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1. 北京航空航天大学计算机学院, 北京 100083;
2. 中航工业西安航空计算技术研究所, 西安 710068

Application of singular spectrum analysis to failure time series analysis
WANG Xin1 , WU Ji1 , LIU Chao1 , NIU Wensheng1,2 , ZHANG Hua1 , ZHANG Kui1
1. School of Computer Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China ;
2. Xi'an Aeronautical Computing Technique Research Institute, Aviation Industry Corporation of China, Xi'an 710068, China
Received: 2015-11-02; Accepted: 2016-02-29; Published online: 2016-04-13
Foundation item: China Civil Aviation Special Research Project (MJ-S-2013-10)
Corresponding author. Tel.:010-82317624, E-mail:wuji@buaa.edu.cn
Abstract: Due to significant industrial demands toward flight safety andairplane maintenance quality, improving airplane's reliability in usage stage has become an important activity and the research domain is rapidly evolving. In this paper, eighteen years' field data gathered from the maintenance phase of two Boeing 737 aircrafts are prepared as time-to-failure series. Then singular spectrum analysis (SSA) is usedto cope with this data for modeling and forecasting. Furthermore, a SSA parameter optimization algorithm is proposed by minimizing root mean square error (RMSE) of the prediction results. Based on this, a broader method of model combination is raised by utilizing different time series models to the components obtained from SSA decomposition, which represent trend, period, residuals, etc.The combination model and detailed algorithm are designed. The experimental results are compared with those of cubic exponential smoothing (Holt-Winters) and autoregressive integrated moving average (ARIMA), which verifies that the proposed models and algorithms have better fitting and prediction accuracyin failure time series analysis.
Key words: singular spectrum analysis (SSA)     failure time series     prediction     parameter optimization     model combination

SSA[10-11]是一种时域和频域相结合的非参数方法，可以用于处理非线性、非平稳性以及包含噪声的时间序列[1]，能够有效提取时间序列中的主要成分，适用性广泛且操作灵活。文献[1]应用SSA对柴油机涡轮增压器和汽车发动机2个故障时间序列[4]进行了建模和预测，结果表明比其他方法更为有效。然而，该文献没有清晰地给出故障时间序列应用SSA方法的数学模型，也没有进一步分析模型参数的优选问题。文献[12]提出了一种基于SSA和ARIMA的组合模型，并用该模型预测水库中长期年径流，实验结果表明,组合模型要优于单一模型。然而，该组合方法仅通过ARIMA方法对SSA分解出的各个成分建立预测模型，没有针对每个成分的特点应用不同的分析方法，也没有清晰地给出建立该组合模型的具体算法。

1 相关理论和技术 1.1 奇异谱分析

SSA是一种广义功率谱分析[12]，其主要思想是提取时间序列中的有效成分进行建模和预测。SSA包括分解和重构2个过程，分解过程又分为嵌入和奇异值分解(Singular Value Decomposition，SVD)，重构过程又分为分组和对角平均[11, 13-14]

1.1.1 嵌 入

 (1)

1.1.2 奇异值分解

 (2)

1.1.3 分 组

 (3)

1.1.4 对角平均

 (4)

1.1.5 预 测

SSA的预测使用了线性递归公式(Linear Recurrent Formulae,LRF)，即用前L-1个数据点的线性组合作为当前数据点的预测值，如下：

 (5)

1.1.6 参数选择

SSA的参数选择仅涉及嵌入和分组这2个步骤。在嵌入过程中，窗口长度L的选择主要依据2个原则：① 不能大于时间序列长度的一半；② 可以整除时间序列的已知周期。此外，较小的L有利于提取趋势成分，较大的L有利于分析周期成分[14, 16]。在分组过程中，需要确定截取的子矩阵个数r和分组数p，以及子矩阵在每个分组内的分配。r主要根据奇异值的贡献率来确定，通过比较前r个子矩阵的贡献率之和同预先设定的阈值η(0.85~0.95)的大小，得到满足阈值的子矩阵个数；子矩阵的分组一般是综合考虑贡献率、谐波和噪声等因素，将子矩阵分为趋势组、周期组和噪声组，可以采用一些定量和统计的手段，如用特征向量、重构序列等的散点图来检测谐波成分等[15]，用Kendal非参数检验的方法判断某个重构序列是否属于趋势成分[17]，用加权相关系数来判断重构序列之间的可分性[18]，根据周期图或者加权相关系数对重构序列自动分组[19]等。

1.2 三次指数平滑

 (6)

1.3 自回归移动平均

ARIMA是20世纪70年代Box和Jenkins[23]提出的时间序列分析方法，其模型可以表示为ARIMA(b，c，q)，其中，b、cq分别为自回归项数、差分次数和移动平均项数，当序列平稳即c=0时，对应的模型为

