Data driven approach for performance assessment of linear andnonlinear Kalman filters

Laya Das1, Babji Srinivasan2 and Raghunathan Rengaswamy3

Abstract— A new technique is developed for assessing theperformance of linear and nonlinear Kalman filter based stateestimators. The proposed metric will indicate the performanceof these state estimators which will be primarily influencedby: (i) difference between the model dynamics and processdynamics and, (ii) various approximations of the nonlinearplant dynamics used in nonlinear Kalman filters. Currently,there exists no such quantification method to analyze theperformance of linear and nonlinear Kalman filters, a keyrequirement for improvement and a practical benchmark forcomparison of these state estimation algorithms. The proposedtechnique uses the generalized Hurst exponent of the predictionerrors (difference in measured output and a posteriori estimates)obtained from the state estimators to quantify the performance.This technique could be implemented on-line as it requiresonly plant operating data and the predicted outputs (from thelinear and nonlinear Kalman filters) to assess the performance.Several simulation studies demonstrate the applicability of theproposed performance metric to both linear and non-linearKalman filters.

I. INTRODUCTION

State estimation is a crucial task for all industries wherethere is large number of variables, internal to the system,whose knowledge is required for efficient control and faultdiagnosis & monitoring. Accurate knowledge of the systemstate is necessary for knowing the margins of operating limitsand safety of the overall plant. The celebrated Kalman Filterand its variants are largely used for the purpose of stateestimation. Though these estimators produce highly reliableestimates, the filters themselves depend heavily on the modelof the plant. When there is Model Plant Mismatch (MPM) [1]( i.e., a mismatch between plant and model dynamics), thesefilter provide poor estimates, or at times, divergent estimates,giving incorrect information to the plant operator. In practicesuch a mismatch can occur mainly due to: (i) changes in plantdynamics over time, a phenomenon primarily because of thechange in plant operating conditions motivated by processeconomics, (ii) approximations made in various non-linearKalman filters (usually in the covariance propagation step)[2] and, (iii) wear and tear of the process equipments. This

*This work was partly supported by funds from Department of Scienceand Technology Fast track Project SERB/ET-26/2013, Government of India

1Laya Das is a postgraduate student in Department of Electri-cal Engineering, Indian Institute of Technology Gandhinagar, Indialaya 12210019@iitgn.ac.in

2Corresponding author; Babji Srinivasan is with Depart-ments of Chemical Engineering (affiliated with Electricalengineering), Indian Institute of Technology Gandhinagar, Indiababji.srinivasan@iitgn.ac.in

1Raghunathan Rengaswamy is with Department of ChemicalEngineering, Indian Institute of Technology Madras, Indiaraghur@iitm.ac.in

MPM problem has been identified in several applications,such as the Model Predictive Controller [3], [4] and is aninevitable problem to the reliability of all model based stateestimators.

Several techniques have been proposed to avoid mismatchbetween plant and model by careful selection of parametersand by suitably incorporating unmodelled effects in the noisecovariance matrices [2]. There are also techniques such asinnovation sequence test [5] that have been proposed todetect MPM from plant operating data and filter outputs.Such techniques use the properties of the optimal filteringdynamics to detect deviation from optimality. However, theapplicability of these tests has been questioned [6], whenthe system matrix is not exactly known, which is preciselythe problem in case of MPM. Moreover, mere detectionof a mismatch will not help the plant operator, except forinformative purposes. Quantification of the mismatch, on theother hand, would give an idea of the extent of deteriorationof filter performance, and help the operator to take suitableactions. Recently, the application of scaled Hurst exponent asa measure of performance for linear control loops has beenproposed in [7]. It was shown that the Hurst exponent basedlinear control loop measure compares favourably with theother existing linear control loop performance measures. Thecurrent work explores the use of generalized Hurst exponentfor developing a quantification methodology to assess theperformance of linear and nonlinear Kalman filter based stateestimation techniques.

This paper is organised as follows: Section II introducesthe framework of analysis and defines the problem statement.The effect of model plant mismatch on filter performanceis also introduced. Section III introduces the Hurst expo-nent computed using Detrended Fluctuation analysis anddescribes the proposed methodology. Section IV providesresults and discussion of the simulation studies performedon various benchmark systems (used in other articles). Thearticle ends with a few concluding remarks along with thescope for future work in Section V.

