METHOD AND APPARATUS FOR CREATING STATE ESTIMATION
MODELS IN MACHINE CONDITION MONITORING
Claim of Priority
This application claims priority to, and incorporates by reference herein in
its entirety, pending United States Provisional Patent Application Serial Number
61/106,699, filed October 20, 2008, and entitled "Method and Apparatus for Creating
Slate Estimation Models in Machine Condition Monitoring."
Field of the Disclosure
The present invention relates generally to machine condition monitoring
for the purpose of factory automation. More specifically, the invention relates to
techniques for building state estimation models describing a relation among a set of
machine sensors.
Background
The task of machine condition monitoring is to detect faults as early as
possible to avoid further damage to a machine. This is usually done by analyzing data
from a set of sensors, installed on different parts of a machine, for measuring
temperature, pressure, vibrations, etc. When a machine is operating normally, all sensors
obey a certain relationship. That relation can be described by the dependency of one .
sensor against other related sensors During monitoring, violation of that relationship or
dependency may indicate a fault. For example, in a gas turbine, given certain system
inputs such as gas flow, inlet temperature and air humidity, the power output should be
close to a predicted value. If the actual observed value deviates from that predicted
value, the observation may indicate a system failure.
A fundamental step in machine condition monitoring is to build state
estimation (SE) models that describe the relation among a set of sensors. During training,
the SE model is trained to learn the sensor relationships from historical training data.
During testing, for observed sensor values, the trained SE model is used to estimate the
values that sensors should have if they operate normally.
One challenge in creating the SE model is that there are usually many
sensors. In many circumstances, the relation among sensors is unknown. Sensors may
monitor totally independent parts of the machine so that some sensors are not correlated
with other sensors. If one simply builds a single SE model using all sensors, and
estimates one sensor using the remaining sensors including unrelated sensors,
performance of the SE model will be adversely affected.
In one approach, the SE model is constructed in two steps. First, pair-wise
correlation scores of sensors are computed. The scores may be computed by standard
correlation coefficients for linear cases, or by more sophisticated mutual information for
nonlinear cases. In the second step, based on the correlation scores, a clustering method
such as hierarchical clustering is applied to cluster sensors into groups That approach is
limited in that only pair-wise correlation between two sensors is used, and the approach
thus cannot capture correlation involving more than two sensors, which exists extensively
in complex machines.
Mutual information can be extended for multiple sensors, but that is at the
cost of an exponential increase in computation time. In addition, mutual information
usually requires discretization of continuous sensor signals, leading to a loss of precision.
There is therefore presently a need for an improved technique to partition
sensors into groups, and to monitor machines using such groups. The technique should
create groups wherein, within each group, sensors are correlated, but between groups,
sensors are not correlated. By using such groups, one SE model can be trained for each
group.
Summary of the Invention
In the present disclosure, a method is presented for grouping sensors by
analyzing the dependency of one sensor against all remaining sensors. In particular, a
Gaussian process regression method is employed to predict the target sensor (as output)
from the remaining sensors (as inputs). A kernel function with automatic relevance
determination is used such that each input sensor has its own kernel width. Those kernel
widths are parameters and are learned from training data.
After training the SB model, two indications reveal information about
sensor dependency. First, the noise variance of this Gaussian process model represents
the overall dependency of the output against the inputs. The smaller the error is, the more
dependent the output is.
Secondly, the kernel widths associated with different input sensors
indicate the relative dependency of the output, sensor against each input sensor That is
because the input sensors that are more relevant or related to the output sensor tend to
have smaller kernel widths (and thus larger effects in kernel functions) than less relevant
input sensors.
If the overall dependency is smaller than a threshold, it is determined that
the output or target sensor does not depend on other sensors If the relative dependency
of an input sensor is smaller than a threshold, it is determined that this output or target
sensor does not depend this input sensor. That dependency analysis is performed for
every sensor, against all other sensors. Two sensors are correlated if one depends on the
other. A new grouping algorithm is presented accordingly
One embodiment of the invention is a method for grouping interrelated
sensors of a set of sensors into clusters for use in state estimation models. In a computer,
a separate Gaussian Process Regression is trained for each sensor in the set of sensors.
wherein in a Gaussian Process Regression for a sensory, the sensory is a target sensor
and d remaining sensors of the set are input sensors. The training uses a training set of
signal values from the sensors to determine a noise variance v for the target sensory and
d kernel widths sk. Each kerne! width sk represents a relevance of a respective sensor k of
the d input sensors in predicting a value of the sensor y.
A dependency analysis is then performed on each sensor of the sensor set
by using the noise variance v and the kernel widths sk of the sensor to determine whether
or not the sensor is correlated to each of the d other sensors. The sensors of the set of
sensors are then grouped into clusters based on the dependency analysis.
In that method, the Gaussian Process Regression may be performed using
a kernel function defined as:

wherein k(xi,xj) is an element of a covariance matrix for input samples (xi, xj), f is a
signal variance, xfk and xfk are kth elements of the vectors xi and xj respectively, and d = 1
if i = j and 0 otherwise. The step of training a Gaussian process regression may utilize
conjugate gradient methods.
The dependency analysis further include comparing the noise variance v of
the sensor y to a first threshold T1, and concluding that the sensor is not dependent on
other sensors if v exceeds the threshold T1, and that the sensor is dependent on other
sensors if v is smaller than the threshold Tt; determining relative dependency weights wk
of the sensor y on each of the remaining sensors k, the relative dependency weights wk
being a function of a corresponding kernel width sk, and determining whether pairs of
sensors in the set of sensors are correlated by comparing the relative dependency weights
wk between sensors of the sensor pairs to a second threshold T2.
The relative dependency weight wk may be defined by
The relative dependencies of a sensor y may be determined only if the
sensor y is determined to be dependent on other sensors.
The step of grouping the sensors into clusters may include initializing a
sensor index set Q = {1,2, ...,d}. The following is then performed until Q is empty:
removing a first element i from Q, initializing two new index sets Z:=: {i} and G:=: {i},
each containing a single index i, and performing the following: deleting a first element j
from Z; identifying all sensors correlated to j; if an identified correlated sensor is not in
G, then adding its index to Z and G and removing its index from Q; repeating the
deleting, identifying and adding steps until Z is empty; and then outputting G as contents
of a sensor cluster. The steps of removing a first element i, initializing two new index
sets and performing are repeated until Q is empty.
Another embodiment of the invention is a method for monitoring a
condition of one or more machines via a set of sensors installed on the machines. A
training set of sensor signals is acquired, the signals comprising a series of simultaneous
readings of the sensors. The above steps are then performed to group interrelated sensors
of the set of sensors into clusters.
Cluster state estimation models are trained, each model having a target
sensor and all input sensors in the same cluster. Target sensor signals are then predicted
based on input sensor signals, using the trained cluster state estimation models. An alarm
is generated if a predicted target sensor signal is sufficiently different from an actual
sensor signal.
Another embodiment of the invention is a computer-usable medium
having computer readable instructions stored thereon for execution by a processor to
perform methods as described above.
Brief Description of the Drawings
FIG. 1 is a schematic view showing a system according to the present
disclosure.
FIG. 2 is a flow diagram showing a technique for grouping sensors
according to one embodiment of the invention.
FIG 3 is a flow diagram showing a technique for creating a slate
estimation model according to one embodiment of the invention.
Description
The present invention may be embodied in a system for creating a state
estimation model, which may be included in a machine monitoring system or may be a
stand-alone system. FIG. 1 illustrates a system 100 for creating a state estimation model
according to an exemplary embodiment of the present invention. As shown in FIG I, the
system 100 includes a personal or other computer 110. The computer 110 may be
connected to one or more sensors 171-174 over a wired or wireless network 105.
The sensors 171 -174 are arranged to acquire data representing various
characteristics of one or more machines or systems i 80. The sensors measure
characteristics of the machine 180 and its environment, such as temperature, pressure,
humidity, rotational or linear speed, vibration, force, strain, power, voltage, current,
resistance, flow rate, proximity, chemical concentration or any other characteristic As
noted above, groups of sensors may be related, in which case sensor signals from a group
are predictors of signals of other sensors in the group. Some of the sensors may be
independent, having no relationship with other sensors.
The sensors may be connected directly with the computer 110, or signals
from the sensors may be conditioned by a signal conditioner 160 before being transmitted
to the computer. Signals from sensors monitoring many different machines and their
environments may be connected through the network 105 to the computer 110.
The computer 130, which may be a portable or laptop computer or a
mainframe or other computer configuration, includes a central processing unit (CPU) 125
and a memory 130 connected to an input device 150 and an output device 155. The CPU
125 includes a state estimation model creation module 145 and that includes one or more
methods for creating a state estimation model as discussed herein. Although shown
inside the CPU 125, the module 145 can be located outside the CPU 125. The CPU may
also contain a machine monitoring module 146 that utilizes the state estimation model in
monitoring the machine 180. The machine monitoring module 146 may also be used in
acquiring training data from the sensors 171-174 for use in creating the state estimation
model.
The memory 130 includes a random access memory (RAM) 135 and a
read-only memory (ROM) 140. The memory 130 can also include a database, disk drive,
tape drive, etc., or a combination thereof. The RAM 135 functions as a data memory mat
stores data used during execution of a program in the CPU 125 and is used as a work-
area. The ROM 140 functions as a program memory for storing a program executed in
the CPU 125. The program may reside on the ROM 140 or on any other computer-usable
medium as computer readable instructions stored thereon for execution by the CPU 125
or other processor to perform the methods of the invention. The ROM 140 may also
contain data for use by the programs, such as training data that is acquired from the
sensors 171-174 or created artificially
The input 150 may be a keyboard, mouse, network interface, etc, and the
output 155 may be a liquid crystal display (LCD), cathode ray tube (CRT) display,
printer, etc.
The computer 110 can be configured to operate and display information
by using, e.g., the input 150 and output 155 devices to execute certain tasks. Program
inputs, such as training data, etc, may be input through the input 150, may be stored in
memory 130, or may be received as live measurements from the sensors 171-174.
Described herein is a method for creating a. state estimation model for
machine condition monitoring. A general procedure for creating the model is shown in
FIG. 2, and described in more detail below. At step 210, a Gaussian process regression
(GPR) analysis is performed. For each of d sensors, a GPR model is trained to predict
the sensor using all the remaining sensors. A total of d GPR models are thereby created.
At step 220, a dependency analysis is performed. For every sensor i.
based on its GPR model, an overall dependency is determined based on its noise variance
v. If v is smaller than a threshold T1, then a relative dependency wk is determined for all
other sensors k relative to sensor i. If wk i s greater than a threshold T2 for any sensor k,
then sensor i and sensor k are considered correlated.
The sensors are then grouped at step 230 according to their correlation.
The groups are then used to create separate state estimation models for use in monitoring
the condition of the subject machine or machines.
Each of the above steps will now be described in more detail
Gaussian Process Regression Analysis
In initial step 210, a Gaussian process regression (GPR) is used to predict
a sensor denoted by scalar y from all other of sensors denoted by a vector
Suppose that there are N training samples
The GPR assumes that all training outputs, or an N-dimensional vector
have a Gaussian distribution with zero mean and the N x N
covariance matrix C whose element. is referred to as a kernel
function between two input samples xi and xj.
The form of the kernel function is defined as follows:

In the above equation, there are d + 2 parameters including the signal
variance/, noise variance v and kernel vvidth sk for the kth input sensor (where
k = 1,2,...,d ). xik and xjk are the kth component of the velcors xi,xj, respectively, d is the
delta function, which takes 1 when i = j, and 0 otherwise.
The goal of training such GPRs is to maximize the log likelihood of the
probability of Y over the parameters of f, v and s1,s2....,sd . This is usually done by
conjugate gradient methods. The estimated value of sk is usually quite different for
different input sensors. If an input sensor is more relevant to predict the output sensor,
the corresponding sk is usually small, so that sensor has a large effect in the kernel
function (1). On the other hand, if an input sensor is not relevant to predict the output
sensor, the corresponding kernel width sk is likely to be large and this input sensor
becomes negligible in the kernel function. That behavior is often referred to as automatic
relevance determination. During testing, given the input sensor values x, it is possible to
estimate the corresponding y value or easily. An example of that estimation
process is given with reference to the test results below.
Dependency Analysis
The dependency analysis of step 220 (FIG. 2) is now described. The
predictive error of a GPR is indicated by the noise variance v learned from training. The
larger v is, less accurate that GPR is. If v can be accurately predicted from other sensors
via the GPR or the noise variance is small, it may be concluded that v is dependent on
other sensors. Such dependency is referred to as overall dependency. If v is smaller than
a threshold T1, it is claimed that y is dependent on at least some other sensors.
passes the overall dependency check, the relative dependency of v on
an input sensor k may be revealed by inspecting the corresponding kernel width sk. Since
the effect of a sensor on the kernel function (1) relies on the inverse of its kernel width sk,
the following relative dependency weight wk defined as;

