-
Notifications
You must be signed in to change notification settings - Fork 1
/
index.html
2531 lines (2498 loc) · 238 KB
/
index.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.2.269">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<meta name="author" content="庄亮亮、吴温慧">
<title>退化过程模型综述</title>
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1.6em;
vertical-align: middle;
}
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { color: #008000; } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { color: #008000; font-weight: bold; } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style>
<script src="index_files/libs/clipboard/clipboard.min.js"></script>
<script src="index_files/libs/quarto-html/quarto.js"></script>
<script src="index_files/libs/quarto-html/popper.min.js"></script>
<script src="index_files/libs/quarto-html/tippy.umd.min.js"></script>
<script src="index_files/libs/quarto-html/anchor.min.js"></script>
<link href="index_files/libs/quarto-html/tippy.css" rel="stylesheet">
<script src="index_files/libs/bootstrap/bootstrap.min.js"></script>
<link href="index_files/libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
<link href="index_files/libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
</head>
<body>
<div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
<nav id="TOC" role="doc-toc" class="toc-active">
<h2 id="toc-title">目录</h2>
<ul>
<li><a href="#介绍" id="toc-介绍" class="nav-link active" data-scroll-target="#介绍">1. 介绍</a></li>
<li><a href="#wiener-退化过程模型" id="toc-wiener-退化过程模型" class="nav-link" data-scroll-target="#wiener-退化过程模型">2. Wiener 退化过程模型</a></li>
<li><a href="#经典的-wiener-过程" id="toc-经典的-wiener-过程" class="nav-link" data-scroll-target="#经典的-wiener-过程">2.1 经典的 Wiener 过程</a>
<ul>
<li><a href="#统计推断" id="toc-统计推断" class="nav-link" data-scroll-target="#统计推断">2.1.1 统计推断</a>
<ul>
<li><a href="#极大似然估计" id="toc-极大似然估计" class="nav-link" data-scroll-target="#极大似然估计">2.1.1.1 极大似然估计</a></li>
<li><a href="#贝叶斯分析" id="toc-贝叶斯分析" class="nav-link" data-scroll-target="#贝叶斯分析">2.1.1.2 贝叶斯分析</a></li>
</ul></li>
</ul></li>
<li><a href="#带随机效应情形" id="toc-带随机效应情形" class="nav-link" data-scroll-target="#带随机效应情形">2.2 带随机效应情形</a>
<ul>
<li><a href="#失效时间分布和-rul" id="toc-失效时间分布和-rul" class="nav-link" data-scroll-target="#失效时间分布和-rul">失效时间分布和 RUL</a></li>
<li><a href="#统计推断-1" id="toc-统计推断-1" class="nav-link" data-scroll-target="#统计推断-1">统计推断</a></li>
</ul></li>
<li><a href="#多应力情形" id="toc-多应力情形" class="nav-link" data-scroll-target="#多应力情形">2.3 多应力情形</a>
<ul>
<li><a href="#统计推断-2" id="toc-统计推断-2" class="nav-link" data-scroll-target="#统计推断-2">2.3.1 统计推断</a>
<ul>
<li><a href="#极大似然估计-1" id="toc-极大似然估计-1" class="nav-link" data-scroll-target="#极大似然估计-1">2.2.1.1 极大似然估计</a></li>
</ul></li>
</ul></li>
<li><a href="#gamma-退化过程模型" id="toc-gamma-退化过程模型" class="nav-link" data-scroll-target="#gamma-退化过程模型">3. Gamma 退化过程模型</a>
<ul>
<li><a href="#经典-gamma-退化过程模型" id="toc-经典-gamma-退化过程模型" class="nav-link" data-scroll-target="#经典-gamma-退化过程模型">3.1 经典 Gamma 退化过程模型</a>
<ul>
<li><a href="#统计推断-3" id="toc-统计推断-3" class="nav-link" data-scroll-target="#统计推断-3">3.1.1 统计推断</a>
<ul class="collapse">
<li><a href="#极大似然估计-2" id="toc-极大似然估计-2" class="nav-link" data-scroll-target="#极大似然估计-2">3.1.1.1 极大似然估计</a></li>
<li><a href="#贝叶斯分析-1" id="toc-贝叶斯分析-1" class="nav-link" data-scroll-target="#贝叶斯分析-1">3.1.1.2 贝叶斯分析</a></li>
</ul></li>
</ul></li>
<li><a href="#加速-gamma-退化过程" id="toc-加速-gamma-退化过程" class="nav-link" data-scroll-target="#加速-gamma-退化过程">3.2 加速 Gamma 退化过程</a>
<ul>
<li><a href="#统计推断-4" id="toc-统计推断-4" class="nav-link" data-scroll-target="#统计推断-4">3.2.1 统计推断</a>
<ul class="collapse">
<li><a href="#极大似然估计-3" id="toc-极大似然估计-3" class="nav-link" data-scroll-target="#极大似然估计-3">3.2.1.1 极大似然估计</a></li>
</ul></li>
</ul></li>
</ul></li>
<li><a href="#ig-退化过程模型" id="toc-ig-退化过程模型" class="nav-link" data-scroll-target="#ig-退化过程模型">4. IG 退化过程模型</a>
<ul>
<li><a href="#经典的-ig-过程" id="toc-经典的-ig-过程" class="nav-link" data-scroll-target="#经典的-ig-过程">4.1 经典的 IG 过程</a>
<ul>
<li><a href="#统计推断-5" id="toc-统计推断-5" class="nav-link" data-scroll-target="#统计推断-5">4.1.1 统计推断</a>
<ul class="collapse">
<li><a href="#极大似然估计-4" id="toc-极大似然估计-4" class="nav-link" data-scroll-target="#极大似然估计-4">4.1.1.1 极大似然估计</a></li>
<li><a href="#贝叶斯估计" id="toc-贝叶斯估计" class="nav-link" data-scroll-target="#贝叶斯估计">4.1.1.2 贝叶斯估计</a></li>
</ul></li>
</ul></li>
<li><a href="#加速-ig-退化过程" id="toc-加速-ig-退化过程" class="nav-link" data-scroll-target="#加速-ig-退化过程">4.2 加速 IG 退化过程</a>
<ul>
<li><a href="#统计推断-6" id="toc-统计推断-6" class="nav-link" data-scroll-target="#统计推断-6">4.2.1 统计推断</a>
<ul class="collapse">
<li><a href="#极大似然估计-5" id="toc-极大似然估计-5" class="nav-link" data-scroll-target="#极大似然估计-5">4.2.1.1 极大似然估计</a></li>
</ul></li>
</ul></li>
</ul></li>
<li><a href="#模拟数据分析" id="toc-模拟数据分析" class="nav-link" data-scroll-target="#模拟数据分析">5. 模拟数据分析</a>
<ul>
<li><a href="#经典维纳过程" id="toc-经典维纳过程" class="nav-link" data-scroll-target="#经典维纳过程">5.1 经典维纳过程</a></li>
<li><a href="#经典-gamma-过程" id="toc-经典-gamma-过程" class="nav-link" data-scroll-target="#经典-gamma-过程">5.2 经典 Gamma 过程</a></li>
<li><a href="#经典-ig-过程" id="toc-经典-ig-过程" class="nav-link" data-scroll-target="#经典-ig-过程">5.3 经典 IG 过程</a></li>
</ul></li>
<li><a href="#案例分析" id="toc-案例分析" class="nav-link" data-scroll-target="#案例分析">6 案例分析</a>
<ul>
<li><a href="#维纳过程相关" id="toc-维纳过程相关" class="nav-link" data-scroll-target="#维纳过程相关">6.1 维纳过程相关</a>
<ul>
<li><a href="#nasa-锂电池数据" id="toc-nasa-锂电池数据" class="nav-link" data-scroll-target="#nasa-锂电池数据">6.1.1 NASA 锂电池数据</a></li>
<li><a href="#激光器件-laser-数据集" id="toc-激光器件-laser-数据集" class="nav-link" data-scroll-target="#激光器件-laser-数据集">6.1.2 激光器件 Laser 数据集</a></li>
<li><a href="#炭膜电阻器-carbon-film-resistors" id="toc-炭膜电阻器-carbon-film-resistors" class="nav-link" data-scroll-target="#炭膜电阻器-carbon-film-resistors">6.1.3 炭膜电阻器 carbon-film resistors</a></li>
<li><a href="#惯导平台-interial-navigition-数据" id="toc-惯导平台-interial-navigition-数据" class="nav-link" data-scroll-target="#惯导平台-interial-navigition-数据">6.1.4 惯导平台 interial navigition 数据</a></li>
<li><a href="#铝合金疲劳裂纹长度-2017-t4-aluminum-alloy" id="toc-铝合金疲劳裂纹长度-2017-t4-aluminum-alloy" class="nav-link" data-scroll-target="#铝合金疲劳裂纹长度-2017-t4-aluminum-alloy">6.1.5 铝合金疲劳裂纹长度 2017-T4 Aluminum Alloy</a></li>
</ul></li>
<li><a href="#伽马过程相关" id="toc-伽马过程相关" class="nav-link" data-scroll-target="#伽马过程相关">6.2 伽马过程相关</a>
<ul>
<li><a href="#wear-data" id="toc-wear-data" class="nav-link" data-scroll-target="#wear-data">6.2.1 wear data</a></li>
<li><a href="#crack-of-titanium" id="toc-crack-of-titanium" class="nav-link" data-scroll-target="#crack-of-titanium">6.2.2 Crack of Titanium</a></li>
<li><a href="#lithium-ion-batteries" id="toc-lithium-ion-batteries" class="nav-link" data-scroll-target="#lithium-ion-batteries">6.2.3 Lithium-ion batteries</a></li>
<li><a href="#fatigue-crack-data" id="toc-fatigue-crack-data" class="nav-link" data-scroll-target="#fatigue-crack-data">6.2.4 fatigue crack data</a></li>
</ul></li>
<li><a href="#逆高斯过程相关" id="toc-逆高斯过程相关" class="nav-link" data-scroll-target="#逆高斯过程相关">6.3 逆高斯过程相关</a>
<ul>
<li><a href="#防辐射-anti-radiation-数据集" id="toc-防辐射-anti-radiation-数据集" class="nav-link" data-scroll-target="#防辐射-anti-radiation-数据集">6.3.1 防辐射 Anti-radiation 数据集</a></li>
