Patent Number: 062020382
Section: description

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS A system 10 constructed in accordance with the invention is set forth in general in the flow chart of FIG. 1A. In describing various preferred embodiments, specific reference will be made throughout to application of the surveillance methodologies to specific industrial systems, such as nuclear reactors; however, the inventions are equally applicable to any system which provides signals or other data over time which describe attributes or parameters of the system. Therefore, the inventions herein are, for example, applicable to analysis, modification and termination of processes and systems involving physical, chemical, biological and financial sources of data or signals. The system 10 is made up of three methodologies which, as appropriate, can be used separately, and possibly together, to monitor or validate data or signals. A series of logical steps can be taken to choose one or more of the methods shown in detail in FIGS. 1B-1D. Initialization of the system 10 is shown in FIG. 1A. The first step in the initialization is to obtain the user specified parameters; the Sample Failure Magnitude (SFM), the false alarm probability (.alpha.), and the missed alarm probability (.beta.). The next step in the initialization is to query the monitored system to obtain the sensor configuration information. If the system has a single sensor the method selected for monitoring will be the MONOSPRT approach described immediately hereinafter. For the single sensor case, that is all that needs to be done to complete the initialization. If the system has exactly two sensors, then information about the relationship between the two sensors is required. First, are the two sensors linearly related? If so, the regression SPRT algorithm is selected for monitoring, and this will be discussed in detail hereinafter. If the two sensors aren't linearly related, the next step is to check to see if they are non-linearly related. If so, the BART algorithm (described hereinafter) is used for monitoring. Otherwise, each sensor is monitored separately using the MONOSPRT method. In a first preferred embodiment (MONOSPRT) involving surveillance and analysis of systems having only one source of signals or data, such as, non-safety grade nuclear reactors and many industrial, biological and financial processes, a highly sensitive methodology implements a sequential analysis technique when the decision process is based on a single, serially correlated stochastic process. This form of the invention is set forth in detail in FIG. 1B on the portion of the flow diagram of FIG. 11A directed to "one sensor" which activates a MONOSPRT methodology. Serial correlation can be handled by a vectorized type of SPRT (sequential probability ratio test) method which is based on a time series analysis, multivariate statistics and the parametric SPRT test (see, for example, U.S. Pat. Nos. 5,223,207; 5,410,492; 5,586,066 and 5,629,872 which describe details of various SPRT features and are incorporated by reference herein for such descriptions). The MONOSPRT method is described in FIG. 1B. The method is split into two phases, a training phase and a monitoring phase. During the training phase N samples are collected from the single sensor (or data source) that are representative of normal operation. Next, a covariance matrix is constructed from the representative data that is p.sub.x p, where p is the user specified number of lags to consider when characterizing the autocorrelation structure of the sensor signal. The final steps in the training phase of the MONOSPRT method are to calculate the SPRT parameters; SDM, L and U. The SDM (System Disturbance Magnitude) is calculated by multiplying the standard deviation of the sensor signal with the SFM specified during the system initialization. The standard deviation of the sensor signal is the square root of the diagonal elements of the covariance matrix. L and U are the lower and upper thresholds used for to compare the MONOSPRT indexes to in order to make a failure decision. Both L and U are functions of .alpha. and .beta. specified during system initialization. During the monitoring phase of MONOSPRT a data vector of length p is acquired at each time step t and is used in the calculation of the MONOSPRT index .lambda.. The index is then compared to L and H. If the MONOSPRT index is greater than or equal to U, then the sensor signal is not behaving normally and a failure alarm is annunciated. If the MONOSPRT index is less than or equal to L then the decision that the sensor is good is made. In either case, after a decision is made in the MONOSPRT index is reset to zero and the process continues. In this vectorized SPRT methodology, (hereinafter "MONOSPRT"), suppose there exists the following stationary, a periodic sequence of serially correlated random variables: {X'}.sub.t where t=1, 2, 3 . . . , N. It is conventional that a periodic sequence can be handled by removing the periodic component of the structural time series model, and a non-stationary sequence can be differenced to produce a stationary sequence. The stationary assumption provides constant mean, constant variance and covariances that depend only on the separation of two variates in time and not the actual times at which they were recorded. The mean, .mu., is given by EQU .mu.