Soft ComputDOI 10.1007/s00500-014-1281-1

FOCUS

A fuzzy-filtered grey network technique for systemstate forecasting

Dezhi Li · Wilson Wang · Fathy Ismail

© Springer-Verlag Berlin Heidelberg 2014

Abstract A fuzzy-filtered grey network (FFGN) techniqueis proposed in this paper for time series forecasting and mate-rial fatigue prognosis. In the FFGN, the fuzzy-filtered rea-soning mechanism is proposed to formulate fuzzy rules cor-responding to different data characteristics; grey models areused to carry out short-term forecasting corresponding todifferent rules. A novel hybrid training method is proposedto adaptively update model parameters and improve train-ing efficiency. The effectiveness of the developed FFGN isdemonstrated by a series of simulation tests. It is also imple-mented for material fatigue prognosis. Test results show thatthe developed FFGN predictor can capture data characteris-tics effectively and forecast data trend accurately.

Keywords Fuzzy-filtered neural networks · Grey model ·Material fatigue prognosis

1 Introduction

System state forecasting is a process to predict future statesof a dynamic system based on its past observations. Reliableforecasting information is very useful in many real-worldapplications such as weather forecasting (Jiang et al. 2013;Liu et al. 2013) and stock market prediction (Ticknor 2013;

Communicated by E. Lughofer.

D. Li · F. IsmailDepartment of Mechanical and Mechatronics Engineering,University of Waterloo, Waterloo, ON N2L 3G1, Canadae-mail: d45li@uwaterloo.ca; fmismail@uwaterloo.ca

W. Wang (B)Department of Mechanical Engineering, Lakehead University,Thunder Bay, ON P7B 5E1, Canadae-mail: wwang3@lakeheadu.ca

Kazem et al. 2013). In industrial applications, predictor canbe used to foresee the future health state of a machine andforecast the remaining useful life of damaged equipment.Reliable prognostic information can be applied to machin-ery health condition monitoring, production quality control,and predictive maintenance scheduling rather than periodi-cally shutdown machines for manual inspection as commonlypracticed in a wide array of industries.

The classical time series forecast tools are based on sto-chastic models (Pourahmadi 2001). However, it is difficult toderive accurate analytical models for many complex dynamicsystems (Li and Lee 2005). The alternative approach is basedon the use of data-driven paradigms such as neural networks(NNs) (Wu and Liu 2012; Li et al. 2013b; Kulkarni et al.2013) and neuro-fuzzy (NF) systems (Svalina et al. 2013; Wuet al. 2014; Wang et al. 2012). Although these data-driven par-adigms do not require analytical system models, their reason-ing structure may become clumsy when the number of inputsbecome large. Bodyanskiy and Viktorov (2009a) presents acascade neo-fuzzy system with online learning algorithm;however, the singleton (a single weight) is used for each ruleoperation in the consequent, which may limit its capacity tomodel complex data features. The cascade growing NNs arepresented to learn data characteristics using quadratic neu-rons (Bodyanskiy et al. 2009b); however, its reasoning struc-ture is opaque to users. In addition, the online learning algo-rithms used in these two cascade systems may cause trainingprocess to be trapped in local optima, because both the gra-dient method and least square estimate are local searchingalgorithms and are sensitive to initial conditions.

The authors’ research team has proposed an enhancedfuzzy filtered neural network (EFFNN) (Li et al. 2013a) fortime series forecasting. The EFFNN aims to extract somerepresentative features from large input data sequence, soas to induce relatively simple reasoning architecture by the

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use of fuzzy filters. The fuzzy filters in EFFNN can com-bine duplicated features and properly address sample vari-ations. Although EFFNN has several advantages over clas-sical NFs using fuzzy filters, it may not fully utilize datacharacteristics because of its limitation in modelling com-plex systems’ dynamics using simple linear functions in itsconsequent reasoning so that forecasting results may not besufficiently accurate in some applications.

The grey model method proposed by Deng (1982) can beused for time series prediction based on limited data informa-tion, which has been used in many applications such as airlinepassenger growth prediction (Benitez et al. 2013), financialindices prediction (Chen et al. 2010), and polluted gas emis-sion prediction (Lin et al. 2011). However, the grey modelsare mainly used for short-term prediction because of their rel-atively simple modelling structure. Their prediction accuracymay degrade when training data size becomes large (Wu etal. 2013). Furthermore, a grey model cannot fully utilize thewhole historical data, which may limit its prediction accu-racy.

