## Principal components analysis (PCA) using a sequential method

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# Limitations: does not handle missing data
import numpy as np
# Download the CSV data file from:
# http://datasets.connectmv.com/info/silicon-wafer-thickness
raw = np.genfromtxt('silicon-wafer-thickness.csv', delimiter=',', skip_header=1)
N, K = raw.shape
# Preprocessing: mean center and scale the data columns to unit variance
X = raw - raw.mean(axis=0)
X = X / X.std(axis=0)
# Verify the centering and scaling
X.mean(axis=0) # array([ -3.92198351e-17, -1.74980803e-16, ...
X.std(axis=0) # [ 1. 1. 1. 1. 1. 1. 1. 1. 1.]
# We are going to calculate only 2 principal components
A = 2
# We could of course use SVD ...
u, d, v = np.linalg.svd(X)
# Transpose the "v" array from SVD, which contains the loadings, but retain
# only the first A columns
svd_P = v.T[:, range(0, A)]
# Compute the scores from the loadings:
svd_T = np.dot(X, svd_P)
# But what if we really only wanted calculate A=2 components (imagine SVD on
# a really big data set where N and K >> 1000). This is why will use the NIPALS,
# nonlinear iterative partial least squares, method.
nipals_T = np.zeros((N, A))
nipals_P = np.zeros((K, A))
tolerance = 1E-10
for a in range(A):
t_a_guess = np.random.rand(N, 1)*2
t_a = t_a_guess + 1.0
itern = 0
# Repeat until the score, t_a, converges, or until a maximum number of
# iterations has been reached
while np.linalg.norm(t_a_guess - t_a) > tolerance or itern < 500:
# 0: starting point for convergence checking on next loop
t_a_guess = t_a
# 1: Regress the scores, t_a, onto every column in X; compute the
# regression coefficient and store it in the loadings, p_a
# i.e. p_a = (X' * t_a)/(t_a' * t_a)
p_a = np.dot(X.T, t_a) / np.dot(t_a.T, t_a)
# 2: Normalize loadings p_a to unit length
p_a = p_a / np.linalg.norm(p_a)
# 3: Now regress each row in X onto the loading vector; store the
# regression coefficients in t_a.
# i.e. t_a = X * p_a / (p_a.T * p_a)
t_a = np.dot(X, p_a) / np.dot(p_a.T, p_a)
itern += 1
# We've converged, or reached the limit on the number of iteration
# Deflate the part of the data in X that we've explained with t_a and p_a
X = X - np.dot(t_a, p_a.T)
# Store result before computing the next component
nipals_T[:, a] = t_a.ravel()
nipals_P[:, a] = p_a.ravel()
# PCA also has two very important outputs we should calculate:
# The SPE_X, squared prediction error to the X-space is the residual distance
# from the model to each data point.
SPE_X = np.sum(X**2, axis=1)
# And Hotelling's T2, the directed distance from the model center to
# each data point.
inv_covariance = np.linalg.inv(np.dot(nipals_T.T, nipals_T)/N)
Hot_T2 = np.zeros((N, 1))
for n in xrange(N):
Hot_T2[n] = np.dot(np.dot(nipals_T[n,:], inv_covariance), nipals_T[n,:].T)
# You can verify the NIPALS and SVD results are the same:
# (you may find that the signs have flipped, but this is still correct)
nipals_T / svd_T
nipals_P / svd_P
# But since PCA is such a visual tool, we should plot the SPE_X and
# Hotelling's T2 values
from matplotlib import pylab
pylab.plot(SPE_X, 'r.-') # see how observation 154 is inconsistent
pylab.plot(Hot_T2, 'k.-') # observations 38, 39,110, and 154 are outliers
# And we should also plot the scores:
pylab.figure()
pylab.plot(nipals_T[:,0], nipals_T[:,1], '.')
pylab.grid()
# Confirm the outliers in the raw data, giving one extra point above and below
raw[37:41,:]
raw[109:112,:]
raw[153:156,:]
# Next step for you: exclude observation 38, 39, 110 and 154 and
# rebuild the PCA model. Can you interpret what the loadings, nipals_P, mean?
``` |

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The singular value decomposition is usually presented as the way to calculate the PCA decomposition of a data matrix.

The NIPALS algorithm is a very computationally tractable way of calculating PCA for large data sets, since we only calculate the components we actually need; whereas SVD calculates all components in one go.

The nonlinear iterative partial least squares (NIPALS) method is more informative, as the interpretation of what the loadings and scores really mean becomes apparent when examining the above code.

For example, in step 1 of the `while` loop we see the loading, \(p_a\) contains the regression coefficients when regressing the score vector, \(t_a\), onto each column in \(\mathbf{X}\). So at convergence of the while loop, any columns in \(\mathbf{X}\) that are strongly correlated, will have similar loading values, \(p_a\), for those columns.

In step 3 of the `while` loop the loading, \(p_a\), is regressed onto each row in \(\mathbf{X}\) and the regression coefficient is stored at the relevant row in the score, \(t_a\). At convergence of the while loop, any rows in \(\mathbf{X}\) that are strongly correlated (aligned with) that loading will have have a large positive or negative score value. Row entries in \(\mathbf{X}\) that are not explained (i.e. unrelated) to that \(p_a\)
loading will have a near-zero score value in \(t_a\).

*Still to come*

- Calculating confidence limits for SPE and Hotelling’s \(T^2\) to determine which points are likely outliers

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