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Variance is a measure of the variability of the data you have. Potentially the number of components is infinite, so you want to "squeeze" the most information in each component of the finite set you build.
If, to exaggerate, you were to select a single principal component, you would want it to account for the most variability possible: hence the search for maximum variance, so that the one component collects the most "uniqueness" from the data set.
Maximizing the component vector variances is the same as maximizing the 'uniqueness' of those vectors. The vectors are as distant from each other as possible (orthogonal to each other).
Take for example a situation where you have 2 lines that are orthogonal in a 3D space. You can capture the environment much more completely with those orthogonal lines than 2 lines that are parallel (or nearly parallel). When applied to very high dimensional states using very few vectors, this becomes a much more important relationship among the vectors to maintain. In a linear algebra sense you want independent rows to be produced by PCA, otherwise some of those rows will be redundant.
The principal component analysis deals with the problem of fitting a low-dimensional affine subspace \( S \) of dimension \( d \) much smaller than the total dimension \( D \) of the problem at hand (our data set). Mathematically it can be formulated as a statistical problem or a geometric problem. In our discussion of the theorem for the classical PCA, we will stay with a statistical approach. Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable.
We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see below for its definition)
A good read is for example Vidal, Ma and Sastry.
Before we discuss the PCA theorem, we need to remind ourselves about the definition of the covariance and the correlation function. These are quantities
Suppose we have defined two vectors \( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
\end{bmatrix},
$$
where for example
$$
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
$$
With this definition and recalling that the variance is defined as
$$
\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
$$
we can rewrite the covariance matrix as
$$
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\
\end{bmatrix}.
$$
The covariance takes values between zero and infinity and may thus lead to problems with loss of numerical precision for particularly large values. It is common to scale the covariance matrix by introducing instead the correlation matrix defined via the so-called correlation function
$$
\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
$$
The correlation function is then given by values \( \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \in [-1,1] \). This avoids eventual problems with too large values. We can then define the correlation matrix for the two vectors \( \boldsymbol{x} \) and \( \boldsymbol{y} \) as
$$
\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\
\end{bmatrix},
$$
In the above example this is the function we constructed using pandas.
In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression we defined the design/feature matrix \( \boldsymbol{X} \) as
$$
\boldsymbol{X}=\begin{bmatrix}
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
\dots & \dots & \dots & \dots \dots & \dots \\
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
\end{bmatrix},
$$
with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) refering to the column numbers and the entries \( n \) being the row elements. We can rewrite the design/feature matrix in terms of its column vectors as
$$
\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix},
$$
with a given vector
$$
\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}.
$$
With these definitions, we can now rewrite our \( 2\times 2 \) correlation/covariance matrix in terms of a moe general design/feature matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). This leads to a \( p\times p \) covariance matrix for the vectors \( \boldsymbol{x}_i \) with \( i=0,1,\dots,p-1 \)
$$
\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots & \dots & \dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots & \dots & \dots \\
\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\
\end{bmatrix},
$$
and the correlation matrix
$$
\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots & \dots & \dots & \dots & \dots & \dots \\
\dots & \dots & \dots & \dots & \dots & \dots \\
\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\
\end{bmatrix},
$$
The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
$$
\boldsymbol{W}^T = \begin{bmatrix} x_0 & y_0 \\
x_1 & y_1 \\
x_2 & y_2\\
\dots & \dots \\
x_{n-2} & y_{n-2}\\
x_{n-1} & y_{n-1} &
\end{bmatrix},
$$
which in turn is converted into into the \( 2\times 2 \) covariance matrix \( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function.
# Importing various packages
import numpy as np
n = 100
x = np.random.normal(size=n)
print(np.mean(x))
y = 4+3*x+np.random.normal(size=n)
print(np.mean(y))
W = np.vstack((x, y))
C = np.cov(W)
print(C)
The previous example can be converted into the correlation matrix by simply scaling the matrix elements with the variances. We should also subtract the mean values for each column. This leads to the following code which sets up the correlations matrix for the previous example in a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors).
import numpy as np
n = 100
# define two vectors
x = np.random.random(size=n)
y = 4+3*x+np.random.normal(size=n)
#scaling the x and y vectors
x = x - np.mean(x)
y = y - np.mean(y)
variance_x = np.sum(x@x)/n
variance_y = np.sum(y@y)/n
print(variance_x)
print(variance_y)
cov_xy = np.sum(x@y)/n
cov_xx = np.sum(x@x)/n
cov_yy = np.sum(y@y)/n
C = np.zeros((2,2))
C[0,0]= cov_xx/variance_x
C[1,1]= cov_yy/variance_y
C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
C[1,0]= C[0,1]
print(C)
We see that the matrix elements along the diagonal are one as they should be and that the matrix is symmetric. Furthermore, diagonalizing this matrix we easily see that it is a positive definite matrix.
