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I. INTRODUCTION AND MOTIVATION

The moving horizon estimation (MHE) concepts date back to the early 1990s [1]. The fundamental philosophy of MHE is to estimate the temporal evolution of states by solving a (nonlinear) least-squares problem, while penalizing the de-viation between the measurements and predicted outputs of the system. The observations considered for optimization lie in a fixed finite length time window. The MHE problem is particularly appealing due to the ability to integrate non-linearities and constraints into states and disturbances. The viability and stability of MHE for state estimation of linear and nonlinear systems, along with its ability to explicitly in-corporate constraints, was shown to perform better than oth-er strategies, such as extended Kalman filtering and output error linearization. The accuracy with which the old data is approximated by the arrival cost (analogous to cost-to-arrive concept in deterministic dynamic programming) determines the performance of MHE algorithms. Hence, the old measure-ments can be discarded from the estimation window when new measurements become accessible, and subsequently, the current state estimate is computed online by solving a finite horizon optimization problem. Two major advantages of a moving horizon approach over an increasing horizon or batch

estimation approach are that it solves a problem of fixed size at each time epoch and summarizes the past data by the ar-rival cost. In summary, MHE algorithms are ideal for practical implementation because they amount to problems of finite di-mension and have the capability to incorporate nonlinearities and constraints on states and disturbances [2].

Prognosis deals with early detection of anomalies, root cause isolation, and remaining useful life (RUL) estimation. It plays a pivotal role in condition-based maintenance (CBM) of contemporary systems. The prognostic methods can be broadly classified into model-based and data-driven approaches [3]. Model-based techniques assume an accurate mathemati-cal model of the system and track the residuals, using observ-ers (such as Kalman filters, interacting multiple models, par-ticle filters) and parity relations (dynamic consistency checks among measured variables). On the other hand, data-driven techniques are derived from routinely monitored system op-erating data and are based on statistical regression techniques (such as nonlinear least squares, partial least squares, princi-pal component analysis, support vector machines, multilayer perceptrons) to graphical models (Bayesian networks, hidden Markov models), and signal analysis (wavelets, fast Fourier transforms, correlation analysis).

The time series-based approaches to prognosis are compo-nent centric and do not make use of widely available data in archived databases of equipment, such as historical usage pat-terns, error codes, observed failure modes, repair and inspec-tion intervals, environmental factors, skill levels of personnel, and status parameters collected periodically or at the onset of error codes. Consequently, the time series-based prognos-tic health management approaches are both incomplete and inaccurate for coupled systems with cross-subsystem fault propagation. On the other hand, the classical survival theory-based approaches rely on Weibull and other nonlinear regres-sion models to infer the time to failure, and these estimates are used to optimize the time to maintain or time to repair/replace; these techniques do not consider condition indicators of equipment and cross-subsystem fault propagation. Conse-quently, the survival theory-based techniques result in large variability in the time-to-failure estimates. Evidently, the two disparate methodologies need to be reconciled and unified un-der a common modeling framework that combines archived failure time data and static and dynamic parametric data [4].

Multiple Model Moving Horizon Estimation Approach to Prognostics in Coupled SystemsBharath Pattipati, Chaitanya Sankavaram, Krishna Pattipati, Yilu Zhang, Mark Howell, Mutasim Salman University of Connecticut General Motors Company

Authors’ current addresses: B. Pattipati, C. Sankavaram, and K. Pattipati, Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Road, U-2157, Storrs, CT 06269, USA. E-mail: Krishna, Bharath@engr.uconn.edu. Y. Zhang, M. Howell, and M. Salman, GM Global R&D, General Motors Company, 30500 Mound Road, Warren, MI 48090, USA.

The work reported in this paper was supported by the Na-tional Science Foundation (NSF) under grants ECCS-0931956 (NSF CPS) and ECCS-1001445 (NSF GOALI).

