 Research
 Open Access
Prognostic for hydraulic pump based upon DCTcomposite spectrum and the modified echo state network
 Jian Sun^{1}Email author,
 Hongru Li^{1} and
 Baohua Xu^{1}
 Received: 5 May 2016
 Accepted: 27 July 2016
 Published: 8 August 2016
Abstract
Prognostic is a key step of the conditionbased maintenance (CBM). In order to improve the predicting performance, a novel method for prognostic for the hydraulic pump is proposed in this paper. Based on the improvement of the traditional composite spectrum, the DCTcomposite spectrum (DCS) fusion algorithm is initially presented to make fusion of multichannel vibration signals. The DCS composite spectrum entropy is extracted as the feature. Furthermore, the modified echo state networks (ESN) model is established for prognostic using the extracted feature. The reservoir is updated and the elements of the neighboring matrix are redefined for improving predicting accuracy. Analysis of the application in the hydraulic pump degradation experiment demonstrates that the proposed algorithm is feasible and is meaningful for CBM.
Keywords
 Prognostic
 Composite spectrum analysis
 ESN
 Small world networks
Background
Conditionalbased maintenance (CBM) is one of the modern maintenance concepts. The open system architecture for CBM organization divides a standard CBM system into several various layers with technical modules (Ahmad and Kamaruddin 2012). The core functions, corresponding to maintenance decisions among the architecture, can be summarized as condition monitoring, health assessment and prognostics (Coraddu et al. 2015; Sun et al. 2015). With the fast development of industry and the international market, especially the areas of complex plants such as shipbuilding and aircraft, CBM recommends timely and accurate maintenance actions based on the information obtained from condition monitoring (Altosole et al. 2014). Therefore, more and more advanced machine learning approaches have been applied in CBM for better improve the reliability, such as support vector machine and information fusion algorithms (Coraddu et al. 2014; Niu et al. 2010; Widodo and Yang 2007). Furthermore, some realtime online diagnostics have also been employed to extract sensitive information of the current operational status for identify the detailed failures (Gulen et al. 2002). In order to meet the requirement of CBM for better improving the hydraulic system reliability, it is necessary to search for an effective method for accurate prognostics of the hydraulic pump.
Normal algorithms for prognostics mainly contains two categories (EIThalji and Jantunen 2015): the one are algorithms based on models, such as the physics model algorithm (Jin et al. 2013), the autoregressive and moving average algorithm (ARMA) (Yu et al. 2013), and the autoregressive integrated moving average algorithm (ARIMA) (Li and Hu 2012). Others are algorithms based on neural intelligence, such as the BP neural network (Shuran and Shujin 2011), the recurrent neural network (RNN) (Cao et al. 2012), the extreme learning machine (ELM) (Wang et al. 2011), and the echo state networks (ESN) (Park et al. 2014). The physics model algorithm is capable of making prediction of the operating trends and the possible failures. However, it requires the mathematic model to be established extremely accurately, which can hardly be fulfilled in engineering application. ARMA is one of the typical random time series predicting methods. Limited by its computing theory, ARMA performs effectively only for stationary series (Zhu et al. 2014). Being the improvement of ARMA, ARIMA can be applied in nonstationary series. However, it is lack of the ability to uncover the system inner mechanism (Liu et al. 2012). Compared with the algorithms based on models, BP neural network performs better nonlinear fitting ability, whereas, the predicting for nonlinear and timevarying series is limited (Ren et al. 2014). Considering time series correlation, the prognostics by RNN are better than the BP. Since the back propagation is still employed, RNN is easy to trap in local minimums (Wu and Zhu 2014). ELM is the highspeed learning algorithm with the onelayer feedforward neural networks. Instead of the iteration strategy of gradient descent, ELM only requires setting the number of hidden layer nodes and the driving function. It has advantages such as simple parameters, fast learning and better searching ability. Influenced by the singular values decomposition in ELM, if the number of hidden layer nodes is too large, the computing complexity will obviously increase leading to the reduction of the learning efficiency. Additionally, the random selection of input weights vectors and hidden layer nodes’ thresholds may also affect the network stability and the prediction accuracy (Wang and Han 2014). ESN is a novel recurrent neural network algorithm proposed by Jaeger and Haas (2004). It applies the reservoir as the inner network to inspire complex nonlinear state space for increasing the nonlinear fitting ability. In the ESN training process, the connecting weights keep the same. The computing complexity is reduced and the issue of trapping into local minimums is avoided. Therefore, ESN can fundamentally meet the predicting requirements of hydraulic pumps with nonlinear and nonstationary vibration signals. Since neurons in the reservoir are randomly and sparsely connected, the guidance and purpose of the network are poor. To solve this problem, the small world networks (Friedkin 2011) are introduced to make modification on the structure of the ESN reservoir in this paper. Meanwhile, the elements in neighboring matrix are redefined based on distance and random factor to further improve the generalization ability.