II. PRELIMINARIES

Extensive research on linear as well as non-linear stateestimation has resulted in an exhaustive inventory of modelbased state estimators. The linear Kalman filter proposed byKalman [8] in 1960 set the stage for model based filteringand state estimation algorithms. It was extended to non-linearsystems, resulting in the Extended Kalman Filter (EKF).Several other filters [9], [10], [11] have been proposed inthe literature, which are summarised in Figure 1.

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Fig. 1. State Estimation Techniques

This article deals with the performance analysis of lin-ear Kalman filter, extended & unscented Kalman filters(nonlinear state estimators) under the presence of model-plant mismatch. For linear systems with Gaussian noisecorrupting the process and measurements, the linear Kalmanfilter is the best linear unbiased estimator (BLUE)[8]. For theclass of non-linear systems, on the other hand, there is notheoretically optimal filter that can handle all kinds of non-linearity. However, the non-linear versions of the Kalmanfilter such as the Extended Kalman Filter (EKF) and theUnscented Kalman Filter (UKF) [12] are very commonlyused for non-linear state estimation. A brief account of thesethree filtering algorithms on sampled discrete time systemsalong with the framework of analysis is given below.

Consider a continuous time system with measurementstaken at discrete time instants. The discrete state spacerepresentation of such as system is as follows:

xk+1 = f(xk, uk, k) + vk (1)yk+1 = h(xk+1, uk+1, k + 1) + wk+1 (2)

where, x ∈ Rn, u ∈ Rp, y ∈ Rm are the states, inputs andoutputs of the plant respectively; v ∈ Rn and w ∈ Rm arethe process and observation noises that corrupt the systemrespectively. The noises v(k) ∼ N (0, Q), w(k) ∼ N (0, R)are assumed to be identically and independently distributedwhite noises with zero mean and covariances Q and Rrespectively. In case of linear systems, the functions f and hare linear, and the above equations reduce to the following:

xk+1 = Axk +Buk + vk (3)yk+1 = Cxk+1 +Duk+1 + wk+1 (4)

A. Kalman Filter based State Estimation Techniques

The linear Kalman filter operates in two steps - pre-diction and updation/correction. In the prediction step, ituses the model and current estimate to predict the futureestimate, called the a priori estimate, xk+1|k. In the upda-tion/correction step, after the measurement is available, thepredicted value and the plant output measurement are suit-ably weighed to obtain the a posteriori estimate, xk+1|k+1.The weighing is done so as to minimise the error covarianceof the state estimates. This predictor-corrector frameworkis followed by all Kalman filter based state estimationtechniques.

The linear Kalman filter requires that the plant dynamicsbe completely linear in addition to the noise assumptionsmentioned above. The linearity assumption prevents theapplication of this algorithm to non-linear systems. For suchsystems, (EKF) is one of the commonly used techniques [13].EKF algorithm uses a first order Taylor series expansion ofthe non-linear functions f and h (Jacobaian matrices) forapproximating the error covariance matrices in the updationstep. When the non-linearity of the system is very high, thelinearisation of the model introduces large losses leadingto poor performance of EKF. Higher order approximationsrequire the calculation of the Hessian matrices at everyiteration leading to increased computational complexity andare therefore not preferred.

One of the popularly used nonlinear Kalman filters calledUnscented Kalman Filter (UKF) is based on the idea that itis easier to approximate a probability distribution functionthan an arbitrary non-linear function [9]. To that end, theUKF approximates the probability distribution function ofthe state estimates. It generates a set of sigma points and usesthe non-linear model to obtain an estimate as a weighted sumof the individual outputs. The weights can be tuned so as tocapture higher order moments of the probability distributionfunction of the state estimates. This approach makes errors inestimating the mean and covariance at the fourth and higherorders in the Taylor series [9] for a continuous function. TheEKF, on the other hand estimates the mean only upto thefirst order and the covariance upto third order. Given thecomparable computational expenses of both the techniques,the UKF is expected to have a better performance than theEKF.

B. Effect of Model Plant Mismatch on State Estimator Per-formance

In Kalman filter based state estimation techniques, it isassumed that the model used in the filter is the best approx-imation of the plant dynamics (shown in functions in equa-tions 1 and 2) and ideally simulates the plant behaviour. Thefilter uses these simulated data and measurement data withnoise to produce estimates that are closer to the true values.Under these conditions, the prediction errors (difference inthe measured output and textita posteriori estimates) from thefilter are completely random, with no predictable component.This signifies that whatever component of the output that thefilter is not able to predict using a posterior estimates is trulyrandom & unpredictable, therefore indicating the optimalityof the filter.

