Robust sensor correlation analysis for machine condition monitoring
View Patent ↗A method for monitoring machine conditions is based on machine learning through the use of a statistical model. A correlation coefficient is calculated using weights assigned to each sample that indicate the likelihood that that sample is an outlier. The resulting correlation coefficient is more robust against outliers. The calculation of the weight is based on the Mahalanobis distance from the sample to the sample mean. Additionally, hierarchical clustering is applied to intuitively reveal group information among sensors. By specifying a similarity threshold, the user can easily obtain desired clustering results.
1. A method for machine condition monitoring, comprising the steps of:
in a machine condition monitoring system processor, determining a robust correlation coefficient ρ xy between pairs of sensors using data from a group of samples (x i , y i ) from the pairs of sensors, the robust correlation coefficient ρ xy being determined by:
initializing a weight w i for each sample (x i , y i ), wherein 0≦w i ≦1 and Σw i =1, each weight w i being proportional to an inverse of a distance between the sample (x i , y i ) and a sample mean;
estimating a mean μ and covariance matrix Ω of the sample as
μ=Σw i z i and Ω=Σw i (z i −μ)(z i −μ) T ,
wherein z i =[x i y i ] T ;
updating the weight w i for each observation (x i ,y i ) as
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i
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wherein
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=
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;
and
f 0 is a predetermined threshold probability; and
repeating the estimating and updating steps until convergence;
calculating the robust correlation coefficient as
ρ
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∑
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=
1
n
w
i
(
x
i
-
x
_
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(
y
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2
where x =Σw i x i and y =Σw i y i ; and
predicting sensor readings using the robust correlation coefficient.
2. The method of claim 1 , further comprising the step of:
clustering the sensors based on the robust correlation coefficient.
3. The method of claim 2 , wherein the clustering is a hierarchical clustering.
4. The method of claim 2 , wherein the step of clustering the sensors further comprises:
initializing a cluster list by placing each sensor in its own cluster C i ;
determining distances between pairs of clusters
d
avg
(
C
i
,
C
j
)
=
1
C
i
C
j
∑
x
∈
C
∑
y
∈
C
d
xy
,
wherein d xy =1−abs(ρ xy ); and
if a lowest of the distances d avg (C i ,C j ) is smaller than a threshold, combining the respective clusters C i , C j , updating the cluster list and continuing with the determining step.
5. A method for machine condition monitoring, comprising the steps of:
receiving a group of readings from a plurality of sensors;
in a machine condition monitoring system processor, for at least one pair of sensors (x, y) of the plurality of sensors, determining a robust correlation coefficient ρ xy , using a plurality of samples (x i , y i ) from the group of readings, and using a weight w i for each sample (x i , y i ) based on how closely the sample obeys a joint distribution of the readings of the pair of sensors (x, y), the robust correlation coefficient being calculated as
ρ
xy
=
∑
i
=
1
n
w
i
(
x
i
-
x
_
)
(
y
i
-
y
_
)
∑
i
=
1
n
w
i
(
x
i
-
x
_
)
2
∑
i
=
1
n
w
i
(
y
i
-
y
_
)
2
where x =Σw i x i and y =Σw i y i ; and
clustering the sensors in a hierarchical cluster scheme using distances calculated from the robust correlation coefficient ρ xy .
6. The method of claim 5 , wherein the weights w i are determined by:
initializing the weight w i for each sample (x i , y i ), wherein 0≦w i ≦1 and Σw i =1, each w i being proportional to an inverse of a distance between the sample (x i , y i ) and a sample mean;
estimating a mean μ and covariance matrix Ω of the sample as
μ=Σw i z i and Ω=Σw i (z i −μ)(z i −μ) T ,
wherein z i =[x i y i ] T ;
updating the weight w i for each sample (x i ,y i ) as
w
i
=
w
(
x
i
,
y
i
)
∑
w
(
x
j
,
y
j
)
wherein
w
(
x
i
,
y
i
)
=
{
1
f
(
x
i
,
y
i
)
<
f
0
f
0
/
f
(
x
i
,
y
i
)
f
(
x
i
,
y
i
)
≥
f
0
,
wherein
f
(
x
i
,
y
i
)
=
(
z
i
-
μ
)
T
Ω
-
1
(
z
i
-
μ
)
;
and
f 0 is a predetermined threshold probability; and
repeating the estimating and updating steps until convergence.