</ul></li>
</ul></li>
</ul>
</nav>
</div>
<main class="content" id="quarto-document-content">
<header id="title-block-header" class="quarto-title-block default">
<div class="quarto-title">
<div class="quarto-title-block"><div><h1 class="title">退化过程模型综述</h1><button type="button" class="btn code-tools-button" id="quarto-code-tools-source"><i class="bi"></i> Code</button></div></div>
</div>
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">Author</div>
<div class="quarto-title-meta-contents">
<p>庄亮亮、吴温慧 </p>
</div>
</div>
</div>
</header>
<section id="介绍" class="level2">
<h2 class="anchored" data-anchor-id="介绍">1. 介绍</h2>
<p>近年来，基于退化的可靠性技术在模型、方法和应用等方面得到快速发展。其中，在<strong>基于退化的可靠性模型</strong>方面，以随机过程理论为基础，根据工程需要，并考虑模型的简明性、实用性和适用性，已经提出了多种类型的模型，包括：</p>
<!-- 1. 退化轨道模型（如 Paris 模型、随机斜率/截距模型、幂律模型、反应论模型） -->
<ol type="1">
<li><p>基于 Wiener 过程的模型</p></li>
<li><p>基于 Gamma 过程的模型</p></li>
<li><p>基于逆高斯过程的模型</p></li>
</ol>
<p>在基于退化的可靠性建模方法，以数理统计理论为基础，针对模型和退化数据的类型，研究<strong>矩估计、极大似估计及 EM 算法、Bayes 估计及 MCMC算法、基于滤波的状态估计方法</strong>，以及<strong>非参数和半参数方法</strong>等，解决模型辨识和修正问题;</p>
<p>采用<strong>似然比检验、Bayes 因子分析</strong>等方法进行模型验证;</p>
<p>采用<strong>均方误差（MSE）、Akaike 信息准则(AIC)、Bayes 信息准则(BIC)、偏差信息准则(DIC)</strong>等进行模型优良性检验。</p>
<p>在基于退化的可靠性技术应用方面，针对<strong>机械零部件（如轴承、润滑系统）、半导体器件（如功率MOS器件）、机电部件（如感应电动机）、光电器件(如太阳电池、激光器、LED)、电子器件（如电容、蓄电池）</strong>等的退化失效过程，开展<strong>退化过程建模、可靠性评估、剩余寿命预测、可靠性试验设计</strong>：特别是加速退化试验方案设计的研究。</p>
</section>
<section id="wiener-退化过程模型" class="level2">
<h2 class="anchored" data-anchor-id="wiener-退化过程模型">2. Wiener 退化过程模型</h2>
<p>Wiener 过程由于其简单的结构和比较丰富的研究成果，成为目前退化过程建模中应用最为广泛的一种模型。Wiener 退化过程的<strong>首达时间分布具有解析形式</strong>，便于对<strong>产品寿命</strong>和<strong>可靠度</strong>的分析和计算。</p>
<p>Wiener 退化过程可以描述产品退化过程的时间不确定性，而且比较容易处理测量数据存在误差的情况。</p>
<p>通过对经典 Wiener 过程的参数引入<strong>随机性</strong>，使得 Wiener 退化过程能够描述个体差异，并且一般不会给模型参数的估计带来更多的困难。</p>
<blockquote class="blockquote">
<p>对 Wiener 退化过程及其观测过程所构成的<strong>状态空间模型</strong>，可以采用 <strong>Kalman 滤波技术</strong>进行处理，为产品<strong>在线寿命预测、视情维修决策</strong>等提供了可行的算法和实现途径。</p>
</blockquote>
</section>
<section id="经典的-wiener-过程" class="level2">
<h2 class="anchored" data-anchor-id="经典的-wiener-过程">2.1 经典的 Wiener 过程</h2>
<p>称 <span class="math inline">\(\{X(t), t \geqslant 0\}\)</span> 是漂移参数为 <span class="math inline">\(\mu\)</span> 、扩散参数为 <span class="math inline">\(\sigma\)</span> 的 (一元) Wiener 过程, 若</p>
<ol type="1">
<li><p><span class="math inline">\(X(0)=0\)</span>。</p></li>
<li><p><span class="math inline">\(\{X(t), t \geqslant 0\}\)</span> 有平稳独立增量。</p></li>
<li><p><span class="math inline">\(X(t)\)</span> 服从均值为 <span class="math inline">\(\mu t\)</span>，方差为 <span class="math inline">\(\sigma^2 t\)</span> 的正态分布。</p></li>
</ol>
<p>根据上面的定义, 可以将带漂移的 Wiener 过程表示为下面的形式: <span class="math display">\[
X(t)=\mu t+\sigma B(t)
\]</span> 式中: <span class="math inline">\(\{B(t)\}, t \geqslant 0\)</span> 是标准 Wiener 过程或标准布朗运动过程。</p>
<p>根据定义, 对带漂移的 Wiener 过程, 显然有如下的性质成立:</p>
<p>（1）时刻 <span class="math inline">\(t \sim t+\Delta t\)</span> 之间的增量服从正态分布, 即 <span class="math inline">\(\Delta X=X(t+\Delta t)-X(t) \sim\)</span> <span class="math inline">\(N\left(\mu \Delta t, \sigma^2 \Delta t\right)\)</span> 。</p>
<p>（2）对任意两个不相交的时间区间 <span class="math inline">\(\left[t_1, t_2\right],\left[t_3, t_4\right], t_1<t_2 \leqslant t_3<t_4\)</span>, 增量 <span class="math inline">\(X\left(t_4\right)-X\)</span> <span class="math inline">\(\left(t_3\right)\)</span> 与 <span class="math inline">\(X\left(t_2\right)-X\left(t_1\right)\)</span> 相互独立。</p>
<p><strong>可靠度</strong></p>
<p>如果产品的性能退化过程是一元 Wiener 过程，失效阈值为<span class="math inline">\(l\)</span>。 则可靠度为 <span class="math display">\[
R(t)= 1-F(t ; \mu, \sigma)=\Phi\left(\frac{l-\mu t}{\sigma \sqrt{t}}\right)-\exp \left(\frac{2 \mu l}{\sigma^2}\right) \Phi\left(\frac{-l-\mu t}{\sigma \sqrt{t}}\right)
\]</span></p>
<p><strong>剩余使用寿命</strong></p>
<p>假设其运行到时刻 <span class="math inline">\(\tau\)</span> 仍末失效, 且当前性能退化量为 <span class="math inline">\(x_\tau\left(x_\tau<l\right)\)</span>, 则产品的剩余寿命 <span class="math inline">\(T_1\)</span> 可以表示为 <span class="math display">\[
T_1=\inf \left\{t \mid X(t+\tau) \geqslant l, X(\tau)=x_\tau, t \geqslant 0\right\}
\]</span> 由一元 Wiener 过程的独立增量性质和齐次马尔可夫性得到 <span class="math display">\[
\begin{aligned}
T_1 &=\inf \left\{t \mid X(t+\tau)-X(\tau) \geqslant l-x_\tau, t \geqslant 0\right\} \\
&=\inf \left\{t \mid X(t) \geqslant l-x_\tau, t \geqslant 0\right\}
\end{aligned}
\]</span> 可知剩余寿命 <span class="math inline">\(T_1\)</span> 也是逆 Gaussian 分布, 其密度函数只需将寿命 <span class="math inline">\(T\)</span> 密度函数中的失效阈值 <span class="math inline">\(l\)</span> 替换为 <span class="math inline">\(l-x_\tau\)</span>, 即 <span class="math display">\[f_{T_1}(t)=\frac{l-x_\tau}{\sqrt{2 \pi \sigma^2 t^3}} \exp \left[-\frac{\left(l-x_\tau-\mu t\right)^2}{2 \sigma^2 t}\right]\]</span></p>
<section id="统计推断" class="level3">
<h3 class="anchored" data-anchor-id="统计推断">2.1.1 统计推断</h3>
<p>假设共有 <span class="math inline">\(n\)</span> 个样品进行性能退化试验。样品 <span class="math inline">\(i\)</span> 初始时刻 <span class="math inline">\(t_{i 0}\)</span> 退化量取值为 <span class="math inline">\(X_{i 0}=0\)</span>, 在 时刻 <span class="math inline">\(t_{i 1}, \cdots, t_{i m_i}\)</span> 样品 <span class="math inline">\(i\)</span> 退化量为 <span class="math inline">\(X_{i 1}, \cdots, X_{i m_i}\)</span> 。记 <span class="math inline">\(\Delta X_{i j}=X_{i j}-X_{i, j-1}\)</span> 是样品 <span class="math inline">\(i\)</span> 在时刻 <span class="math inline">\(t_{i, j-1} \sim t_{i j}\)</span> 之间的退化增量, <span class="math inline">\(\Delta t_{i j}=t_{i j}-t_{i j-1}, j=1,2, \cdots, m_i, i=1,2, \cdots, n\)</span> 为各样品的测量间隔。</p>
<section id="极大似然估计" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计">2.1.1.1 极大似然估计</h4>
<!-- 在简单情形, 认为退化数据的测量不存在误差, 并且总体退化过程不存在随机效应。 -->
<p>由 Wiener 过程的性质, 容易知道 <span class="math display">\[
\Delta X_{i j} \sim N\left(\mu \Delta t_{i j}, \sigma^2 \Delta t_{i j}\right)
\]</span></p>
<p>于是由退化数据 <span class="math inline">\(X_{i j}=x_{i j}\)</span> <span class="math inline">\(j=1,2, \cdots, m_i, i=1,2, \cdots, n\)</span>, 得到模型参数的似然函数为 <span class="math display">\[
L\left(\mu, \sigma^2\right)=\prod_{i=1}^n \prod_{j=1}^{m_i} \frac{1}{\sqrt{2 \sigma^2 \pi \Delta t_{i j}}} \exp \left[-\frac{\left(\Delta x_{i j}-\mu \Delta t_{i j}\right)^2}{2 \sigma^2 \Delta t_{i j}}\right]
\]</span> 由式可以直接求得漂移参数 <span class="math inline">\(\mu\)</span> 和扩散参数 <span class="math inline">\(\sigma^2\)</span> 的极大似然估计如下: <span class="math display">\[
\hat{\mu}=\frac{\sum_{i=1}^n X_{i m_i}}{\sum_{i=1}^n t_{i m_i}}, \hat{\sigma}^2=\frac{1}{\sum_{i=1}^n m_i}\left[\sum_{i=1}^n \sum_{j=1}^{m_i} \frac{\left(\Delta X_{i j}\right)^2}{\Delta t_{i j}}-\frac{\left(\sum_{i=1}^n X_{i m_i}\right)^2}{\sum_{i=1}^n t_{i m_i}}\right]
\]</span></p>
</section>
<section id="贝叶斯分析" class="level4">
<h4 class="anchored" data-anchor-id="贝叶斯分析">2.1.1.2 贝叶斯分析</h4>
<!-- 基于 rstan 进行贝叶斯估计。详细的参考文献如下： -->
<!-- - 最新、入门级别 [rstan](https://mirrors.sjtug.sjtu.edu.cn/cran/web/packages/rstan/vignettes/rstan.html) 资料。 -->
<!-- - [官网资料](https://mc-stan.org/users/interfaces/rstan) -->
<p>基于前面所提的经典线性维纳过程，利用 rstan 进行贝叶斯分析。</p>
<p><span class="math display">\[
\begin{aligned}
&\Delta X_{ij} = N(\mu \Delta t_{ij}, \sigma^2 \Delta t_{ij}),\\
&w = \frac{1}{\sigma^2} \sim Gam(a,b),\\
& \mu|w \sim N(d,\frac{c}{w}).