The realization of prognostic lays foundation of the extraction of effective features. However, the fault information in onechannel vibration signal is incomplete. The effective way to obtain accurate failure information is to make fusion of multichannel vibration signals for extracting appropriate features (Safizadeh and Latifi 2014). As a novel spectrum analysis algorithm proposed recently, composite spectrum (CS) is able to realize information fusion of various signals by calculating correlative index and mutual power spectrum of neighboring signals (Elbhbah and Sinha 2013). It can effectively restrain noises, and obtain sensitive feature information of signals simultaneously (Akilu et al. 2014). Considering the structural characters of hydraulic pumps, some modification is made on the CS algorithm in this paper to improve feature sensitivity.
Consequently, a novel method for prognostic for hydraulic pump based on discrete cosine transform—composite spectrum (DCS) and the modified ESN is proposed. Contributions of this article are summarized hereinafter. In “Feature extraction based on DCS fusion algorithm” section, the DCS fusion algorithm is presented and the method for the extracting DCS composite spectrum entropy is illustrated in detail; in “Fault predicting based on INW–ESN” section, we make modification on ESN and propose the INW–ESN model for failure prediction; in “Experimental validation” section, we confirm the results through the experiment of hydraulic pump; and in “Conclusions” section, we draw some conclusions.
Feature extraction based on DCS fusion algorithm
The proposed DCS fusion algorithm
However, according to Eq. (3), the multiplication of the Fourier coefficient and its complex conjugate is applied in CS. If we take Eq. (3) into Eq. (2), there will be terms like \(X_{2}^{r *} (f_{k} )X_{2}^{r} (f_{k} ),X_{3}^{r *} (f_{k} )X_{3}^{r} (f_{k} ), \ldots.\) These terms can be merged together resulting in possibly loss of some important information.
Compared with Eq. (1), DCS fusion algorithm applies DCT instead of Fourier transform. All coefficients are real numbers. Consequently, Eq. (5) is able to combine Eq. (2) with Eq. (3) to avoid information losing caused by multiplication of Fourier coefficient and its complex conjugate. Furthermore, high sensitivity of DCT coefficients is also inherited. Theoretically, feature performance may be effectively improved.
Extraction of the DCS composite spectrum entropy
 1.
Divide each signal into N parts. Conduct DCT on each part and achieve \(X_{i}^{j} ,i = 1,2,3,j = 1, \ldots ,N\).
 2.
Calculate the correlation coefficient between the ith component and the (i + 1)th component in the frequency \(f_{k}\):
where, \(S_{{x_{i} x_{i + 1} }}^{j} (f_{k} )\) is defined as:$$\gamma_{i(i + 1)}^{2} (f_{k} ) = \left {\sum\limits_{j = 1}^{N} {S_{{x_{i} x_{i + 1} }}^{j} (f_{k} )} } \right^{2} /\sum\limits_{j = 1}^{N} {S_{{x_{i} x_{i} }}^{j} (f_{k} )} \sum\limits_{j = 1}^{N} {S_{{x_{i + 1} x_{i} }}^{j} (f_{k} )}$$(6)where, \(X_{i}^{j} (f_{k} )\) denotes the jth part’s DCT coefficient in the ith signal.$$S_{{x_{i} x_{i + 1} }}^{j} (f_{k} ) = X_{i}^{j} (f_{k} )X_{{i{ + }1}}^{j} (f_{k} )$$(7)  3.
Calculate the composite spectrum \(S_{CCS} (f_{k} )\):
$$S_{CS} (f_{k} ) = \left( {\sum\limits_{j = 1}^{N} {X_{1}^{j} (f_{k} )\gamma_{12}^{2} X_{2}^{j} (f_{k} )\gamma_{23}^{2} X_{3}^{j} (f_{k} )} } \right)^{1/3} /N$$(8)  4.