Note that the sum of wK is one (1). If wk is larger than a threshold T2, it
may be concluded that the output sensor depends on the input sensor denoted by k, or this
input sensor is relevant to y.
If the sensor i depends on sensor j or sensory depends on sensor i, it is
concluded that sensor i and sensor j are correlated and connect an edge between those
two sensors
Sensor Grouping Algorithm
In accordance with step 230 of FIG. 2, the sensors are now grouped using
the results of the dependency analysis. All connected sensors are placed in one group,
and no pair of connected sensors is separated into two groups. Sensors that are
unconnected are placed in different groups. In one embodiment, connected sensors may
be directly correlated, may each be correlated to another common sensor, or may be
separated by several degrees by a chain of correlated sensors.
In one embodiment of the invention, a technique 300 having a work flow-
as shown in FIG. 3 is used to group the sensors. A sensor index set Q is initialized at step
310. The sensor index set Q is used to indicate unprocessed sensors. If every sensor is
processed and the sensor index set Q is empty at decision block 320, the algorithm is
terminated at step 390.
Z is the sensor set representing unprocessed sensors for the current sensor
group, and G includes all sensor indices which should be in the current sensor group
Those sets are initialized at step 330 for each new sensor from Q. The initialized sets Z
and G each contain a single index for sensor i.
Once Z is empty, at decision block 340, G is output, at step 350 for the
current sensor group, and work flow returns to decision block 320.
If Z is not empty, the first element j is removed from Z at step 360. Based
on the dependency analysis described above, ail sensors that correlate with sensor j are
found. If a correlated sensor s index is not in G, that index is added to both G and Z, and
that index is removed from Q. The work flow then returns to step 340
The resulting dusters of sensors may then be used in monitoring the
condition of a machine or system. State estimation models are constructed for clusters
containing related sensors. In a preferred embodiment, Gaussian process regression is
used to construct a state estimation model for each sensor in a multi-sensor cluster, using
other sensors in the cluster as input sensors. The state estimation models are then trained
using the same training data as above, or using different training data. The models are
then used to predict values for the sensors, and those predictions are compared to actual
sensor signals. The machine condition monitoring system may output an alarm when a
predicted value deviates from an actual value by more than some threshold quantity
Other modeling techniques, such as a trend analysis, may be used in cases where only a
single sensor is contained in a cluster.
Test Results
To test the efficiency of the proposed algorithm, the following artificial
data sets were created with nine variables representing nine sensors The two variables x1
and x2 are independent and both have uniform distributions from [0,t]. The third x3 is
defined as:

In addition, another three independent variables x4, x5, x6 are added with
uniform distributions from [0,1] The seventh variable x7 is defined by:

The noise terms in both (3) and (4) have a Gaussian distribution with zero
mean and 0.1 standard deviation. Finally, two extra independent variables x8 and x9 are
added with uniform distribution from [0,1], Two hundred (200) training samples were
randomly generated based on the above description. Each variable-was normalized to
zero mean and unit standard deviation.
It is clear that the first three variables have a linear relation while the next
four variables have a complex nonlinear relation. Thus, ideally, there should be four
groups and therefore four state estimation models. The first three sensors should be in
one group; the next four sensors should be in another group; each of the remaining two
sensors should form a separate new group.
The standard group methods-based pair-wise correlation does not work in
this test because the first and second groups in this case both involve a higher dimension
(i.e., greater than 2) correlation. For example, the correlation coefficients of x1 against x2
and x3 are 0.0569, 0.2915, respectively. Since those numbers are very small, x1 will not
be included in the same group as x2 and x3 if traditional methods are used.
After the regression analysis of step 210 (FIG. 2), the following noise
variances v. are obtained for x1, x2,..., x9:
0.5538 0.1604 . 0.158S 0.9950 0.0316 0.0307 0.0039 0.9950 0.9942
As shown, the noise variances for x3 and x7 are relatively small, because, .
based on equations (3) and (4), those variables should be able to be predicted by other
variables. On the other hand, the noise variances for x8 and x9 are very large because they
are independent from other variables.
After the step 2 dependency analysis, the following relative dependency
wk matrix is obtained:

The ith row of the above matrix represents the corresponding relative
dependency for the remaining variables. For example, the second row indicates results of
predicting x2 from all other variables. x3 has the largest relative dependency (0.7712); x1
also has a relatively large value (0.2264). All the other variables have very small relative
dependencies. That is expected because x1 is correlated with x1 and x3, but independent
from the remaining variables.
In this test, the threshold for overall dependency was set to T1 = 0.3 and
the threshold for relative dependency was set to T2 = 0.01. With those settings, x1, x4, x8
and x9 are not dependent on other variables because they fail the overall dependency test.
Thus the corresponding rows of the relative dependency matrix are ignored. After the
overall dependency and relative dependency tests, the following correlation matrix is
produced.

If there is a "1" at the ith row and the jth column, the variable i and
variable j are correlated.
After performing the sensor grouping algorithm using the correlation
matrix, the variables are clustered into the following four groups.
Group 1: {1, 2, 3}
Group 2: {4, 5, 6, 7}
Group 3: {8}
Group 4: {9}
Those groups exactly match the ground truth.
Conclusion
The foregoing Detailed Description is to be understood as being in every
respect illustrative and exemplary, but not restrictive, and the scope of the invention
disclosed herein is not to be determined from the Description of the Invention, but rather
from the Claims as interpreted according to the full breadth permitted by the patent laws.
It is to be understood that the embodiments shown and described herein are only
illustrative of the principles of the present invention and that various modifications may
be implemented by those skilled in the art without departing from the scope and spirit of
the invention.
We claim:
1. A method for grouping interrelated sensors of a set of sensors into
clusters for use in state estimation models, the method comprising:
(A) in a computer, training a separate Gaussian Process Regression for each
sensor in the set of sensors, wherein in a Gaussian Process Regression for a sensor v, the
sensory is a target sensor and d remaining sensors of the set are input sensors, the
training using a training set of signal values from the sensors to determine a noise
variance v for the target sensor y and d kernel widths sk, each kernel width sk representing
arefevance of a respective sensor k of the d input sensors in predicting a value of the
sensor y;
(B) performing a dependency analysis on each sensor of the sensor set by
using the noise variance v and the kernel widths sk of the sensor to determine whether or
not the sensor is correlated to each of the d other sensors; and
(C) grouping sensors of the set of sensors into clusters based on the
dependency analysis
2. The method of claim 1, wherein the Gaussian Process Regression is
performed using a kernel function defined as:
wherein k(xi,xj) is an element of a covariance matrix for input samptes (xi,xj),f
is a signal variance, xik and xjk are kth elements of the vectors xi and xj, respectively, and d
=1 if i = j and 0 otherwise.
3. The method of claim 1, wherein the step of training a Gaussian process
regression utilizes conjugate gradient methods.
4. The method of claim 1, wherein the dependency analysis further
comprises
(1) comparing the noise variance v of the sensor v to a first threshold
T|, and concluding that the sensor is not dependent on other sensors if v exceeds
the threshold T1, and that the sensor is dependent on other sensors if v is smaller
than the threshold T1;
(2) determining relative dependency weights wk of the sensor v on
each of the remaining sensors k, the relative dependency weights wk being a
function of a corresponding kerne! width sk; and
(3). determining whether pairs of sensors in the set of sensors are
correlated by comparing the relative dependency weights wk between sensors of
the sensor pairs to a second threshold T2.
5. The method of claim 4, wherein the relative dependency weight wk is
defined by.