=E[X'.sub.t ] where E[.] is the expectation operator. If we let EQU X.sub.t =X'.sub.t -X PA1 where, ##EQU1## PA1 Decision Rule: if .lambda..sub.t &lt;L, then ACCEPT H.sub.0 : PA1 where S is either 0 or A. Therefore: ##EQU7## PA1 H.sub.0 : D.sub.1,D.sub.2, . . . have Gaussian distribution with mean M.sub.0 and variance .sigma..sup.2 PA1 H.sub.F : X.sub.1,X.sub.2, . . . have Gaussian distribution with mean M.sub.F and variance .sigma..sup.2 PA1 A) The similarity between the maximum and minimum values in the similarity domain is 0, and PA1 B) the similarity between equal values is 1. PA1 if x.sub.1 =[X.sub.11 X.sub.12 X.sub.13 . . . X.sub.1n ]and x.sub.2 =[X.sub.21 X.sub.22 X.sub.23 . . . X.sub.2n ] and n.sub.s is the sample size, then E[X.sub.t ]=0. The autocovariance of two time points, X.sub.t and X.sub.s is .sigma..sub..Arrow-up bold.t-s.Arrow-up bold. =E[X.sub.t X.sub.s ], where s and t are integers in the set {[1, N]} and .sigma..sub.0 is the variance. Suppose there exists p&lt;N such that for every m.gtoreq.p: .sigma..sub.n &lt;.delta., where .delta.is arbitrarily close to 0. ##EQU2## Therefore, we have constructed a stationary sequence of random vectors. The mean of the sequence {{character pullout}}.sub.t is {character pullout}.sub.p where {character pullout}.sub.p is the zero vector with p rows. The variance of the sequence is the covariance matrix .SIGMA..sub.y. ##EQU3## The SPRT type of test is based on the maximum likelihood ratio. The test sequentially samples a process until it is capable of deciding between two alternatives: H.sub.0 :.mu.=0; and H.sub.A :.mu.=M. It has been demonstrated that the following approach provides an optimal decision method (the average sample size is less than a comparable fixed sample test). A test statistic, .lambda..sub.t, is computed from the following formula: ##EQU4## where 1n(.) is the natural logarithm, .function..sub.H.sub..sub.s ( ) is the probability density function of the observed value of the random variable Y.sub.i under the hypothesis H.sub.s and j is the time point of the last decision. In deciding between two alternative hypotheses, without knowing the true state of the signal under surveillance, it is possible to make an error (incorrect hypothesis decision). Two types of errors are possible. Rejecting H.sub.0 when it is true (type I error) or accepting H.sub.0 when it is false (type II error). We would like to control these errors at some arbitrary minimum value, if possible. We will call the probability of making a type I error, .alpha., and the probability of making a type II error .beta.. The well-known Wald's Approximation defines a lower bound, L, below which one accepts H.sub.0. and an upper bound, U beyond which one rejects H.sub.0. ##EQU5## else if .lambda., &lt;U, then REJECT H.sub.0 : PA2 otherwise, continue sampling. To implement this procedure, this distribution of the process must be known. This is not a problem in general, because some a priori information about the system exists. For our purposes, the multivariate Normal distribution is satisfactory. Multivariate Normal: ##EQU6## The equation for .lambda..sub.t can be simplified into a more computationally efficient form as follows: ##EQU8## For the sequential test the equation is written as ##EQU9## In practice, we implement two separate tests. One test is for M greater than zero and the second test for M less than zero. Here, M is chosen by the evaluating, EQU M=[1 1 1 . . . 