To tackle the aforementioned problems in the EFFNN andthe grey model, a fuzzy filtered grey network (FFGN) tech-nique is proposed in this work for time series forecasting.The fuzzy filters are used in the antecedent part of the FFGNto partition the data space. The grey models are adopted toexplore the short-term data relationship corresponding to dif-ferent subspaces. The proposed FFGN technique is new in thefollowing aspects: (1) it uses a composite network architec-ture with fuzzy filters and grey models to improve predictionaccuracy. The fuzzy filters are used to generate rules, andthe grey models are applied to conduct short-term predictionfor each rule; (2) a new hybrid training method is proposedto adaptively optimize model parameters offline and improvetraining efficiency; (3) the proposed predictor is implementedfor material fatigue prognosis.

The remainder of this paper is organized as follows. Thedeveloped FFGN is described in Sect. 2. The proposed hybridtraining scheme is discussed in Sect. 3. In Sect. 4, the effec-tiveness of the proposed FFGN predictor is first examined bysimulations, and then it is implemented for material fatigueprognosis. Finally, some concluding remarks of this studyare summarized in Sect. 5.

2 Fuzzy filtered grey network system

The EFFNN can aggregate a large number of inputs into afew fuzzy channels; however, its capacity in capturing systemdynamics may be limited due to its simplified linear modelsused in the consequent part of the network. The proposedFFGN employs the grey models to enhance the modellingability of the EFFNN so as to further improve predictionaccuracy.

2.1 Discussion of the classical grey models

Grey models conduct forecast based on a small set of mostrecent data. The data to be used in the grey model must bepositive (Deng 1989) and the data are collected with a fixedsampling frequency (Kayacan et al. 2010). In grey modeltheory, GM(n, m) represents a grey model, where n is theorder of the difference equation and m denotes the numberof variables. GM(1, 1) is the most widely used grey modelbecause of its simple structure.

Given a time series data set, X0 = {x0(1), x0(2), . . . , x0

(q)}, where X0 is a non-negative sequence and q is the samplesize of the data. To fit a GM(1, 1) model to the given timeseries, the accumulating generation operator (AGO) is usedto formulate a monotonically increasing data sequence X1:

X1 = {x1(1), x1(2), . . . , x1(q)} (1)

where

x1(k) =k∑

i=1

x0(i), k = 1, 2, . . . , q. (2)

The generated mean sequence Z1 of X1 is defined as

Z1 = {(z1(1), z1(2), . . . , z1(q)} (3)

where z1(k) is the mean value of two adjacent data, and iscomputed as

z1(k) = 1

2x1(k)+ 1

2x1(k + 1), k = 1, 2, . . . , q − 1. (4)

The difference equation of GM(1, 1) will be

x0(k)+ αz1(k) = β (5)

The whitening equation is given as

dx1(t)

dt+ αx1(t) = β (6)

The parameter set {a, β} can be calculated from (5) using theleast square estimate (LSE) (Pratama et al. 2013; Pratamaet al. 2014; Rubio 2014; Rubio and Pérez-Cruz 2014;Buchachia 2012),

[α β]T = (AT A)−1 AT B (7)

where

A =

⎡

⎢⎢⎢⎣

−z1(2) 1−z1(3) 1...

...

−z1(q) 1

⎤

⎥⎥⎥⎦

B = [x0(2), x0(3), . . . , x0(q)]T

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According to (6), x1(k + 1) can be computed as

x1(k + 1) =[

x0(1)− β

α

]e−αk + β

α(8)

The predicted value of the original data at time k + 1 can bedetermined using the inverse AGO,

x0(k + 1) =[

x0(1)− β

α

]e−αk(1 − eα) (9)

2.2 The proposed FFGN technique

The architecture of the proposed FFGN is illustrated in Fig. 1.It is a 4-layer feedforward network. Layer 1 is the inputlayer with each node representing an input data {Xi }, i =1, 2, . . . , n. The synaptic weights between layer 1 and layer2 are unity. The fuzzy filters are employed in layer 2.

Gaussian functions (Pratama et al. 2013; Rubio 2014;Rubio and Pérez-Cruz 2014; Buchachia 2012) have beenemployed as bandpass filters to preprocess input data, whichare given by

μB(xi ) = e− 1

2 (xi −ψiωi

)2

, (10)

where ψi and ωi represent the center and spread of theGaussian function, respectively. Sigmoid functions (Pérez-Cruz et al. 2013; Bordignon and Gomide 2014; Rubio 2012),as given in the following equation, are used in this case aslowpass and highpass filters to ensure that fuzzy rules cancover the whole feasible rule space:

μB(xi ) = 1

1 + e−εi (xi −γi ), (11)

where γi and εi are the center and spread of the Sigmoidfunctions, respectively. If εi < 0, μB(xi ) represents a low-pass filter; otherwise, if εi > 0, μB(xi ) corresponds to ahighpass filter. The initial conditions of the Gaussian andSigmoid functions are shown in Fig. 2.