The above procedure with numpy can be made more compact if we use pandas.
We whow here how we can set up the correlation matrix using pandas, as done in this simple code
import numpy as np
import pandas as pd
n = 10
x = np.random.normal(size=n)
x = x - np.mean(x)
y = 4+3*x+np.random.normal(size=n)
y = y - np.mean(y)
X = (np.vstack((x, y))).T
print(X)
Xpd = pd.DataFrame(X)
print(Xpd)
correlation_matrix = Xpd.corr()
print(correlation_matrix)
We expand this model to the Franke function discussed above.
# Common imports
import numpy as np
import pandas as pd
def FrankeFunction(x,y):
term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
return term1 + term2 + term3 + term4
def create_X(x, y, n ):
if len(x.shape) > 1:
x = np.ravel(x)
y = np.ravel(y)
N = len(x)
l = int((n+1)*(n+2)/2) # Number of elements in beta
X = np.ones((N,l))
for i in range(1,n+1):
q = int((i)*(i+1)/2)
for k in range(i+1):
X[:,q+k] = (x**(i-k))*(y**k)
return X
# Making meshgrid of datapoints and compute Franke's function
n = 4
N = 100
x = np.sort(np.random.uniform(0, 1, N))
y = np.sort(np.random.uniform(0, 1, N))
z = FrankeFunction(x, y)
X = create_X(x, y, n=n)
Xpd = pd.DataFrame(X)
# subtract the mean values and set up the covariance matrix
Xpd = Xpd - Xpd.mean()
covariance_matrix = Xpd.cov()
print(covariance_matrix)
We note here that the covariance is zero for the first rows and columns since all matrix elements in the design matrix were set to one (we are fitting the function in terms of a polynomial of degree \( n \)). We would however not include the intercept and wee can simply drop these elements and construct a correlation matrix without them by centering our matrix elements by subtracting the mean of each column.
We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \( \boldsymbol{X} \) as
$$
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}].
$$
To see this let us simply look at a design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{2\times 2} \)
$$
\boldsymbol{X}=\begin{bmatrix}
x_{00} & x_{01}\\
x_{10} & x_{11}\\
\end{bmatrix}=\begin{bmatrix}
\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\
\end{bmatrix}.
$$
If we then compute the expectation value
$$
\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix}
x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\
x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\
\end{bmatrix},
$$
which is just
$$
\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\
\end{bmatrix},
$$
where we wrote
$$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$
to indicate that this the covariance of the vectors \( \boldsymbol{x} \) of the design/feature matrix \( \boldsymbol{X} \).
It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \).
We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as
$$
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}].
$$
Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \). These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).
Assume also that there is a transformation \( \boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).
That is we have
$$
\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}^T\boldsymbol{X}^T\boldsymbol{X}T\boldsymbol{S}]=\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S},
$$
since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S} \) from the left we have
$$
\boldsymbol{S}\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S},
$$
and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that
$$
\boldsymbol{S}_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}_i.
$$
In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
The eigenvalues tell us then how much we need to stretch the corresponding eigenvectors. Dimensions with large eigenvalues have thus large variations (large variance) and define therefore useful dimensions. The data points are more spread out in the direction of these eigenvectors. Smaller eigenvalues mean on the other hand that the corresponding eigenvectors are shrunk accordingly and the data points are tightly bunched together and there is not much variation in these specific directions. Hopefully then we could leave it out dimensions where the eigenvalues are very small. If \( p \) is very large, we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) features/predictors.
Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here.
$$
\boldsymbol{X}=\begin{bmatrix}
x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
\dots & \dots & \dots & \dots \dots & \dots \\
x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
\end{bmatrix},
$$
We will use a simple example first with two-dimensional data drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below):
$$
\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
2 & 2
\end{bmatrix}
$$
Note that the mean refers to each column of data. We will generate \( n = 10000 \) points \( X = \{ x_1, \ldots, x_N \} \) from this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \). This is our design matrix where we have forced the covariance and mean values to take specific values.