A preliminary version of the paper was published as Pat-tipati, B., Sankavaram, C., Pattipati, K., Zhang, Y., Howell, M., and Salman, M. Multiple model moving horizon estimation approach to prognostics in coupled systems. In Proceedings of the IEEE AUTOTESTCON, Baltimore, MD, September 12–15, 2011, 149–157. Manuscript number SYSAES-2012-0054rr re-ceived March 11, 2012, revised June 1, 2012, July 23, 2012, and ready for production October 3, 2012. 0885/8985/13/ $26.00 © 2013 IEEE

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This article focuses on a data-driven prognostics ap-proach for estimating the states of component survival functions and cluster weights of an electronic throttle control (ETC) system, given noisy test measurements from a mixed com-ponent cluster. The main contributions of this article can be summarized as follows:

1. Novel multiple-model MHE (MM-MHE) algorithm for predicting the survival functions of components based on their usage profiles.

2. A comprehensive prognostic framework based on offline and online data for RUL prediction.

The article presents a general model predictive estima-tion model and previous work in this area; a unified prog-nostics framework that combines the failure time data, as well as static and dynamic parametric data based on Cox proportional hazards model (PHM) [5],[6]; a description of the MM-MHE algorithm used for predicting the survival functions and cluster weights; experimental results; and di-rections for future work.

II. MODEL PREDICTIVE ESTIMATION

Model predictive control (MPC) is a form of control in which, at each sampling instant, the current control action is ob-tained online by solving a finite horizon open-loop optimal control problem, using the current state of the system as the initial state. The process yields an optimal control sequence, and the first control in this sequence is applied to the system [7]. An important advantage of this approach is its ability to cope with equality and inequality constraints on controls and states. The essence of MPC is to optimize predictions of the system behavior over the manipulatable inputs. Hence, the MPC strategy generally comprises (i) a model to predict the process output over a finite future time horizon, (ii) the computation of the control sequence to optimize a perfor-mance index over this horizon, (iii) the application of the first control signal of the sequence, and (iv) the moving the horizon by one step toward the future.

The moving horizon MPC approach is depicted in Figure 1. Based on this figure, the moving horizon approach can be summarized by the following three fundamental ideas:

1. The process model (linear or nonlinear) computes the predicted future outputs ( )ˆ | , 1,..., 1y k j k j N+ = − for the prediction horizon N at each time instant k based on the known values up to instance k (past inputs and outputs), including the current output (or initial con-dition) ( )y k and the calculated future control signals

( )| , 0,..., 1u k j k j N+ = − .

2. The sequence of future control signals is computed to optimize a performance criterion over the prediction ho-rizon.

3. At the next sampling instant, y(k + 1) is measured, Step 1 is repeated, and all sequences are brought up to date. Hence, ( )1| 1u k k+ + is calculated using the moving ho-rizon concept, and the predicted state estimates are com-puted by minimizing the predicted error or a finite hori-zon cost function.

Originally developed to meet the specialized control needs of power plants and petroleum refineries [9], MPC technology can now be found in a wide variety of application areas, including chemical, food processing, automotive, and aerospace applications. Several publications provide a good introduction to the theoretical and practical issues associated with MPC technology. Rawlings et al. [10] presents an excel-

Figure 1. Moving horizon strategy of MPC [8].

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MM-MHE Approach to Prognost ics

lent tutorial aimed at control practitioners. An overview of commercially available linear and nonlinear MPC technology, its applications, and a brief history is provided in Qin et al. [11].

Rao et al. [12] investigated MHE as an online constrained optimization strategy for estimating the state variables of constrained discrete-time systems and demonstrated the superior performance and modeling accuracy of MHE com-pared to other strategies, such as extended Kalman filtering and output error linearization. Rawlings and Bakshi [13] have also proposed a combination of powerful state estima-tion techniques, such as particle filters and MHE for detect-ing and tracking multi-modal densities, which are common in chemical process control applications.

III. PROGNOSTIC FRAMEWORK

As shown in Figure 2, three types of data are considered in our prognostic framework:

1. Archived failure data (or Type I data): age of the equip-ment at the time of failure, i.e., age when an error code or symptom is observed or a component is replaced;

2. Static environmental and status parameter data (or Type II data), such as those collected during take-off and cruise conditions of an aircraft; and

3. Dynamic data (or Type III data): time series data and pe-riodic status data.

The proposed data-driven prognostic framework is shown in Figure 3. It is divided into two phases, namely, the train-ing or model learning phase (offline module) and testing or deployment phase (online module).

