Chapter 6 Conclusions
A. CUSUM Algorithm
The detection of a change-point is performed by the cumulative-sum (CUSUM) algorithm on the generated squared residuals k(6). The principle of CUSUM stems from a stochastic hypothesis testing method (Chen, 1999).
Cumulative sum: S) = s«r i ¬ ó k 6 ó k 6 # )# A1
In this expression, % denotes the present time instant; k(6) is the residual signal; õ and õ are the mean values under the two hypothesis and , assuming that the variance &' is the same. (k(6)) is a log-likelihood ratio of the residual.
A1. Principle of the CUSUM algorithm
The diagnostic signal k(6) has the nature of the Gaussian distribution, due to inclusion of measurement noises that normally obey the Gaussian distribution. Residuals generally are stochastic. Thus, according to this reasonable assumption, the residual k(6) has the Gaussian probability density with mean µ and variance σ:
ó(k) = 1
√2À&exp ÷−
(k − õ)'
2&' ø ù2
The residual signal k(6) carries the information of whether there is a change-point or not. The following hypothesis can be used to evaluate the residuals:
: õ = õ i d : õ = õ ù3
A zero hypothesis is for the case where there is no change, and an alternative hypothesis for the case where there is a change-point. The log-likelihood ratio of the residual k(6) is defined as:
s«r i ¬ ó k 6
ó k 6 #
100
Substituting the expressions of ó (k(6)) and ó (k(6)) from equation A2 into equation A4 yields s«r i ¬ ln 1 √2À&exp ª− (k(6) − õ ) ' 2&' ¯ 1 √2À&exp ª− (k(6) − õ ) ' 2&' ¯ s«r i ¬ µ σ− µ' ªr i − µ < µ2 ¯ A5 The next step is to calculate the accumulative sum equation A1
S) = s«r i ¬ )#
The valid test consists of the following decisions: if g) ≤ h, accept H ;
if g) > h, accept H
where the threshold h is a positive number. g) is the decision function with the expression as
g) S0,), SS)) > 0 < 0 A6
An efficient way to implement the decision function g) is to use the recursive form that can be written into a maximum-choosing function as follows:
g) = max«g)Q + s«r i ¬, 0¬ A7
The idea of setting a threshold is motivated because g) will increase when a change- point occurs and decrease when the change disappears. Only the contributions to the cumulative sum that add up to the threshold h must be taken into account in order to determine the decision g). During the no change period, the value of g) is around zero. An alarm is set at the time ta when the fault is detected.
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A2. Enhanced Algorithm for the CUSUM Test
In the real system, the change-point may happen either with a positive µ or a negative µ . The CUSUM algorithm is improved to test the residual signals in two sides – both positive and negative:
H : μ = μ = 0; μ = μ = ±υ A9
This means that the zero hypothesis is still μ and the alternative hypothesis is two- sided μ = ±υ.
In the two-sided CUSUM algorithm, the decision functions are two equations - one for the positive mean of the residual and the other for the negative mean of the residual signal, both under a change-point.
g)D = max«g)QD + s«r i ¬, 0¬, where s«r i ¬ συ'-r i −υ2® A10 g)Q = max«g)QQ + s«r i ¬, 0¬, where s«r i ¬ −συ'-r i −υ2® A11 It is noted that the decision is made when the prior fault happens. This means that
both g)D and g)Q will increase when the change-point occurs; however, the detection time depends on which reaches the threshold earlier.
ta = min«i ∶ (g)D ≥ h) ∨ (g)Q ≥ h)¬ A12 where ta is the time of detecting the fault and an alarm is set. h is the threshold both
for g)D and g)Q.
A3. Initialization of Parameters in the CUSUM Test
Known from the algorithm presented above, μ , μ , σ and h are the parameters designed before the test. First, the variance σ' has been assumed the same in both hypotheses. For the no change-point case H , the mean μ and the variance σ' can be calculated from the residual signal r i . Normally, μ can be chosen from the change of the residual ri with respect to the time when the change happens.
The threshold h can be chosen either by an empirical value if drawing the figure of g) or by a theoretical method called an average-run-length (ARL) function. In the CUSUM algorithm, the theoretical method aims at make a trade-off between the time interval of two change-point detections and the interval time of two false alarms. One
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can expect the shortest interval between two detections of change-points and longest interval between two false alarms. The ARL function is the expected value of the alarm time instant
L = E(t ) A13 where L is the value of ARL function and t is the alarm time defined in the equation
A12 for the enhanced CUSUM algorithm.
An approximation for the ARL function calculates the mean μ and the variance σ' of the cumulative sum increment s(r(i))
when H : μ = μ , Eμ (r(i)) = μ then μ = Eμ (s(r(i))) = − 'QîÆ when H : μ = μ , Eμ (r(i)) = μ then μ = Eμ (s(r(i))) = Q Æ
' î
In another way, the mean μ is written as μ = μ = E Æ-s«r(i)¬® = −
μ − μ
2σ' when H : μ = μ
μ = E -s«r(i)¬® = −μ − μ2σ' when H : μ = μ A14 In both cases, the variance σ' is derived from
σ' = (μ − μ )'
2σ' A15
A4. Implementation of the CUSUM Algorithm
The CUSUM algorithm for detection of a change in the mean of a Gaussian distributed residual signal k(6) can be implemented as follows:
At each sample time 6
1. Acquire the new data of the residual signal k(6).
2. Compute * using equation A7 for a one-sided hypothesis or equations A10 and A11 for a two sided hypothesis.
3. Issue an alarm if g) (g)D or g)Q ) > h at time t and initialize the decision function g) (or g)D and g)Q ) back to zero.
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A5. Initialization
1. The mean μ and variance σ' of the residual signal are calculated in the nominal working mode.
2. The mean μ of the residual signal can be chosen based on the experiment data of the residual signal under a fault.
3. The threshold h is estimated to meet either the specified mean time for detection τ or the specified mean time between false alarms T.