1]'.sigma..sub.0 k (11) where k is a user specified constant that is multiplied by the standard deviation of y. M is then used in equation (10) to determine the amount of change in the mean of y that is necessary to accept the alternative hypothesis. FIGS. 2A-2F show results after applying the MONOSPRT embodiment to a sinusoid containing no disturbance, a step disturbance, and a linear drift. In these examples the noise added to the sinusoid is Gaussian and white with a variance of 2. The sinusoid has an amplitude of 1, giving an overall signal-to-noise ratio ("SNR" hereinafter) of 0.25 (for a pure sinusoid SNR=0.5A.sup.2.sigma..sup.2, where .sigma..sup.2 is the variance of the noise and A is the amplitude of the sinusoid). The autocorrelation matrix used in MONOSPRT for these examples were calculated using 30 lags. The false alarm probability .alpha. and missed alarm probability .beta. are both specified to be 0.0001 for MONOSPRT, and the sample failure magnitude (SFM) is set to 2.5. FIG. 2A shows the sinusoid with noise without any disturbance being present. FIG. 2B is the resulting MONOSPRT when applied to the signal. FIGS. 2C and 2D illustrate the response of MONOSPRT to a step change in the sinusoid. The magnitude of the step is 2.sigma..sub.s, where .sigma..sub.s is the standard deviation of the sinusoid plus the noise. The step begins at time 500 seconds. Due to the low SNR, MONOSPRT takes 25 samples to alarm, indicating that the signal is not at a peak in the sinusoid but rather that the mean of the overall signal has changed. In FIGS. 2E and 2F analogous MONOSPRT results are shown for a linear drift introduced into the noisy sinusoid signal. Here, the drift starts at time 500 seconds at a value of 0 and increases linearly to a final value of 4.sigma..sub.s at 1000 seconds. MONOSPRT detects the drift when it has reached a magnitude of approximately 1.5.sigma..sub.s. In FIGS. 3A-3F the results of running the same experiment are shown except this time the SNR is 0.5 and the SFM is changed to 1.5. The degree of autocorrelation is much higher in this case, but MONOSPRT can detect the disturbances more quickly due to the increased SNR. To test MONOSPRT on an actual sensor signal exhibiting non-white characteristics a sensor signal was selected from the primary pump #2 of the EBR-II nuclear reactor at Argonne National Laboratory (West) in Idaho. The signal is a measure of the pump's speed over a 1000 minute interval. FIG. 4A shows the sensor signal under normal operating conditions. The MONOSPRT results are shown in FIG. 4B. For this example .alpha. and .beta. are specified to be 0.0001 and the SFM is 2.5. The autocorrelation matrix was calculated using 10 lags. In FIGS. 5A and 5B MONOSPRT results are shown when a very subtle sensor drift is simulated. FIG. 5A is the sensor signal with a linear drift starting at time 500 min and continuing through the rest of the signal to a final value of -0.10011% of the sensor signal magnitude. MONOSPRT detects this very small drift after about only 50 min, i.e. when the drift has reached a magnitude of approximately 0.01% of the signal magnitude. The MONOSPRT plot is shown in FIG. 5B with the same parameter settings as were used in FIG. 4B. FIG. 5B illustrates the extremely high sensitivity attainable with the new MONOSPRT methodology. In another preferred embodiment (the regression SPRT method of FIG. 1C), a methodology provides an improved method for monitoring redundant process signals of safety- or mission-critical systems. In the flow diagram shown in FIG. 1C, the method is split into two phases, a training phase and a monitoring phase. During the training phase N data samples are collected from both sensors when the system is operating normally. The two data sets are then used to calculate the regression coefficients m and b using the means of both sensor signals (.mu..sub.1 and .mu..sub.2), the autocorrelation coefficient of one of the sensors (.sigma..sub.22), and the cross-correlation coefficient (.sigma..sub.12) between both sensors. The SPRT parameters are also calculated in the same manner as was calculation of the SDM is from the regression difference function. During the monitoring phase of the regression SPRT method, a regression based difference (D.sub.t) is generated at each time point t. The regression based difference is then used to calculate the SPRT index and to make a decision about the state of the system or sensors being monitored. The logic behind the decision is analogous to the decision logic used in the MONOSPRT method. Further details are described hereinafter. In this method, known functional relationships are used between process variables in a SPRT type of test to detect the onset of system or sensor failure. This approach reduces the probability of false alarms while maintaining an extremely high degree of sensitivity to subtle changes in the process signals. For safety- or mission-critical applications, a reduction in the number of false alarms can save large amounts of time, effort and money due to extremely conservative procedures that must be implemented in the case of a failure alarm. For example, in nuclear power applications, a failure alarm could cause the operators to shut down the reactor in order to diagnose the problem, an action which typically costs the plant a million dollars per day. In this preferred embodiment shown schematically in flow diagram FIG. 1C (two sensors, linearly related), highly redundant process signals