In the training and validation (or model learning) phase, we compute the static data-modulated component survival functions, error codes, symptoms, and any observable test outcomes via Cox PHM and cluster the survival functions of each component via clustering techniques, such as k means, learning vector quantization (LVQ), Gaussian mixture mod-

els (GMM), or hierarchical clustering [14], [15]. These clusters represent the different usage profiles for the components, de-pending on the usage conditions, environmental factors, etc. The Cox PHM assumes a hazard function of the form

( ) ( ) ( )0, exp Ti ih t z h t r z= , (1)

where i denotes the component, diagnostic error code, or any failure mode of interest, z is a vector of covariates (Type II static data such as freeze frame data in automotive vehicles or health and usage monitoring data in helicopters), ir is the vector of regression parameters, and h0(t) is the baseline haz-ard function (with z =0). The baseline hazard function can be from any of the standard failure time distributions (e.g., exponential, Weibull, normal, log normal, gamma, etc.) or it can be nonparametric. The baseline hazard function and regression parameters are estimated via a maximum likeli-hood method [16].

The survival function, also known as the reliability func-tion, represents the probability that the system will survive beyond a specified time. The component survival functions, Si(t,z) and the associated failure density functions, fi(t,z) can be computed from the hazard functions hi(t,z) as follows:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

0

0

, exp ,

,,

, exp ,

, ,

t

i i

ii

t

i i

i i

S t z h z d

dS t zf t z

dt

h t z h z d

h t z S t z .

τ τ

τ τ

= −

= −

= − =

∫

∫

(2)

Subsequently, in the testing (or deployment) phase, on obtain-ing new measurements via online data acquisition systems, the survival functions and cluster weights are estimated via the MM-MHE algorithm.

The RUL of a component at any time epoch t can be com-puted from the survival function by defining an application-dependent threshold on the survival probability.

IV. MM-MHE APPROACH

The key problem is to estimate the states of the compo-nent survival functions and cluster weights (posterior prob-abilities of discrete states denoting usage profile), given noisy test measurements (prognostic indicators) from a mixed com-ponent cluster. Mathematically, the dynamics of the survival function can be represented in discrete time as [17],

A k l a r l((

( ((( ( ( ( (

)

) ))

)0

1, , , , 0

, exp ,

, ,

i i i i i

i i

k

i ir

S k l a k l S k l A k l S

a k l h k l

=

+ = =

= − ∆

= ,

∏

)))))

(3)

Figure 2. Prognostic data categorization.

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where Δ is the time step in a suitable timescale (e.g., days). The terms Si(k) and hi(k) are, respectively, the survival func-tion and hazard rate of component { }1,2,....,i m∈ at time epoch kΔ. The cluster number is denoted by l, and each com-ponent may have different numbers of clusters Li, where

{ }1,2,...., il L∈ for component i. This implies that we need to infer the discrete-state usage profile from the prognostic indicators.

The measurements are generated based on a nonlinear function (negative logarithm) of the test survival functions (soft test outcomes), which represent a nonlinear function of mixed component survival functions with noise to mimic observations from an unknown user profile as shown below.

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1 1

ln

ln 0 , 1i

j j

Lm

i i i ij ij ij ji l

y k T k

S A k l l PD PF PD v kα= =

= −

= − − + − + ∑ ∑ , (4)

where ( ) ( )2~ 0,d

jv k N σ . Equation (4) corresponds to a noisy-OR dependency between the test outcome and the set of components monitored by that test [4]. The noise variance

( ) 22jy kσ δ= (d is varied from 0.01–5.12 in our experiments

to assess the robustness of the algorithm to measurement noise. This corresponds to a signal-to-noise ratio (SNR) range of 20 dB to −7 dB). Hence, Rv is a diagonal n × n mea-surement noise covariance matrix. The survival function of test { }1,2,....,j n= at time kΔ is denoted by Tj (k). The prob-ability of detection and the probability of the false alarm of test { }1,2,....,j n= for failure in component { }1,2,....,i m= are denoted by PDijand PFij, respectively.