When there is MPM, the plant dynamics no longer satisfyequations 1 and 2. However, the model used in the filterstill remains the same without accounting for changes inthe plant. So, with time, the model falls short of the plantby an amount (δf , δh), depending upon how much theplant has deviated. It is the task of quantifying δf and δhthat are addressed in this work, under the assumption thatthe initial estimates of the system are correct, , though theassumption is not necessary for the proposed technique tomeasure the performance of the filter. Since the model is

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no more representative of the plant, the prediction errors (asdefined earlier) have a predictable component (time seriesfor which a parametric model of type AR (Auto-Regressive)or MA (Moving Average) could be used to predict) withoutbeing truly random. This key fact is used subsequently fordevelopment of performance metric for Kalman filter basedstate estimators. Notice that in this work, it assumed thatreasonable initial estimates of the system are availabe, failingwhich will result in performance degradation of state esti-mators and presence of predictable component in predictionerrors.

In signal processing applications, the Auto-CorrelationFunction (ACF) is used as a tool for detecting the presenceof repeating patterns in a signal [17]. It is well known thatfor a true random sequence, the ACF has a value only atzero lag while this is not obeyed for a time series whichcould be predicted using AR or MA model and their variants.However, in order for this approach to be useful, it should beable to provide quantitative information regarding the MPMto the plant operator. With ACF being a series of numbers,it will be difficult to obtain a single quantitative measure ofperformance. So, we propose to use the scaling exponent ofthe prediction errors as an index to quantify MPM.

III. THE PROPOSED METHODOLOGY

The on-line performance assessment of control loops wasrecently addressed in [7]. The quantification of model plantmismatch in this paper is done along similar lines. A briefoverview of the scaling exponent is provided followed by theproposed approach for quantifying MPM.

A. The Scaling Exponent

The scaling exponent is a measure used in time seriesanalysis to quantify the long term correlations in time series.It is used in several applications such as river data analysisfor dam construction [14], fractal analysis[17], and biologicalsignal processing [18]. The scaling exponent α of a timeseries y(t) calculated using y(t) ≡ aαy

(ta

)is an indicative

of the statistical self-similarity of the time series. A smallunit of actual series is expanded in time and scaled along themagnitude by an appropriate factor (aα) and the statisticalproperties of the resultant time series are compared withthose of the original time series. Typically for a stationarytime series, the value of α ranges from 0 to 1. A value lessthan 0.5 indicates that the series is anti-persistent, i.e., a high(low) value at an instant is more likely to be followed by alow (high) value. If the value of α is between 0.5 and 1, thenit signifies a persistent time series, i.e., a high (low) value atan instant is more likely to be followed by a higher (lower)value. Both these types of series can be predicted by usinga suitable AR and/or MA model. For a random series, itsvalue is 0.5. So, a value of α other than 0.5 will indicatepredictability.

Although the rescaled range analysis was originally pro-posed by Hurst [14], there are several other techniques[15], [16], [17] which have been proposed for estimation ofthis exponent. Of these, the Detrended Fluctuation Analysis

(DFA) [18] is one of the most commonly used methods. TheDFA algorithm is as follows:

1) The given time series y(k) is integrated after mean

removal: Y (k) =

k∑i=1

(y(k)− 〈y〉).

2) Y (k) is then divided into windows of length n and ineach window, a least squares line y is fit.

3) The fluctuation is then defined as:

F (n) =

√1

n

n∑i=1

(Y (i)− yi)2

This is a function of the window size n.4) Steps 2 and 3 are repeated for varying window sizes

n5) A log-log plot of the fluctuation with the corresponding

window size is made, and the slope of this plot iscalculated, which gives the estimate of α.

The technique gets its name from the removal of trendsin each window before the calculation of the fluctuation.This technique could also be used for estimation of scalingexponents for certain type of non-stationary time series [19].

B. Proposed approach to quantify MPMAs mentioned in subsection II-B, the existence of pre-

dictable component in the prediction errors of the filter iskey to the detection of MPM. The proposed approach toquantify MPM is as follows:

1) Compute the prediction errors ε(t), difference betweenthe measured outputs from the sensor and the corre-sponding a posterior estimates from the filter

2) Estimate the scaling exponent (α) of the time seriesε(t) using DFA.