can be monitored when the signals have a known functional relationship given by EQU X.sub.t =f(X.sub.2) (12) where f( ) is some function determined by physical laws or by known (or empirically determined) statistical relationships between the variables. In principle, if either of the process signals X.sub.1 or X.sub.2 have degraded (i.e. fallen out of calibration) or failed, then (12) will no longer hold. Therefore, the relationship (12) can be used to check for sensor or system failure. In practice, both monitored process signals, or any other source of signals, contain noise, offsets and/or systematic errors due to limitations in the sensors and complexity of the underlying processes being monitored. Therefore, process failure cannot be detected simply by checking that (12) holds. More sophisticated statistical techniques must be used to ensure high levels of noise or offset do not lead to false and missed failure alarms. This preferred embodiment involves (a) specifying a functional relationship between X.sub.1 and X.sub.2 using known physical laws or statistical dependencies and linear regression when the processes are known to be in control, and (b) using the specified relationship from (a) in a sequential probability ratio test (SPRT) to detect the onset of process failure. For example, in many safety- or mission-critical applications, multiple identical sensors are often used to monitor each of the process variables of interest. In principle, each of the sensors should give identical readings unless one of the sensors is beginning to fail. Due to measurement offsets and calibration differences between the sensors, however, the sensor readings may be highly statistically correlated but will not be identical. By assuming that the sensor readings come from a multivariate normal distribution, a linear relationship between the variables can be specified. In particular, for two such sensor readings it is well-known that the following relationship holds EQU E[X.sub.1.vertline.X.sub.2 ]=.sigma..sub.12 /.sigma..sub.12 (X.sub.2 -u.sub.2)+u.sub.1 (13) where E[X.sub.1.vertline.X.sub.2 ] is the conditional expectation of the signal X.sub.1 given X.sub.2, .sigma..sub.12 is the square root of the covariance between X.sub.1 and X.sub.2. The .sigma..sub.22 is the standard deviation of X.sub.2, and u.sub.1 and u.sub.2 are the mean of X.sub.1 and X.sub.2 respectively. Equation (13) is simply a linear function of X.sub.2 and can therefore be written EQU X.sub.1 =mX.sub.2 +b (14) In practice, the slope m=.sigma..sub.12 /.sigma..sub.22 and intercept b=-.sigma..sub.12 /.sigma..sub.22 u.sub.2 +u.sub.1 can be estimated by linear regression using data that is known to have no degradation or failures present. Once a regression equation is specified for the relationship between X.sub.1 and X.sub.2, then the predicted X.sub.1 computed from (14) can be compared to the actual value of X.sub.1 by taking the difference EQU D.sub.1 =X.sub.1 -(mX.sub.2 +b) (15) Under normal operating conditions, D.sub.1, called the regression-based difference, will be Gaussian with mean zero and so me fixed standard deviation. As one of the sensors begins to fail or degrade, the mean will begin to chance. A change in the mean of this regression-based difference can be detected using the SPRT methodology. The SPRT approach is a log-likelihood ratio based test for simple or composite hypothesis (also see the incorporated patents cited hereinbefore). To test for a change in the mean of the regression-based difference signal D.sub.1, D.sub.2, . . . , the following two hypotheses are constructed: where H.sub.0 refers to the probability distribution of the regression-based difference under no failure and H.sub.F refers to the probability distribution of the regression-based difference under system or process failure. The SPRT is implemented by taking the logarithm of the likelihood ratio between H.sub.0 and H.sub.F. In particular, let f.sub.0 (d.sub.i) represent the probability density function for D.sub.1, D.sub.2, . . . under H.sub.0, and f.sub.1 (d.sub.i) represent the probability density function for D.sub.1, D.sub.2, . . . under H.sub.F. Let Z.sub.i =log [.function..sub.1 (X.sub.i)/.function..sub.0 (X.sub.i)] the log-likelihood ratio for this test. Then ##EQU10## Defining the value S.sub.n to be the sum of the increments Z.sub.i up to time n where S.sub.n =.SIGMA..sub.1.ltoreq.i.ltoreq.n Z.sub.i, then the SPRT algorithm can be specified by the following: If S.sub.n .ltoreq. B terminate and decide H.sub.0 If B &lt; S.sub.n &lt; A continue sampling If S.sub.n .gtoreq. A terminate and decide H.sub.F The endpoints A and B are determined by the user specified error probabilities of the test. In particular, let .alpha.