Calculate the DCSE:
where, K means the number of frequency bands. Based on fusion of various signals, DCSE is sensitive to the energy changing. When the fault degradation degree is light, the energy distributes in frequency bands in balance and the DCSE will be higher. When the degree is heavy, the energy mainly centered on unique feature frequency bands and the DCSE will be lower.$$\left\{ {\begin{array}{*{20}l} {{\text{DCSE}} =  \left( {\sum\limits_{k = 1}^{K} {p_{i} \ln p_{i} } } \right)/\ln K} \hfill \\ {p_{k} = S_{CS} (f_{k} )/\sum\limits_{k = 1}^{K} {S_{CS} (f_{k} )} } \hfill \\ {\sum\limits_{k = 1}^{K} {p_{k} } = 1} \hfill \\ \end{array} } \right.$$(9)
Fault predicting based on INW–ESN
Since neurons in the reservoir of ESN are randomly and sparsely connected, this character may lead to the poor guidance ability. It may even affect the generalization and prognostic accuracy (Koryakin et al. 2012). Small world networks possess shorter feature route length, so they perform like the random networks. Furthermore, the polymerization coefficient is higher, small world networks also operate like the regular networks (Zippo et al. 2013; Quan and Zhu 2010). If the small world networks are applied in the reservoir, the generalization may be possibly improved. However, in the NW small world model, the connection weights between nodes are only 0 or 1. Elements in the neighboring matrix are also the inherent 0 and 1, which belongs to the determinacy connection. Limited by this kind of connection, the updating ability of the structure can hardly meet the predicting requirements of the nonlinear and timevarying series. To solve the problem, the improved NW small world network–ESN (INW–ESN) is proposed.

Step 1 Fuse the threechannel vibration signals by the proposed DCS fusion algorithm and extract the DCSE as the feature (Specific algorithm is indicated in “Extraction of the DCS composite spectrum entropy” section).

Step 2 Select the training section and the predicting section from DCSE series and carry out INW–ESN training.

Step 2.1 Make optimization of the reservoir scale EN and the inner connection weights matrix spectral radius ER by the fruit fly optimization algorithm (FOA) (Pan 2012). Achieve the optimal parameters of the INW–ESN.

Step 2.2 Conduct the training of INW–ESN according to Eqs. (10), (11).

Step 3 Make prediction by the trained INW–ESN based upon the dynamic multisteps strategy.

Step 3.1 After one predicting step, select part of the output to be the predicting results of this step of prediction and update the input vector for the prediction in next step.

Step 3.2 Decide whether it meets the termination condition. If it is, combine prediction results of all steps to be the final result. Otherwise, return to the Step 3.1 and go on to the next step of prediction.

Step 4 Make analysis and comparison of the predicting result and the real DCSE series. Statistics the errors.
The improvement of the node connection in the proposed INW–ESN ensures that the reservoir is able to update the topology structure dynamically according to various input series. This character not only keeps the sparsity of reservoir neurons, but also reduces the blindness of random connection. Therefore, the predicting accuracy and the network generalization ability are effectively improved in the proposed INW–ESN.
Experimental validation
Experimental rig
The loose slipper shown in Fig. 6 is the failure most easily to occur in the pump degradation. Influenced by the stress during long time operating, the abrasion between the slipper hat and the piston bulb becomes more and more severe. As a result, the distance between the slipper hat and the piston bulb keeps increasing. Additionally, the metal particles caused by abrasion in the pump embed into the distance. When the distance is wide enough, the loose slipper occurs. In this failure condition, the space between the slipper hat and the piston bulb is filled with hydraulic oil. During the movement of the piston, there may be the relative motion between the two parts leading to the strikes on swash plate. If the loose slipper is severe, the piston may be sticking and even the piston bulb happens to break, resulting in catastrophic accident of the hydraulic system. Therefore, it is meaningful to apply the proposed method for the prediction for loose slipper degradation.