T1

X1

X2

X3

Xn

YC

.

.

.

.

.

.

.

.

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.

.

Layer 1 Layer 2 Layer 3 Layer 4

T2

T3

Tl

.

.

.

Fig. 1 The architecture of the proposed FFGN

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Fig. 2 The initial conditions of the Gaussian functions and Sigmoidfunctions

The node output in layer 2 is formulated as

O j =∑

i μB(xi ) f (xi )∑i μB(xi )

, j = 1, 2, . . . , l; i = 1, 2, . . . , n

(12)

where l is the number of nodes in layer 2 and n is the numberof inputs. The output of each node O j denotes the firingstrength of the corresponding rule. f (xi ) is the distribution ofthe training data, which can be calculated using kernel densityestimation method (Bowman and Azzalini 1997). Each nodein layer 2 triggers only one rule/node in layer 3. The firingstrength O j represents the contribution of the correspondingrule/node in layer 3 to the final output.

Layer 3 represents the rule base. Each node in layer 3receives its inputs from the corresponding nodes in layer 2.The operation of each node/rule in layer 3 is a combinationof a grey model and a linear function represented by

Tj = G j + L j , (13)

where

G j =[

x(i − r + 1)− v1, j

v2, j

]e−αi (1 − eα) (14)

L j = w1, j x(i)+ w2, j x(i − 1)+ w3, j , (15)

where v1, j and v2, j are the parameters of grey models; wk, j

are the linear weights (k = 1, 2, 3) and j = 1, 2, . . . , l, and lis the number of nodes in layer 3. The data sequence input togrey models is {x(i − r + 1), x(i − r + 2), . . . , x(i)}, wherer is the number of the data sequence.

Defuzzification is performed in layer 4. If centroiddefuzzification method is used, the overall output will be

Y =∑

j O j Tj∑j O j

(16)

The parameters of fuzzy filters and weights of rules in layer 3are optimized by the proposed training technique discussedin the following section.

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To improve prediction performance, the input data are par-titioned into subspaces by using fuzzy filters (10)–(11), andeach subspace will be addressed using a grey model in (14)and a linear function in (15). The output of each node in layer2 is the firing strength to trigger the corresponding grey modelin layer 3. When the firing strength is high, the correspond-ing grey model contributes more to the final prediction. Eachgrey model only processes a small size of the data sequence(e.g., r = 4 in this case) so as to conduct short-term prediction.

3 The proposed hybrid training algorithm

The proposed hybrid training algorithm is used offline tooptimize the system parameters of the developed FFGN.In system training, the genetic algorithm (GA) can be usedfor global system optimization. However, the classical GAusually suffers slow convergence. To overcome this prob-lem, a two-stage strategy is used in this work for optimiza-tion: first, the GA is used to locate the area containing theglobal optima. The second stage is to apply the conjugateLevenberg-Marquardt (CLM) technique to quickly find theoptima. The CLM has been proved to be an efficient super-vised local training algorithm, which can integrate the advan-tages of both conjugate gradient algorithm and Levenberg-Marquardt method. The details of the GA and CLM can befound from the authors’ prior work in (Li and Wang 2011,Li et al. 2013a).

Specifically, in the first stage, a real-coded GA is applied toprimarily optimize the fuzzy filters’ parameters [i.e., {ψi , ωi }in (10) and {γi , εi } in (11)], and linear weights wk, j in (15).A set of parameters {ψi , ωi , γi , εi , wk, j } corresponding toFFGN is considered as an individual at each generation ofGA optimization. The fitness of each set of parameters (orindividual) is tested by computing the root mean square valueof the prediction errors at all time steps using training dataset. At each time step, the firing strengths in (12) are cal-culated, and the grey model corresponding to the largest fir-ing strength is updated using least square estimate (LSE).The other grey models remain unchanged except the termx(i − r + 1) in (14).