The goal is to accurately predict the component degra-dation according to the current operating conditions. The cost function for the current window is computed based on a nonlinear least-squares approach minimizing the square of the residual between the current measurements and pre-dicted outputs of the system. The arrival cost is calculated by penalizing the deviations based on previous predicted state estimates and covariances. Hence, the overall cost is formu-lated as a summation of the arrival cost and the current win-dow cost. Consequently, the MM-MHE algorithm predicts the component survival functions and cluster weights based on the optimal cost and constraints.

The overall cost for the augmented vector S

xa

=

of sur-

vival functions and cluster weights is as follows:

Figure 3. Cox PHM multiple model prognostics approach.

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MM-MHE Approach to Prognost ics

, (5)

where, b is the arrival cost scale factor (tuning parameter) and vj denotes the residual, which is the difference between the current and predicted measurements. For implementation purposes, the augmented vector x is set up as an augmented column vector of S and a, where ( ) ( )1 0 . . . 0

TmS S S= and

[ ]11 12 1 21 22.. ... Tl mlα α α α α α α= . In the initial

window, the cluster weights a are assumed to be uniformly distributed for all components, and the component survival functions are initialized with ones. The survival functions are constrained to be between 0 and 1, and the sum of all cluster weights for each component is constrained to be equal to 1.

The moving horizon or online estimation algorithms require us to keep track of the covariance and update it at each time window to compute the arrival cost, which summarizes the past information. The information matrix for each window 1

LK−∑ is updated as a summation of the

information matrix from the previous window, 10−∑ , and

the information matrix from the new measurements. The information matrix is initialized with zeros for the 0th time window.

V. APPLICATION TO ETC CONTROL SYSTEM

Here, we apply the MM-MHE approach to an automotive ETC system. The ETC, also termed drive-by-wire tech-nology, replaces the conventional mechanical linkage be-tween accelerator pedal and throttle body. The function of an ETC system is to determine the necessary throttle opening using sensors (such as accelerator pedal position, engine RPM, and vehicle speed) and drive the actuator to obtain the required throttle position via a closed-loop control algorithm in the electronic control module (ECM). The ECM monitors the health of the subsystems by pro-cessing parameter identifier data (PIDs) collected from various sensors and generates diagnostic trouble codes (DTCs or error codes) when a failure in a component is observed.

The dataset derived from the ETC simulator consisted of 11 error codes (DTCs), 479 status parameters (PIDs) collected at the time of DTC firing, age of the vehicle, and the repair/replacement actions (i.e., repair codes [RCs]) performed on the system. A total of five different RCs (replaceable compo-nents) were present in our training data. However, due to ambiguity in two of the RCs, they were grouped into a single RC [4]. Mutual information gain is employed to select the minimal number of PIDs, and the top 16 PIDs are selected for our analysis [4], [14], [15].

The survival functions for components and tests are ini-tially learned using the Cox PHM model as described in the previous discussion of prognostic framework. Then, k-means clustering technique is employed to group the survival func-

tions for RCs as well as DTCs. For the datasets provided, there appear to be three clusters of survival functions for each RC. To validate the prognostic framework, the detec-tion and false alarm probabilities of tests are initially learned from the averaged DTC survival function and averaged RC survival functions by minimizing the objective function [Eq. (6)] shown below [4].

.

(6)

This implies that for test j to pass all the components moni-tored by test j, it must be healthy, and test j must not have a false alarm, or if a component has failed, then test j must have missed it. This is the so-called “noisy-OR” model of evidence in Bayesian networks [4].

Taking the negative logarithm and rewriting, we get,

(7)

The nonlinear least-squares minimization is implemented using the optimization toolbox function of MATLAB® fmin-con to determine the optimal detection and false alarm prob-abilities {PDij, PFij}.

The MM-MHE algorithm was also implemented in MAT-LAB using the fmincon function, which finds the minimum of the constrained nonlinear multivariate objective function based on sequential quadratic programming using quasi-Newton line search. The optimization was terminated when the magnitude of directional derivative in the search direc-tion was less than a sufficiently small tolerance (of the order of 10−9) on the function value and the maximum constraint violation was less than a specified tolerance (of the order of 10−9). The scale factor b is a tuning parameter and was found by experimentation to lie between 10−6 and 10−9 for the ETC dataset. The RUL of a component at any time can be com-puted by defining an application-specific threshold on the survival probability.

