ACS6124 Multisensor and Decision Systems Part I: Multisensor Systems
Lecture 4: Sensor Signal Detection
George Konstantopoulos g.konstantopoulos@sheffield.ac.uk
Automatic Control and Systems Engineering The University of Sheffield
(lecture notes produced by Dr. Inaki Esnaola and Prof. Visakan Kadirkamanathan)
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Change Detection for Quality Control
Manufacturing processes produce batches of parts and products that must satisfy stringent dimensional and other functional requirements.
Bosch: RoMulus Project
Material characteristics, manufacturing process parameters and conditions will affect the particular product functional parameter.
Product functional parameter is measured from random samples in the batch.
Using the measurements, a decision is required whether to accept reject the batch of products as satisfying quality requirements.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Change Detection with Collection of Measurements
The measurement model is yt = xt + vt , and our assumptions are:
xt = θ0 when the system is normal
vt ∼ N(0,σ02) when the system is normal
vt is independent and identically distributed (iid)
H0: xt = θ0 i.e. the system is normal H1: xt = θ1 i.e. the system is faulty
Measurements:
Y1:T ={y1,y2,…,yT}
Similarly to the decision theory on single measurement, a decision on joint
likelihood ratio test can be formulated:
S(Y1:T)= p(Y1:T|H0)
p (Y1:T |H1 ) T p(y |H )
p(yt|H1) Neyman-Pearson test using this likelihood ratio can be used for detection.
Given that the noise is independent,
T
∏ p(yt|H0)
T
p(y |H ) t 0
lnS(Y1:T)=lnt=1 T
=ln∏ t 0 t=1 p(yt|H1)
= ∑ln t=1
∏ p(yt|H1) t=1
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Likelihood Ratio Test
Recall that the Neyman-Pearson test involving the likelihood ratio is:
Decide H0 Decide H1
lnS(Y1:T ) =
With vt ∼ N(0,σ02) and ∆ = θ1 −θ0, the term,
ln
> −ln(τ) ≤ −ln(τ)
p(yt|H0) ∆
=−2 yt−θ0−
∆ p(yt|H1) σ0 2
Which results in the expression for the likelihood ratio as,
T∆∆ lnS(Y1:T)=−∑σ2 yt −θ0− 2
t=1 0
When the system has not changed, ln S(Y1:T ) has a positive trend. When the system has changed, ln S(Y1:T ) has a negative trend.
In a batch setting, can we detect if the system has changed given Y1:T ? In a sequential setting, how soon can we detect a change?
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Batch Test
The likelihood ratio test for H0 is:
lnS(Y1:T)=−∑σ2 yt −θ0− 2 >−ln(τ)
T∆∆ t=1 0
which can be re-written as a two-sided test:
T
yt −θ0 T |∆| σ0
∑ < + ln(τ) t=1 σ0 2 σ0 |∆|
Defining st = yt −θ0 and ∆ ̃ = ∆ , σ0 σ0
T T1
̃ ∑st< ∆+ ln(τ)=δ
2 ̃
t = 1 ∆
The term ∑Tt=1 st is computed for the decision making.
The threshold depends on the batch size and the level of change.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Decision statistic
Assume that we have only the information concerning H0 (system is normally operating).
Knowledge is limited to θ0 and σ0.
Threshold δ can be chosen using ROC curve or level of confidence (probability
of false alarm).
We have seen that st =yt−θ0 ∼N(0,1) if the system is normal.
σ0
Let gT = ∑Tt =1 st . The statistical distribution of gT can also be used to determine the
threshold for the decision rule,
TT E[gT]=E ∑st = ∑E[st]=0
t=1 t=1 T2TT T
E[gT2]=E ∑st =E ∑∑stsτ =∑E[st2]=T t=1 t=1τ=1 t=1
Note:E[stsτ]=0ift̸=τasvt isindependent. If the system is normal, g ∼ N(0, T ).
