right by $X^T$, we get the follwoing simplification
\begin{align}
X &= U S V^T \
X X^T &= U S V^T (U S V^T)^T \
 &= U S V^T V S U^T \
 &= U S^2 U^T
\end{align}
Compare with the eigendecomposition of a matrix $A = W \Lambda W^{-1}$, we see that SVD gives us the eigendecomposition of the matrix $XX^T$, which as we have just seen, is basically a scaled version of the covariance for a data matrix with zero mean, with the eigenvectors given by $U$ and eigenvealuse by $S^2$ (scaled by $n-1$)..
End of explanation
"""


