Abstract
In order to achieve the nonstationary denoising signal effectively, and to solve the prediction of less sample, a hybrid model composed of FCCA (Fuzzy Cmeans clustering algorithm) and FOM (Fractional Order Model) was constructed. The degree of each data point was determined by FCCA to denoise and the $p$ order cumulative matrix was extended to $r$ fractional cumulative matrix, so that the fractional order cumulative grey model was established to make forecasting. The results of numerical example showed that the hybrid model can obtain better prediction accuracy.
1. Introduction
In the analysis of vibration abrasion, noise factor is a significant aspect to reduce the prediction accuracy. In order to discard noise in signal, scholars domestic and abroad have carried on many years of research, and the denoising methods such as Fourier transform [13] and wavelet transform [4, 5] were proposed. However, although the stationary signal can be denoised by Fourier transform well, but this method can hardly identify the high frequency part and noise of the signal, which is not conducive to the analysis of nonstationary signals. Compared with the Fourier transform, the wavelet transform can improve the performance of denoising of nonstationary nonlinear timevarying signals, but the effect of wavelet filtering is unsatisfactory when the noise is large, or the energy is limited, and also the wavelet basis is not easy to select.
For improving the situation mentioned above, a fuzzy Cmeans clustering algorithm was established. After distinguishing large scale noise and sharp features, fuzzy Cmeans clustering algorithm was utilized to cluster the sharp features and obtain more accurate data. Besides, considering the characteristics of less samples, the fractional order cumulative model was established to reduce the perturbation bounds and improve the stability of the prediction model.
2. Fuzzy Cmeans clustering algorithm
Fuzzy Cmean clustering (FCM) is a method which utilizes the membership degree to determine the degree of each data point belonging to a certain cluster. A set of data can be divided into two or more cluster centers by FCM. Suppose that $P=\left\{{p}_{1},{p}_{2},\dots ,{p}_{N}\right\}$ is a set of data points with $s$dimensional which contains$N$data points. Then the set can be divided into $C$ clusters: $V=\left\{{v}_{1},{v}_{2},\dots ,{v}_{C}\right\}$, where ${v}_{i}$ is center of the cluster. Thus, the objective function of the fuzzy Cmeans clustering algorithm is defined as:
where:
${u}_{ki}$ – The membership degree of data point ${p}_{k}$ relative to cluster center ${v}_{i}$;
$m$ – Fuzzy control degree parameter;
${d}_{ki}^{2}\left({p}_{k},{v}_{i}\right)$ – Euclidean distance between data point ${p}_{k}$ and cluster center ${v}_{i}$, and ${d}_{ki}^{2}\left({p}_{k},{v}_{i}\right)={\Vert {p}_{k}{v}_{i}\Vert}^{2}$.
The constraints of Eq. (1) are as follows:
In order to reflect the influence of different data points, the improved fuzzy clustering method was established, which defined the fuzzy weight coefficient $\omega $ of fuzzy weight distance, based on the different contribution of different data points to the same cluster center. And then the distance between the data points ${p}_{k}$ and the cluster centers ${v}_{i}$ is defined as:
In contrast, the fuzzy weight coefficient makes the distance farther more farther and closer, which can reduce the influence of noise, and make the clustering results better.
The basic steps of the algorithm are as follows:
1) Set the radius of the bounding sphere is $R$, and the threshold $\xi $ of the number of data points in the sphere is determined according to the density of the point cloud;
2) Select a data point ${p}_{k}$ and calculate the number of the adjacent points $\epsilon $ in the sphere enclosed of which the center is ${p}_{k}$ and the radius is $R$;
3) Compare the number of data in sphere enclosed and the threshold value $\xi $, if $\epsilon <\xi $, the data point would be determined as noise and deleted directly;
4) If $\epsilon \ge \xi $ the data point ${p}_{k}$ is regarded as the qualified point, the improved FCM algorithm is utilized to cluster the enclosing sphere contains ${p}_{k}$. Meanwhile, the data in the cluster is recorded, and all data in the sphere will be replaced by the data of the cluster center;
5) Return to step (2) and determine the remaining points in turn.
3. Fractional order model
In order to define the fractional order operation, the ascending order function was introduced:
where $\left[\begin{array}{l}x\\ 0\end{array}\right]=$ 1.