In the second stage, the parameter optimization is carriedout using both CLM and LSE to accelerate training conver-gence. At each time step, the CLM is first used to optimizethe nonlinear system parameters {ψi , ωi } in (10) and {γi , εi }in (11). If the j th node in layer 2 generates the highest firingstrength, only the parameters of the j th grey model, {α β} in(14), are updated using the LSE in (7). Meanwhile the termx(i − r + 1) in (14) is updated for all grey models, wherei is a time index. The training data for the j th grey modelis the most recent r data in the time series. The predictionerror of the grey model G j is compensated by a linear func-tion Tj in (15). Only one grey model and the corresponding

Grey models (LSE)

Satisfying stop criterion ?

End

N

Fuzzy filters & linear weighs (GA)

Fuzzy filters (CLM)Linear weights

(LSE)

Y

Satisfying stop criterion ?

N

Y

Initialization

Grey models (LSE)

Fig. 3 The flowchart of the hybrid training process

linear function will be updated at each time step so that thetraining efficiency can be improved. The hybrid method canreduce the search space dimension compared to an individualtraining method, and reduce the possibility of trapping dueto local optima (Wang 2008).

The flowchart of the two-stage training process is illus-trated in Fig. 3. Specific implementation is discussed as fol-lows:

1. A population of m individuals is randomly generated.Each individual is a parameter vector of the fuzzy fil-ters’ parameters {ψi , ωi , γi , εi } in (10)–(11) and linearweights {wk, j } in (15).

2. The primary parameters are optimized by using GA toexplore the parameters space and search for the area con-taining the global optima. At each time step, the parame-ters of the grey model corresponding to the largest firingstrength are updated using the LSE.

3. Once the specified maximum number of generation isreached (stop criterion), the GA operation is terminated.Otherwise, step (2) is repeated until the stop criterion issatisfied. The individual with the least training error ischosen for further optimization.

4. After GA training, detailed search is conducted usinga hybrid method of CLM and LSE: the parameters offuzzy filters {ψi , ωi , γi , εi } in (10)–(11) are optimizedby CLM; the parameters of the grey model {v1, j , v2, j }in (14) and linear weights {wk, j } in (15) are updated byLSE at each training iteration.

5. The stop criterion of CLM and LSE training is eithera specified maximum number of iteration or a threshold

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of training errors over two successive training processes.Otherwise, step (4) is repeated until the stop criterion isachieved.

4 Performance evaluation and applications

The effectiveness of the proposed FFGN predictor is firstexamined by simulation tests using three common bench-mark data sets, and then it is implemented for material fatigueprognosis.

To compare the performance of the proposed FFGN, thefollowing related techniques are applied:

1. EFFNN is used to compare the proposed reasoning struc-ture of the FFGN, which does not contain grey modelsin its architecture. The EFFNN has the same number ofnodes in layers 1-4 (i.e., 10-5-5-1) as in the FFGN inFig. 1, but it does not contain grey models. It is used tocompare the proposed reasoning structure of the FFGN.

2. DENFIS method (Kasabov and Song 2002) is used tocompare the prediction accuracy of the FFGN. It formsthe antecedent of rules using evolving clustering method,and trains consequent of the rules using LSE.

3. FFGN-F is used to compare the suggested hybrid trainingstrategy in the FFGN technique. FFGN-F has the samestructure as in the FFGN, but its grey models are updatedtogether with nonlinear parameters in fuzzy filters ratherthan optimized independently.

The prediction performance is evaluated using normalizedmean square error (NRMSE). The NRMSE is given by

NRMSE = RMSE

xmax − xmin

=√∑n

i=1 (xi −xdi )

2

n

xmax − xmin, (17)

where xi is the predicted value; xdi is the real value; n is the

data length; xmax is the maximum of xi ; xmin is the minimumof xi .

4.1 Mackey–Glass data forecast

The Mackey–Glass data set (Farmer 1982; Bodyanskiy andViktorov 2009a; Li et al. 2013b; Wang et al. 2012; Kasabovand Song 2002) is a commonly used benchmark data fortime series forecasting due to its chaotic, aperiodic and non-convergence properties, which is from the following equa-tion:

ds(t)

dt= 0.2s(t − τ)

1 + s10(t − τ)− 0.1s(t) (18)

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Interval

Dis

trib

utio

n

Fig. 4 The distribution of Mackey–Glass training data

Table 1 The mean square error of Mackey–Glass data forecastingbased on different predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.0489 0.0507 5 58.507

DENFIS 0.0074 0.0079 24 0.594

FFGN-F 0.0072 0.0073 5 53.892

FFGN 0.0061 0.0066 5 50.174

In this simulation test, a data set is generated with the initialcondition of τ = 30, s(0) = 1.2, dt = 1 and s(t) = 0 fort < 0. 1,000 data are used for training, whose distribution isshown in Fig. 4. Another 1,000 data are used for testing. Themaximum number of generations in GA is set at 100.