Results and analysis
Figure 8 shows that the whole process is divided into 6 parts: during 0–25,329 min (the 1–1101 groups of samples), it is the normal status. In this period, the variety of DCSE is faint and stable; During 25,330–29,284 min (the 1102–1273 groups of samples), the period is the mild wear status (F1). In this stage, the friction pairs in the hydraulic pump had been working for a long time in the accelerated condition. Signs of abrasion occurred because of the continuous friction. Therefore, DCSE tends to decrease; During 29,285–31,548 min (the 1274–1372 groups of samples), it is the slow development status (F2). In this period, the decreasing trend of DCSE become more obvious caused by the aggravation of the continuous friction between the friction pair; during 31,549–33,416 min (the 1373–1453 groups of samples), it is the initial degradation status (F3). There had been signs of some failure mode along with the aggravation of friction. The variety of DCSE is violent and it tends to be momentary steady; During 33,417–34,439 min (the 1454–1497 groups of samples), it is the accelerated degradation status (F4). Since the moment coefficient of the friction pairs became larger, the oil slick became thinner and the fitting clearance became longer. As a result, there were oil leakage and the performance parameters of the pump changed severely. Therefore, DCSE continues decreasing with obvious fluctuation; during 34,440–37,214 min (the 1498–1619 groups of samples), it is the loose slipper status (F5). The metallic particles in the hydraulic fluid had entered into the fitting clearance between the piston cap and the plunger head, which greatly aggravated the friction of the slipper pair. The hydraulic pump performance in this stage can hardly meet the engineering requirements. Therefore, DCSE in this period appears with severe fluctuations.
Figure 11 shows the prognostic results by the INW–ESN with the static strategy. The input vector is not updated by the predicting results in each step. The errors between the predicting data and the real data are obvious. The effect is severely worse than those with the dynamic strategy, and the RUL can hardly be predicted based on the current DCSE data. Figure 12 shows the results by the RNN. It can only roughly fit the trend of the stage of F5. Influenced by the back propagation algorithm, RNN is easy to be trapped in local minimums and the convergence rate is also low. Furthermore, there has been large deviation from the actual. In the 38th sample (for the 1491th sample in Fig. 8), the algorithm reaches the threshold. The error of RUL is 6 data points, which is 138 min. Figure 13 shows the results of the traditional ESN. Compared with RNN, the ESN applies the reservoir as the inner network, so that the prognostic effect is improved. However, limited by the sparsely connecting of inner neurons in the reservoir, parts of the predicting results still deviate a little far from the real data. In the 37th sample (for the 1490th sample in Fig. 8), the ESN algorithm meets the termination condition. The RUL predicting error is 7 data points, which is 161 min. Figure 14 shows the results of ELM prognostic. By the abandon of the iteration strategy of gradient descent, ELM has characteristics as simple parameters, fast learning and better searching ability. As a result, the predicting curve is closer to the real one. However, influenced by the input weights vector and the random selection of hidden nodes’ weights, the ELM algorithm reaches the termination condition in the 41th sample (for the 1494th sample in Fig. 8). The RUL predicting error is 3 data points, which is 69 min.
MAPE and RMSPE of various algorithms
Algorithms  MAPE (%)  RMSPE (%)  Error of RUL prediction 

INW–ESN (dynamica)  9.32  7.27  23 min 
INW–ESN (static)  17.14  14.64  Failed 
RNN (dynamica)  16.38  13.44  138 min 
ESN (dynamica)  14.60  10.87  161 min 
ELM (static)  13.91  9.48  69 min 
Table 1 shows that, because of the updating of the input vector, the algorithms with the dynamic strategy perform better than those with the static strategy. Since the reservoir is modified and the element in the neighboring matrix is redefined, the prognostic performance of INW–ESN has been effectively improved. Therefore, the error for RUL prediction is only 23 min, and the MAPE and RMSPE are also the lowest compared with other four algorithms.
Results of the application in other degradation experiments
Algorithms  Loose slipper failure  Loose slipper failure  Slipper abrasion failure  

MAPE (%)  RMSPE (%)  Error  MAPE (%)  RMSPE (%)  Error  MAPE (%)  RMSPE (%)  Error  
INW–ESN (dynamica)  10.17  7.89  23 min  13.11  9.03  46 min  6.25  4.69  0 min 
INW–ESN (static)  20.43  16.97  Failed  25.92  19.56  Failed  14.35  10.08  Failed 
RNN (dynamica)  18.29  15.11  161 min  21.05  16.47  184 min  11.17  8.83  69 min 
ESN (dynamica)  16.08  12.14  207 min  19.38  13.92  253 min  8.41  6.05  92 min 
ELM (static)  14.87  10.93  92 min  16.77  12.01  115 min  7.86  5.33  46 min 
In Table 2, compared with other algorithms, the proposed method is also able to achieve the best prognostic performance. The MAPE and the RMSPE are the highest and the error is the lowest. Furthermore, the prediction accuracy of the proposed INW–ESN for the slipper abrasion failure is obviously higher than that for the loose slipper. The reason can be explained that the slipper abrasion failure mechanism is much simpler and the degradation time is less. Combined with Table 1, results shows that the proposed method is feasible in the prediction for various failure modes.