The following Python code aids in setting up the data and writing out the design matrix. Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from IPython.display import display
n = 10000
mean = (-1, 2)
cov = [[4, 2], [2, 2]]
X = np.random.multivariate_normal(mean, cov, n)
Now we are going to implement the PCA algorithm. We will break it down into various substeps.
The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
$$
\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
$$
and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
$$
\bar{x}_i = x_i - \mu_n.
$$
When you are done with these steps, print out \( \mu_n \) to verify it is close to \( \mu \) and plot your mean centered data to verify it is centered at the origin! The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function.
df = pd.DataFrame(X)
# Pandas does the centering for us
df = df -df.mean()
# we center it ourselves
X_centered = X - X.mean(axis=0)
Alternatively, we could use the functions we discussed earlier for scaling the data set. That is, we could have used the StandardScaler function in Scikit-Learn, a function which ensures that for each feature/predictor we study the mean value is zero and the variance is one (every column in the design/feature matrix). You would then not get the same results, since we divide by the variance. The diagonal covariance matrix elements will then be one, while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our specific case.
Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation
$$
\begin{equation*}
\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
\end{equation*}
$$
where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \). We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows
print(df.cov())
print(np.cov(X_centered.T))
Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix.
# extract the relevant columns from the centered design matrix of dim n x 2
x = X_centered[:,0]
y = X_centered[:,1]
Cov = np.zeros((2,2))
Cov[0,1] = np.sum(x.T@y)/(n-1.0)
Cov[0,0] = np.sum(x.T@x)/(n-1.0)
Cov[1,1] = np.sum(y.T@y)/(n-1.0)
Cov[1,0]= Cov[0,1]
print("Centered covariance using own code")
print(Cov)
plt.plot(x, y, 'x')
plt.axis('equal')
plt.show()
Depending on the number of points \( n \), we will get results that are close to the covariance values defined above. The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed.
Now we are ready to solve for the principal components! To do so we diagonalize the sample covariance matrix \( \Sigma \). We can use the function np.linalg.eig to do so. It will return the eigenvalues and eigenvectors of \( \Sigma \). Once we have these we can perform the following tasks:
$$
\begin{equation*}
x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
\end{equation*}
$$
where \( v_0 \) is the first principal component.
Collecting all these steps we can write our own PCA function and compare this with the functionality included in Scikit-Learn.
The code here outlines some of the elements we could include in the analysis. Feel free to extend upon this in order to address the above questions.
# diagonalize and obtain eigenvalues, not necessarily sorted
EigValues, EigVectors = np.linalg.eig(Cov)
# sort eigenvectors and eigenvalues
#permute = EigValues.argsort()
#EigValues = EigValues[permute]
#EigVectors = EigVectors[:,permute]
print("Eigenvalues of Covariance matrix")
for i in range(2):
print(EigValues[i])
FirstEigvector = EigVectors[:,0]
SecondEigvector = EigVectors[:,1]
print("First eigenvector")
print(FirstEigvector)
print("Second eigenvector")
print(SecondEigvector)
#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2Dsl = pca.fit_transform(X)
print("Eigenvector of largest eigenvalue")
print(pca.components_.T[:, 0])
This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?
We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{s}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
We are almost there, we have obtained a relation between minimizing the reconstruction error and the variance and the covariance matrix. Minimizing the error is equivalent to maximizing the variance of the projected data.
We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by \( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize
$$
J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0).
$$
Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain
$$
\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0,
$$
meaning that
$$
\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0.
$$
The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is
$$
\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0.
$$
If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \).
The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be established by applying the above arguments and using the fact that our basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein.
For more details, see for example Vidal, Ma and Sastry, chapter 2.
For a detailed demonstration of the geometric interpretation, see Vidal, Ma and Sastry, section 2.1.2.
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
The following Python code uses NumPy’s svd() function to obtain all the principal components of the training set, then extracts the first two principal components. First we center the data using either pandas or our own code
import numpy as np
import pandas as pd
from IPython.display import display
np.random.seed(100)
# setting up a 10 x 5 vanilla matrix
rows = 10
cols = 5
X = np.random.randn(rows,cols)
df = pd.DataFrame(X)
# Pandas does the centering for us
df = df -df.mean()
display(df)
# we center it ourselves
X_centered = X - X.mean(axis=0)
# Then check the difference between pandas and our own set up
print(X_centered-df)
#Now we do an SVD
U, s, V = np.linalg.svd(X_centered)
c1 = V.T[:, 0]
c2 = V.T[:, 1]
W2 = V.T[:, :2]
X2D = X_centered.dot(W2)
print(X2D)
PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t forget to center the data first.