The training error, prediction error, number of rules andexecution time of Mackey–Glass data forecasting are listed inTable 1. From Table 1, it can be seen that the proposed FFGN(as well as FFGN-F) outperforms EFFNN and DEFNISbecause of its more efficient modelling capability using greymodels. The grey models in FFGN and FFGN-F are strongerin capturing data features than linear models in DENFIS sothat FFGN and FFGN-F can achieve less NRMSE than DEN-FIS with fewer rules. In addition, DENFIS trains its nonlin-ear parameters only using clustering method, which may alsolead to inaccurate modelling. The DENFIS employs an onlinelearning algorithm; however, EFFNN, FFGN-F and FFGNutilize the offline training with GA, CLM and LSE. ThusDENFIS consumes the least execution time among all pre-dictors. The FFGN performs better than the FFGN-F becauseit upgrades grey models by extracting data characteristicsfrom the most recent r data samples rather than only the cur-rent data sample at each training epoch. So the modellingaccuracy can be improved.

4.2 Lorenz data forecast

Lorenz attractor (Chandra and Zhang 2012; Ardalani-Farsaand Zolfaghari 2010) is another commonly used chaotic, ape-

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

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Dis

trib

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n

Fig. 5 The distribution of Lorenz training data

Table 2 The mean square error of Lorenz data forecasting based ondifferent predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.0711 0.0738 5 69.263

DENFIS 0.0381 0.0395 33 0.788

FFGN-F 0.0337 0.0358 5 67.194

FFGN 0.0244 0.0252 5 63.128

riodic data set, which can be described by the following equa-tions:

dx(t)

dt= σ(y(t)− x(t)) (19)

dy(t)

dt= x(t)(ρ − z(t))− y(t) (20)

dz(t)

dt= x(t)y(t)− βx(t)) (21)

The system parameters are selected as σ = 10, β = 8/3, andρ = 28. 1,000 data are used for training, with distributiondescribed in Fig. 5. Another 1,000 data samples are used fortesting. The maximum number of generations in GA is set at100.

Table 2 summarizes the test results of Lorenz data predic-tion. It can be seen from Table 2 that both FFGN-F and FFGNgenerate less NRMSE than EFFNN because of the effectivemodelling using grey models. DENFIS trains nonlinear para-meters using the clustering algorithm, which may not reachhigh accuracy. The FFGN-F and FFGN outperform DEN-FIS because their nonlinear parameters are updated using ahybrid training method with GA and CLM rather than a clus-tering algorithm. The FFGN generates the highest accuracy(with least NRMSE) because of its advanced grey modellingupdate mechanism.

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Inte

nsity

Fig. 6 The distribution of Sunspot training data

Table 3 The mean square error of sunspot data prediction based ondifferent predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.1349 0.1654 5 10.673

DENFIS 0.2135 0.3013 32 0.190

FFGN-F 0.0826 0.1230 5 9.712

FFGN 0.0868 0.1070 5 8.798

Fig. 7 The experimental setup for material fatigue testing

4.3 Sunspot data forecast

Sunspot activity record (Chandra and Zhang 2012; Ardalani-Farsa and Zolfaghari 2010; Li et al. 2013a) is another com-monly used benchmark data set to evaluate the performanceof predictors. 262 data sets are used for test in this case. Thefirst 190 sunspot data (from years 1750 to 1939) are used for

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Fig. 8 The performance of material fatigue prognosis in the first sce-nario. The blue solid line is the real data to estimate; the red dotted lineis the prediction performance of different schemes: a by EFFNN; b byDENFIS; c by FFGN-F; d by FFGN (color figure online)

training, and the remaining 72 data pairs (from years 1940to 2011) are used for testing. The distribution of the sunspottraining data is shown in Fig. 6.

The forecasting performance of the related predictors issummarized in Table 3. The DEFNSI has larger NRMSE thanEFFNN because its rules cannot be effectively extracted withthe clustering algorithm to improve prediction accuracy. TheFFGN-F predictor outperforms the EFFNN due to its effec-tive employment of the grey models in short-term prediction.But the developed FFGN generates the best forecasting per-formance and outperforms the other three counterparts dueto its enhanced modelling mechanism and its effective hybridtraining strategy.

