In order to achieve the non-stationary de-noising signal effectively, and to solve the prediction of less sample, a hybrid model composed of FCCA (Fuzzy C-means clustering algorithm) and FOM (Fractional Order Model) was constructed. The degree of each data point was determined by FCCA to de-noise 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

Chenchen Yu

Qi Gao

Xiaobo Su

Yukun Chen

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2017, VOL. 16. ISSN 2345-0533 129 
Research on consumption prediction of spare parts 
based on fuzzy C-means clustering algorithm and 
fractional order model 
Yukun Chen1, Qi Gao2, Xiaobo Su3, Chenchen Yu4 
1, 2, 3Shijiazhuang Division of Army Engineering University, Shijiazhuang, China 
3Shijiazhuang Division of PLAA Infantry Academy, Shijiazhuang, China 
4The Third Office of Troop 63916, Beijing, China 
1Corresponding author 
E-mail: 1heianzhihou123@126.com, 2b351571411@163.com, 3giantsu030700@sina.com, 
4heianzhihou123456@126.com 
Received 15 November 2017; accepted 25 November 2017 
DOI https://doi.org/10.21595/vp.2017.19465 
Abstract. In order to achieve the non-stationary de-noising signal effectively, and to solve the 
prediction of less sample, a hybrid model composed of FCCA (Fuzzy C-means clustering 
algorithm) and FOM (Fractional Order Model) was constructed. The degree of each data point 
was determined by FCCA to de-noise and the ݌  order cumulative matrix was extended to ݎ 
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. 
Keywords: signal de-noising, prediction model, fuzzy C-means clustering algorithm, fractional 
order model. 
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 de-noising methods such as Fourier transform [1-3] and wavelet 
transform [4, 5] were proposed. However, although the stationary signal can be de-noised 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 non-stationary signals. Compared with the 
Fourier transform, the wavelet transform can improve the performance of de-noising of  
non-stationary nonlinear time-varying 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 C-means clustering algorithm was 
established. After distinguishing large scale noise and sharp features, fuzzy C-means 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 C-means clustering algorithm 
Fuzzy C-mean 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 ܲ = ሼ݌ଵ, ݌ଶ, … , ݌ேሽ is a set of data points 
with ݏdimensional which containsܰdata points. Then the set can be divided into ܥ  clusters:  
ܸ = ሼݒଵ, ݒଶ, … , ݒ஼ሽ, where ݒ௜ is center of the cluster. Thus, the objective function of the fuzzy  
C-means clustering algorithm is defined as: 
RESEARCH ON CONSUMPTION PREDICTION OF SPARE PARTS BASED ON FUZZY C-MEANS CLUSTERING ALGORITHM AND FRACTIONAL ORDER 
MODEL. YUKUN CHEN, QI GAO, XIAOBO SU, CHENCHEN YU 
130 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2017, VOL. 16. ISSN 2345-0533  
ܬ௠ሺݑ, ݒሻ = ෍ ෍ሺݑ௞௜ሻ௠݀௞௜ଶ ሺ݌௞, ݒ௜ሻ
஼
௜ୀଵ
ே
௞ୀଵ
, (1)
where: 
ݑ௞௜ – The membership degree of data point ݌௞ relative to cluster center ݒ௜; 
݉ – Fuzzy control degree parameter; 
݀௞௜ଶ ሺ݌௞, ݒ௜ሻ  – Euclidean distance between data point ݌௞  and cluster center ݒ௜ , and 
݀௞௜ଶ ሺ݌௞, ݒ௜ሻ = ‖݌௞ − ݒ௜‖ଶ. 
The constraints of Eq. (1) are as follows: 
෍ ݑ௞௜
஼
௜ୀଵ
= 1,   ∀݇, 0 < ෍ ݑ௞௜
ே
௞ୀଵ
< ܰ, ∀݅, 0 ≤ ෍ ݑ௞௜
஼
௜ୀଵ
≤ 1,   ∀݇, ݅. (2)
In order to reflect the influence of different data points, the improved fuzzy clustering method 
was established, which defined the fuzzy weight coefficient ߱ 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 ݌௞ and the cluster centers ݒ௜ is defined as: 
݀௞௜ଶ ሺ݌௞, ݒ௜ሻ = ‖ሺ1/߱௞௜ሻሺ݌௞ − ݒ௜ሻ‖ଶ, (3)
߱௞௜ =
ݑ௞௜
௜ܹ
, ௜ܹ = ෍ ݑ௞௜.
ே
௞ୀଵ
 (4)
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 ܴ, and the threshold ߦ of the number of data points 
in the sphere is determined according to the density of the point cloud; 
2) Select a data point ݌௞  and calculate the number of the adjacent points ߝ  in the sphere 
enclosed of which the center is ݌௞ and the radius is ܴ; 
3) Compare the number of data in sphere enclosed and the threshold value ߦ, if ߝ < ߦ, the data 
point would be determined as noise and deleted directly; 
4) If ߝ ≥ ߦ the data point ݌௞ is regarded as the qualified point, the improved FCM algorithm 
is utilized to cluster the enclosing sphere contains ݌௞ . 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: 
ሺݔሻሺ௡ሻ ≜ ݔሺݔ + 1ሻ ⋯ ሺݔ + ݊ − 1ሻ ,   ݊ ∈ ܈଴ା,   ݔ ∈ ܀, (5)
ቂݔ݊ቃ ≜
ݔሺݔ + 1ሻ ⋯ ሺݔ + ݊ − 1ሻ
݊! , (6)
where ቂݔ0ቃ = 1. 
The fractional cumulative grey model ܩܯሺݎ, 1ሻ  is obtained after making the order of 
ܩܯሺ݌, 1ሻ extended to the general positive real number ݎ . Then ܥ௞ି௜ା௣ିଵ௞ି௜  can be obtained by 
combining with Eq. (1): 
RESEARCH ON CONSUMPTION PREDICTION OF SPARE PARTS BASED ON FUZZY C-MEANS CLUSTERING ALGORITHM AND FRACTIONAL ORDER 
MODEL. YUKUN CHEN, QI GAO, XIAOBO SU, CHENCHEN YU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2017, VOL. 16. ISSN 2345-0533 131 
ܥ௞ି௜ା௣ିଵ௞ି௜ =
ሺ݇ − ݅ + ݌ − 1ሻ!
ሺ݌ − 1ሻ! ሺ݇ − ݅ሻ! =
݌ሺ݌ + 1ሻሺ݌ + ݇ − ݅ − 1ሻ
ሺ݇ − ݅ሻ! = ቂ
 ݌
݇ − ݅ቃ. (7) 
Make ݌  extended to ݎ  which is general positive real number and ݎ ∈ ܀ା . The ݎ  order 
cumulative generating series can be defined as: 
ݔሺ௥ሻሺ݇ሻ = ෍ ቂ ݎ݇ − ݅ቃ
௞
௜ୀଵ
ݔሺ଴ሻሺ݅ሻ. (8) 
When ݎ ∈ ሺ0, +∞ሻ, ݔሺ௥ሻሺ݇ሻ is ݎ fractional order accumulation, which can be written in matrix 
form: 
܆ሺ௥ሻ = ܆ሺ଴ሻ ෍
௥
, (9) 
where: 
෍ =
௥
ۏ
ێ
ێ
ێ
ێ
ێ
ۍ1 ቂݎ1ቃ ⋯ ቂ
 ݎ
݊ − 2ቃ ቂ
 ݎ
݊ − 1ቃ
0 1 ⋯ ቂ  ݎ݊ − 3ቃ ቂ
 ݎ
݊ − 2ቃ
⋮ ⋮ ⋱ ⋮ ⋮
0 0 ⋯ 1 ቂݎ1ቃ
0 0 ⋯ 0 1 ے
ۑ
ۑ
ۑ
ۑ
ۑ
ې
. (10) 
∑௥  is ݎ fractional order cumulative matrix. 
Similar to the integer order cumulative matrix, the transposed matrix of ∑௥  is a ݎ fractional 
Inverse cumulative matrix. 
The reverse fractional order cumulative generation of series can be obtained by 
܆ሺ௥ሻ = ܆ሺ଴ሻሺ∑௥ ሻ், then the reverse cumulative gray model ܩܱܯሺݎ, 1ሻ can be established. 
The discrete form of fractional order model ܩܯሺݎ, 1ሻ can be expressed as: 
ݔሺ௥ሻሺ݇ + 1ሻ = ߙݔሺ௥ሻሺ݇ሻ + ߚ, (11) 
where ݎ ∈ ሺ0, +∞ሻ. 
Because ݀݁ݐሺ∑௥ ሻ = 1 ≠ 0, so the ∑௥  is nonsingular matrices. Therefore, the ∑  ିଵ௥ exists, and ∑ ·௥ ∑ = ିଵ௥ ∑  ·ିଵ௥ ∑௥ = ۷, where ۷ 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 1. The 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 non-stationary 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. 
RESEARCH ON CONSUMPTION PREDICTION OF SPARE PARTS BASED ON FUZZY C-MEANS CLUSTERING ALGORITHM AND FRACTIONAL ORDER 
MODEL. YUKUN CHEN, QI GAO, XIAOBO SU, CHENCHEN YU 
132 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2017, VOL. 16. ISSN 2345-0533  
4.2. Simulation 
In order to eliminate the influence of noise in the series, the original data was utilized as input 
to Fuzzy C-means clustering algorithm. Before the prediction, the residual was discarded, and the 
de-noised data were constructed by FCCA. The original data and the de-noised data were shown 
in Fig. 1. 
 