\end{aligned}
\]</span></p>
<p>其中，<span class="math inline">\(a = b = 1, d = 0, c =100\)</span>。</p>
<ul>
<li>数据准备</li>
</ul>
<p>准备<span class="math inline">\(\Delta t_{ij}\)</span> 和 <span class="math inline">\(\Delta x_{ij}\)</span>的数据，并且标志出矩阵的维数。</p>
<ul>
<li>构建模型（<code>wiener_linear.stan</code>）</li>
</ul>
<div class="sourceCode" id="cb1"><pre class="sourceCode numberSource markdown number-lines code-with-copy"><code class="sourceCode markdown"><span id="cb1-1"><a href="#cb1-1"></a>data {</span>
<span id="cb1-2"><a href="#cb1-2"></a> int<lower=0> I;</span>
<span id="cb1-3"><a href="#cb1-3"></a> int<lower=0> J;</span>
<span id="cb1-4"><a href="#cb1-4"></a> matrix<span class="co">[</span><span class="ot">I,J</span><span class="co">]</span> x;</span>
<span id="cb1-5"><a href="#cb1-5"></a> matrix<span class="co">[</span><span class="ot">I,J</span><span class="co">]</span> t;</span>
<span id="cb1-6"><a href="#cb1-6"></a>}</span>
<span id="cb1-7"><a href="#cb1-7"></a></span>
<span id="cb1-8"><a href="#cb1-8"></a>parameters {</span>
<span id="cb1-9"><a href="#cb1-9"></a> real mu;</span>
<span id="cb1-10"><a href="#cb1-10"></a> real<lower=0> w;</span>
<span id="cb1-11"><a href="#cb1-11"></a>}</span>
<span id="cb1-12"><a href="#cb1-12"></a></span>
<span id="cb1-13"><a href="#cb1-13"></a>model {</span>
<span id="cb1-14"><a href="#cb1-14"></a> w ~ gamma(1,1);</span>
<span id="cb1-15"><a href="#cb1-15"></a> mu ~ normal(0, 100/w);</span>
<span id="cb1-16"><a href="#cb1-16"></a> for (i in 1:I){</span>
<span id="cb1-17"><a href="#cb1-17"></a> for (j in 1:J) {</span>
<span id="cb1-18"><a href="#cb1-18"></a> x<span class="co">[</span><span class="ot">i,j</span><span class="co">]</span> ~ normal(mu * t<span class="co">[</span><span class="ot">i,j</span><span class="co">]</span>, 1/w * t<span class="co">[</span><span class="ot">i,j</span><span class="co">]</span>);</span>
<span id="cb1-19"><a href="#cb1-19"></a> }</span>
<span id="cb1-20"><a href="#cb1-20"></a> }</span>
<span id="cb1-21"><a href="#cb1-21"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
</section>
</section>
<section id="带随机效应情形" class="level2">
<h2 class="anchored" data-anchor-id="带随机效应情形">2.2 带随机效应情形</h2>
<p>从性能退化过程的角度看，采用上述带漂移的 Wiener 过程模型，相当于认为同一批产品的退化过程相同，即参数心和口相同。实际也可能由于产品的原材料、生产过程等的差异，退化过程参数<span class="math inline">\(\mu\)</span>和<span class="math inline">\(\sigma\)</span>可能因产品个体差异而不同。为了刻画这种情景，我们给出带随机效应的 Wiener 过程模型。给出几种常用的模型：</p>
<p><span class="math display">\[
X(t)=\mu t+\sigma B(t)
\]</span></p>
<ul>
<li><strong>模型一</strong></li>
</ul>
<p>假定 <span class="math inline">\(\mu\)</span>为随机变量，并服从<span class="math inline">\(N(\mu_\beta,\sigma^2_\beta)\)</span>，来描述个体退化速率的差异，<span class="math inline">\(\sigma\)</span>是对所有个体相同的扩散参数。</p>
<ul>
<li><strong>模型二</strong></li>
</ul>
<p>文献 [5] 提出一种不同个体具有不同退化速率和扩散行为的随机效应模型，其中，认为不同单元的漂移和扩散参数都是随机变量，且服从正态-逆Gamma分布。假设</p>
<p><span class="math display">\[
\sigma^{2} \sim InvGama(\alpha, \beta), \quad
\mu \mid \sigma^{2} \sim N(v, \eta \sigma^2).
\]</span></p>
<section id="失效时间分布和-rul" class="level3">
<h3 class="anchored" data-anchor-id="失效时间分布和-rul">失效时间分布和 RUL</h3>
<ul>
<li><strong>模型一</strong> [4]</li>
</ul>
<p><span class="math display">\[
f_T(t)=\sqrt{\frac{l^2}{2 \pi t^3\left(\sigma_\beta^2 t+\sigma_B^2\right)}} \exp \left\{-\frac{\left(l-\mu_\beta t\right)^2}{2 t\left(\sigma_\beta^2 t+\sigma_B^2\right)}\right\}, t \geqslant 0
\]</span> 对应的不可靠度函数为 <span class="math display">\[
F_T(t)=\Phi\left(\frac{\mu_\beta t-l}{\sqrt{\sigma_\beta^2 t^2+\sigma_B^2 t}}\right)+\exp \left\{\frac{2 \mu_\beta l}{\sigma_\beta^2}+\frac{2 \sigma_B^2 l^2}{\sigma_B^2}\right\} \Phi\left(\frac{2 \sigma_\beta^2 l t+\sigma_B^2\left(\mu_\beta t+l\right)}{\sigma_B^2 \sqrt{\sigma_\beta^2 t^2+\sigma_B^2 t}}\right)
\]</span></p>
<ul>
<li><strong>模型二</strong> [4]</li>
</ul>
<p><span class="math display">\[
\begin{aligned}
f_T(t) & =\frac{\alpha^\beta \Gamma\left(\beta+\frac{1}{2}\right)}{\Gamma(\beta)} \frac{l}{\sqrt{2 \pi t^3(1+\eta t)}}\left[\frac{(l-v t)^2}{2 \alpha t(1+\eta t)}+\alpha\right]^{-\beta-\frac{1}{2}} \\
& =\frac{\Gamma\left(\beta+\frac{1}{2}\right)}{\Gamma(\beta)} \frac{l}{\sqrt{2 \pi \alpha t^3(1+\eta t)}}\left[\frac{(l-v t)^2}{2 \alpha t(1+\eta t)}+1\right]^{-\beta-\frac{1}{2}}
\end{aligned}
\]</span></p>
</section>
<section id="统计推断-1" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-1">统计推断</h3>
<ul>
<li><strong>模型一</strong></li>
</ul>
<p>由Wiener过程的性质，知道<span class="math inline">\(\Delta X_{i j}| \mu \sim N\left(\mu \Delta t_{i j}, \sigma^2 \Delta t_{i j}\right)\)</span>，如上面模型类似，似然函数为：</p>
<p><span class="math display">\[
\begin{aligned}
l &= \sum_{i=1}^{n} \sum_{j=1}^{m_i} \log \int_{-\infty}^{\infty} f(x;\mu,\sigma) f(\mu) d \mu\\
& = \sum_{i=1}^{n} \sum_{j=1}^{m_i} \log \int_{-\infty}^{\infty}N\left(\Delta x_{i j}; \mu \Delta t_{i j}, \sigma^2 \Delta t_{i j}\right) \times N(\mu;\mu_\beta,\sigma^2_\beta) d\mu \\
\end{aligned}
\]</span></p>
<ul>
<li><strong>模型二</strong></li>
</ul>
<p>对数似然函数为</p>
<p><span class="math display">\[
\begin{aligned}
l &= \sum_{i=1}^{n} \sum_{j=1}^{m_i} \log \int_{-\infty}^{\infty} f(x;\mu,\sigma) f(\mu) f(\sigma) d \mu d\sigma \\
\end{aligned}
\]</span></p>
</section>
</section>
<section id="多应力情形" class="level2">
<h2 class="anchored" data-anchor-id="多应力情形">2.3 多应力情形</h2>
<p>恒定应力加速Wiener退化过程考虑以下三个假设：</p>
<ol type="1">
<li><p>确定正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，且满足如下关系 <span class="math display">\[{S_0<S}_1<S_2<\ldots<S_r.\]</span></p></li>
<li><p>在正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，产品退化量服从漂移参数为<span class="math inline">\(\mu_i\)</span>，扩散参数为<span class="math inline">\(\sigma\)</span>的Wiener过程。</p></li>
<li><p>漂移参数为<span class="math inline">\(\mu_i\)</span>和应力水平存在以下关系：<span class="math display">\[\ln \mu_i = a + b \varphi (S_i).\]</span></p></li>
</ol>
<p>注： 在加速退化试验情况下进行统计推断后，可以外推正常应力水平，包括对可靠度函数、RUL等进行推断，具体表达式和经典Wiener过程相同，只是表达式中的<span class="math inline">\(\mu\)</span>用<span class="math inline">\(\mu_0= \exp(a+b \varphi (S_0))\)</span>代替。</p>
<section id="统计推断-2" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-2">2.3.1 统计推断</h3>
<p>假设在加速应力水平<span class="math inline">\(S_i\)</span>下有<span class="math inline">\(n_i\)</span> 个样品进行性能退化试验。对第<span class="math inline">\(i\)</span>个应力水平的第<span class="math inline">\(j\)</span>个样品测量<span class="math inline">\(n_{ij}\)</span>次，取每个样品的初始退化量为<span class="math inline">\(X_{ij0}=0\)</span>，在时刻<span class="math inline">\(t_{ijk}\)</span>样品相应的退化量为<span class="math inline">\(X_{ijk}\)</span>。记 <span class="math inline">\(\Delta X_{i j k}=X_{i j k}-X_{i, j, k-1}\)</span> 是第<span class="math inline">\(i\)</span>个应力水平下第<span class="math inline">\(j\)</span>个样品在时刻 <span class="math inline">\(t_{i j k} \sim t_{i j k-1}\)</span> 之间的退化增量, <span class="math inline">\(\Delta t_{i j k}=t_{i j k }-t_{i j k-1}, i = 1,2 \cdots,r, j=1,2, \cdots, n_i, k=1,2, \cdots, n_{i j}\)</span> 为各样品的测量间隔。</p>
<section id="极大似然估计-1" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计-1">2.2.1.1 极大似然估计</h4>
<p>由 Wiener 过程的性质, 容易知道 <span class="math display">\[
\Delta X_{i j k} \sim N\left(\mu_i \Delta t_{i j k}, \sigma^2 \Delta t_{i j k}\right)
\]</span></p>
<p>于是由退化数据 <span class="math inline">\(X_{i j k}=x_{i j k}\)</span> <span class="math inline">\(i = 1,2 \cdots,r, j=1,2, \cdots, n_i, k=1,2, \cdots, n_{i j}\)</span>, 得到模型参数的似然函数为 <span class="math display">\[
\begin{aligned}
L\left(a,b, \sigma^2\right)&=\prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{i j}} \frac{1}{\sqrt{2 \sigma^2 \pi \Delta t_{i j k}}} \exp \left[-\frac{\left(\Delta x_{i j k}-\mu_i \Delta t_{i j k}\right)^2}{2 \sigma^2 \Delta t_{i j k}}\right]\\
&=\prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{i j}} \frac{1}{\sqrt{2 \sigma^2 \pi \Delta t_{i j k}}} \exp \left[-\frac{\left(\Delta x_{i j k}-\exp(a+b\varphi(S_i)) \Delta t_{i j k}\right)^2}{2 \sigma^2 \Delta t_{i j k}}\right]
\end{aligned}
\]</span></p>
<p>对应的对数似然函数为 <span class="math display">\[