=P{conclude H.sub.F.vertline.H.sub.0 true} be the type I error probability (false alarm probability) and .beta.=P{conclude H.sub.0.vertline.H.sub.F true} be the type II error probability (missed alarm probability) for the SPRT. Then ##EQU11## For real time applications, this test can be run repeatedly on the computed regression-based difference signal as the observations are collected so that every time the test concludes H.sub.0, the sum S.sub.n is set to zero and the test repeated. On the other hand, if the test concludes H.sub.F, then a failure alarm is sounded and either the SPRT is repeated or the process terminated. An illustration of this preferred form of bivariate regression SPRT method can be based on the EBR-II nuclear reactor referenced hereinbefore. This reactor used redundant thermocouple sensors monitoring a subassembly outlet temperature, which is the temperature of coolant exiting fuel subassemblies in the core of the reactor. These sensors readings are highly correlated, but not identical. The method of this embodiment as applied to this example system was performed using two such temperature sensors; X.sub.1 =channel 74/subassembly outlet temperature 4E1, and X.sub.2 =channel 63/subassembly outlet temperature 1A1. For 24 minutes worth of data during normal operation on Jul. 7, 1993, a regression line is specified for X.sub.1 as a function of X.sub.2 according to equation (14). The predicted X.sub.1 from (14) is then compared to the actual X.sub.1 by taking the regression-based difference (15) in our new regression-SPRT algorithm. The results of this experiment are then compared to the results of performing a prior-art SPRT test on the difference X.sub.2 -X1 according to U.S. Pat. No. 5,410,492. Plots of subassembly outlet temperature 1A1 and 4E1 under normal operating conditions are given in FIG. 7A. The relationship between the two variables when no failure is present is illustrated in FIG. 8. In FIG. 8, the slope and intercept of the regression line from equation (14) are given. FIGS. 9A and 9B illustrate the regression-based difference signal along with the difference signal of the prior art proposed by U.S. Pat. No. 5,223,207. It is easy to see that the regression-based difference signal tends to remain closer to zero than the original difference signal under normal operating conditions. FIGS. 9A and 9B plot the results of a SPRT test on both the regression-based difference signal and the original difference signal. In both cases, the pre-specified false- and missed-alarm probabilities are set to 0.01, and the threshold for failure (alternate hypothesis mean) is set to 0.5.degree. F. In both subplots, the circles indicate a failure decision made by the SPRT test. Note that under no failure or degradation modes, the new regression-based SPRT gives fewer false alarms than the original difference. The calculated false alarm probabilities are given in Table I for these comparative SPRT tests plotted in FIGS. 9A and 9B. TABLE I Empirical False Alarm Probability for the SPRT test to Detect Failure of an EBR-II Subassembly Outlet Temperature Sensor Regression-Based Original Difference Difference False Alarm Probability 0.025 0.0056 The empirical false alarm probability for the SPRT operated on the regression-based difference (see FIG. 9A) is significantly smaller than the for the SPRT performed on the original difference signal (see FIG. 9B), indicating that it will have a much lower false-alarm rate. Furthermore, the regression-based difference signal yields a false alarm probability that is significantly lower than the pre-specified false alarm probability, while the original difference function yields an unacceptably high false alarm probability. To illustrate the performance of the regression-based difference method in a SPRT methodology under failure of one of the sensors, a gradual trend is added to the subassembly outlet temperature 4E1 to simulate the onset of a subtle decalibration bias in that sensor. The trend is started at 8 minutes, 20 seconds, and has a slope of 0.005.degree. F. per second. The EBR-II signals with a failure injected in the 4E1 sensor are plotted in FIGS. 10A and 10B. The regression-based difference signal and the original difference signal are plotted in FIGS. 11A and 11B. FIGS. 12A and 12B plot the results of the SPRT test performed on the two difference signals. As before, the SPRT has false and missed alarm probabilities of 0.01, and a sensor failure magnitude of 0.5.degree. F. In this case, the regression-based SPRT annunciated the onset of the disturbance even earlier than the conventional SPRT. The time of failure detection is given in Table II. TABLE II Time to Detection of Gradual Failure of EBR-II Subassembly Outlet Temperature Regression-Based Original Difference Difference Time to Failure