The measurements for the simulations were generated from clusters randomly for each component. For the Fig-ures 4–8 shown below, the true clusters numbers were 3, 2, 2, and 1, respectively, for the four components. The results for MHE are presented next. Figure 4 shows the estimated and true survival function probabilities, the measurements with and without noise, and the predicted measurements for the first window of observations. The results show good accuracy (MSE and R2) in the very first window due to the monotonically decreasing trend of the component survival functions. Figures 5–8 show a gradual improvement in the

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Figure 5. True and estimated states for observations 9 to 18.

Figure 4. True and estimated states for observations 1 to 8.

Figure 6. True and estimated states for observations 19 to 28.

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performance of the MM-MHE algorithm with each window. The survival probabilities and cluster numbers in the final win-dow (Figure 8) show good agreement with the truth indicating the robustness and computational accuracy of the MM-MHE algorithm.

The simulation results presented here were generated for a noise level of approximately 8 dB (i.e., d = 0.16). How-ever, experiments were conducted with d varying from 0.01 to 5.12 (an SNR range from 20 dB to −7 dB, i.e., low to very high level noise). As shown in Figure 9, the MM-MHE al-gorithm performs very well, with R2 approximately 99% or higher for deltas in the range of 0.1 to 0.32, and MSE of the order of 10-3, for all windows, in the presence of significant measurement noise. The plot in Figure 9 computes the R2 by averaging over all windows and components.

R2-statistic(i) = [0.9938; 0.9953; 0.9834; 0.9958]MSE(i) = [0.00086; 0.00053; 0.0015; 0.00048]

Figure 9. R2 statistic versus d.

Figure 8. True and estimated states for observations 39 to 48.

Figure 7. True and estimated states for observations 29 to 38.

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1e-03 1e-03 0.99801e-03 0.9980 1e-031e-03 0.95722 0.041770.9980 1e-03 1e-03

ila

= ,

R2-statistic(i) = [0.9971; 0.9974; 0.9956; 0.9967]MSE(i) = [0.0004; 0.00029; 0.00038; 0.000371]

1e-03 0.001 0.99801e-03 0.9980 1e-031e-03 0.9980 1e-030.9620 1e-03 0.0369

ila

= ,

R2-statistic(i) = [0.9975; 0.9977; 0.9980; 0.9973]MSE(i) = [0.00034; 0.00025; 0.00017; 0.0003]

1e-03 1e-03 0.99801e-03 0.9980 1e-031e-03 0.9927 0.006230.9721 0.0268 1e-03

ila

= .

The simulation results presented in this section are based on experiments for a fixed finite length window, and the survival function estimates are determined online by solving a finite horizon state estimation problem as new measurements ar-rive. However, further experimentation showed that the ef-

Figure 10. Illustrative example for switching clusters.

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MM-MHE Approach to Prognost ics

fects of window size and length are negligible, and we can have either fixed or variable length windows (for example, based on calendar days).

R2-statistic(i) = [0.9974; 0.9978; 0.99860; 0.9973]MSE(i) = [0.00036; 0.000247; 0.00035; 0.0003]

0.0026 0.011 0.99631e-03 0.9980 1e-030.009 0.9434 0.0475

0.9980 1e-03 1e-03

ila

= ,

R2-statistic(i) = [0.9974; 0.9978; 0.9981; 0.9973]MSE(i) = [0.00036; 0.000247; 0.000169; 0.0003]

1e-03 1e-03 0.99801e-03 0.9980 1e-030.007 0.99134 1e-030.9980 1e-03 1e-03

ila

= .

An illustrative example was run to demonstrate the robust-ness of the MM-MHE algorithm to switching clusters, thus modeling switching modes (similar to that modeled in Inter-acting Multiple Models) and varying usage conditions. The MM-MHE algorithm was implemented over five windows (similar to the previous example). The observations for the first window were generated from clusters 3, 2, 2, and 1. Sub-sequently, the cluster mode changes to 1, 1, 1, and 3 for the second and third windows and, finally, switches back to 3, 2, 2, and 1 for the fourth and fifth windows. The survival function probabilities (final window) of the MHE along with cluster tracking via the MM-MHE algorithm (over all win-dows) are shown in Figure 10.