The fractional cumulative grey model $GM\left(r,1\right)$ is obtained after making the order of $GM\left(p,1\right)$ extended to the general positive real number $r$. Then ${C}_{ki+p1}^{ki}$ can be obtained by combining with Eq. (1):
Make $p$ extended to $r$ which is general positive real number and $r\in {\mathbf{R}}^{+}$. The $r$ order cumulative generating series can be defined as:
When $r\in \left(0,+\infty \right)$, ${x}^{\left(r\right)}\left(k\right)$ is $r$ fractional order accumulation, which can be written in matrix form:
where:
${\sum}_{r}\u200d$ is $r$ fractional order cumulative matrix.
Similar to the integer order cumulative matrix, the transposed matrix of ${\sum}_{r}\u200d$ is a $r$ fractional Inverse cumulative matrix.
The reverse fractional order cumulative generation of series can be obtained by ${\mathbf{X}}^{\left(r\right)}={\mathbf{X}}^{\left(0\right)}{\left({\sum}_{r}\u200d\right)}^{T}$, then the reverse cumulative gray model $GOM\left(r,1\right)$ can be established.
The discrete form of fractional order model $GM\left(r,1\right)$ can be expressed as:
where $r\in \left(0,+\infty \right)$.
Because $det\left({\sum}_{r}\u200d\right)=1\ne 0$, so the ${\sum}_{r}\u200d$ is nonsingular matrices. Therefore, the ${\sum}_{r}^{1}$exists, and ${\sum}_{r}\u200d\xb7{\sum}_{r}^{1}={\sum}_{r}^{1}\xb7{\sum}_{r}\u200d=\mathbf{I}$, where $\mathbf{I}$ is unit matrix.
4. Numerical example
As shown in Table 1, in order to verify the validity and reliability of the established hybrid model, the data from [6] was used as original signal for analyzing.
Table 1The original signal from 2007 to 2015
Time  2007  2008  2009  2010  2011  2012  2013  2014  2015 
Consumption  33  35  25  34  38  31  26  35  36 
4.1. Data analysis
According to Table 1, the original series shows nonstationary characteristics. In order to test the prediction ability of established model, the consumption from 2007 to 2014 were utilized as input to forecast consumption in 2015.
4.2. Simulation
In order to eliminate the influence of noise in the series, the original data was utilized as input to Fuzzy Cmeans clustering algorithm. Before the prediction, the residual was discarded, and the denoised data were constructed by FCCA. The original data and the denoised data were shown in Fig. 1.
Fig. 1The denoised and original consumption data
The prediction model was established by fractional order grey model GM($r$,1) and the optimal order $r$ was 0.703, which calculated by genetic algorithm. And then the prediction results of denoised signal were calculated. Moreover, for facilitate analyzing the predictive capabilities of FCCAFOM, GM(1,1), FOM model and EWTBP neural network were employed for comparison. The relative error and mean relative error(MRE) were utilized to measure the forecasting accuracy, and the results were shown in Table 2.
Table 2The prediction results and performance evaluations of forecasting results by different models
Time and MRE  Actual data  FCCAFOM  GM(1,1)  FOM  EWTBP neural network  
Prediction data  Relative error/%  Prediction data  Relative error/%  Prediction data  Relative error/%  Prediction data  Relative error/%  
2007  33  32.1  –2.73  35.9  8.79  36.7  11.21  34.6  4.85 
2008  35  36.5  4.29  33.3  –4.86  33.4  –4.57  36.9  5.43 
2009  25  25.9  3.60  27.1  8.40  26.5  6.00  26.3  5.20 
2010  34  35.5  4.41  32.1  –5.59  31.7  –6.76  36.1  6.18 
2011  38  39.4  3.68  40.3  6.05  39.8  4.74  36.4  –4.21 
2012  31  30.3  –2.26  28.8  –7.10  29.9  –3.55  29.5  –4.84 
2013  26  27.5  5.77  28.9  11.2  29.1  11.92  27.6  6.15 
2014  35  35.8  2.29  36.9  5.43  37.5  7.14  36.5  4.29 
2015  36  36.9  2.50  38.6  7.22  38.8  7.78  37.7  4.72 
MRE /%  3.50  7.18  7.07  5.10 
4.3. Comparisons and discussion
According to the results of the Table 1, the relative error and the average relative error of FCCAFOM are all smaller than the other 3 models, which can reflect the better accuracy and practicability of the model. Furthermore, it can be seen that the Fuzzy Cmeans clustering algorithm can effectively improve the prediction accuracy through comparison between FCCAFOM and FOM.
5. Conclusions
Considering the noise in consumption series, a hybrid forecasting method combining FCCA and Fractional order was proposed, which can eliminate the noise in the original series and improve the prediction accuracy. At the same time, the method requires less data and no complicated integral calculation, which is easier to implement and convenient for engineering application. The numerical example showed that the relative error and average relative error of FCMFOM model are less than GM(1,1), FOM and EWTBP neural network, which reflect the accuracy of the model established.
References

