Combining estimates using fuzzy sets

A system for obtaining a estimate of a physical parameter. The system is a multiple point system, in that estimates are obtained from a number of distributed sensors at a single observation time, or alternatively, from a single sensor at a number of observation times. Each sensor point is associated with a processor that calculates an estimate pair, consisting of an estimate of the parameter and a variance. Each estimate pair is used to construct a membership function. The membership functions are combined, and the combined function is optimized to determine a final estimate.

TECHNICAL FIELD OF THE INVENTION 
This invention relates to electronic measuring and analysis systems, and 
more particularly to a system for estimating a value from data obtained at 
multiple sensing points. 
BACKGROUND OF THE INVENTION 
Many of the phenomena that occur in nature are best characterized by random 
fluctuations. For example, meteorological phenomena such as fluctuations 
in air temperature and pressure, are characterized as random processes. 
The various types of sensors used to detect these processes generate 
random signals. As another example, thermal noise voltages generated in 
electronic devices generate random signals. A third example is a sonar or 
a radar signal, whose angle of arrival may occur at random with respect to 
a detection unit. 
Because of the random fluctuations in such processes, the measurement and 
analysis techniques used for determinate signals are not suitable. Instead 
statistical methods are used, which are derived from a branch of 
statistics known as estimation theory. Essentially, it is assumed that a 
plausible estimate can be made from a finite number of observations. 
A specific application of signal estimation is the use of distributed 
sensors, each associated with a processor, such that each sensor 
determines its own estimate of the parameter being measured. These 
estimates are communicated to a central unit, which determines a combined, 
or "fusion" estimate 
Another example of signal estimation uses a single sensor, but multiple 
estimates are made over a period of time. The observed process is 
represented as a stationary signal, such as from the ambient noise in a 
room. The estimates are combined to obtain a combined estimate. 
Distributed sensor and single sensor estimation systems are both 
"multi-point" systems in the sense that a number of estimates are taken 
for subsequent combination. For these multi-point systems, a number of 
methods for combining estimates have been proposed. A simple method is the 
use of averages. Another relatively simple method uses a least squares 
approach. More innovative methods include "robust methods". Each of these 
methods is essentially "linear" in the sense that the combined estimate is 
a summation of estimates multiplied by a weighting coefficient. 
When random signals having more than one frequency are being detected, the 
methods used to measure the total signal are referred to as spectral 
estimation methods. Similarly, sonar or radar signals may have more than 
one angle of arrival. These multi-component signal estimations can be 
accomplished with equipment and techniques similar to that used for single 
component estimates, except that additional processing is required to 
collect and correlate, from all sensors, those estimates that are 
associated with each frequency or angle of arrival, before combining them. 
This processing involves various ranking and refitting algorithms. 
Existing methods of combining estimates from multi-point systems, whether 
for single component or for multi-component signals, have not successfully 
overcome the problem of sensitivity to "bad" estimates, known as 
"outliers". For example, an estimate from a malfunctioning sensor that has 
a significant error may have a substantial adverse affect on accuracy of 
the combined estimate. 
A need exists for a measurement system that will combine estimates with 
reduced sensitivity to bad estimates. 
SUMMARY OF THE INVENTION 
One aspect of the invention is as a distributed sensor system for 
estimating the value of a physical parameter. A number of sensors observe 
and detect a physical quantity, which is represented as a parameter. A 
sensor processor associated with each sensor is programmed to calculate an 
estimate and a variance for each parameter. This data is communicated to a 
central unit, which is programmed to construct a membership function for 
each estimate. The central unit then combines the membership functions to 
determine a final estimate. In the combination process, the combined 
membership function is optimized and its maximum point determined. 
A technical advantage of the invention is that estimates obtained by a 
multi-point sensor system can be combined in a manner that is not 
sensitive to malfunctioning sensors. The use of fuzzy set theory to 
represent and combine estimates results in a diminishment of the effect of 
bad estimates.