Once you have identified all the principal components, you can reduce the dimensionality of the dataset down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components. Selecting this hyperplane ensures that the projection will preserve as much variance as possible.
W2 = V.T[:, :2]
X2D = X_centered.dot(W2)
Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note that it automatically takes care of centering the data):
#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2D = pca.fit_transform(X)
print(X2D)
After fitting the PCA transformer to the dataset, you can access the principal components using the components variable (note that it contains the PCs as horizontal vectors, so, for example, the first principal component is equal to
pca.components_.T[:, 0]
Another very useful piece of information is the explained variance ratio of each principal component, available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s variance that lies along the axis of each principal component.
We can now repeat the above but applied to real data, in this case our breast cancer data. Here we compute performance scores on the training data using logistic regression.
import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
logreg = LogisticRegression()
logreg.fit(X_train, y_train)
print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
# We scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Then perform again a log reg fit
logreg.fit(X_train_scaled, y_train)
print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2D_train = pca.fit_transform(X_train_scaled)
# and finally compute the log reg fit and the score on the training data
logreg.fit(X2D_train,y_train)
print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.
Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). Unless, of course, you are reducing dimensionality for data visualization — in that case you will generally want to reduce the dimensionality down to 2 or 3. The following code computes PCA without reducing dimensionality, then computes the minimum number of dimensions required to preserve 95% of the training set’s variance:
pca = PCA()
pca.fit(X)
cumsum = np.cumsum(pca.explained_variance_ratio_)
d = np.argmax(cumsum >= 0.95) + 1
You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
pca = PCA(n_components=0.95)
X_reduced = pca.fit_transform(X)
One problem with the preceding implementation of PCA is that it requires the whole training set to fit in memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new instances arrive).
Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic algorithm that quickly finds an approximation of the first d principal components. Its computational complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the previous algorithms when \( d \) is much smaller than \( n \).
The kernel trick is a mathematical technique that implicitly maps instances into a very high-dimensional space (called the feature space), enabling nonlinear classification and regression with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature space corresponds to a complex nonlinear decision boundary in the original space. It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a twisted manifold. For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
from sklearn.decomposition import KernelPCA
rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
X_reduced = rbf_pca.fit_transform(X)
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
Here are some of the most popular:
In general terms cluster analysis, or clustering, is the task of grouping a data-set into different distinct categories based on some measure of equality of the data. This measure is often referred to as a metric or similarity measure in the literature (note: sometimes we deal with a dissimilarity measure instead). Usually, these metrics are formulated as some kind of distance function between points in a high-dimensional space.
The simplest, and also the most common is the Euclidean distance.
The simplest of all clustering algorithms is the k-means algorithm , sometimes also referred to as Lloyds algorithm. It is the simplest and also the most common. From its simplicity it obtains both strengths and weaknesses. These will be discussed in more detail later. The \( k \)-means algorithm is a centroid based clustering algorithm.
Assume, we are given \( n \) data points and we wish to split the data into \( K < n \) different categories, or clusters. We label each cluster by an integer
$$ k\in\{1, \cdots, K \}.
$$
In the basic k-means algorithm each point is assigned to only one cluster \( k \), and these assignments are non-injective i.e. many-to-one. We can think of these mappings as an encoder \( k = C(i) \), which assigns the \( i \)-th data-point \( \bf x_i \) to the \( k \)-th cluster.
\( k \)-means algorithm in words:
We assume we have \( n \) data-points
$$
\begin{equation}\tag{1}
\boldsymbol{x_i} = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p.