4.4 Real-world application: material fatigue prognosis

As an application example, the proposed FFGN predictor isimplemented for material fatigue prognosis. Material fatigue

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Fig. 9 The performance of material fatigue prognosis in the secondscenario. The blue solid line is the real data to estimate; the red dottedline is the prediction performance of different schemes: a by EFFNN;b by DENFIS; c by FFGN-F; d by FFGN (color figure online)

tests are usually very time-consuming. So an efficient pre-dictor is very helpful in industries to forecast the trend ofthe material’s dynamic property and estimate the material’sfatigue properties without using full-scale fatigue testing. Asa demonstration, a 3-mm thick aluminum specimen is testedin this case using the experimental setup shown in Fig. 7. Bothends of the specimen are clamped to the testing machine. Asmall hole is drilled in the middle of the specimen to ini-tiate crack propagation. The crack dimension is measuredeither directly (by calipers) or indirectly (by relative voltagemeasurement). The testing frequency is set at 20 Hz. Themeasurement is taken every 3,000 load cycles. The test isbased on an indirect measurement.

In material fatigue prognosis test, experiments are car-ried out corresponding to three different initial crack dimen-sions. In each testing scenario with a specific initial crackdimension, the tests are repeated over six times (i.e., six data

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sets). Among the six data sets in each scenario, two datasets are randomly selected for training and another data setis randomly chosen for testing. The maximum number ofgenerations in GA is set at 100. Figures 8, 9 and 10 showthe one-step-ahead forecasting results of three test scenarios,respectively, using these three predictors. The correspondingtest results are summarized in Tables 4, 5 and 6.

By comparing graph (a) and graph (c) in Figs. 8, 9 and 10,it can be seen that the FFGN-F predictor can track materialfatigue propagation trend more accurately than the EFFNNpredictor, because the grey models in FFGN-F can extractinformation from most recent data sequence to improve pre-diction accuracy. Comparing graph (b) and graph (c) in Figs.8, 9 and 10, it is seen that the FFGN-F predictor is superiorto the DENFIS predictor because of the more accurate mod-elling of the grey models, and the more effective nonlinearparameter training method. On the other hand, the FFGN

Table 4 The mean square error of material fatigue prognosis in the firstscenario based on different predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.0936 0.0120 5 51.523

DENFIS 0.0773 0.0076 87 0.405

FFGN-F 0.0736 0.0036 5 50.663

FFGN 0.0627 0.0011 5 48.833

Table 5 The mean square error of material fatigue prognosis in thesecond scenario based on different predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.0853 0.0174 5 58.380

DENFIS 0.0472 0.0067 27 0.114

FFGN-F 0.0447 0.0025 5 56.376

FFGN 0.0438 0.0010 5 54.743

Table 6 The mean square error of material fatigue prognosis in thethird scenario based on different predictors

NRMSE Trainingerror

Predictionerror

Rules Executiontime (s)

EFFNN 0.0695 0.0271 5 57.036

DENFIS 0.0729 0.0424 59 0.224

FFGN-F 0.0631 0.0037 5 53.164

FFGN 0.0584 0.0017 5 51.395

predictor (graph (d) in Figs. 8, 9 and 10) gives better predic-tion results than the FFGN-F predictor, because its adaptivegrey model training strategy can improve the dominant greymodel prediction accuracy as well as the overall forecastingperformance. It can catch the dynamic characteristics of thematerial crack propagation trend accurately.

5 Conclusion

A fuzzy-filtered grey network, FFGN, technique is developedin this work for time series forecasting and material fatigueprognosis. In the FFGN, the fuzzy filters extract features fromdata. The feature processing is enhanced using grey modelsto extract information from most recent data sequence. Theadaptive grey model training strategy can improve the train-ing efficiency of the FFGN. The effectiveness of the proposedFFGN predictor is verified using simulation of two bench-mark data sets. The proposed predictor is also implementedfor material fatigue prognosis. Test results have shown thatthe FFGN scheme is an efficient predictor. It is able to capture

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A fuzzy-filtered grey network technique

the dynamic behaviour of the material properties quickly andtrack the crack propagation trend accurately.

Further research is under way to propose an analyticalmethod to properly select system orders. In addition, anonline training algorithm will be suggested to adaptively tuneFFGN model structures and optimize system parameters inreal-time.

Acknowledgments This work was partly funded by the Natural Sci-ences and Engineering Research Council of Canada (NSERC) andeMech Systems Inc.

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