DETAILED DESCRIPTION OF THE INVENTION 
System Overview 
FIG. 1 is a block diagram of a basic distributed-sensor estimation system, 
having a number, M, of sensor/processor sites (S/P's) 11, in communication 
with a central unit 12. With respect to hardware, the components 
illustrated in FIG. 1 are similar to those of conventional distributed 
multi-sensor systems. However, as explained below, each S/P 11 calculates 
an estimate pair, upon which central unit 12 operates, using fuzzy set 
theory, to obtain a final estimate value. 
At each S/P 11, a sensor 11a observes and quantifies a physical process. 
Each sensor 11a observes the physical process, with the use of multiple 
S/P's 11 being a feature of estimation systems that assure redundancy and 
accuracy. 
The significant characteristic of sensors 11a is that they detect values 
useful for estimating at least one measurand, referred to herein as a 
parameter. Thus, sensors 11a could be any type of sensor, such as a 
transducer that generates an electrical signal representing an amplitude 
or angle of arrival. 
For purposes of example in this description, sensors 11a are frequency 
sensors. Each sensor 11a may detect multiple frequencies, and the true 
number of source frequencies is assumed to be unknown. The source signal 
is considered to be the sum of sinusoidal signals and a white noise 
process. Thus, the observed signal at the i-th sensor 11a at time, t, may 
be expressed as: 
##EQU1## 
where p is the number of sinusoids. The total number of observations is N, 
where t=1 . . . N. The unknown parameters, a.sub.j,k and b.sub.i,k, are 
the coefficients of the sinusoidal signal, and the unknown parameter, 
f.sub.j,k, represents the signal's central frequencies. 
The noise factor, n, is assumed to be uncorrelated, but is a contaminated 
Gaussian random process. For purposes of this description, the following 
noise model is used: 
##EQU2## 
where P[e.sub.1 (t)]=Gaussian (o,.rho.), P[e.sub.2 (t)]=Gaussian 
(o,.eta..rho.), .epsilon.&lt;&lt;1, and .eta.&gt;&gt;1. 
Each sensor 11a is associated with a processor 11b, which is programmed to 
generate at least two items of data for every observation: a parameter 
estimate and an estimate of the variance of the parameter estimate. In the 
example of this description, the estimated parameter is frequency, 
represented by f. The variance is represented as .mu.. Each estimate, f, 
and its variance, .mu., are referred to herein as an estimate pair. Also, 
where more than one frequency is being observed and estimated, each S/P 11 
provides an estimate of the number of frequencies, p. 
In the preferred embodiment, the frequency estimate, f at each sensor 11a, 
is obtained using a robust estimation method. This method is preferred 
because it is insensitive to changes in the underlying distribution. 
The processor 11b at each S/P 11 may be a general processor, such as are 
commercially available, programmed to calculate f, .mu., and p. 
Alternatively, processors 11b may be dedicated processors. The programming 
may be in the form of loadable software or may be permanently stored in a 
memory, such as a read only memory. 
Each S/P 11 communicates its estimates to central unit 12 via a 
communications link 13. This link is provided by means of well known data 
communications techniques, and need only be a one way link. An example of 
such communication is a simplex connection. 
Central unit 12 combines the estimates gathered from each S/P 11, in the 
manner described in the following sections of this patent application. 
Central unit 12 is often referred to as a "fusion unit" in other 
distributed estimation systems. Like sensor processors 11b, central unit 
may be a general processor or may be a dedicated processor, and may 
execute its programming from permanent or temporary memory. 
Multiple Component Estimation 
As stated above, estimation systems can be configured to measure a vast 
diversity of parameters. A common application is spectral estimation, 
where the parameter is one or more frequencies of a source signal. 
However, the invention is not limited to spectral applications. For 
example, another common application might be a system for measuring 
multiple angles of arrival, such as might be present in a sonar signal. 
For purposes of this description, multiple frequency signals and multiple 
angle signals are referred to collectively as "multiple component" 
signals. Although this description is in terms of frequency measurements, 
the same concepts may be used for any other type of measurement system. 