l \left(a,b, \sigma^2\right) \propto N \ln \sigma - \frac{1}{2 \sigma^2} \sum_{i=1}^r \sum_{j=1}^{n_i} \sum_{k=1}^{n_{i j}} \frac{\left(\Delta x_{i j k}-e^{a+b\varphi(S_i)} \Delta t_{i j k}\right)^2}{\Delta t_{i j k}}
\]</span></p>
<p>还有一种基于加速因子的似然函数。定义<span class="math display">\[\mu_i=\mu_0 \theta^{h_i},\]</span>其中<span class="math inline">\(h_i = \frac{\varphi(S_0)-\varphi(S_i)}{\varphi(S_0)-\varphi(S_1)}\)</span>。 此时，相应的对数似然函数为 <span class="math display">\[
l \left( \mu_0 ,\theta, \sigma^2\right) \propto N \ln \sigma - \frac{1}{2 \sigma^2} \sum_{i=1}^r \sum_{j=1}^{n_i} \sum_{k=1}^{n_{i j}} \frac{\left(\Delta x_{i j k}- \mu_0 \theta^{h_i} \Delta t_{i j k}\right)^2}{\Delta t_{i j k}}
\]</span></p>
</section>
</section>
</section>
<section id="gamma-退化过程模型" class="level1">
<h1>3. Gamma 退化过程模型</h1>
<p>Gamma 过程描述的退化过程是严格单调非负的，并且Gamma 过程是纯跳过程，其样本路径不连续，既可描述连续的微小冲击导致的缓慢退化，也可以描述大的冲击导致的大的损伤。 当产品的退化过程是严格单调非负时, Wiener 过程不再适用。Gamma 过程具有独立非负增量,可以对该类退化过程进行建模。</p>
<section id="经典-gamma-退化过程模型" class="level2">
<h2 class="anchored" data-anchor-id="经典-gamma-退化过程模型">3.1 经典 Gamma 退化过程模型</h2>
<p>称连续时间随机过程 <span class="math inline">\(\{X(t), t \geqslant 0\}\)</span> 是平稳 Gamma 过程 <span class="math inline">\({ }^{[2]}\)</span>, 若它满足以下性质:</p>
<ol type="1">
<li><p><span class="math inline">\(X(0)=0\)</span> 以概率 1 成立。</p></li>
<li><p><span class="math inline">\(X(t)\)</span> 具有平稳独立增量。</p></li>
<li><p>对任意 <span class="math inline">\(t \geqslant 0\)</span> 和 <span class="math inline">\(\Delta t, X(t+\Delta t)-X(t) \sim G a(\alpha \Delta t, \beta)\)</span> 。</p>
<p>其中 <span class="math inline">\(G a(\alpha, \beta)\)</span> 是形状参数为 <span class="math inline">\(\alpha>0\)</span> 、尺度参数为 <span class="math inline">\(\beta>0\)</span> 的 Gamma 分布, 分布密度函数为 <span class="math display">\[
f(x \mid \alpha, \beta)=\frac{1}{\Gamma(\alpha) \beta^\alpha} \alpha^{\alpha-1} \mathrm{e}^{-x / \beta} I_{(0, \infty)}(x)
\]</span> 并且 <span class="math inline">\(\{X(t), t \geqslant 0\}\)</span> 称为形状参数为 <span class="math inline">\(\alpha>0\)</span> 、尺度参数为 <span class="math inline">\(\beta>0\)</span> 的平稳 Gamma 过程。</p></li>
</ol>
<p><strong>产品寿命</strong></p>
<p>因为Gamma 过程的退化是单调递增的，即<span class="math inline">\(X(t) \sim Ga(\alpha t,\beta)\)</span> ,因此<strong>产品寿命</strong>的分布可以直接通过退化量的转换获得。当给定产品退化失效阈值 <span class="math inline">\(\ell\)</span> 时，产品寿命 <span class="math inline">\(T\)</span> 的CDF和PDF为</p>
<p><span class="math display">\[
\begin{aligned}
{F}_{{T}}(t) &={P}({T} \leqslant {t})={P}({X}({t}) \geqslant \ell) \\
&=\int_{\ell}^{\infty} \frac{1}{\Gamma(\alpha {t}) \beta^{\alpha {t}}} {x}^{\alpha {t}-1} \mathrm{e}^{-\frac{{x}}{\beta}} {dx} \\
&=\frac{1}{\Gamma(\alpha {t})} \int_{\frac{\ell}{\beta}}^{\infty} \xi^{\alpha {t}-1} \mathrm{e}^{-\xi} {d} \xi ,\\
{f}_{{T}}(t) &=\frac{{d}}{{dt}} \frac{\Gamma(\alpha {t}, 1 / \beta)}{\Gamma(\alpha {t})} \\
&=\frac{\alpha}{\Gamma(\alpha {t})} \int_0^{\nu / \beta}\left[\ln (\xi)-\frac{\Gamma^{\prime}(\alpha {t})}{\Gamma(\alpha {t})}\right] \xi^{\alpha {t}-1} \mathrm{e}^{-\xi} {d} \xi.
\end{aligned}
\]</span></p>
<p>由于<span class="math inline">\({f}_{{T}}({t})\)</span> 相当复杂，因此一般用B-S分布来逼近：</p>
<p><span class="math display">\[
{F}(t ; \ell)=\Phi\left[\frac{1}{{v}}\left(\sqrt{\frac{{t}}{{u}}}-\sqrt{\frac{{u}}{{t}}}\right)\right], \quad {t}>0.
\]</span> 其中，<span class="math inline">\(\Phi(\cdot)\)</span> 为标准正态分布，<span class="math inline">\(v=\sqrt{\frac{\beta}{\ell}}, u=\frac{\ell}{\beta \alpha}\)</span>. 相应的 PDF 为 <span class="math display">\[
{f}(t ; \ell)=\frac{1}{2 \sqrt{2 \pi} {uv}}\left[\left(\frac{{u}}{{t}}\right)^{\frac{1}{2}}+\left(\frac{{u}}{{t}}\right)^{\frac{3}{2}}\right] \exp \left[-\frac{1}{2 {v}^2}\left(\frac{{t}}{{u}}-2+\frac{{u}}{{t}}\right)\right], {t}>0.
\]</span></p>
<p><strong>剩余寿命</strong></p>
<p>根据Gamma 过程的增量独立性，可以得到剩余寿命的分布 <span class="math display">\[
\begin{aligned}
{F}(t \mid s) &={P}\left({T} \leqslant t \mid {X}(s)={x}_{s}\right) \\
&={P}\left(X(t)-X(s) \geqslant \ell-x_{s}\right) \\
&=\frac{\Gamma\left(\alpha(t-s),\left(\ell-x_{s}\right) / \beta\right)}{\Gamma(\alpha(t-s))} .
\end{aligned}
\]</span> 这与将失效阈值由 <span class="math inline">\(\ell\)</span> 变为 <span class="math inline">\(\ell-{x_s}\)</span> 、时间 <span class="math inline">\(t\)</span> 变为 <span class="math inline">\(t-s\)</span> 情形的寿命分布是一样的. 因此，在给定当前状态情况下，可以按照类似的途径更新剩余寿命.</p>
<section id="统计推断-3" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-3">3.1.1 统计推断</h3>
<p>假设共有 <span class="math inline">\({n}\)</span> 个样品进行性能退化试验. 对于样品 <span class="math inline">\(i\)</span>, 初始时刻 <span class="math inline">\(t_0\)</span> 性能退化量为 <span class="math inline">\(X_{i 0}=0\)</span>, 在时刻 <span class="math inline">\(t_1, \cdots, t_{m_i}\)</span> 测量产品的性能退化量, 得到其测量值为 <span class="math inline">\(X_{i 1}, \cdots, X_{i m_i}\)</span>. 记 <span class="math inline">\(\Delta x_{i j}=X_{i j}-X_{i(j-1)}\)</span> 是样品 <span class="math inline">\(i\)</span> 在时刻 <span class="math inline">\(t_{j-1}\)</span> 和 <span class="math inline">\(t_j\)</span> 之间的性能退化量, 由 Gamma 过程的独立增量性得到<span class="math inline">\(\Delta X(t) \sim Ga(\alpha \Delta t_{ij},\beta)\)</span>。</p>
<!-- #### 3.1.1 矩估计 -->
<!-- 令 $R_{i j}=\Delta X_{i j} / \Delta t_{i j}$, 则诸 $R_{i j}, j=1,2, \cdots, m_i, i=1,2, \cdots, n$ 服从形状参数为 $\alpha$ 、尺度参数为 $\beta$ 的 Gamma 分布, 且相互独立(根据 Gamma 过程独立增量特性, 以及各样品退化过程的独立性)。记 $M=\sum_{i=1}^n m_i$, 样本均值和样本方差为 $$ -->
<!-- \begin{align} -->
<!-- \bar{R}& =\frac{1}{M} \sum_{i=1}^n \sum_{j=1}^{m_i} R_{i j} \\ -->
<!-- S_R^2 & =\frac{1}{M-1} \sum_{i=1}^n \sum_{j=1}^{m_i}\left[R_{i j}-\bar{R}\right]^2 -->
<!-- \end{align} -->
<!-- $$ 易知 $$ -->
<!-- \begin{aligned} -->
<!-- E[\bar{R}] &=\alpha \beta \\ -->
<!-- Var[\bar{R}]&=\frac{1}{M^2} \alpha \beta^2 \sum_{i=1}^n \sum_{j=1}^{m_i}\left[1 / \Delta t_{i j}\right]\\ -->
<!-- \end{aligned} -->
<!-- $$ 所以， -->
<!-- $$ -->
<!-- \begin{aligned} -->
<!-- E\left[S_R^2\right]&=\frac{1}{M-1} \sum_{i=1}^n \sum_{j=1}^{m_i} E\left[R_{i j}-\bar{R}\right]^2\\ -->
<!-- &=\frac{1}{M-1} \sum_{i=1}^n \sum_{j=1}^{m_i} E\left[R_{i j}-\alpha \beta-(\bar{R}-\alpha \beta)\right]^2 \\ -->
<!-- &=\frac{1}{M-1} \sum_{i=1}^n \sum_{j=1}^{m_i}\left\{Var\left[R_{i j}\right]^2-\frac{2}{m_i} Var\left[R_{i j}\right]+ Var[\bar{R}]\right\} \\ -->
<!-- &=\frac{1}{M-1} \alpha \beta^2 \sum_{i=1}^n \sum_{j=1}^{m_i}\left[\frac{1}{\Delta t_{i j}}\right] -->
<!-- \end{aligned} -->
<!-- $$ 将待估参数和样本矩联系起来, 即令 $$ -->
<!-- \left\{\begin{array}{l} -->
<!-- \alpha \beta=\bar{R} \\ -->
<!-- \alpha \beta^2=M S_R^2 / \sum_{i=1}^n \sum_{j=1}^{m_i}\left[1 / \Delta t_{i j}\right] -->
<!-- \end{array}\right. -->
<!-- $$ 可得如下矩估计: $$ -->
<!-- \begin{gathered} -->
<!-- \hat{\alpha}=\frac{\sum_{i=1}^n \sum_{j=1}^{m_i}\left[1 / \Delta t_{i j}\right] \bar{R}^2}{M S_R^2} \\ -->
<!-- \hat{\beta}=\frac{M S_R^2}{\sum_{i=1}^n \sum_{j=1}^{m_i}\left[1 / \Delta t_{i j}\right] \bar{R}} -->
<!-- \end{gathered} -->
<!-- $$ -->
<!-- 上述矩估计方法是基于退化速率的，金光还给出了基于退化量的矩估计（详见P128）,注意到当测量间隔都相同时，两个矩估计方法相同。 -->
<section id="极大似然估计-2" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计-2">3.1.1.1 极大似然估计</h4>
<p>由 Gamma 过程独立增量特性, 以及 <span class="math display">\[
\Delta X_{i j} \sim G a\left(\alpha \Delta t_{i j}, \beta\right)=\frac{\left(\Delta x_{i j} / \beta\right)^{\alpha \Delta t_{i j}-1}}{\beta \Gamma\left(\alpha \Delta t_{i j}\right)} \mathrm{e}^{-\Delta x_{i j} / \beta}
\]</span> 可以获得对数似然函数为 <span class="math display">\[
l(\alpha, \beta)=\sum_{i=1}^n\left(\sum_{j=1}^{m_i}\left(\alpha \Delta t_{i j}-1\right) \ln \Delta x_{i j}-\alpha t_{i m_i} \ln \beta-\sum_{j=1}^{m_i} \ln \Gamma\left(\alpha \Delta t_{i j}\right)-\frac{x_{i m_i}}{\beta}\right)
\]</span> 由极大似然估计原理, 令 <span class="math display">\[
\left\{\begin{array}{l}
\frac{\partial l}{\partial \alpha}=\sum_{i=1}^n \sum_{j=1}^{m_i} \Delta t_{i j}\left(\ln x_{i j}-\psi\left(\alpha \Delta t_{i j}\right)-\ln \beta\right)=0 \\
\frac{\partial l}{\partial \beta}=\sum_{i=1}^n\left(\frac{x_{i m_i}}{\beta^2}-\frac{\alpha t_{i m_i}}{\beta}\right)=0
\end{array}\right.