6. The method of claim 4, wherein the relative dependencies of a sensor y
are determined only if the sensor y is determined to be dependent on other sensors
7. The method of claim I, wherein the step of grouping die sensors into
clusters comprises:
. initializinga sensor index set Q = {1, 2,..., d},
until Q is empty, removing a first element i from Q, initializing two new index
sets Z = {i} and G = {i}, each containing a single index i, and performing the following,
deleting a first element j from Z;
identifying all sensors correlated to j;
if an identified correlated sensor is not in G. then adding its index-
to Z and G and removing its index from Q;
repeating the deleting, identifying and adding steps until Z is
empty; and then outputting G as contents of a sensor cluster; and
repeating the steps of removing a first element i, initializing two new index sets
and performing until Q is empty.
8 A method for monitoring a condition of one or more machines via a
set of sensors installed on the machines, the system comprising.
(A) acquiring a training set of sensor signals comprising a series of
simultaneous readings of the sensors;
(B) in a computer, training a separate Gaussian Process Regression for each
sensor in the set of sensors, wherein in a Gaussian Process Regression for a sensor y, the
sensor v is.a target sensor and d remaining sensors of the set are input sensor values, the
training using the training set of signals from the sensors to determine a noise variance v
for the target sensor y and d kernel widths sk, each kernel width sk representing a
relevance of a respective sensor k of the d input sensors in predicting a value of the
sensory,
(C) performing a dependency analysis on each sensor of the sensor set by
using the noise variance v and the kernel widths sk of the sensor to determine whether or
not the sensor is correlated to each of the d other sensors;
(D) grouping sensors of the set of sensors into clusters based on the
dependency analysis;
(E) training cluster state estimation models having a target sensor and all input
sensors in the same cluster,
(F) predicting target sensor signals based on input sensor signals, using the
trained cluster state estimation models; and
(G) generating an alarm if a predicted target sensor signal is sufficiently
different from an actual sensor signal.
9. The method of claim 8, wherein the steps of training cluster state
estimation models, predicting and generating are performed by the same computer as the
step of training a separate Gaussian Process Regression for each sensor in the set of
sensors
10, The method of claim 8, wherein the cluster state estimation models are
Gaussian Process Regressions.
11. The method of claim 8, wherein a predicted target sensor signal is
sufficiently different from an actual sensor signal when a difference between those values
exceeds a threshold.
12. The method of claim 8, further comprising the following step:
predicting a signal of a target sensor in a cluster containing no other sensors using
historical data from the target sensor.
13. The method of claim 8, wherein the Gaussian Process Regression is
performed using a kernel function defined as:

wherein k(xi, xj) is an element of a covariance matrix for input samples (xi, xj), f
is a signal variance, xik and xjk are kth elements of the vectors xi, and xj, respectively, and
d = 1 if i = j and 0 otherwise.
14. The method of claim 8, wherein the step of training a Gaussian
Process Regression utilizes conjugate gradient methods. .
15. The method of claim 8, wherein the dependency analysis further
comprises
(1) comparing the noise variance v of the sensor y to a first threshold
T1, and concluding that the sensor is not dependent on other sensors if v exceeds
the threshold T1, and that the sensor is dependent on other sensors if v is smaller
than the threshold T1;
(2) determining relative dependency weights wk of the sensor y on
each of the remaining sensors k, the relative dependency weights wk being a
function of a corresponding kernel width .sk; and
(3) determining whether pairs of sensors in the set of sensors are
correlated by comparing the relative dependency weights wk between sensors of
the sensor pairs to a second threshold T2.
16. The method of claim 15, wherein the relative dependency weight wk is
defined by:

17. The method of claim 15, wherein the relative dependencies of a sensor
v are determined only if the sensor v is determined to be dependent on other sensors.
18. The method of claim 8, wherein die step of grouping the sensors into
clusters comprises:
initializing a sensor index set Q = {1,2,..., d};
until Q is empty, removing a first element i from Q, initializing two new index
sets Z = {i} and G = {i}, each containing a single index i, and performing the following.
deleting a first element j from Z;
identifying all sensors correlated to j;
if an identified correlated sensor is not in G, then adding its index
to Z and G and removing its index from Q,
repeating the deleting, identifying and adding.steps until Z is
empty; and then outputting G as contents of a sensor cluster; and
repeating the steps of removing the first element i, initializing two new index-
steps and performing until Q is empty. .
19. A computer-usable medium having computer readabl e instructions
stored thereon for execution by a processor to perform a method for grouping interrelated
sensors of a set of sensors into clusters for use in state estimation models, the method
comprising:
(A) training a separate Gaussian Process Regression for each sensor in the set
of sensors, wherein in a Gaussian Process Regression for a sensor y, the sensor y is a
target sensor and d remaining sensors of the set are input sensors, the training using a
training set of signal valuess from the sensors to determine a noise variance v for the
target sensor y and d kernel widths sk, each kernel width sk representing a relevance of a
respective sensor k of the d input sensors in predicting a value of the sensor v;
(B) performing a dependency analysis on each sensor of the sensor set by
using the noise variance v and the kernel widths sk of the sensor to determine whether or
not the sensor is correlated to each of the d other sensors; and
(C) grouping sensors of the set of sensors into clusters based on the
dependency analysis.
20. The computer-usable medium of claim 19, wherein the Gaussian
Process Regression is performed using a kernel function defined as:

wherein k(xt,xf) is an element of a covariance matrix for input samples (xi,xj),f
i s a signal variance, xik and xjk are kth elements of the vectors xi and xj, respectively, and d
= 1 if i = j and 0 otherwise.
21. The computer-usable medium of claim 19, wherein the step of training
a Gaussian process regression utilizes conjugate gradient methods
22. The computer-usable medium of claim 19, wherein the dependency
analysis further comprises
(l) comparing the noise variance v of the sensor y to a first threshold
T1, and concluding that the sensor is not dependent on other sensors if v exceeds
the threshold T1, and that the sensor is dependent on other sensors if v is smaller
than the threshold T1;
(2) determining relative dependency weights wk of the sensor y on
each of the remaining sensors k, the relative dependency weights wk being a
function of a corresponding kernel width sk; and
(3) determining whether pairs of sensors in the set of sensors are
correlated by comparing the relative dependency weights wk between sensors of
the sensor pairs to a second threshold T2.
23. The computer-usable medium of claim 22, wherein the relative
dependency weight wk is defined by:

24. The computer-usable medium of claim 22, wherein the relative
dependencies of a sensor y are determined only if the sensor y is determined to be
dependent on other sensors.
25. The computer-usable medium of claim 19, wherein the step of
grouping the sensors into clusters comprises:
initializing a sensor index set Q = {1,2, ...,d};
until Q is empty, removing a first element i from Q and initializing two new index
sets Z = {i} and G = {i}, each containing a single index i, and performing the following:
deleting a first element j from Z;
identifying all sensors correlated to j;
if an identified correlated sensor is not in G, then adding its index
to Z and G and removing its index from Q;
repeating the deleting, identifying and adding steps until Z is
empty; and then outputting G as contents of a sensor cluster; and
repeating the steps of removing the first element i, initializing two new index sets
and performing until Q is empty.

In a machine condition monitoring technique, related
sensors are grouped together in clusters to improve the
performance of state estimation models. To form the
clusters, the entire set of sensors is first analyzed
using a Gaussian process regression (GPR) to make a
prediction of each sensor from the others in the set. A
dependency analysis of the GPR then uses thresholds to
determine which sensors are related. Related sensors
are then placed together in clusters. State estimation
models utilizing the clusters of sensors may then be
trained.