From the above analysis, we conclude that the proposed DCS and INW–ESN algorithm performs much better in prognostic. The feature information is effectively extracted by the fusion of multichannel vibration signals based on DCS. Furthermore, the reservoir is better improved in the INW–ESN for making prognostic accurately and effectively. However, there are also some problems that should be the focus of further study: (1) only one feature was extracted in this paper, and this may lead to the loss of some important information. To solve this problem, other typical features could also be extracted and the information fusion algorithm will be introduced to make fusion of all the extracted features so as to better obtain fault information and to improve the feature performance; (2) The research in this paper represents the initial application of the presented method to make fault prediction. And the research on combination prognostic method to further improve the predicting effects was not involved. To make fault prediction of the hydraulic pump more accurately and meaningful, the research that addresses the problems mentioned above will be reported in our future work.
Conclusions
 1.
The DCS algorithm is presented and the multichannel vibration signals are fused. Meanwhile, the DCS spectrum entropy with high sensitivity is extracted to be the degradation feature.
 2.
The INW–ESN is proposed for prognostic for hydraulic pump. The reservoir is modified and the disadvantages caused by the sparsely connection are solved. The generalization ability and predicting accuracy of the network have been effectively improved.
 3.
Results of the application in the hydraulic pump degradation experiment show that the proposed method is feasible and the predicting accuracy is satisfactory, which is meaningful for CBM.
Declarations
Authors' contributions
JS: Research concept and design, conduction of experiment, collection and assembly of data; data analysis and interpretation. HL: Critical reversion of the article. BX: Critical reversion of the article. All authors read and approved the final manuscript.
Acknowledgements
This project is supported by National Natural Science Foundation of China (Grant No. 51275524). We also appreciate the AVIC Liyuan Hydraulic Corporation for their support to our experiment. At the same time, we are grateful to the Mechanical Engineering College, China, for providing the experimental situation. At the end, we’d like to express sincere appreciation to the anonymous.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Ahmad R, Kamaruddin S (2012) An overview of timebased and conditionbased maintenance in industrial application. Comput Ind Eng 63(1):135–149View ArticleGoogle Scholar
 Akilu YK, Sinha JK, Elbhbah K (2014) An improved data fusion technique for faults diagnosis in rotating machines. Measurement 58:27–32View ArticleGoogle Scholar
 Altosole M, Campora U, Martelli M et al (2014) Performance decay analysis of a marine gas turbine propulsion system. J Ship Res 58(58):117–129View ArticleGoogle Scholar
 Cao Q, Ewing BT, Thompson MA (2012) Forecasting wind speed with recurrent neural networks. Eur J Oper Res 221(1):148–154View ArticleGoogle Scholar
 Coraddu A, Oneto L, Ghio A et al (2014) Machine learning approaches for improving conditionbased maintenance of naval propulsion plants. Proc Inst Mech Eng Part M J Eng Marit Environ 4:53–56Google Scholar
 Coraddu A, Oneto L, Ghio A et al (2015) Machine learning for wear forecasting of naval assets for conditionbased maintenance applications. In: International conference on electrical systems for aircraftGoogle Scholar
 EIThalji I, Jantunen E (2015) A summary of fault modeling and predictive health monitoring of rolling element bearings. Mech Syst Signal Process 60(1):252–272View ArticleGoogle Scholar
 Elbhbah K, Sinha JK (2013) Vibrationbased condition monitoring of rotating machines using a machine composite spectrum. J Sound Vib 332:2831–2845View ArticleGoogle Scholar
 Friedkin NE (2011) Spine segments in small world networks. Social Netw 33(1):88–97View ArticleGoogle Scholar
