# Advanced numerical computation of χ2 tests for fault detection

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```					                                                                                                          Introduction
Advanced numerical computation of χ2 tests                     • Typical fault detection and isolation (FDI) procedure:

for fault detection and isolation                               ◦ residual generation

◦ residual evaluation
`
Qinghua Zhang and Michele Basseville
• Evaluation of Gaussian residuals
IRISA (INRIA & CNRS)
Rennes, France
◦ for parametric change in linear systems

◦ for small parametric change in nonlinear systems
(local asymptotic approach to change detection)

• Changes in the mean of a Gaussian vector — χ2-tests

4th June 2003                                                page 1   4th June 2003                                                            page 2

Basic formulas for the χ2-tests
Gaussian residual evaluation — χ2-tests                                                                      ηa                        Iaa Iab
z ∼ N (M η, Σ) I = M T Σ−1M η =                       M = [Ma Mb] I =
ηb                        Iba Ibb
Consider m-dimensional residual z ∼ N (M η, Σ)
with M ∈ Rm×n, η ∈ Rn, Σ ∈ Rm×m, m ≥ n.                               Fault detection (global test)

Fault detection
T
t = zT Σ−1M (M T Σ−1Ma)−1Ma Σ−1z
H0 : η = 0 against H1 : η = 0
Fault isolation by sensitivity test
ηa
Fault isolation for some partition η =                                                ˜
ηb                                                          T          T
ta = zT Σ−1Ma(Ma Σ−1Ma)−1Ma Σ−1z
H0 : ηa = 0 against H1 : ηa = 0
or equivalently
2
These hypothesis testing problems lead to χ -tests.                                                    ˜       T
ζa = Ma Σ−1z
˜    ˜T     ˜
ta = ζa I−1ζa
aa

4th June 2003                                                page 3   4th June 2003                                                            page 4
Basic formulas of the χ2-tests (contd.)                                    Advanced numerical computation – Global test

Fault isolation by min-max test                                                               t = zT Σ−1M (M T Σ−1M )−1M T Σ−1z
• Use pseudo-inverse for Σ−1 if badly conditioned
˜
ζa =       T
Ma Σ−1z                                                                                             1
• Compute t as a square : t = (M T Σ−1M )− 2 M T Σ−1z   2
˜
ζb =       T
Mb Σ−1z
• Avoid the inverse involving M .
∗
ζa =   ˜
ζa − IabI−1ζb  ˜
bb                                        Proposed solution:
I∗ =
a     Iaa − IabI−1Iba
bb                                                    1
• let Γ = Σ− 2 (with pseudo-inverse if necessary),
t∗ =
a     ζa T I∗ −1ζa
∗
a
∗

• QR decomposition of ΓM : ΓM = QR with QT Q = I,
• Then
Numerical difﬁculties: when the matrices to be inverted are badly                                  t = zT ΓQR(RT QT QR)−1RT QT Γz
conditioned, these basic formulas can lead to large numerical er-                                    = zT ΓQR(RT R)−1RT QT Γz
rors. It is thus important to develop advanced numerical algo-
rithms.                                                                                              = QT Γz 2

4th June 2003                                                          page 5   4th June 2003                                                        page 6

Advanced numerical computation – Minmax test

More computations are involved.
Advanced numerical computation – Sensitivity test
QR decompositions:
˜              T          T
ta = zT Σ−1Ma(Ma Σ−1Ma)−1Ma Σ−1z                                                                   ΓMa = QaRa
ΓMb = QbRb
Same as global test (M being replaced by Ma).
T
(I − QbQb )Qa = QcRc
t∗ = QT Γz
a    c
2
QR decomposition of ΓMa: ΓMa = QaRa with        QT Qa
a      = I,
then
A non-trivial step for deriving the algorithm:
˜
ta = QT Γz
a
2
(I − QT QbQT Qa) = QT (I − QbQT )(I − QbQT )Qa
a      b           a           b b

Remark: SVD can be used instead of QR decomposition.
4th June 2003                                                          page 7   4th June 2003                                                        page 8
A numerical example
Gaussian vector: dim(z) = 9, dim(η) = 5, dim(ηa) = 2.
Condition number of Σ: 3.4 × 1010.                                              0.06

Histograms are based on 10000 random realizations.                              0.05

0.04
0.06
0.03
0.05
0.02
0.04
0.01
0.03
0
0.02                                                                                     0   10   20   30   40    50   60    70    80    90   100

0.01

0
Advanced minmax test (solid line) and χ 2(2, 44.955) density
0   10   20     30    40     50    60     70    80     90   100              function (dashed line).
Basic minmax test (solid line) and χ 2(2, 44.955) density function
(dashed line).

4th June 2003                                                            page 9   4th June 2003                                                     page 10

Conclusion

• χ2-tests are frequently used for residual evaluation.

• Advanced numerical algorithms can signiﬁcantly improve the
numerical accuracy of badly conditioned problems.

4th June 2003                                                           page 11

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