 (7)

1.4 故 障 率

 (8)

1.5 度量指标

 (9)

2 研究方法

 图 1 故障时间序列分析框架 Fig. 1 Analysis framework of failure time
2.1 SSA模型

 (10)
 (11)
 (12)

 (13)
 (14)

 (15)
 (16)
 (17)

2.2 参数优选

 (18)

 (19)
 (20)
 (21)

1. execute Ossa.window(TFS,trn)

2. for all L:2,3,…,min(trn/2,50)

3. execute Ossa.groups(TFS,trn,L)

4. for all p:1,2,…,min ((L-1),50)

5.execute Ossa.unit(TFS,trn,L,p)

6. get TR and TE

7.s←Ossa(TR,L)

9f←Fssa(s,p,ten,Fssa.type)

10.execute VRMSE(f.output,TFS,trn)

11.return lG←VRMSE.output

12.end(for)

13.return lL←min(list(lG))

14.end(for)

15.return l←min(list(lL))

2.3 组合模型

 图 2 模型组合流程 Fig. 2 Process of model combination

 (22)
 (23)
 (24)

1. execute Dssa(TFS,nd)

2.get TRcom and TE by TFS,nd

3.s1←Cssa(TRcom,A)

5. TRRC1r1 $F1 6. s2←Cssa(TRcom-TRRC1,Lmax) 7. get p by the features of s2 8. r2←Rssa(s2,p) 9. for each RC in r2 10. TRRCjr2$ Fj-1

11. end(for)

12.

13.execute Mcom,Fcom,input:TRRCjn1,n2,…,nm

14.for each TRRCj

15. get Ml by the features of TRRCj

16. execute Mj(TRRCj,nj),Fj(TRRCj,nj)

17. end(for)

18.

19. execute VRMSE(Fcom.output,TFS,Mcom)

20.return Mcom,Fcom,VRMSE

3 案例分析

 图 3 A、B飞机故障数据集散点图 Fig. 3 atter diagram of failure dataset for aircraft A and B

 图 4 A、B飞机预处理后的故障序列 Fig. 4 Failure series of aircraft A and B after preprocessing
3.1 基本模型实验

 图 5 FA序列的前50个奇异值 Fig. 5 First 50 singular values of FA series
 图 6 FA序列前12个奇异值对应的特征向量 Fig. 6 First 12 singular values’ eigenvectors of FA series
 图 7 FA序列前12个奇异值对应的重构序列 Fig. 7 First 12 singular values’ reconstructed series of FA series

 图 8 第2个和第3个奇异值对应的特征对 Fig. 8 Eigenvector pairs of 2nd and 3rd singular values
 图 9 A飞机重构序列的拟合图 Fig. 9 Fitting chart of reconstructed series for aircraft A
 图 10 A飞机2种预测方法的实验结果 Fig. 10 Experimental results of two forecast methods for aircraft A

 Holt-Winters样本容量 α β γ 拟合RMSE 预测RMSE 1 2 3 6 12 180a 0.0073 0.4348 0.2233 2.6059 0.1180 0.2874 2.5912 2.7752 2.4159 180m 0.0101 0.2824 0.3502 2.8859 0.9178 0.7694 2.6128 2.8313 2.5968 204a 0.0440 0.0728 0.2233 2.6172 0.0884 0.0687 2.2783 2.8825 2.5951 204m 0.0000 0.0000 0.6000 3.1987 1.7044 1.2054 2.8371 3.1791 3.0663 ARIMA样本容量 b c q 拟合RMSE 预测RMSE 1 2 3 6 12 180 2 1 3 2.3169 1.4440 1.2108 2.7102 2.8438 2.5292 204 0 1 1 2.3644 1.4123 1.2243 2.7922 2.7590 2.3416 SSA样本容量 L p 拟合RMSE 预测RMSE 1 2 3 6 12 180r 84 list(1:50) 0.6600 3.7181 2.6824 2.4439 2.9820 3.2423 180v 84 list(1:50) 0.6600 2.3112 1.7406 2.2802 2.8875 2.8722 204r 96 list(1:50) 0.7696 2.4873 1.7811 1.8713 2.4372 2.6217 204v 96 list(1:50) 0.7696 2.5093 1.8427 2.1750 2.5000 2.2947 注：a和m分别表示Holt-Winters加法和乘法季节性模型；r和v分别表示SSA递归和向量预测模型。