Alieva T., Bastiaans M., Stankovic L. Signal reconstruction from two close fractional Fourier power spectra. IEEE Transactions on Signal Processing, Vol. 51, Issue 1, 2003, p. 112123.

Zhu B. H., Liu S. T., Ran Q. W. Optical image encryption based on multifractional Fourier transforms. Optics Letters, Vol. 25, Issue 16, 2000, p. 11591161.

Ran Q., Yeung D., Tsang E., et al. General multifractional Fourier transform method based on the generalized permutation matrix group. IEEE Transactions on Signal Processing, Vol. 53, Issue 1, 2005, p. 8398.

Catalao J. P. S., Pousinho H. M. I. VMF Mendes. Shortterm wind power forecasting in Portugal by neural networks and wavelet transform. Renew Energy, Vol. 36, 2011, p. 12451251.

Partal Turgay, Kisi Ozgur Wavelet and neurofuzzy conjunction model for precipitation forecasting. Journal of Hydrology, Vol. 342, 2007, p. 199212.

Zhou Hao, Huang Shanzhong Equipment consumption based on GM(1,1) and gray prediction Markov model. Journal of Wuhan University of Technology, Transportation Science and Engineering, Vol. 39, Issue 6, 2015, p. 11661168.

Zhao Jianzhong, Xu Tingxue, Liu Yong, et al. Consumption forecasting of missile spare parts based on rough set, entropy weight and improved SVM. Acta Armamentarii, Vol. 33, Issue 10, 2012, p. 12581265.

Zhou Yimin, Wang Qian, Yang Shuo The application of dynamic gray forecast model on the consumption rule of aviation ammunition spares. Logistics Science Technology, Vol. 7, 2017, p. 135141.

Chen Yan, Wang Zijian, Zhao Ze, et al. Gaussian process modeling and multistep prediction for time series data in wireless sensor network environmental monitoring. Journal on Communications, Vol. 36, Issue 10, 2015, p. 111.

Liang Lihua, Xun Panpan, Wang Baohua Shock response spectrum prediction of underwater explosion based on an ARMA model. Journal of Vibration and Shock, Vol. 32, Issue 13, 2013, p. 2326.

Cheng Jinjun, Xia Zhixun, Hu Leigang Prognostics for aeronautic equipment based on genetic neural network. Journal of Air Force Engineering University: Natural Science Edition, Vol. 12, Issue 1, 2011, p. 1519.

Qi Deqing, Li Weimin, Wang Huanxiang, et al. Prediction of the ballistic missile position information based on improved GM(1,1) model. Journal of Air Force Engineering University: Natural Science Edition, Vol. 13, Issue 1, 2012, p. 3741.

Li Dawei, Zhang Zhihua, Liu Tianhua A regulated method of initial spare parts based on Bayes method. System Engineering Theory and Practice, Vol. 33, Issue 11, 2013, p. 29672971.