7. The method of claim 5 , wherein the step of clustering the sensors further comprises:
initializing a cluster list by placing each sensor in its own cluster C i ;
determining distances between pairs of clusters
d
avg
(
C
i
,
C
j
)
=
1
C
i
C
j
∑
x
∈
C
∑
y
∈
C
d
xy
,
wherein d xy =1−abs(ρ xy ); and
if a lowest of the distances d avg (C i ,C j ) is smaller than a threshold, combining the respective clusters C i , C j , updating the cluster list and continuing with the determining step.
8. The method of claim 7 , further comprising the step of:
adjusting the threshold to adjust a dissimilarity of sensors in each cluster.
9. The method of claim 5 , wherein the weight w i for each sample (x i , y i ) is further based on a Mahalanobis distance from the sample to a sample mean.
10. A non-transitory computer-usable medium having computer readable instructions stored thereon for execution by a processor to perform a method comprising:
receiving a group of readings from a plurality of sensors;
for at least one pair of sensors (x, y) of the plurality of sensors, determining a robust correlation coefficient ρ xy , using a plurality of samples (x i , y i ) from the group of readings, and using a weight w i for each sample (x i , y i ) based on how closely the sample obeys a joint distribution of the readings of the pair of sensors (x, y), the robust correlation coefficient being calculated as
ρ
xy
=
∑
i
=
1
n
w
i
(
x
i
-
x
_
)
(
y
i
-
y
_
)
∑
i
=
1
n
w
i
(
x
i
-
x
_
)
2
∑
i
=
1
n
w
i
(
y
i
-
y
_
)
2
where x =Σw i x i and y =Σw i y i ; and
clustering the sensors in a hierarchical cluster scheme using distances calculated from the robust correlation coefficient ρ xy .
11. The non-transitory computer-usable medium of claim 10 , wherein the weights w i are determined by:
initializing the weight w i for each sample (x i , y i ), wherein 0≦w i ≦1 and Σw i =1, each w i being proportional to an inverse of a distance between the sample (x i , y i ) and a sample mean;
estimating a mean μ and covariance matrix Ω of the sample as
μ=Σw i z i and Ω=Σw i (z i −μ)(z i −μ) T ,
wherein z i =[x i y i ] T ;
updating the weight w i for each sample (x i ,y i ) as
w
i
=
w
(
x
i
,
y
i
)
∑
w
(
x
j
,
y
j
)
wherein
w
(
x
i
,
y
i
)
=
{
1
f
(
x
i
,
y
i
)
<
f
0
f
0
/
f
(
x
i
,
y
i
)
f
(
x
i
,
y
i
)
≥
f
0
,
wherein
f
(
x
i
,
y
i
)
=
(
z
i
-
μ
)
T
Ω
-
1
(
z
i
-
μ
)
;
and
f 0 is a predetermined threshold probability; and
repeating the estimating and updating steps until convergence.
12. The non-transitory computer-usable medium of claim 10 , wherein the step of clustering the sensors further comprises:
initializing a cluster list by placing each sensor in its own cluster C i ;
determining distances between pairs of clusters
d
avg
(
C
i
,
C
j
)
=
1
C
i
C
j
∑
x
∈
C
∑
y
∈
C
d
xy
,
wherein d xy =1−abs(ρ xy ); and
if a lowest of the distances d avg (C i ,C j ) is smaller than a threshold, combining the respective clusters C i , C j , updating the cluster list and continuing with the determining step.