3) Under the assumption that the initial estimates andcovariance matrices are reasonably correct, ideallywith no MPM the α value of ε(t) is expected tobe 0.5. Deviations of α values from 0.5 indicate thepresence of MPM, with larger deviations indicatinghigher MPM. Thus, deviations from α = 0.5 serveas a metric for quantifying MPM, in turn a measureof performance of Kalman filter based state estimationtechniques.

A few noteworthy features of the method are listed below:• As mentioned earlier, the model used in the filter is only

the best approximation of the actual plant dynamics. So,in practice, one would never expect the value of α tobe exactly 0.5. This is where one can bring in processinformation and set suitable thresholds αl and αu onthe value of α. Any deviation of the estimated scalingexponents from these thresholds (α−αl or α−αu) canbe used as an index indicating the extent of MPM.

• An important feature of this technique is that it requiresonly the plant output y(k) and the a posteriori estimateof the filter yk|k to detect and quantify model plant mis-match. This allows for on-line performance monitoringof the model based state estimators.

The proposed technique is demonstrated using both linearand non-linear systems in the next section.

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IV. RESULTS AND DISCUSSION

Performance analysis of linear Kalman filter, the bestunbiased estimator for all linear systems is first discussed.The demonstration of the proposed metric on EKF and UKFis considered next using non-linear systems. Without loss ofgenerality, the free response of the system is considered. Ineach of the cases, the model fed to the filter was modified,which has the same effect as changing the plant dynamics.Such modifications were introduced in steps with increasingorder of mismatch.

A. Linear System

With linear plant dynamics, equations 1 and 2 simplify to3 and 4. Let us consider a spring mass damper (SMD) systemwith one mass, one spring and one damper, governed by thefollowing equations:

x1 =x2

x2 =− k

mx1 −

b

mx2

y =x1

(5)

The mass was m = 1.8kg, the spring stiffness wask = 0.25Nm−1, and the damping coefficient was b =0.15Nm−1sec−1. A sampling frequency of 4 Hz wasused. Noise was added to the process and observationequations for plant simulation, with covariances Q =blkdiag(0.005, 0.005) (blkdiag refers to block diagonal ma-trix) and R = 0.005. The system was initalized with trueinitial states x0 = [3, 5]T and error covariance P0 =blkdiag(5, 5). To introduce mismatch, the mass, spring stiff-ness and the damping coefficient fed to the filter werechanged. Two such sets of deviation were used. The scalingexponents of the filter errors were estimated for all thecases. The results are summarised in Table I. It can be seenclearly that with increasing MPM, the deviation of α from0.5 (for white noise) is higher indicating poor performanceof the filter. This also compares favourably with the MSE(mean square squared error between the measured outputand a posteriori estimate), a quantity which is minimized inKalman based state estimators. The tracking of the filter forthe cases of no mismatch and highest mismatch are shownin Figure 2, indicating the deteroriation in tracking due topresence of MPM.

TABLE IMPM DETECTION RESULTS FOR LINEAR KALMAN FILTER

Mismatch Mean Squared Error (MSE) αNo MPM 0.0015 0.5154

Small MPM 0.0018 0.6374Large MPM 0.0035 0.7130

B. Non-Linear System

The first non-linear system considered is a batch reactorprocess with two gas phase reversible reactions, originallypublished in [20] and recently used in [2] for analysing andpreventing failure of EKF by proper selection of parameters

and covariance matrices. The states are the concentrations ofthe gases while the measurement is the total pressure in thechamber. The governing equations of the system are:

x1 =− k1x1 + k2x2x3

x2 =k1x1 − k2x2x3 − 2k3x22 + 2k4x3

x3 =k1x1 − k2x2x3 + k3x22 − k4x3

y =RT (x1 + x2 + x3)

(6)

The system was initialized with true initial states ofx0 = [05, 0.05, 0]T and state error covariance P0 =blkdiag(0.52, 0.052, 42). The true parameters k1 = 0.5, k2 =0.05, k3 = 0.2, k4 = 0.01, RT = 32.84 were used in thesimulations. Noise was added to the system with covariancesQ = blkdiag(0.0012, 0.0012, 0.0012) and R = 0.252. Asampling frequency of 4 Hz was used. The same parameterswere fed to both extended and unscented Kalman filters. Thereaction rate constants were changed to introduce mismatch.The results are summarised in Table II. Notice that whenthere is no MPM, as expected UKF performs better comparedto EKF based on the MSE and α values. However, in thiscase, it turns out that UKF is much sensitive to MPMcompared to that of EKF in terms of MSE and α values(Table II). Thus, the proposed approach could possibly helpidentify a better state estimator for a process under variouslevels of MPM. The tracking of both EKF and UKF for thecases of no and highest mismatch are shown and comparedin Figure 3.