Detection 9 min. 44 sec. 9 min. 31 sec. These results indicate that the regression-based SPRT methodology yields results that are highly sensitive to small changes in the mean of the process. In this case, using the regression-based SPRT gave a failure detection 13 seconds before using the prior art method. A problem that is endemic to conventional signal surveillance methods is that as one seeks to improve the sensitivity of the method, the probability of false alarms increases. Similarly, if one seeks to decrease the probability of false alarms, one sacrifices sensitivity and can miss the onset of subtle degradation. The results shown here illustrate that the regression-based SPRT methodology for systems involving two sensors simultaneously improves both sensitivity and reliability (i.e. the avoidance of false alarms). It is also within the scope of the preferred embodiments that the method can be applied to redundant variables whose functional relationship is nonlinear. An example of this methodology is also illustrated in FIG. 1 branching off the "sensors are linearly related" to the "monitor separately" decision box which can decide to do so by sending each signal to the MONOSPRT methodology or alternatively to the BART methodology described hereinafter. In particular for a nonlinear relation, if the monitored processes X.sub.1 and X.sub.2 are related by the functional relationship EQU X.sub.1 =f(X.sub.2) (18) where f( ) is some nonlinear function determined by physical laws (or other imperical information) between the variables, then the relationship (18) can be used to check for sensor or system failure. In this case, the relationship (18) can be specified by using nonlinear regression of X.sub.1 on X.sub.2. The predicted X.sub.1 can then be compared to the actual X.sub.1 via the regression-based SPRT test performed on the resulting nonlinear regression-based difference signal. In another form of the invention shown in FIG. 1D in systems with more than two variables one can use a nonlinear multivariate regression technique that employs a bounded angle ratio test (hereinafter BART) in N Dimensional Space (known in vector calculus terminology as hyperspace) to model the relationships between all of the variables. This regression procedure results in a nonlinear synthesized estimate for each input observation vector based on the hyperspace regression model. The nonlinear multivariate regression technique is centered around the hyperspace BART operator that determines the element by element and vector to vector relationships of the variables and observation vectors given a set of system data that is recorded during a time period when everything is functioning correctly. In the BART method described in FIG. 1D., the method is also split into a training phase and a monitoring phase. The first step in the training phase is to acquire a data matrix continuing data samples from all of the sensors (or data sources) used for monitoring the system that are coincident in time and are representative of normal system operation. Then the BART parameters are calculated for each sensor (Xmed, Xmax and Xmin). Here Xmed is the median value of a sensor. The next step is to determine the similarity domain height for each sensor (h) using the BART parameters Xmed, Xmax and Xmin. Once these parameters are calculated a subset of the data matrix is selected to create a model matrix (H) that is used in the BART estimation calculations. Here, H is an NxM matrix where N is the number of sensors being monitored and M is the number of observations stored from each sensor. As was the case in both the MONOSPRT and regression SPRT method, the last steps taken during the training phase are the SPRT parameters calculations. The calculations are analogous to the calculations in the other methods, except that now the standard deviation value used to calculate SDI is obtained from BART estimation errors from each sensor (or data source) under normal operating conditions. During the BART monitoring phase a sample vector is acquired at each time step t, that contains a reading from all of the sensors (or data sources) being used. Then the similarity angle (SA) between the sample vector and each sample vector stored in H is calculated. Next an estimate of the input sample vector Y is calculated using the BART estimation equations. The difference between the estimate and the actual sensor values is then used as input to the SPRT. Each difference is treated separately so that a decision can be made on each sensor independently. The decision logic is the same as is used in both MONOSPRT and the regression SPRT methods. This method is described in more detail immediately hereinafter. In this embodiment of FIG. 1D of the invention, the method measures