\]</span> 式中 <span class="math inline">\(\psi(u)\)</span> 是 digamma 函数。 <span class="math display">\[
\begin{aligned}
&\hat{\beta}=\frac{\sum_{i=1}^n X_{i m_i}}{\alpha \sum_{i=1}^n t_{i m_i}}\\
&\sum_{i=1}^n\left[\sum_{j=1}^{m_i} \Delta t_{i j} \ln \left(\Delta x_{i j}\right)-t_{i m_i} \ln \frac{x_{i m_i}}{\alpha t_{i m_i}}-\sum_{j=1}^{m_i} \Delta t_{i j} \psi\left(\alpha \Delta t_{i j}\right)\right]=0
\end{aligned}
\]</span></p>
</section>
<section id="贝叶斯分析-1" class="level4">
<h4 class="anchored" data-anchor-id="贝叶斯分析-1">3.1.1.2 贝叶斯分析</h4>
</section>
</section>
</section>
<section id="加速-gamma-退化过程" class="level2">
<h2 class="anchored" data-anchor-id="加速-gamma-退化过程">3.2 加速 Gamma 退化过程</h2>
<p>恒定应力加速Gamma退化过程考虑以下三个假设：</p>
<ol type="1">
<li><p>确定正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，且满足如下关系 <span class="math display">\[{S_0<S}_1<S_2<\ldots<S_r.\]</span></p></li>
<li><p>在正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，产品退化量服从漂移参数为<span class="math inline">\(\alpha_i\)</span>，扩散参数为<span class="math inline">\(\beta\)</span>的Gamma过程。</p></li>
<li><p>漂移参数为<span class="math inline">\(\alpha_i\)</span>和应力水平存在以下关系：<span class="math display">\[\ln \alpha_i = a + b \varphi (S_i).\]</span></p></li>
</ol>
<p>注： 在加速退化试验情况下进行统计推断后，可以外推正常应力水平，包括对可靠度函数、RUL等进行推断，具体表达式和经典Gamma过程相同，只是表达式中的<span class="math inline">\(\alpha\)</span>用<span class="math inline">\(\alpha_0= \exp(a+b \varphi (S_0))\)</span>代替。</p>
<section id="统计推断-4" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-4">3.2.1 统计推断</h3>
<p>假设在加速应力水平<span class="math inline">\(S_i\)</span>下有<span class="math inline">\(n_i\)</span> 个样品进行性能退化试验。对第<span class="math inline">\(i\)</span>个应力水平的第<span class="math inline">\(j\)</span>个样品测量<span class="math inline">\(n_{ij}\)</span>次，取每个样品的初始退化量为<span class="math inline">\(X_{ij0}=0\)</span>，在时刻<span class="math inline">\(t_{ijk}\)</span>样品相应的退化量为<span class="math inline">\(X_{ijk}\)</span>。记 <span class="math inline">\(\Delta X_{i j k}=X_{i j k}-X_{i, j, k-1}\)</span> 是第<span class="math inline">\(i\)</span>个应力水平下第<span class="math inline">\(j\)</span>个样品在时刻 <span class="math inline">\(t_{i j k} \sim t_{i j k-1}\)</span> 之间的退化增量, <span class="math inline">\(\Delta t_{i j k}=t_{i j k }-t_{i j k-1}, i = 1,2 \cdots,r, j=1,2, \cdots, n_i, k=1,2, \cdots, n_{i j}\)</span> 为各样品的测量间隔。</p>
<section id="极大似然估计-3" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计-3">3.2.1.1 极大似然估计</h4>
<!-- 在简单情形, 认为退化数据的测量不存在误差, 并且总体退化过程不存在随机效应。 -->
<p>由 Gamma 过程独立增量特性, 以及 <span class="math display">\[
\Delta X_{i j k} \sim G a\left(\alpha_i \Delta t_{i j k}, \beta\right)=\frac{\left(\Delta x_{i j k} / \beta\right)^{\alpha_i \Delta t_{i j k}-1}}{\beta \Gamma\left(\alpha_i \Delta t_{i j k}\right)} \mathrm{e}^{-\Delta x_{i j k} / \beta}
\]</span> 可以获得对数似然函数为 <span class="math display">\[
\begin{aligned}
l(a,b, \beta)&=\sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \alpha_i \Delta t_{i j k} \ln \Delta x_{i j k}-\ln \beta \sum_{i=1}^r \alpha_i T_i - \sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \ln \Gamma \left(\alpha_i \Delta t_{i j k}\right)-\sum_{i=1}^r X_i/ \beta\\
&=\sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} e^{a+b\varphi(S_i)} \Delta t_{i j k} \ln \Delta x_{i j k}-\ln \beta \sum_{i=1}^r e^{a+b\varphi(S_i)} T_i - \sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \ln \Gamma \left(e^{a+b\varphi(S_i)} \Delta t_{i j k}\right)-\sum_{i=1}^r X_i/ \beta
\end{aligned},
\]</span> 其中<span class="math inline">\(T_i=\sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \Delta t_{i j k} ,X_i=\sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \Delta x_{i j k}\)</span>。</p>
<p>还有一种基于加速因子的似然函数。定义<span class="math display">\[\alpha_i=\alpha_0 \theta^{h_i},\]</span>其中<span class="math inline">\(h_i = \frac{\varphi(S_0)-\varphi(S_i)}{\varphi(S_0)-\varphi(S_1)}\)</span>。 此时，相应的似然函数为 <span class="math display">\[
l(\alpha_0,\theta, \beta)=\sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \alpha_0 \theta^{h_i} \Delta t_{i j k} \ln \Delta x_{i j k}-\ln \beta \sum_{i=1}^r \alpha_0 \theta^{h_i} T_i - \sum_{i=1}^r \sum_{j=1}^n \sum_{k=1}^{n_{ i j}} \ln \Gamma \left(\alpha_0 \theta^{h_i} \Delta t_{i j k}\right)-\sum_{i=1}^r X_i/ \beta
\]</span></p>
</section>
</section>
</section>
</section>
<section id="ig-退化过程模型" class="level1">
<h1>4. IG 退化过程模型</h1>
<!-- > 来源：Ye, Z.-S. and M. Xie (2015). "Stochastic modelling and analysis of degradation for highly reliable products." Applied Stochastic Models in Business and Industry 31(1): 16-32. -->
<p>虽然 Wiener 过程和 Gamma 过程在退化建模中得到了广泛的应用，但这两种模型在实际应用中无法拟合大量退化数据。当这两个过程失败时，另一个选择是逆高斯(inverse Gaussian, IG)过程。</p>
<section id="经典的-ig-过程" class="level2">
<h2 class="anchored" data-anchor-id="经典的-ig-过程">4.1 经典的 IG 过程</h2>
<p>设<span class="math inline">\(X(t)\)</span>为产品在<span class="math inline">\(t\)</span>时刻的退化特性。假设退化路径<span class="math inline">\(X(t)\)</span>可以用参数<span class="math inline">\(\mu\)</span>和<span class="math inline">\(\lambda\)</span>的IG过程来建模。此时，<span class="math inline">\(X(t)\)</span>具有以下属性:</p>
<ul>
<li><span class="math inline">\(X(0)=0\)</span>，概率为1;</li>
<li>对于 <span class="math inline">\(t>s>u, X(t)-X(s) \geq 0, X(s)-X(u) \geq 0\)</span>, <span class="math inline">\(X(t)-X(s)\)</span> 与 <span class="math inline">\(X(s)-X(u)\)</span> 相互独立;</li>
<li>对于 <span class="math inline">\(t>s \geq 0\)</span>, 每个增量 <span class="math inline">\(X(t)-X(s)\)</span> 都服从 IG 分布 <span class="math inline">\(I G\left(\mu (t-s), \lambda(t-s)^2\right)\)</span>。</li>
</ul>
<p>IG 分布的概率密度函数(pdf) <span class="math inline">\(I G(a, b)\)</span>为 <span class="math display">\[
f_{I G}(x, a, b)=\left[\frac{b}{2 \pi x^3}\right]^{1 / 2} \exp \left\{-\frac{b(x-a)^2}{2 a^2 x}\right\}
\]</span> 累计分布函数（cdf）为 <span class="math display">\[
F_{IG}(x, a, b)=\Phi\left[\sqrt{\frac{b}{x}}\left(\frac{x}{a}-1\right)\right]+\exp \left(\frac{2 b}{a}\right) \Phi\left[-\sqrt{\frac{b}{x}}\left(\frac{x}{a}+1\right)\right], X>0 .