TABLE IIMPM DETECTION RESULTS FOR BATCH REACTOR PROBLEM WITH

EKF AND UKF

Mismatch MSE αEKF UKF EKF UKF

No MPM 0.0435 0.0418 0.5487 0.5220Small MPM 0.0460 0.0483 0.5984 0.7114Large MPM 0.0498 0.0518 0.6630 0.7988

The second system is again a gas phase reversible reactiontaken from [20]. The states are the partial pressures of thereactant and product while the measurement is the totalpressure. The system is governed by the following equations:

x1 = −2kx21

x2 = kx21

y = x1 + x2

(7)

The true initial states of the system (used for initialization)were x0 = [3, 1]T with k = 0.16 and the state error covari-ance matrix is P0 = blkdiag(4, 4). Noise added to the pro-cess had covariance matrices Q = blkdiag(0.0012, 0.0012)and R = 0.022. A sampling frequency of 100 Hz was used.The same parameters were fed to both the state estimators.Mismatch was introduced by manipulating the reaction ratesin the model. The parameter k was varied to result in adeviation from plant dynamics. The results are summarisedin Table III. The tracking of both the filters for the cases ofno and highest mismatch are shown and compared in Figure4.

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(a) (b)

Fig. 2. Performance of Linear Kalman Filter with and without MPM (a) Output of plant and filter, i.e., distance of the mass (same as the first state ofthe system) (b) Second state of the plant and filter, i.e., velocity of the mass

(a) (b)

(c) (d)

Fig. 3. Performance of EKF and UKF with and without MPM for the Batch reactor process governed by equation 6. (a) Output of plant and filters, i.e.,total pressure (b) Concentration of species A (c) Concentration of species B (d) Concentration of species C

TABLE IIIMPM DETECTION RESULTS FOR REVERSIBLE GAS PHASE REACTION

PROBLEM WITH EKF AND UKF

Mismatch MSE αEKF UKF EKF UKF

No MPM 3.75 ∗ 10−4 3.60 ∗ 10−4 0.5496 0.4779Small MPM 3.76 ∗ 10−4 3.63 ∗ 10−4 0.5608 0.5476Large MPM 3.82 ∗ 10−4 3.78 ∗ 10−4 0.5854 0.7916

C. Key outcomes from proposed methodology

As expected, introducing mismatch into a model basedstate estimator affects its performance, which reflects in theprediction errors. This is true for linear and nonlinear Kalmanfilters. The predictability property in the prediction errors isexploited to detect as well as quantify mismatch using thescaling exponent. It is noteworthy that upon increasing themismatch in the model, the performance degrades further

and the scaling exponents also consistently deviate furtheraway from 0.5 (value for white noise). This can be clearlyseen from the simulation results summarized in Tables I, IIand III. Further, the mean squared error values also comparefavourably with the trends of the scaled exponent. Sincethe proposed method only depends on the routine operatingdata, it is easy to do on-line performance assessment andmonitoring of the state estimation algorithms.

Another important outcome of the proposed method liesin the comparison of various nonlinear state estimators underdifferent levels of MPM, since there is no optimal filter forsuch systems. For example, in the systems used in this work,the UKF performed better than the EKF for the case of nomismatch, but when mismatch was introduced, in one ofthe cases EKF performed better than the UKF. This featureis specific to the systems considered in this work and theproposed methodology can be used to assess and compare

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(a)

(b)

(c)

Fig. 4. Performance of EKF and UKF with and without MPM for thereversible gas phase reaction process governed by equation 7. (a) Outputof plant and filters, i.e., total pressure (b) Partial Pressure of species A (c)Partial Pressure of species B

the performance of the Kalman filter based state estimationalgorithms for given non-linear systems.

V. CONCLUSIONS AND FUTURE WORK

A new technique was proposed for assessing the perfor-mance of Kalman filter based state estimation approaches.The proposed metric computes the scaling exponent of theoutput errors to determine the predictability component in theoutput error data. The plant operating data and the model arethe only information needed for the technique. This allowsfor on-line performance assessment as well as monitoring.Another important application is the comparison of the per-formance of various model based state estimation algorithmsfor non-linear systems. Our future work is focussed towardsextending the technique to various other constrained non-linear state estimators. Further, the possibility of improvingthe state estimates, after having detected and quantified themismatch shall also be explored.