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Control Chart Decision Rule
Equivalently √ ∼N(0,1), T
Decision Rule
gT
√√
WedecideinfavourofH0 if−zα Decision statistic, gT can be computed as:
T T y t − θ 0 ∑ Tt = 1 y t − T θ 0 T y ̄ − T θ 0 y ̄ − θ 0
T
time
break; End If
%γ is a
g(t) = g(t-1) + s(t) + γ small +ve value
If g(t) > 0 Then g(t) = 0
End If
% Decision Rule If g(t) < δ Then
time
break; End If
%γ is a
End For
End For
% Threshold
% Set alarm and note fault on-set
% Threshold
% Set alarm and note fault on-set
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
CUSUM Statistic
CUSUM statistic is designed to look for consistent trends. Recall that,
T∆∆ lnS(Y1:T)=−∑σ2 yt −θ0− 2
t=1 0
When the system has not changed, ln S(Y1:T ) has a positive trend.
When the system has changed, ln S(Y1:T ) has a negative trend. Define
T
GT =lnS(Y1:T)− min lnS(Y1:t)= max [lnS(Y1:T)−lnS(Y1:t)]= max ∑ lnS(yτ)
0≤t≤T 0≤t≤T 0≤t≤T τ=t+1 which can be implemented as,
GT = [GT−1]+ +lnS(yT ), G0 = 0 where [·]+ indicates function with all negative values mapped to 0.
The above recursion translates into gT = [gT −1 ]+ + sT , and g0 = 0 with appropriate scaling.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Generalised Likelihood Ratio Test (GLRT)
The proper formulation of the change detection problem requires consideration of the change on-set time tf , which leads to the following hypothesis testing:
H0: System has not changed : tf > T
H1: System changed at time tf : 0≤tf ≤T
The likelihood of all observations under H0 being true:
T
S(H0 ) = ∏ p (yt |H0 ) t=1
The likelihood of all observations under H1 being true:
tf T S(H1)= ∏p(yt|H0)∏p(yt|H1)
t=1 t=tf The generalised likelihood ratio (GLR) is given by,
̃ S(H0)
T
= min ∏S(yt)
0≤tf ≤T t=tf
ST =
max0≤tf ≤T S(H1)
This can be computed recursively by taking ln and reducing products to sums as, L ̃T = [L ̃T−1]+ +lnS(yT ), L ̃0 = 0
L ̃T =lnS ̃T.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Detection of Signal Patterns
In some cases, the change in the system is associated with a known dynamic signal pattern. The measurement model is yt = xt + vt , and our assumptions are:
xt = z(0) when the system is normal t
xt = z(1) when the system is faulty t
vt ∼ N(0,σ02), independent and identically distributed (iid) Measurements:
Y1:T ={y1,y2,…,yT}
The likelihood ratio is given by,
S(Y1:T)=p(Y |H)= (1)=∏ (1) 1:T 1 p Y1:T|x1:T =z1:T t=1 p yt|xt =zt
The algebra follows in a similar way to the analysis in statistical decision theory, with ∆t = zt1 − zt0 leading to:
pY |x =z(0) T py|x =z(0) p(Y1:T|H0) 1:T 1:T 1:T t t t
T ∆ yt −z0 ∆2 lnS(Y1:T)=−∑ t t − t
t = 1 σ 0 σ 0 2 σ 02
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Matched Filtering based Detection
̃ ∆ t Define ∆t = σ0 , st =
y t − z t0
σ0 , the expression for the log likelihood ratio reduces to:
Decision Rule:
Note that, with z ̃
z(i) t
(i)
t σ0
,
=
T ̃ ∑Tt=1∆2t lnS(Y1:T)=−∑∆tst + 2σ2
t=1
T ̃ <δ DecideH0 ∑ ∆t st ≥ δ DecideH1
t=1
0
TTT
∆ ̃ s = z ̃(1)s − z ̃(0)s
∑tt∑tt∑tt t=1 t=1 t=1
Defining filter pulse response functions h(i) = z ̃(i) , the correlation decision statistic
becomes
t T−t
[h(1) ∗ s]t − [h(0) ∗ s]t , where ∗ indicates convolution.
This is known as matched filtering
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems
Residual Generation for Detection
The decision rules investigated can be summarised as:
The detection problems considered have all been reduced to computing a
residual signal st .
The residual signal is then processed (aggregated, match filtered). The processed residual is checked against a threshold.
Allows separation of residual generation and decision rule design.
G. Konstantopoulos
ACS6124 Multisensor and Decision Systems