When multiple frequency signals are being observed, central unit 12 may 
receive different numbers of estimate pairs from different S/P's 11 at a 
single observation time. As explained above, each S/P 11 also provides an 
estimate of the number of frequencies, p.sub.k. 
Because the true number of source frequencies is not known, central unit 12 
performs a ranking or refitting method to combine the estimates. In 
general, the ranking method distinguishes among estimates on the basis of 
reliability, by ranking them according to their variances, so that a 
determination of which estimates to ignore can be made. Various ranking 
methods and refitting methods are known in the art of statistics. 
In the preferred embodiment, a two stage ranking method is used. The first 
stage combines frequency estimates from S/P's 11 that obey p.sub.k, 
.gtoreq.p, where p is an estimate calculated by central unit 12. For 
example, if there are eight S/P's 11, of which five deliver two estimates, 
two deliver three estimates, and one delivers one estimate, a reasonable 
value for p is 2. The second stage combines frequency estimates, f, from 
all S/P's 11. 
Thus, the first stage has the following steps: 
1) Rank estimates in ascending order of their variance values. 
2) Eliminate p.sub.k -p estimates whose rank is greater than p. 
3) Rank the estimates again, but in ascending order of estimate values 
rather than variance values. 
4) Group the estimates from various S/P's 11 such that the members of each 
group have the same rank. 
5) Combine estimates for each group using a single component estimate 
combination method. 
The result of stage one is a set of temporary combined estimates, t.sub.k 
where k=1 . . . p. 
The second stage of the ranking method combines the frequency estimates 
from all S/P's 11. It has the following steps: 
1) Represent S/P's 11 that obey p.sub.k &lt;p with a set of indices, 
.GAMMA.={k.vertline.p.sub.k &lt;p where k=1 . . . p}. 
2) Compute distances between f.sub.ij where i=1 . . . p, and t.sub.k where 
k=1 . . . p, for each j .epsilon..GAMMA.. The rank of f.sub.ij is 
determined by the minimum distance between two estimates. In other words, 
if a difference is the smallest of all i's, the rank of that f.sub.ij is 
n. 
3) Group the estimates from various S/P's 11 such that the members of each 
group have the same rank. 
4) Combine estimates from each group, including the estimates obtained in 
the first stage, using a single component estimate combination method. 
As indicated above, both stages of the ranking method include a single 
component estimate combination step. Thus, in the example of this 
description, where the parameter being measured is frequency, the multiple 
frequency problem is solved by obtaining a set of single frequency 
estimates. The calculation of each single frequency estimate is described 
in the following section. 
Single Component Estimation 
Because the frequency estimate, f, from a S/P 11 is a function of the 
observations, a frequency estimate may be expressed as: 
EQU f.sub.ij =f.sub.i.sup.o +.epsilon..sub.ij (3) 
where f.sub.i.sup.o is the true value of the i-th component frequency and 
.epsilon..sub.ij is a perturbation term whose distribution is 
asymptotically Gaussian. The variance of .epsilon..sub.ij, denoted by 
.mu..sub.ij, is asymptotically the same as a diagonal component of a 
covariance matrix. However, due to physical conditions of sensors 11a, the 
distributions of a frequency estimate indicating the same component are 
not identical; in other words, 
Pr(.epsilon..sub.ij).noteq.Pr(.epsilon..sub.i,k) if j.noteq.k. Thus, the 
problem of combining frequency estimates may be treated as an estimation 
problem. As stated in the background section, conventional methods of 
frequency estimate combining are linear, and include averaging, weighted 
least squares, and robust estimates methods. 
A basic concept of the invention is the use of fuzzy set theory for 
combining estimates. The method "fuzzifies" the problem by constructing a 
fuzzy membership function for each frequency estimate. These membership 
functions are then combined, so as to minimize the area under the curve of 
the combined function. The problem is "defuzzified" by searching the 
argument of the maximum point of the combined membership function to 
obtain the combined frequency estimate. 
For the case of an estimate of a single frequency, the fuzzy set estimation 
can be expressed as: 
##EQU3## 
where h.sub.k (f) is a membership function of the k-th sensor. 
