Stochastic Process Prediction of Clutch Remaining Life Based on Oil Spectral Data

Copula	分布函数表达式
Frank	$C (u, v) = - \frac{1}{α} \ln (1 + \frac{[\exp (- αu) - 1] [\exp (- αv) - 1]}{\exp (- α) - 1})$
Caucasian	$\begin{array}{l} C (u, v) = msubsup msubsup \frac{1}{2 π \sqrt[]{1 - α^{2}}} \cdot \\ \exp (- \frac{x^{2} - 2 αxy + y^{2}}{2 (1 - α^{2})}) d x d y \end{array}$
Gumbel	$C (u, v) = e xp (- ({(- \ln u)}^{α} + {(- \ln v)}^{α})^{\frac{1}{α}})$
Clayton	$C (u, v) = \max ((u^{- α} + v^{- α} {- 1)}^{- \frac{1}{α}}, 0)$

Table 2. AIC values of common copula functions
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Table 2. AIC values of common copula functions
Copula Frank Caucasian Gumbel Clayton
AIC -19.22 -11.42 -7.36 -8.24

Table 3. Estimated value of model coefficient

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Table 3. Estimated value of model coefficient

模型	η¹	η²	σ¹	σ²	α
30	0.548 148 148	0.496 296 296	0.649 8	0.953 5	0.842
60	0.291 228 070	0.228 070 175	0.904 0	1.378 8	0.839
90	0.298 850 575	0.232 183 908	1.241 2	1.681 6	0.822
120	0.261 538 462	0.229 914 530	1.387 7	2.008 1	0.847
150	0.219 727 891	0.159 863 946	1.608 5	2.226 2	0.854
180	0.175 706 215	0.206 779 661	1.969 6	2.172 8	0.874
210	0.147 342 995	0.138 164 251	2.014 0	2.440 5	0.822
240	0.158 649 789	0.127 426 160	1.953 2	2.480 8	0.814

Table 4. Comparison between prediction results and test results

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Table 4. Comparison between prediction results and test results

预测时刻 /h	试验值	一元模型		二元模型
预测时刻 /h	试验值	预测/h	偏差/%	预测/h	偏差/%
30	245	165	32.7	188	23.3
60	215	120	44.2	131	39.0
90	185	87	52.9	117	36.8
120	155	78	49.7	110	29.0
150	125	67	46.4	99	20.8
180	95	60	35.2	77	18.9
210	65	57	12.3	61	6.2
240	35	39	11.4	41	17.1

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Jiang ZHANG, Jun-jie CUI, Chang-song ZHENG, Yong LIU, Ya-jun LIU, Jian SHEN. Stochastic Process Prediction of Clutch Remaining Life Based on Oil Spectral Data[J]. Spectroscopy and Spectral Analysis, 2022, 42(8): 2631

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