similarity between scalar values. BART uses the angle formed by the two points under comparison and a third reference point lying some distance perpendicular to the line formed by the two points under comparison. By using this geometric and trigonometric approach, BART is able to calculate the similarity of scalars with opposite signs. In the most preferred form of BART an angle domain must be determined. The angle domain is a triangle whose tip is the reference point (R), and whose base is the similarity domain. The similarity domain consists of all scalars which can be compared with a valid measure of similarity returned. To introduce the similarity domain, two logical functional requirements can be established: Thus we see that the similarity range (i.e. all possible values for a measure of similarity), is the range 0 to 16) inclusive. BART also requires some prior knowledge of the numbers to be compared for determination of the reference point (R). Unlike a ratio comparison of similarity, BART does not allow "factoring out" in the values to be compared. For example, with the BART methodology the similarity between 1 and 2 is not necessarily equal to the similarity between 2 and 4. Thus, the location of R is vital for good relative similarities to be obtained. R lies over the similarity domain at some distance h, perpendicular to the domain. The location on the similarity domain at which R occurs (Xmed) is related to the statistical distribution of the values to be compared. For most distributions, the median or mean is sufficient to generate good results. In or preferred embodiment the median is used since the median provides a good measure of data density, and is resistant to skewing caused by large ranges of data. Once Xmed has been determined, it is possible to calculate h. In calculating h, it is necessary to know the maximum and minimum values in the similarity domain. (Xmax and Xmin respectively) for normalization purposes the angle between Xmin and Xmax is defined to be 90.degree.. The conditions and values defined so far are illustrated in FIG. 13. From this triangle it is possible to obtain a system of equations and solve for h as shown below: EQU c=Xmed-Xmin EQU d-Xmax-Xmin EQU a.sup.2 -c.sup.2 +h.sup.2 EQU b.sup.2 -d.sup.2 +h.sup.2 (19) EQU (c+d(.sup.2 -.sup.2 +b.sup.2 EQU (c+d).sup.2 c.sup.2 -d.sup.2 +2h.sup.2 EQU h.sup.2 =cd EQU h=cd Once h has been calculated the system is ready to compute similarities. Assume that two points: X.sub.0 and X.sub.1 (X.sub.0.ltoreq.X.sub.1) are given as depicted in FIG. 14 and the similarity between the two is to be measured. The first step in calculating similarity is normalizing X.sub.0 and X.sub.1 with respect to Xmed. This is done by taking the euclidean distance between Xmed and each of the points to be compared. Once X.sub.0 and X.sub.1 have been normalized, the angle .angle.X.sub.0 RX.sub.1 (hereinafter designated .theta.) is calculated by the formula: EQU .theta.=Arc Tan(X.sub.1.vertline.h)=Arc Tan(X.sub.0.vertline.h) (20) After .theta. has been found, it must be normalized so that a relative measure of similarity can be obtained that lies within the similarity range. To ensure compliance with functional requirements (A) and (B) made earlier in this section, the relative similarity angle (SA) is given by: ##EQU12## Formula (21) satisfies both functional requirements established at the beginning of the section. The angle between Xmin and Xmax was defined to be 90.degree., so the similarity between Xmin and Xmax is 0. Also, the angle between equal values is 0.degree.. The SA therefore will be confined to the interval between zero and one, as desired. To measure similarity between two vectors using the BART methodology, the average of the element by element SAs are used. Given the vectors x.sub.1 and x.sub.2 the SA is found by first calculating S.sub.i for i=1,2,3 . . . n for each pair of elements in x.sub.1 and x.sub.2 i.e., The vector SA .GAMMA. is found by averaging over the S.sub.i 's and is given by the following equation. ##EQU13## In general, when given a set of multivariate observation data from a process (or other source of signals), we could use linear regression to develop a process model that relates all of the variables in the process to one another. An assumption that must be made when using linear regression is that the cross-correlation information calculated from the process data is defined by a covariance matrix. When the cross-correlation between the process variables is nonlinear, or when the data are out of phase, the covariance matrix can give misleading results. The BART methodology is a nonlinear technique that measures similarity instead of the traditional cross-correlation between variables. One advantage of the BART method is that it