VI. CONCLUSIONS AND FUTURE WORK

This article presents a novel MM-MHE algorithm for on-line prediction of the component survival functions based on their usage profiles. The framework employs Cox PHM based on offline and online data for the RUL prediction. The proposed approach has been validated by way of ap-plication to data derived from an automotive ETC system simulator. The MM-MHE algorithm shows excellent per-formance (R2 and MSE) in the presence of significant mea-surement noise over all windows and converges to the correct cluster number. The future work includes applica-tion of this approach to continuous PID and to account for the uncertainty in RUL estimation. In the near future, by simple transformations, the authors plan on implement-ing the MM-MHE algorithm for measurements and states between ( ),−∞ ∞ and considering the effects of process noise, hence, modifying the cost function accordingly. A potential extension of the Cox PHM framework for prog-nosis of coupled systems will be to model the coupled survival dynamics as monotone positive linear systems or monotone Markov processes in which the state matrix is a Metzler matrix (i.e., has nonnegative off-diagonal ele-ments).

ACKNOWLEDGMENT

We thank NSF for their support of this work. Any opinions expressed in this article are solely those of the authors and do not represent those of the sponsors.

REFERENCES

[1] Michalska H., and Mayne, D. Q. Moving horizon observers. Presented at the Nonlinear Control Systems Design Sympo-sium (NOLCOS), Bordeaux, France, 1992.

[2] Ferrari-Trecate, G., Mignone, D., and Morari, M. Moving hori-zon estimation for hybrid systems. IEEE Transactions on Auto-

matic Control, Vol. 47, 10 (Oct. 2002), 1663–1676.[3] Chiang, L. H., Russel, E., and Braatz, R. Fault Detection and Diag-

nosis in Industrial Systems. London, United Kingdom: Springer-Verlag, 2001.

[4] Sankavaram, C., Kodali, A., Pattipati, K., Wang, B., Azam, M. S., and Singh, S. A prognostic framework for health management of coupled systems. In Proceedings of the 2011 IEEE Conference

on Prognostics and Health Management, Montreal, Canada, June 20–23, 2011.

[5] Kalbfleisch, J. D., and Prentice, R. L. The Statistical Analysis of

Failure Time Data (2nd ed.). New York: Wiley, 2002.[6] Cox, D. R., and Oakes, D. Analysis of Survival Data. Boca Raton,

FL: Chapman and Hall, 1984.[7] Mayne, D. Q., Rawlings, J. B., Rao, C. V., and Scokaert, P. O. M.

Constrained model predictive control: stability and optimality. Automatica, Vol. 36, 6 (2000), 789–814.

[8] Camacho, E. F., and Bordons, C. Model Predictive Control. Lon-don, United Kingdom: Springer-Verlag, 2004.

[9] Garcia, C. E., Prett, D. M., and Morari, M. Model predictive con-trol: theory and practice a survey. Automatica, Vol. 25, 3 (1989), 335–348.

[10] Rawlings, J. B. Tutorial overview of model predictive control. IEEE Control Systems, Vol. 20, 3 (Jun. 2000), 38–52.

[11] Qin, S. J., and Badgwell, T. A. A survey of industrial model pre-dictive control technology. Control Engineering Practice, Vol. 11, 7 (2003), 733–764.

[12] Rao, C. V., Rawlings, J. B., and Mayne, D. Q. Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations. IEEE Transactions on Automat-

ic Control, Vol. 48, 2 (Feb. 2003), 246–258.[13] Rawlings, J. W., and Bakshi, B. R. Particle filtering and moving

horizon estimation. Computers and Chemical Engineering, Vol. 30, 10–12 (2006), 1529–1541.

[14] Duda, R. O., Hart, P. E., and Stork, D. G. Pattern Classification

(2nd ed.). New York: John Wiley & Sons, 2001.[15] Bishop, C. M. Pattern Recognition and Machine Learning. New

York: Springer-Verlag, 2006.[16] Klein, J. P., and Moeschberger, M. L. Survival Analysis Techniques

for Censored and Truncated Data. New York: Springer-Verlag, 2003.

[17] Jardine, A. K. S., and Tsang, A. H. C. Maintenance, Replacement,

and Reliability: Theory and Applications. New York: CRC Press, 2005.