FIG. 2 illustrates the method of combining estimates, using fuzzy set 
theory. The following paragraphs explain these steps, as well as their 
theoretical justification. 
Step 20 is receiving an estimate pair, f and .mu., from each S/P 11. 
Step 21 is constructing a membership function for each S/P 11. Construction 
of membership functions is based on the assumption that if the variance 
value of f.sub.ij is smaller than that of f.sub.i,k when j.noteq.k, then 
the frequency estimate f.sub.ij is more reliable than f.sub.i,k. 
Similarly, it is assumed that if the area of a membership function of an 
estimate is smaller than that of another estimate, then the former 
estimate is more reliable than the latter. 
By following the above assumptions, membership functions are constructed as 
a function of f and .mu.. Three possible membership functions are a 
triangular function, a bell-shaped function, and a trimmed bell-shaped 
function. 
The triangular function is expressed as: 
##EQU4## 
The bell-shaped function is expressed as: 
EQU h.sub.k (f)=exp {-(f- f.sub.k).sup.2 /.mu..sub.k ] (6) 
The trimmed bell-shaped function is expressed as: 
##EQU5## 
Step 22 is combining the membership functions that were constructed in Step 
21. Membership functions are combined by minimizing the area of the 
combined function. The minimum area, represented as: 
##EQU6## 
is obtained with the following operation: 
##EQU7## 
subject to 
##EQU8## 
and .alpha..sub.k .gtoreq.0.0 where {.alpha..sub.k } is a set of unknown 
weighting parameters. The weighting exponent, m, is a constant greater 
than 1.0, which controls the fuzziness of the combined result. Then, 
determining unknown values for .alpha..sub.k can be treated as an 
optimization problem. 
If, with the above constraints, 
##EQU9## 
the object function becomes: 
##EQU10## 
Then, 
##EQU11## 
A feature of the invention is that f is not a function of .alpha., as in 
the conventional methods. Thus, the following partial derivative may be 
calculated: 
##EQU12## 
which permits the optimization to be solved with the iterative operation 
described below. 
Step 23 is optimizing the combined function to obtain the combined 
estimate. The combined estimate problem is represented by the following 
expression: 
##EQU13## 
The combined function .SIGMA. [.] h(f) is a convex function around the 
critical point. 
To solve the combined estimate, an iteration method is used, which is 
essentially a gradient method. The frequency estimate is initialized as: 
##EQU14## 
where values for w.sub.k =.alpha..sub.k.sup.m are obtained from equation 
(17) above. 
Step 2 is computing j iterations, where at each j-th iteration, the 
frequency estimate is computed as: 
##EQU15## 
where .zeta. is a positive constant that controls the speed of 
convergence. 
Step 3 is determining when a stopping condition is reached as follows: 
EQU .vertline.f.sup.(i) -f.sup.(j-1) .vertline.&lt;.gamma. 
where .gamma. represents a predetermined accuracy value. The iterations of 
Step 2 are continued until the convergence of Step 3 is reached. 
FIG. 3 provides a graphic illustration of the estimate combining method. 
For purposes of example, the combined membership function, h.sub.k (f), is 
a triangular shaped function. Its maximum point is f.sub.k.sup.m, thus the 
final estimate is f=f.sub.k.sup.m. The degree of closeness of the true 
frequency, f, to the estimated frequency, f is the value of h.sub.k (f). 
Thus, as h.sub.k (f) approaches 1, f approaches f. 
Single Sensor Multi-Point Estimation 
Another application of the invention is combining estimates from a single 
S/P 11. Here, although there is only one S/P 11, more than one estimate is 
obtained over time. 
The same techniques as described above may be used to combine the 
estimates. Each estimate is associated with a membership function, and the 
functions are combined to obtain the combined estimate. 
Other Embodiments 
Although the invention has been described with reference to specific 
embodiments, this description is not meant to be construed in a limiting 
sense. Various modifications of the disclosed embodiments, as well as 
alternative embodiments, will be apparent to persons skilled in the art. 
It is, therefore, contemplated that the appended claims will cover all 
modifications that fall within the true scope of the invention.