is independent of the phase between process variables and does not require that relationships between variables be linear. If we have a random observation vector y and a known set of process observation vectors from a process P, we can determine if y is a realistic observation from a process P by combining BART with regression to form a nonlinear regression method that looks at vector SAs as opposed to euclidean distance. If the know observation vectors taken from P are given by, ##EQU14## where H is k by m (k being the number of variables and m the number of observations), then the closest realistic observation vector to y in process P given H is given by EQU y=Hw (24) Here w is a weighting vector that maps a linear combination of the observation vectors in H to the most similar representation of y. The weighting vector w is calculated by combining the standard least squares equation form with BART. Here, .theta.stands for the SA operation used in BART. EQU w=(H'.sym.H).sup.-1 H'.sym.y (25) An example of use of the BART methodology was completed by using 10 EBR-II sensor signals. The BART system was trained using a training data set containing 1440 observation vectors. Out of the 1440 observation vectors 129 were chosen to be used to construct a system model. The 129 vectors were also used to determine the height h of the angle domain boundary as well as the location of the BART reference point R for each of the sensors used in the experiment. To test the accuracy of the model 900 minutes of one minute data observation vectors under normal operating conditions were run through the BART system. The results of the BART system modeling accuracy are shown in FIGS. 15A-15E and FIGS. 16A-16E (BART modelled). The Mean Squared Errors for each of the sensor signals is shown in Table III. TABLE III BART System Modeling Estimation Mean Squared Errors for EBR-II Sensor Signals MSE of Normalized Normalized Sensor Estimation MSE MSE Channel Sensor Description Error (MSE/.mu..sub.3) (MSE/.sigma..sub.3) 1. Primary Pump #1 0.0000190 0.0000002 0.0002957 Power (KW) 2. Primary Pump #2 0.0000538 0.0000004 0.0004265 Power (KW) 3. Primary Pump #1 0.0000468 0.0000001 0.0005727 Speed (RPM) 4. Primary Pump #2 0.0000452 0.0000001 0.0004571 Speed (RPM) 5. Reactor Outlet 8.6831039 0.0009670 0.1352974 Flowrate (GPM) 6. Primary Pump #2 0.0571358 0.0000127 0.0163304 Flowrate (GPM) 7. Subassembly Outlet 0.0029000 0.0000034 0.0062368 Temperature 1A1 (F) 8. Subassembly Outlet 0.0023966 0.0000027 0.0052941 Temperature 2B1 (F) 9. Subassembly Outlet 0.0025957 0.0000029 0.0050805 Temperature 4E1 (F) 10. Subassembly Outlet 0.0024624 0.0000028 0.0051358 Temperature 4F1 (F) A second example shows the results of applying BART to ten sensors signals with three different types of disturbances with their respective BART estimates superimposed followed by the SPRT results when applied to the estimation error signals. The first type of disturbance used in the experiment was a simulation of a linear draft in channel #1. The drift begins at minute 500 and continues through to the end of the signal, reaching a value of 0.21% of the sensor signal magnitude and the simulation is shown in FIG. 17A. The SPRT (FIG. 17B) detects the drift after it has reached a value of approximately 0.06% of the signal magnitude. In FIG. 17C a simulation of a step failure in channel #2 is shown. Here the step has a height of 0.26% of the signal magnitude and begins at minute 500 and continues throughout the signal. FIG. 17D shows the SPRT results for the step failure. The SPRT detects the failure immediately after it was introduced into the signal. The last simulation was that of a sinusoidal disturbance introduced into channel #6 as shown in FIG. 17E. The sinusoid starts at minute 500 and continues throughout the signal with a constant amplitude of 0.15% of the sensor signal magnitude. The SPRT results for this type of disturbance are shown in FIG. 17F. Again the SPRT detects the failure even though the sinusoid's amplitude is within the operating range of the channel #6 sensor signal. In further variations on the above described embodiments a user can generate one or more estimated sensor signals for a system. This methodology can be useful if a sensor has been determined to be faulty and the estimated sensor signal can be substituted for a faulty, or even degrading, sensor or other source of data. This methodology can be particularly useful for a system having at least three sources of data, or sensors. While preferred embodiments of the invention have been shown and described, it will be clear to those skilled in the art that various changes and modifications can be made without departing from the invention in its broader aspects as set forth in the claims provided hereinafter.