Fig. 1. The de-noised and original consumption data  
The prediction model was established by fractional order grey model GM(ݎ,1) and the optimal 
order ݎ was 0.703, which calculated by genetic algorithm. And then the prediction results of  
de-noised signal were calculated. Moreover, for facilitate analyzing the predictive capabilities of 
FCCA-FOM, GM(1,1), FOM model and EWT-BP 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 2. The prediction results and performance evaluations of forecasting results by different models 
Time 
and 
MRE 
Actual 
data 
FCCA-FOM GM(1,1) FOM EWT-BP 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 
FCCA-FOM 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 C-means clustering 
algorithm can effectively improve the prediction accuracy through comparison between  
FCCA-FOM and FOM. 
RESEARCH ON CONSUMPTION PREDICTION OF SPARE PARTS BASED ON FUZZY C-MEANS CLUSTERING ALGORITHM AND FRACTIONAL ORDER 
MODEL. YUKUN CHEN, QI GAO, XIAOBO SU, CHENCHEN YU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2017, VOL. 16. ISSN 2345-0533 133 
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 
FCM-FOM model are less than GM(1,1), FOM and EWT-BP neural network, which reflect the 
accuracy of the model established. 
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Research on consumption prediction of spare parts based on fuzzy C-means clustering algorithm and fractional order model

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