\end{equation}
$$
which we wish to group into \( K < n \) clusters. For our dissimilarity measure we use the squared Euclidean distance
$$
\begin{equation}\tag{2}
d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2
= ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2
\end{equation}
$$
We define the so called within-cluster point scatter which gives us a measure of how close each data point assigned to the same cluster tends to be to the all the others.
$$
\begin{equation}\tag{3}
W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
\sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) =
\sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2
\end{equation}
$$
where \( \boldsymbol{\overline{x_k}} \) is the mean vector associated with the \( k \)-th cluster, and \( N_k = \sum_{i=1}^nI(C(i) = k) \), where the \( I() \) notation is similar to the Kronecker delta (Commonly used in statistics, it just means that when \( i = k \) we have the encoder \( C(i) \)). In other words, the within-cluster scatter measures the compactness of each cluster with respect to the data points assigned to each cluster. This is the quantity that the \( k \)-means algorithm aims to minimize. We refer to this quantity \( W(C) \) as the within cluster scatter because of its relation to the total scatter.
We have
$$
\begin{equation}\tag{4}
T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n
\sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
= \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
\Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
+ \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big).
\end{equation}
$$
This is a quantity that is conserved throughout the \( k \)-means algorithm. It can be thought of as the total amount of information in the data, and it is composed of the aforementioned within-cluster scatter and the between-cluster scatter \( B(C) \). In methods such as principle component analysis the total scatter is not conserved.
Given a cluster mean \( \boldsymbol{m_k} \) we define the total cluster variance
$$
\begin{equation}\tag{5}
\min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2
\end{equation}
$$
Now we have all the pieces necessary to formally revisit the \( k \)-means algorithm.
The \( k \)-means clustering algorithm goes as follows
$$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$
Let us now program the most basic version of the algorithm using nothing but Python with numpy arrays. This code is kept intentionally simple to gradually progress our understanding. There is no vectorization of any kind, and even most helper functions are not utilized.
We need first a dataset to do our cluster analysis on. In our case this is a plain vanilla data set using random numbers using a Gaussian distribution.
import time
import numpy as np
import tensorflow as tf
from matplotlib import image
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from IPython.display import display
np.random.seed(2021)
Next we define functions, for ease of use later, to generate Gaussians and to set up our toy data set.
def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),
sample_variance=1):
"""
Very simple custom function to generate gaussian distributed point clusters
with variable dimension, number of points, means in each direction
(must match dim) and sample variance.
Inputs:
dim (int)
n_points (int)
mean_vector (np.array) (where index 0 is x, index 1 is y etc.)
sample_variance (float)
Returns:
data (np.array): with dimensions (dim x n_points)
"""
mean_matrix = np.zeros(dim) + mean_vector
covariance_matrix = np.eye(dim) * sample_variance
data = np.random.multivariate_normal(mean_matrix, covariance_matrix,
n_points)
return data
def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,
return_data=True):
"""
Toy model to illustrate k-means clustering
"""
data1 = gaussian_points(mean_vector=np.array([5, 5]))
data2 = gaussian_points()
data3 = gaussian_points(mean_vector=np.array([1, 4.5]))
data4 = gaussian_points(mean_vector=np.array([5, 1]))
data = np.concatenate((data1, data2, data3, data4), axis=0)
if plotting:
fig, ax = plt.subplots()
ax.scatter(data[:, 0], data[:, 1], alpha=0.2)
ax.set_title('Toy Model Dataset')
plt.show()
if return_data:
return data
data = generate_simple_clustering_dataset()
With the above dataset we start implementing the \( k \)-means algorithm.
n_samples, dimensions = data.shape
n_clusters = 4
# we randomly initialize our centroids
np.random.seed(2021)
centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]
distances = np.zeros((n_samples, n_clusters))
# first we need to calculate the distance to each centroid from our data
for k in range(n_clusters):
for n in range(n_samples):
dist = 0
for d in range(dimensions):
dist += np.abs(data[n, d] - centroids[k, d])**2
distances[n, k] = dist
# we initialize an array to keep track of to which cluster each point belongs
# the way we set it up here the index tracks which point and the value which
# cluster the point belongs to
cluster_labels = np.zeros(n_samples, dtype='int')
# next we loop through our samples and for every point assign it to the cluster
# to which it has the smallest distance to
for n in range(n_samples):
# tracking variables (all of this is basically just an argmin)
smallest = 1e10
smallest_row_index = 1e10
for k in range(n_clusters):
if distances[n, k] < smallest:
smallest = distances[n, k]
smallest_row_index = k
cluster_labels[n] = smallest_row_index
fig = plt.figure()
ax = fig.add_subplot()
unique_cluster_labels = np.unique(cluster_labels)
for i in unique_cluster_labels:
ax.scatter(data[cluster_labels == i, 0],
data[cluster_labels == i, 1],
label = i,
alpha = 0.2)
ax.scatter(centroids[:, 0], centroids[:, 1], c='black')
ax.set_title("First Grouping of Points to Centroids")
plt.show()
So what do we have so far? We have 'picked' \( k \) centroids at random from our data points. There are other ways of more intelligently choosing their initializations, however for our purposes randomly is fine. Then we have initialized an array 'distances' which holds the information of the distance, or dissimilarity, of every point to of our centroids. Finally, we have initialized an array 'cluster_labels' which according to our distances array holds the information of to which centroid every point is assigned. This was the first pass of our algorithm. Essentially, all we need to do now is repeat the distance and assignment steps above until we have reached a desired convergence or a maximum amount of iterations.