\]</span></p>
<p><strong>产品寿命/首达时分布</strong></p>
<p>令 <span class="math inline">\(T\)</span> 为<span class="math inline">\(X(t)\)</span>越过临界值<span class="math inline">\(l\)</span>的首达时。此时，我们有 <span class="math display">\[
T=\inf \{t \mid X(t) \geq l, t>0\} .
\]</span> 对于给定的<span class="math inline">\(l\)</span>，生命周期<span class="math inline">\(T\)</span>具有以下 cdf</p>
<p><span class="math display">\[
\begin{aligned}
F(t) &=P(T \leq t)=P(X(t)>l)=1-F_{I G}\left(l ; \mu t, \lambda t^2\right) \\
&=\Phi\left[ \sqrt{\frac{\lambda}{l}}(t - \frac{l}{\mu})
\right] - \exp\left({\frac{2 \lambda t}{\mu}}\right) \Phi\left[-\sqrt{\frac{\lambda}{l}}(\frac{l}{\mu} +t)\right]
\end{aligned}\]</span></p>
<p>其中 <span class="math inline">\(\Phi(\cdot)\)</span> 为标准正态分布的cdf。</p>
<p><strong>剩余使用寿命</strong></p>
<p>假设其运行到时刻 <span class="math inline">\(\tau\)</span> 仍末失效, 且当前性能退化量为 <span class="math inline">\(x_\tau\left(x_\tau<l\right)\)</span>, 则产品的剩余寿命 <span class="math inline">\(T_1\)</span> 可以表示为 <span class="math display">\[
T_1=\inf \left\{t \mid X(t+\tau) \geqslant l, X(\tau)=x_\tau, t \geqslant 0\right\}
\]</span> 其密度函数只需将寿命 <span class="math inline">\(T\)</span> 密度函数中的失效阈值 <span class="math inline">\(l\)</span> 替换为 <span class="math inline">\(l-x_\tau\)</span>, 即</p>
<p><span class="math display">\[
\begin{aligned}
F_{T_1}(t) = \Phi\left[ \sqrt{\frac{\lambda}{l-x_\tau}}(t - \frac{l}{\mu})
\right] - \exp\left({\frac{2 \lambda t}{\mu}}\right) \Phi\left[-\sqrt{\frac{\lambda}{l-x_\tau}}(\frac{l-x_\tau}{\mu} +t)\right]
\end{aligned}\]</span></p>
<p><span class="math display">\[
\begin{aligned}
f_{T_1}(t)= & \phi\left(\sqrt{\frac{\lambda}{l-x_\tau}}\left(t-\frac{l}{\mu}\right)\right) \cdot \sqrt{\frac{\lambda}{l-x_\tau}}-\frac{2 \lambda}{\mu} \exp \left(\frac{2 \lambda t}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{l-x_\tau}}\left(\frac{l-x_\tau}{\mu}+t\right)\right) \\
& +\exp \left(\frac{2 \lambda t}{\mu}\right) \phi\left(-\sqrt{\frac{\lambda}{l-x_\tau}}\left(\frac{l-x_\tau}{\mu}+t\right)\right) \cdot \sqrt{\frac{\lambda}{l-x_\tau}}
\end{aligned}
\]</span></p>
<section id="统计推断-5" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-5">4.1.1 统计推断</h3>
<p>令<span class="math inline">\(X\left(t_{ij}\right)\)</span>为第<span class="math inline">\(i\)</span>个测试单元在<span class="math inline">\(t_{ij}\)</span>时刻测量的退化特性，其中<span class="math inline">\(i=1， \ldots, n\)</span>和<span class="math inline">\(j=\)</span> <span class="math inline">\(1， \ldots, m_i\)</span>。</p>
<section id="极大似然估计-4" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计-4">4.1.1.1 极大似然估计</h4>
<p>根据<span class="math inline">\(\Delta x_{ij}=X\left(t_{ij}\right)-X\left(t_{ij -1}\right)， i=1， \ldots n, j=1， \ldots, m_i\)</span>， <span class="math inline">\((\mu, \lambda)\)</span>的似然函数为 <span class="math display">\[
L(\mu, \lambda)=\prod_{i=1}^n \prod_{j=1}^{m_i} f_{I G}\left(\Delta x_{i j}, \mu \Delta t_{i j}, \lambda \Delta^2 t_{i j}\right).
\]</span> 对数似然函数为： <span class="math display">\[
\log L(\mu, \lambda) \propto \sum_{i=1}^n \sum_{j=1}^{m_i}\left(\frac{1}{2} \log (\lambda)+\log \Delta t_{i j}-\frac{3}{2} \log \left(\Delta x_{i j}\right)-\frac{\lambda\left(\Delta x_{i j}-\mu \Delta t_{i j}\right)^2}{2 \mu^2 \Delta x_{i j}}\right)
\]</span></p>
</section>
<section id="贝叶斯估计" class="level4">
<h4 class="anchored" data-anchor-id="贝叶斯估计">4.1.1.2 贝叶斯估计</h4>
</section>
</section>
</section>
<section id="加速-ig-退化过程" class="level2">
<h2 class="anchored" data-anchor-id="加速-ig-退化过程">4.2 加速 IG 退化过程</h2>
<p>恒定应力加速Gamma退化过程考虑以下三个假设：</p>
<ol type="1">
<li><p>确定正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，且满足如下关系 <span class="math display">\[{S_0<S}_1<S_2<\ldots<S_r.\]</span></p></li>
<li><p>在正常应力水平<span class="math inline">\(S_0\)</span>和<span class="math inline">\(k\)</span>个加速应力水平<span class="math inline">\(S_1,S_2,\ldots,S_r\)</span>，产品退化量服从漂移参数为<span class="math inline">\(\mu_i\)</span>，扩散参数为<span class="math inline">\(\lambda\)</span>的Gamma过程。</p></li>
<li><p>漂移参数为<span class="math inline">\(\mu_i\)</span>和应力水平存在以下关系：<span class="math display">\[\ln \mu_i = a + b \varphi (S_i).\]</span></p></li>
</ol>
<p>注： 在加速退化试验情况下进行统计推断后，可以外推正常应力水平，包括对可靠度函数、RUL等进行推断，具体表达式和经典Gamma过程相同，只是表达式中的<span class="math inline">\(\mu\)</span>用<span class="math inline">\(\mu_0= \exp(a+b \varphi (S_0))\)</span>代替。</p>
<section id="统计推断-6" class="level3">
<h3 class="anchored" data-anchor-id="统计推断-6">4.2.1 统计推断</h3>
<p>假设在加速应力水平<span class="math inline">\(S_i\)</span>下有<span class="math inline">\(n_i\)</span> 个样品进行性能退化试验。对第<span class="math inline">\(i\)</span>个应力水平的第<span class="math inline">\(j\)</span>个样品测量<span class="math inline">\(n_{ij}\)</span>次，取每个样品的初始退化量为<span class="math inline">\(X_{ij0}=0\)</span>，在时刻<span class="math inline">\(t_{ijk}\)</span>样品相应的退化量为<span class="math inline">\(X_{ijk}\)</span>。记 <span class="math inline">\(\Delta X_{i j k}=X_{i j k}-X_{i, j, k-1}\)</span> 是第<span class="math inline">\(i\)</span>个应力水平下第<span class="math inline">\(j\)</span>个样品在时刻 <span class="math inline">\(t_{i j k} \sim t_{i j k-1}\)</span> 之间的退化增量, <span class="math inline">\(\Delta t_{i j k}=t_{i j k }-t_{i j k-1}, i = 1,2 \cdots,r, j=1,2, \cdots, n_i, k=1,2, \cdots, n_{i j}\)</span> 为各样品的测量间隔。</p>
<section id="极大似然估计-5" class="level4">
<h4 class="anchored" data-anchor-id="极大似然估计-5">4.2.1.1 极大似然估计</h4>
<!-- 在简单情形, 认为退化数据的测量不存在误差, 并且总体退化过程不存在随机效应。 -->
<p>可以获得似然函数为 <span class="math display">\[
L(a,b, \lambda)=\prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{ij}} f_{I G}\left(\Delta x_{i j k}, \mu_i \Delta t_{i j k}, \lambda \Delta^2 t_{i j k}\right).
\]</span></p>
<p>对数似然函数为： <span class="math display">\[
\begin{aligned}
\log L(a,b, \lambda) & \propto \prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{ij}} \left(\frac{1}{2} \log (\lambda)+\log \Delta t_{i j k}-\frac{3}{2} \log \left(\Delta x_{i j k}\right)-\frac{\lambda\left(\Delta x_{i j k}-\mu_i \Delta t_{i j k}\right)^2}{2 \mu_i^2 \Delta x_{i j k}}\right)\\
& \propto \prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{ij}} \left(\frac{1}{2} \log (\lambda)+\log \Delta t_{i j k}-\frac{3}{2} \log \left(\Delta x_{i j k}\right)-\frac{\lambda\left(\Delta x_{i j k}-e^{a+b\varphi(S_i)} \Delta t_{i j k}\right)^2}{2 e^{2a+2b\varphi(S_i)} \Delta x_{i j k}}\right)\\
\end{aligned}
\]</span></p>
<p>还有一种基于加速因子的似然函数。定义<span class="math display">\[\mu_i=\mu_0 \theta^{h_i},\]</span>其中<span class="math inline">\(h_i = \frac{\varphi(S_0)-\varphi(S_i)}{\varphi(S_0)-\varphi(S_1)}\)</span>。 此时，相应的似然函数为 <span class="math display">\[
\log L(\mu_0, \theta, \lambda) \propto \prod_{i=1}^r \prod_{j=1}^{n_i} \prod_{k=1}^{n_{ij}} \left(\frac{1}{2} \log (\lambda)+\log \Delta t_{i j k}-\frac{3}{2} \log \left(\Delta x_{i j k}\right)-\frac{\lambda\left(\Delta x_{i j k}-\mu_0 \theta^{h_i} \Delta t_{i j k}\right)^2}{2 \mu_0^2 \theta^{2h_i} \Delta x_{i j k}}\right)
\]</span></p>
</section>
</section>
</section>
</section>
<section id="模拟数据分析" class="level1">
<h1>5. 模拟数据分析</h1>
<section id="经典维纳过程" class="level2">
<h2 class="anchored" data-anchor-id="经典维纳过程">5.1 经典维纳过程</h2>
</section>
<section id="经典-gamma-过程" class="level2">
<h2 class="anchored" data-anchor-id="经典-gamma-过程">5.2 经典 Gamma 过程</h2>
</section>
<section id="经典-ig-过程" class="level2">
<h2 class="anchored" data-anchor-id="经典-ig-过程">5.3 经典 IG 过程</h2>
</section>
</section>
<section id="案例分析" class="level1">
<h1>6 案例分析</h1>
<section id="维纳过程相关" class="level2">
<h2 class="anchored" data-anchor-id="维纳过程相关">6.1 维纳过程相关</h2>
<section id="nasa-锂电池数据" class="level3">
<h3 class="anchored" data-anchor-id="nasa-锂电池数据">6.1.1 NASA 锂电池数据</h3>
<blockquote class="blockquote">
<p>数据来源于<a href="https://c3.ndc.nasa.gov/dashlink/resources/133/">NASA Ames Prognostics Center of Excellence (PCoE)</a>，但是找不到数据下载。最后是通过 <a href="https://github.com/psanabriaUC/BatteryDatasetImplementation">Github</a> 获得。</p>
</blockquote>
<p>该数据是 NASA Ames Prognostics Center of Fxcellence (PCoF）对商用锂离子 18650 电池进行充、放电试验获得的一组蓄电池容量变化数据。试验过程中，锂离子在室温下 经历 3 种不同的运行剖面，即<strong>充电</strong>、<strong>放电</strong>和<strong>测量 EIS</strong>。</p>
<p>充电是以<strong>恒流模</strong>式(CC)进行，在 1.5A 电流下直到电池电压达到4.2V，然后以<strong>恒压模式</strong>(CV)继续充电直到电流降到 20mA。</p>
<p>放电也以恒流模式进行，放电电流为2A，直到电池电压降低到2.7V。</p>
<p>重复充电和放电将导致蓄电池老化。 我们直接使用充放电循环次数作为时间刻度。由于随着时间的推移，电池容量呈下降趋势。而用维纳过程来进行 RUL 预测方法是建立在退化过程有增加趋势的基础上的。为了应用该模型，对原始数据进行转换：用初始容量减去每个电池的所有容量数据。此时，失效阈值更改为初始容量减去 1.4 Ahr。</p>
<p>前面对数据进行了导入、处理以及可视化。接下来，使用经典的线性维纳过程对其进行建模。使用处理后的数据集，来自自己创建的R包中<code>sdp</code>中。</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1"></a><span class="co"># devtools::install_github("Liangliangzhuang/sdp",force = TRUE)</span></span>
<span id="cb2-2"><a href="#cb2-2"></a><span class="fu">library</span>(sdp)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1"></a><span class="fu">str</span>(lithium_battery)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>List of 2
$ data :'data.frame': 168 obs. of 5 variables:
..$ Time : int [1:168] 1 2 3 4 5 6 7 8 9 10 ...
