Aerospace and Electronic Systems Magazine April 2017 - 32
Supervised Learning Algorithms for Spacecraft Attitude Determination and Control System Health Monitoring
prior information about the importance of the variables, it may
be desirable to scale some variables up or down, by modifying
their UV weights.
3. Train PLS-DA classifier using the labeled training data. Training samples X = [X1, X2, ....., XN]; class labels Y = [Y1, Y2,.....,
YN]. Then calculate the covariance matrix S as measure of the
spread of data between variables and maximize the separations
between classes (multidimensional concept):
S X ,Y
X Yi Y
Effect of using T2 and Q statistics as fault detection indices.
where X and Y represents the mean of the set X and Y, respectively, n observation number and Λ is a diagonal matrix Λ =
diag(λ1 > λ2 >.... > λm) containing the eigenvalues/coefficients
in a decreasing order (λ1 > λ2 >.... > λm), where m is the original
variables number. The weights on each variable are given by
the indicated loadings vector coefficients (eigenvalues/coefficients). Subsequently, choose how many PCs used. The goodness of the PLS-DA model depends on a good choice of how
many PCs are retained.
4. Construct the model for classification under the diverse operating conditions. Then, compute the ranking of all features with a
certain criterion in terms of their contribution to classification.
As clarified previously, the idea here is that one may consider a
feature discriminative if it significantly influences the width of
the margin of the PLS-DA.
5. Finally, the model is used along with one of the detection
guides of the Hotelling's T2 and the Q statistic, also known as
squared prediction error (SPE), to detect faults for new data
samples (if the detection index falls outside the control limits,
which are defined by the thresholds associated with these indices). The equations for calculating T2 and the Q are itemized in
the following step:
T2 measures variations in the PCs at different time samples (variation of the systematic part of the PLS-DA model) and is defined as :
PLS-DA fault detection algorithm framework.
xT Pˆ 1Pˆ T x
where the matrix Pˆ is the matrix containing the l retained
eigenvectors, the matrix Λ is a diagonal matrix containing
the eigenvalues associated with the l retained PCs. For
new testing data, when the value of T2 exceeds the value
of the threshold, T2μ a fault is declared .
The Q statistic measures the projection of a data sample
on the residual subspace (variation not captured by the
1 Pˆ Pˆ x
When a vector of new data is available, the Q statistic is
calculated and compared with the threshold value Qμ. If
the confidence limit is violated, a fault is declared. The
concept of using the Q and T2 statistics for fault detection can be explained graphically with a simple example
shown in Figure 4 in which two faults occur.
Faults with small magnitudes can easily exceed the Q threshold, but not the T2 threshold, which makes the Q statistic usually
more sensitive than T2 for this type of fault. This explains why the
Q statistic is considered preferable for fault detection than T2 and
in our article only the Q statistic is used as a benchmark for fault
detection. A more detailed discussion, which also highlighs some
mathematical concepts of T2 and Q statistics calculations, can be
found in , . Figure 5 shows the framework of the PLS-DA
fault detection algorithm.
IEEE A&E SYSTEMS MAGAZINE