max_iterations = 100
tolerance = 1e-8
for iteration in range(max_iterations):
prev_centroids = centroids.copy()
for k in range(n_clusters):
# this array will be used to update our centroid positions
vector_mean = np.zeros(dimensions)
mean_divisor = 0
for n in range(n_samples):
if cluster_labels[n] == k:
vector_mean += data[n, :]
mean_divisor += 1
# update according to the k means
centroids[k, :] = vector_mean / mean_divisor
# we find the dissimilarity
for k in range(n_clusters):
for n in range(n_samples):
dist = 0
for d in range(dimensions):
dist += np.abs(data[n, d] - centroids[k, d])**2
distances[n, k] = dist
# assign each point
for n in range(n_samples):
smallest = 1e10
smallest_row_index = 1e10
for k in range(n_clusters):
if distances[n, k] < smallest:
smallest = distances[n, k]
smallest_row_index = k
cluster_labels[n] = smallest_row_index
# convergence criteria
centroid_difference = np.sum(np.abs(centroids - prev_centroids))
if centroid_difference < tolerance:
print(f'Converged at iteration {iteration}')
break
elif iteration == max_iterations:
print(f'Did not converge in {max_iterations} iterations')
We now have a simple , un-optimized \( k \)-means clustering implementation. Lets plot the final result
fig = plt.figure()
ax = fig.add_subplot()
unique_cluster_labels = np.unique(cluster_labels)
for i in unique_cluster_labels:
ax.scatter(data[cluster_labels == i, 0],
data[cluster_labels == i, 1],
label = i,
alpha = 0.2)
ax.scatter(centroids[:, 0], centroids[:, 1], c='black')
ax.set_title("Final Result of K-means Clustering")
plt.show()
def naive_kmeans(data, n_clusters=4, max_iterations=100, tolerance=1e-8):
start_time = time.time()
n_samples, dimensions = data.shape
n_clusters = 4
#np.random.seed(2021)
centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]
distances = np.zeros((n_samples, n_clusters))
for k in range(n_clusters):
for n in range(n_samples):
dist = 0
for d in range(dimensions):
dist += np.abs(data[n, d] - centroids[k, d])**2
distances[n, k] = dist
cluster_labels = np.zeros(n_samples, dtype='int')
for n in range(n_samples):
smallest = 1e10
smallest_row_index = 1e10
for k in range(n_clusters):
if distances[n, k] < smallest:
smallest = distances[n, k]
smallest_row_index = k
cluster_labels[n] = smallest_row_index
for iteration in range(max_iterations):
prev_centroids = centroids.copy()
for k in range(n_clusters):
vector_mean = np.zeros(dimensions)
mean_divisor = 0
for n in range(n_samples):
if cluster_labels[n] == k:
vector_mean += data[n, :]
mean_divisor += 1
centroids[k, :] = vector_mean / mean_divisor
for k in range(n_clusters):
for n in range(n_samples):
dist = 0
for d in range(dimensions):
dist += np.abs(data[n, d] - centroids[k, d])**2
distances[n, k] = dist
for n in range(n_samples):
smallest = 1e10
smallest_row_index = 1e10
for k in range(n_clusters):
if distances[n, k] < smallest:
smallest = distances[n, k]
smallest_row_index = k
cluster_labels[n] = smallest_row_index
centroid_difference = np.sum(np.abs(centroids - prev_centroids))
if centroid_difference < tolerance:
print(f'Converged at iteration {iteration}')
print(f'Runtime: {time.time() - start_time} seconds')
return cluster_labels, centroids
print(f'Did not converge in {max_iterations} iterations')
print(f'Runtime: {time.time() - start_time} seconds')
return cluster_labels, centroids