..$ B0005: num [1:168] 0 0.0102 0.0211 0.0212 0.0218 ...
..$ B0006: num [1:168] 0 0.0102 0.022 0.0221 0.0348 ...
..$ B0007: num [1:168] 0 0.0104 0.0104 0.0103 0.0116 ...
..$ B0018: num [1:168] 0 0.0118 0.0154 0.0243 0.0223 ...
$ threshold: num [1:4] 0.456 0.635 0.491 0.455</code></pre>
</div>
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1"></a><span class="fu">plot_path</span>(lithium_battery[[<span class="dv">1</span>]]) <span class="sc">+</span> </span>
<span id="cb5-2"><a href="#cb5-2"></a> <span class="co"># scale_color_discrete() +</span></span>
<span id="cb5-3"><a href="#cb5-3"></a> <span class="fu">theme_bw</span>() <span class="sc">+</span> </span>
<span id="cb5-4"><a href="#cb5-4"></a> <span class="fu">theme</span>(<span class="at">panel.grid =</span> <span class="fu">element_blank</span>()) </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-2-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1"></a><span class="co"># Inference</span></span>
<span id="cb6-2"><a href="#cb6-2"></a><span class="co"># MLE ========</span></span>
<span id="cb6-3"><a href="#cb6-3"></a>mle_fit <span class="ot">=</span> <span class="fu">sta_infer</span>(<span class="at">method =</span> <span class="st">"MLE"</span>, <span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>,</span>
<span id="cb6-4"><a href="#cb6-4"></a> <span class="at">data =</span> lithium_battery[[<span class="dv">1</span>]])</span>
<span id="cb6-5"><a href="#cb6-5"></a>mle_fit</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> low mean up
[1,] 0.0024 0.0038 0.0051
[2,] 0.0167 0.0176 0.0185</code></pre>
</div>
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1"></a><span class="fu">Reliability</span>(<span class="at">t =</span> <span class="dv">100</span>, <span class="at">threshold =</span> lithium_battery[[<span class="dv">2</span>]][<span class="dv">1</span>],<span class="at">par =</span> mle_fit,</span>
<span id="cb8-2"><a href="#cb8-2"></a> <span class="at">process =</span> <span class="st">"Wiener"</span>,<span class="at">type =</span> <span class="st">"classical"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.8633334</code></pre>
</div>
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1"></a><span class="fu">Reliability_cowplot</span>(<span class="at">R_time =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">300</span>, <span class="at">sum_para =</span> mle_fit, <span class="at">threshold =</span> lithium_battery[[<span class="dv">2</span>]],</span>
<span id="cb10-2"><a href="#cb10-2"></a> <span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-3-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1"></a>rul2 <span class="ot">=</span> <span class="fu">RUL</span>(<span class="at">t =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">100</span>, <span class="at">cur_time =</span> <span class="dv">30</span>, <span class="at">threshold =</span> lithium_battery[[<span class="dv">2</span>]][<span class="dv">1</span>], <span class="at">data =</span> lithium_battery[[<span class="dv">1</span>]], <span class="at">par =</span> mle_fit[,<span class="dv">2</span>], <span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>)</span>
<span id="cb11-2"><a href="#cb11-2"></a></span>
<span id="cb11-3"><a href="#cb11-3"></a><span class="fu">RUL_plot</span>(<span class="at">fut_time =</span> <span class="fu">c</span>(<span class="dv">10</span>,<span class="dv">15</span>,<span class="dv">20</span>,<span class="dv">25</span>,<span class="dv">30</span>), </span>
<span id="cb11-4"><a href="#cb11-4"></a> <span class="at">time_epoch =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">300</span>,</span>
<span id="cb11-5"><a href="#cb11-5"></a> <span class="at">group =</span> <span class="dv">1</span>,</span>
<span id="cb11-6"><a href="#cb11-6"></a> <span class="at">process =</span> <span class="st">"Wiener"</span>, </span>
<span id="cb11-7"><a href="#cb11-7"></a> <span class="at">type =</span> <span class="st">"classical"</span>,</span>
<span id="cb11-8"><a href="#cb11-8"></a> <span class="at">threshold =</span> lithium_battery[[<span class="dv">2</span>]][<span class="dv">1</span>],</span>
<span id="cb11-9"><a href="#cb11-9"></a> <span class="at">dat =</span> lithium_battery[[<span class="dv">1</span>]],</span>
<span id="cb11-10"><a href="#cb11-10"></a> <span class="at">zlim =</span> <span class="fu">c</span>(<span class="dv">0</span>,<span class="fl">0.01</span>),</span>
<span id="cb11-11"><a href="#cb11-11"></a> <span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">0</span>,<span class="dv">300</span>),</span>
<span id="cb11-12"><a href="#cb11-12"></a> <span class="at">para =</span> mle_fit[,<span class="dv">2</span>],</span>
<span id="cb11-13"><a href="#cb11-13"></a> <span class="at">real_RUL=</span><span class="fu">c</span>(<span class="cn">NA</span>,<span class="cn">NA</span>,<span class="cn">NA</span>,<span class="cn">NA</span>,<span class="cn">NA</span>,<span class="cn">NA</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-4-1.png" class="img-fluid" width="672"></p>
</div>
<div class="cell-output cell-output-stdout">
<pre><code>$rect
$rect$w
[1] 0.2499835
$rect$h
[1] 0.1139012
$rect$left
[1] -0.1004324
$rect$top
[1] 0.1683858
$text
$text$x
[1] -0.02995602 -0.02995602 -0.02995602
$text$y
[1] 0.13991046 0.11143515 0.08295984</code></pre>
</div>
</div>
</section>
<section id="激光器件-laser-数据集" class="level3">
<h3 class="anchored" data-anchor-id="激光器件-laser-数据集">6.1.2 激光器件 Laser 数据集</h3>
<blockquote class="blockquote">
<p>数据集来自 Meeker, W. Q. (1998). Statistical Methods for Reliability Data。</p>
</blockquote>
<p>激光器件的质量特征是其<strong>工作电流</strong>。当工作电流达到预定义的阈值水平ω = 10时，该设备被认为是故障。其电流的测量频率为每250小时一次，实验在4000小时时终止。图1(a)显示了15个被测试单元的降解路径及其拟合的平均趋势。</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1"></a>Laser_dat <span class="ot">=</span> <span class="fu">read.csv</span>(<span class="st">"dataset/GaAsLaser.csv"</span>)</span>
<span id="cb13-2"><a href="#cb13-2"></a>knitr<span class="sc">::</span><span class="fu">kable</span>(<span class="fu">head</span>(Laser_dat))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table class="table table-sm table-striped">
<thead>
<tr class="header">
<th style="text-align: right;">Value</th>
<th style="text-align: right;">Unit</th>
<th style="text-align: right;">Hours</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.0000</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.4741</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">250</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.9255</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">500</td>
</tr>
<tr class="even">
<td style="text-align: right;">2.1147</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">750</td>
</tr>
<tr class="odd">
<td style="text-align: right;">2.7168</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">1000</td>
</tr>
<tr class="even">
<td style="text-align: right;">3.5110</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">1250</td>
</tr>
</tbody>
</table>
</div>
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1"></a>Laser_dat<span class="sc">$</span>Unit <span class="ot">=</span> <span class="fu">as.factor</span>(Laser_dat<span class="sc">$</span>Unit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1"></a>Laser_dat <span class="sc">%>%</span> <span class="fu">ggplot</span>(<span class="fu">aes</span>(Hours,Value,<span class="at">fill =</span> Unit)) <span class="sc">+</span></span>
<span id="cb15-2"><a href="#cb15-2"></a> <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="fl">0.1</span>) <span class="sc">+</span> </span>
<span id="cb15-3"><a href="#cb15-3"></a> <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="fl">0.8</span>) <span class="sc">+</span> </span>
<span id="cb15-4"><a href="#cb15-4"></a> <span class="fu">theme_bw</span>() <span class="sc">+</span></span>
<span id="cb15-5"><a href="#cb15-5"></a> <span class="fu">theme</span>(<span class="at">panel.grid.major =</span> <span class="fu">element_blank</span>()) <span class="sc">+</span></span>
<span id="cb15-6"><a href="#cb15-6"></a> <span class="co"># scale_color_viridis(discrete = T) + </span></span>
<span id="cb15-7"><a href="#cb15-7"></a> <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">'none'</span>) <span class="sc">+</span></span>
<span id="cb15-8"><a href="#cb15-8"></a> <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">"Hours"</span>, <span class="at">y =</span> <span class="st">"Degradation characteristic"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-6-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1"></a>Laser_dat <span class="sc">%>%</span> </span>
<span id="cb16-2"><a href="#cb16-2"></a> <span class="fu">pivot_wider</span>(</span>
<span id="cb16-3"><a href="#cb16-3"></a> <span class="at">names_from =</span> <span class="st">"Unit"</span>,</span>
<span id="cb16-4"><a href="#cb16-4"></a> <span class="at">values_from =</span> <span class="st">"Value"</span>) <span class="ot">-></span> Laser_new</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1"></a><span class="co"># Inference</span></span>
<span id="cb17-2"><a href="#cb17-2"></a><span class="co"># MLE ========</span></span>
<span id="cb17-3"><a href="#cb17-3"></a>mle_Laser <span class="ot">=</span> <span class="fu">sta_infer</span>(<span class="at">method =</span> <span class="st">"MLE"</span>, <span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>, <span class="at">data =</span> <span class="fu">as.data.frame</span>(Laser_new))</span>
<span id="cb17-4"><a href="#cb17-4"></a>mle_Laser</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> low mean up
[1,] 0.0019 0.0020 0.0021
[2,] 0.0116 0.0127 0.0138</code></pre>
