Fault Diagnosis for Power System Transmission Line Based on PCA and SVMs - Intelligent Computing for Sustainable Energy and Environment - page 526

An eigenvalue decomposition of the matrix S is

S = VΛV T

(3)

which reveals the correlation structure for the covariance matrix, where Λ is a
diagonal and V is orthogonal. The Principal Components (PCs) are represented
by the loading and score vectors so that PCA can decompose the observation
matrix X as:

X = t 1 p 1 + t 2 p 2 + ... + t k p k + E = TP T + E (4)

t i ,i =1 ,...,k are vectors, also named scores of the data, which contain informa-
tion on how the samples are related to each other. p i ,i =1 ,...,k are the loadings,
also are the eigenvectors of the covariance matrix, E is the residual matrix. The
important statistic for PCA monitoring is given by Hotelling's T 2 , which is the
sum of normalized squared scores defined as [5],
T i = t i λ − 1 t i = x i pλ − 1 p T x i (5)
where t i is the ith row of k score vectors from PCA model, and λ − 1 is a diagonal
matrix containing the inverse eigenvalues associated with the k eigenvectors. The
equation represents the distance in principal component model subspace, which
shows that T 2 is a measure of the variation within the training data set.
However, monitoring the output variable via T 2 based on the first k PCs is
not sucient. If a totally new type of special event occurs which was not present
in the reference data used to develop the PCA model, the new observations
X new will move off the plane. Such new events can be detected by computing
the squared prediction errors (SPE) of the residuals of new observations [6].
SPE X new = i =1 ( X new,i −

X new,i ) 2 (6)
This statistic is referred to as the Q-statistic, or distance to the model. It rep-
resents the squared perpendicular distance of a new observation from the plane.
When the process is normal, this value should be small. Upper control limit for
this statistic can be computed from training data set. Q statistic indicates how
well each sample conforms to the PCA model, it is a measure of the amount of
variation in each sample not captured by the k principal components retained
in the model.

2.2 Support Vector Machine

SVM has become an increasingly popular technique for machine learning activ-
ities including classification, regression, and outlier detection [7]. The idea of
using SVMs for separating two classes is to find support vectors, which refers to
a representative of training data points, to define the bounding planes, in which
the margin between the both planes is maximized [8].
SVM maps the input vectors into some high dimensional feature space through
nonlinear methods. In this space a linear decision surface is constructed with
special properties that ensure high generalization ability of the network. The
number of support vectors increases with the complexity of the problem.