REFERENCES

[1] L. Xie, Y. C. Soh, and C. E. de Souza, “Robust kalman filtering for un-certain discrete-time systems,” Automatic Control, IEEE Transactionson, vol. 39, no. 6, pp. 1310–1314, 1994.

[2] R. Schneider and C. Georgakis, “How to not make the extendedkalman filter fail,” Industrial & Engineering Chemistry Research,vol. 52, no. 9, pp. 3354–3362, 2013.

[3] T. Ferhatbegovic, G. Zucker, and P. Palensky, “An unscented kalmanfilter approach for the plant-model mismatch reduction in hvac systemmodel based control,” in IECON 2012-38th Annual Conference onIEEE Industrial Electronics Society. IEEE, 2012, pp. 2180–2185.

[4] Z. Sun, Y. Zhao, and S. J. Qin, “Improving industrial mpc performancewith data-driven disturbance modeling,” in Decision and Controland European Control Conference (CDC-ECC), 2011 50th IEEEConference on. IEEE, 2011, pp. 1922–1927.

[5] R. Mehra, “On the identification of variances and adaptive kalmanfiltering,” Automatic Control, IEEE Transactions on, vol. 15, no. 2,pp. 175–184, 1970.

[6] D. Boozer and W. McDaniel Jr, “On innovation sequence testing ofthe kalman filter,” Automatic Control, IEEE Transactions on, vol. 17,no. 1, pp. 158–160, 1972.

[7] B. Srinivasan, T. Spinner, and R. Rengaswamy, “Control loop perfor-mance assessment using detrended fluctuation analysis (dfa),” Auto-matica, vol. 48, no. 7, pp. 1359–1363, 2012.

[8] R. E. Kalman, “A new approach to linear filtering and predictionproblems,” Journal of basic Engineering, vol. 82, no. 1, pp. 35–45,1960.

[9] S. Julier, J. Uhlmann, and H. F. Durrant-Whyte, “A new method forthe nonlinear transformation of means and covariances in filters andestimators,” IEEE Transactions on Automatic Control, vol. 45, no. 3,pp. 477–482, 2000.

[10] P. Vachhani, S. Narasimhan, and R. Rengaswamy, “Robust and reliableestimation via unscented recursive nonlinear dynamic data reconcil-iation,” Journal of process control, vol. 16, no. 10, pp. 1075–1086,2006.

[11] R. Rengaswamy, S. Narasimhan, and V. Kuppuraj, “Receding-horizonnonlinear kalman (rnk) filter for state estimation,” IEEE Transactionson Automatic Control, 2013.

[12] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinearestimation,” Proceedings of the IEEE, vol. 92, no. 3, pp. 401–422,2004.

[13] H. Sorenson, “Kalman filtering techniques,” Advances in controlsystems, vol. 3, pp. 219–292, 1966.

[14] H. Hurst, “Long-term storage of reservoirs: an experimental study,”Transactions of the American society of civil engineers, vol. 116, pp.770–799, 1951.

[15] I. Simonsen, A. Hansen, and O. M. Nes, “Determination of the hurstexponent by use of wavelet transforms,” Physical Review E, vol. 58,no. 3, p. 2779, 1998.

[16] M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B.Bassingthwaighte, “Evaluating scaled windowed variance methods forestimating the hurst coefficient of time series,” Physica A: StatisticalMechanics and its Applications, vol. 241, no. 3, pp. 606–626, 1997.

[17] G. M. Raymond and J. B. Bassingthwaighte, “Deriving dispersionaland scaled windowed variance analyses using the correlation functionof discrete fractional gaussian noise,” Physica A: Statistical Mechanicsand its Applications, vol. 265, no. 1, pp. 85–96, 1999.

[18] C.-K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger, “Quantifi-cation of scaling exponents and crossover phenomena in nonstationaryheartbeat time series,” Chaos: An Interdisciplinary Journal of Nonlin-ear Science, vol. 5, no. 1, pp. 82–87, 1995.

[19] Z. Chen, P. C. Ivanov, K. Hu, and H. E. Stanley, “Effect of nonstation-arities on detrended fluctuation analysis,” Physical Review E, vol. 65,no. 4, p. 041107, 2002.

[20] E. L. Haseltine and J. B. Rawlings, “Critical evaluation of extendedkalman filtering and moving-horizon estimation,” Industrial & engi-neering chemistry research, vol. 44, no. 8, pp. 2451–2460, 2005.

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