</div>
</div>
<p>计算可靠度并绘图，阈值设置参考 Wang, X., et al. (2020). “Accurate reliability inference based on Wiener process with random effects for degradation data.” Reliability Engineering & System Safety 193. 第6节案例分析</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1"></a><span class="fu">Reliability</span>(<span class="at">t =</span> <span class="dv">5000</span>, <span class="at">threshold =</span> <span class="dv">10</span>,<span class="at">par =</span> mle_Laser,</span>
<span id="cb19-2"><a href="#cb19-2"></a> <span class="at">process =</span> <span class="st">"Wiener"</span>,<span class="at">type =</span> <span class="st">"classical"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.715017</code></pre>
</div>
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1"></a><span class="fu">Reliability_cowplot</span>(<span class="at">R_time =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">10000</span>, <span class="at">sum_para =</span> mle_Laser, <span class="at">threshold =</span> <span class="fu">c</span>(<span class="dv">10</span>,<span class="dv">8</span>,<span class="dv">9</span>,<span class="dv">11</span>,<span class="dv">5</span>,<span class="dv">12</span>),<span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1"></a>rul2 <span class="ot">=</span> <span class="fu">RUL</span>(<span class="at">t =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">1000</span>, <span class="at">cur_time =</span> <span class="dv">30000</span>, <span class="at">threshold =</span> <span class="dv">10</span>, <span class="at">data =</span> <span class="fu">as.data.frame</span>(Laser_new), <span class="at">par =</span> mle_Laser[,<span class="dv">2</span>], <span class="at">process =</span> <span class="st">"Wiener"</span>, <span class="at">type =</span> <span class="st">"classical"</span>)</span>
<span id="cb22-2"><a href="#cb22-2"></a></span>
<span id="cb22-3"><a href="#cb22-3"></a><span class="co"># RUL_plot(fut_time = c(10,15,20,25,30), </span></span>
<span id="cb22-4"><a href="#cb22-4"></a><span class="co"># time_epoch = 1:1000,</span></span>
<span id="cb22-5"><a href="#cb22-5"></a><span class="co"># group = 1,</span></span>
<span id="cb22-6"><a href="#cb22-6"></a><span class="co"># process = "Wiener", </span></span>
<span id="cb22-7"><a href="#cb22-7"></a><span class="co"># type = "classical",</span></span>
<span id="cb22-8"><a href="#cb22-8"></a><span class="co"># threshold = 10,</span></span>
<span id="cb22-9"><a href="#cb22-9"></a><span class="co"># dat = as.data.frame(Laser_new),</span></span>
<span id="cb22-10"><a href="#cb22-10"></a><span class="co"># zlim = c(0,0.01),</span></span>
<span id="cb22-11"><a href="#cb22-11"></a><span class="co"># xlim = c(0,1000),</span></span>
<span id="cb22-12"><a href="#cb22-12"></a><span class="co"># para = mle_Laser[,2],</span></span>
<span id="cb22-13"><a href="#cb22-13"></a><span class="co"># real_RUL=c(NA,NA,NA,NA,NA,NA))</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="炭膜电阻器-carbon-film-resistors" class="level3">
<h3 class="anchored" data-anchor-id="炭膜电阻器-carbon-film-resistors">6.1.3 炭膜电阻器 carbon-film resistors</h3>
<blockquote class="blockquote">
<p>退化数据来自 Meeker, W. Q. (1998). Statistical Methods for Reliability Data。（常应力下的加速退化数据）。</p>
</blockquote>
<p>在某些电阻器的操作过程中，退化会导致电阻的增加。当电阻的增加百分比达到退化阈值水平时，电阻就失效了。</p>
<p>Zhou2022 只使用了 83 摄氏度下的测试数据。数据中有10个碳膜电阻的测试样品，在83C温度下，在相同的检测时间点452 h, 1030 h, 4341 h, 8084 h测量。10个碳膜电阻的电阻百分比递增路径如下图所示。</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1"></a>resistor_dat <span class="ot">=</span> <span class="fu">read.csv</span>(<span class="st">"dataset/Resistor.csv"</span>)</span>
<span id="cb23-2"><a href="#cb23-2"></a>knitr<span class="sc">::</span><span class="fu">kable</span>(<span class="fu">head</span>(resistor_dat))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table class="table table-sm table-striped">
<thead>
<tr class="header">
<th style="text-align: right;">Percent.Increase</th>
<th style="text-align: left;">Resistor.ID</th>
<th style="text-align: right;">DegreesC</th>
<th style="text-align: right;">Thousands.of.Hours</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.28</td>
<td style="text-align: left;">R1</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">0.452</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.32</td>
<td style="text-align: left;">R1</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">1.030</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.38</td>
<td style="text-align: left;">R1</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">4.341</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.62</td>
<td style="text-align: left;">R1</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">8.084</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.22</td>
<td style="text-align: left;">R2</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">0.452</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.24</td>
<td style="text-align: left;">R2</td>
<td style="text-align: right;">83</td>
<td style="text-align: right;">1.030</td>
</tr>
</tbody>
</table>
</div>
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1"></a><span class="fu">colnames</span>(resistor_dat) <span class="ot">=</span> <span class="fu">c</span>(<span class="st">"Value"</span>,<span class="st">"Unit"</span>,<span class="st">"Degree"</span>,<span class="st">"Hours"</span>)</span>
<span id="cb24-2"><a href="#cb24-2"></a>resistor_dat<span class="sc">$</span>Unit <span class="ot">=</span> <span class="fu">as.factor</span>(resistor_dat<span class="sc">$</span>Unit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1"></a><span class="fu">library</span>(latex2exp)</span>
<span id="cb25-2"><a href="#cb25-2"></a>resistor_dat <span class="sc">%>%</span> <span class="fu">filter</span>(Degree <span class="sc">==</span> <span class="dv">83</span>) <span class="sc">%>%</span> </span>
<span id="cb25-3"><a href="#cb25-3"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(Hours,Value,<span class="at">fill =</span> Unit)) <span class="sc">+</span></span>
<span id="cb25-4"><a href="#cb25-4"></a> <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="fl">0.1</span>) <span class="sc">+</span> </span>
<span id="cb25-5"><a href="#cb25-5"></a> <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="fl">0.8</span>) <span class="sc">+</span> </span>
<span id="cb25-6"><a href="#cb25-6"></a> <span class="fu">theme_bw</span>() <span class="sc">+</span></span>
<span id="cb25-7"><a href="#cb25-7"></a> <span class="fu">theme</span>(<span class="at">panel.grid.major =</span> <span class="fu">element_blank</span>()) <span class="sc">+</span></span>
<span id="cb25-8"><a href="#cb25-8"></a> <span class="co"># scale_color_viridis(discrete = T) + </span></span>
<span id="cb25-9"><a href="#cb25-9"></a> <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">'none'</span>) <span class="sc">+</span></span>
<span id="cb25-10"><a href="#cb25-10"></a> <span class="fu">labs</span>(<span class="at">x =</span> <span class="fu">TeX</span>(r<span class="st">"(Hours($10^3$))"</span>), <span class="at">y =</span> <span class="st">"Degradation characteristic"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="index_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>后续参数估计和可靠度估计就先不做了。</p>
<p>类似的数据集还包括：</p>
</section>
<section id="惯导平台-interial-navigition-数据" class="level3">
<h3 class="anchored" data-anchor-id="惯导平台-interial-navigition-数据">6.1.4 惯导平台 interial navigition 数据</h3>
<blockquote class="blockquote">
<p>数据来自于文献：Si, X. S., et al. (2012). “Remaining Useful Life Estimation Based on a Nonlinear Diffusion Degradation Process.” IEEE Transactions on Reliability <strong>61</strong>(1): 50-67.</p>
</blockquote>
<p>惯导平台是武器系统、空间设备等的惯性导航系统的关键部件，其运行状态直接影响导航精度。惯导平台中的传感器主要包括3个陀螺仪、3 个加速度计，分别用于测量角速度和线加速度。统计分析表明，惯性平台几乎70%的失效来自于陀螺漂移。</p>
<p>下面的例子中,安装在惯导平台中的陀螺是在驱动和传感两个轴上具有两个自由度的机械结构。当惯性平台运行时，陀螺的转子以很高的速度转动，将导致旋转轴的磨损，并最终导致陀螺漂移。随着磨损的累积,漂移也逐渐增大，最终导致陀螺的失效。因此，陀螺漂移是用于评估惯性平台健康状态的一个性能指标。</p>
<p>图给出的是传感轴的漂移退化测量数据，与驱动轴相比,传感轴的漂移在评价陀螺退化时占主要地位。退化试验数据包括5个试验样品，每个产品进行9次测量，试验条件与现场条件相似。</p>
<p><img src="images/paste-40CEA295.png" class="img-fluid"></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1"></a>interial_dat <span class="ot">=</span> <span class="fu">read.csv</span>(<span class="st">"dataset/interial_navigition.csv"</span>)</span>
<span id="cb26-2"><a href="#cb26-2"></a>knitr<span class="sc">::</span><span class="fu">kable</span>(<span class="fu">head</span>(interial_dat))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<table class="table table-sm table-striped">
<thead>
<tr class="header">
<th style="text-align: right;">time</th>
<th style="text-align: right;">Item1</th>
<th style="text-align: right;">Item2</th>
<th style="text-align: right;">Item3</th>
<th style="text-align: right;">Item4</th>
<th style="text-align: right;">Item5</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">2.5</td>
<td style="text-align: right;">0.0507</td>
<td style="text-align: right;">0.1300</td>
<td style="text-align: right;">0.0135</td>
<td style="text-align: right;">0.1052</td>
<td style="text-align: right;">0.0928</td>
</tr>
<tr class="even">
<td style="text-align: right;">5.0</td>
<td style="text-align: right;">0.1789</td>
<td style="text-align: right;">0.2554</td>
<td style="text-align: right;">0.0309</td>
<td style="text-align: right;">0.1996</td>
<td style="text-align: right;">0.2633</td>
</tr>
<tr class="odd">
<td style="text-align: right;">7.5</td>
<td style="text-align: right;">0.2059</td>
<td style="text-align: right;">0.3153</td>
<td style="text-align: right;">0.0077</td>
<td style="text-align: right;">0.2927</td>
<td style="text-align: right;">0.3010</td>
</tr>
<tr class="even">