 Holt-Winters样本容量 α β γ 拟合RMSE 预测RMSE 1 2 3 6 12 180a 0.0447 0.0910 0.2737 3.0459 1.6212 3.5408 3.3524 2.5858 2.4386 180m 0.0000 0.0000 0.6193 3.7322 1.7157 4.5208 3.9449 3.1509 3.0339 204a 0.0105 0.2096 0.1910 2.9067 1.7486 3.6045 3.3526 2.6091 2.4742 204m 0.0000 0.0000 0.4384 3.2315 1.8368 4.2517 3.8066 2.9498 2.8160 ARIMA样本容量 b c q 拟合RMSE 预测RMSE 1 2 3 6 12 180 2 1 3 2.6897 2.4327 3.1074 2.9356 2.5425 2.3812 204 0 1 1 2.5723 2.4483 3.1196 2.9428 2.5448 2.3746 SSA样本容量 L p 拟合RMSE 预测RMSE 1 2 3 6 12 180r 84 list(1:20) 1.6284 1.6903 1.2344 1.6472 3.4426 3.5343 180v 84 list(1:20) 1.6284 0.2107 2.4131 2.7029 2.3903 2.2043 204r 96 list(1:20) 1.7041 0.1117 0.7530 2.0155 3.4850 3.3469 204v 96 list(1:20) 1.7041 1.4923 2.5154 2.5599 2.3032 2.3619 注：a和m分别表示Holt-Winters加法和乘法季节性模型；r和v分别表示SSA递归和向量预测模型。

3.2 参数优选结果

 样本容量 优选参数组合(L,p,类型) 拟合RMSE 预测RMSE 1 2 3 6 12 1 2 3 6 12 180 50,49,r 9,3,v 43,42,v 43,42,v 31,26,v 15,3,v 0.0437 0.0038 0.1385 0.6323 1.5738 2.0527 204 51,50,r 44,40,v 41,39,v 49,45,v 25,23,v 91,49,v 0.0944 0.0009 0.0451 0.5854 1.6240 2.0840

3.3 组合模型实验

 预测模型 参数 拟合RMSE 预测RMSE 趋势 残差 趋势 残差 1 2 3 6 12 Holt-Wintersa Holt-Wintersa 0.9519,0.5533,1 0.0032,1,0.2286 2.5001 0.2846 0.2020 2.4030 2.8151 2.5535 ARIMA ARIMA(1,0,2)(1,0,0)[12] 2.0250 0.8782 1.4507 3.0536 2.7995 2.3433 SSAr 84,50 0.5444 3.1262 2.5566 2.6195 2.4319 2.8765 SSAv 84,50 0.5444 2.3345 1.8197 2.3910 2.3591 2.6257 ARIMA Holt-Wintersa ARIMA(2,1,1)(2,0,2)[12] 0.0032,1,0.2286 2.4889 0.3187 0.2420 2.2304 3.0000 2.8746 ARIMA ARIMA(1,0,2)(1,0,0)[12] 1.9577 0.9122 1.3673 2.8652 2.9376 2.6282 SSAr 84,50 0.5343 3.1602 2.6316 2.5683 2.5903 3.1310 SSAv 84,50 0.5343 2.3685 1.8879 2.2930 2.5419 2.9058 SSAr Holt-Wintersa 84,50 0.0032,1,0.2286 2.4884 0.2580 0.2087 2.5315 2.7176 2.4345 ARIMA ARIMA(1,0,2)(1,0,0)[12] 1.9589 0.8516 1.5196 3.1937 2.7400 2.2743 SSAr 84,50 0.5344 3.0996 2.4989 2.6627 2.3523 2.8309 SSAv Holt-Wintersa 0.0032,1,0.2286 2.4884 0.2678 0.2054 2.5193 2.7005 2.4041 ARIMA ARIMA(1,0,2)(1,0,0)[12] 1.9589 0.8614 1.5026 3.1773 2.7221 2.2442 SSAv 84,50 0.5344 2.3177 1.7834 2.4656 2.2632 2.5669

3.4 讨 论

3.4.1 组合方法的优势

3.4.2 预测结果中的分歧点

3.4.3 样本容量对预测效果的影响

 图 11 不同模型的趋势成分预测结果 Fig. 11 Trend component prediction results of different models
4 结 论

1) SSA方法的建模和预测性能与Holt-Winters和ARIMA相比更优，特别是拟合精度，在A飞机样本容量为204的情况下RMSE达到了0.7696。

2) 参数优选算法的效果显著，在A飞机样本容量为204的情况下，模型的拟合和预测RMSE达到了0.0944和2.0840。

3) 趋势-残差二成分组合模型的总体效果优于单一SSA方法。

4) 组合模型中趋势成分的预测结果对所选定的建模方法，特别是样本容量的大小较为敏感。

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#### 文章信息

WANG Xin, WU Ji, LIU Chao, NIU Wensheng, ZHANG Hua, ZHANG Kui

Application of singular spectrum analysis to failure time series analysis

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(11): 2321-2331
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0712