 Gulen SC, Griffin PR, Paolucci S (2002) Realtime online performance diagnostics of heavyduty industrial gas turbines. J Eng Gas Turbines Power 124(4):58–66View ArticleGoogle Scholar
 Huang H, Xiao L, Liu J (2014) CORDICbased unified architectures for computation of DCT/IDCT/DST/IDST. Circuits Syst Signal Process 33(3):799–814View ArticleGoogle Scholar
 Jaeger H, Hass H (2004) Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304(5667):78–80View ArticleGoogle Scholar
 Jin G, Matthews D, Fan Y et al (2013) Physics of failurebased degradation modeling and lifetime prediction of the momentum wheel in a dynamic covariate environment. Eng Fail Anal 28:222–240View ArticleGoogle Scholar
 Koryakin D, Lohmann J, Bytz MV (2012) Balanced echo state networks. Neural Netw 36:35–45View ArticleGoogle Scholar
 Li C, Hu J (2012) A new AIRMAbased neuronfuzzy approach and swarm intelligence for time series forecasting. Eng Appl Artif Intell 25(2):295–308View ArticleGoogle Scholar
 Liu H, Tian H, Li Y (2012) Comparison of two new ARIMAANN and ARIMAKalman hybrid methods for wind speed prediction. Appl Energy 98:415–424View ArticleGoogle Scholar
 Liu L, Liu B, Hua H (2014) Noreference image quality assessment based on spatial and spectral entropy. Sig Process Image Commun 29(8):856–863View ArticleGoogle Scholar
 Niu G, Yang BS, Pecht M (2010) Development of an optimized conditionbased maintenance system by data fusion and reliabilitycentered maintenance. Reliab Eng Syst Saf 95:786–796View ArticleGoogle Scholar
 Pan W (2012) A new fruit fly optimization algorithm: tacking the financial distress model as an example. Knowl Based Syst 26:69–74View ArticleGoogle Scholar
 Park J, Cho D, Kim S et al (2014) Utilizing online learning based on echostate networks for the control of a hydraulic excavator. Mechatronics 24(8):986–1000View ArticleGoogle Scholar
 Quan H, Zhu C (2010) Behaviors of imitated agents in an evolutionary minority game on NW small world networks. Phys Proc 3(5):1741–1745View ArticleGoogle Scholar
 Ren C, An N, Wang J et al (2014) Optimal parameters selection for BP neural network based on particle swarm optimization: a case study of wind speed forecasting. Knowl Based Syst 56:226–239View ArticleGoogle Scholar
 Safizadeh MS, Latifi SK (2014) Using multisensor data fusion for vibration fault diagnosis of rolling element bearings by accelerometer and load cell. Inform Fusion 18:1–8View ArticleGoogle Scholar
 Shuran L, Shujin L (2011) Applying BP neural network model to forecast peak velocity of blasting ground vibration. Proc Eng 26:257–263View ArticleGoogle Scholar
 Sun J, Li H, Wang W et al (2015) Morphological undecimated wavelet decomposition fusion algorithm and its application on fault feature extraction of hydraulic pump. Trans Nanjing Univ Aeronaut Astronaut 32(3):268–278Google Scholar
 Wang X, Han M (2014) Online sequential extreme learning machine with kernels for nonstationary time series prediction. Neurocomputing 145:90–97View ArticleGoogle Scholar
 Wang Y, Cao F, Yuan Y (2011) A study on effectiveness of extreme learning machine. Neurocomputing 74(16):2483–2490View ArticleGoogle Scholar
 Widodo A, Yang BS (2007) Support vector machine in machine condition monitoring and fault diagnosis. Mech Syst Signal Process 21(6):2560–2574View ArticleGoogle Scholar
 Wu G, Zhu ZW (2014) An enhanced discriminability recurrent fuzzy neural network for temporal classification problems. Fuzzy Sets Syst 337:47–62View ArticleGoogle Scholar
 Yang E, Yu X, Meng J et al (2014) Transparent composite model for DCT coefficients: design and analysis. IEEE Trans Image Process 23(3):1303–1316View ArticleGoogle Scholar
 Yu C, Zhang C, Xie L (2013) Blind identification of nonminimum phase ARMA systems. Automatica 49(6):1846–1854View ArticleGoogle Scholar
 Zhu L, Wang Y, Fan Q et al (2014) MODWTARMA model for time series prediction. Appl Math Model 38(5–6):1859–1865View ArticleGoogle Scholar
 Zippo AG, Gelsomino G, VanDuin P et al (2013) Smallworld networks in neuronal populations: a computational perspective. Neural Netw 44:143–156View ArticleGoogle Scholar