Bayesian Data Analytics for Reliability Modeling Improvement Mingyang Li Department of Industrial and Management Systems Engineering University of South Florida Jan 26 st , 2018 M. Li
DSSI Laboratory M. Li 2
Outline • Background • Part I - Multi-level Data Fusion • Part II - Heterogeneous Data Quantification • Summary M. Li 3
Data Analytics Bayesian Statistics Statistics & Math Data Data Analytics Analytics • Focus: Bayesian Data Analytics for Reliability Modeling Improvement M. Li 4
Key Word: Bayesian • Parameter Learning Classic Statistics Bayesian Statistics ? Data Parameters Data Posterior Parameters Limited Data or No Data Prior • External data sources • Domain knowledge • Non-informative prior Flexible & Coherent …… Methodology I: Multi-level Data Fusion M. Li 5
Key Word: Bayesian (Cont’d) • Model Learning Classic Statistics Bayesian Statistics Model 1 Data Posterior Parameters Data Parameters & Model Model m Para. Prior Model Prior Data Parameters Efficient & Effective • Underfitting/Overfitting Methodology II: • Inefficient Heterogeneity Quantification M. Li 6
Key Word: Reliability Modeling • Reliability: product quality over time [1] Pr(T>t) Time-to-failure • Reliability modeling Product Sample: Reliability Data: t 2 t 4 t 1 t 5 Modeling T t 0 • Data Feature • • Non-negative and asymmetric Censoring s 2 t s 1 • • Covariates Others: availability , heterogeneity , etc. M. Li 7
Lifecycle View of Reliability Modeling Requirements Marketing Functional Maintenance Relationship policy Design and Reliability Maintenance Development Modeling Repair Logs Evaluation, allocation, etc. Production Testing M. Li 8
Part I - Multi-level Data Fusion: Bayesian Multi-level Information Aggregation for Hierarchical Systems Reliability Modeling Improvement M. Li 9
Vision EEG / MEG ( high-temporal-resolution ) [3] Heating Ventilating & Air-Conditioning Brain (HVAC) System [2] Crowd Surveillance System [4] fMRI (high-spatial-resolution) [3] Data-rich Environment: Unmanned Data Fusion Unmanned Aerial Ground Vehicle Vehicle(UAV) (UGV) Crowd M. Li 10
Focus: System Reliability • Performance index : system reliability • Modeling Challenges: • Expensive system-level tests • Scarce/absent engineering knowledge Missile • Complex failure relationship ($10 3 k - $10m) • High requirement on reliability assessment • Research Goal: Improve system-level reliability modeling by utilizing all reliability information throughout the system in a systematic and coherent manner. M. Li 11
Opportunity I: Hierarchical System Structure Electro-Mechanical-Actuator (EMA) System Power Supply (PS) Actuator Servo Drive (ASD) DC Motor Elements in Divide & Conquer System Hierarchy EMA System PS Sub-system ASD Sub-system Motor Logic PS Controller Bridge DC Motor PS M. Li 12
Opportunity II: Multi-source Multi-level Data • Multi-source reliability information : prior knowledge (e.g., domain knowledge, historical studies, etc.) + ongoing reliability test data. Elements Prior knowledge Reliability Test Data Lower-level Familiar (1) Abundant (2) Limited but easy to collect Upper-level Unfamiliar or unknown (1) Absent (2) Limited and/or expensive/hard to collect • Multi-level information imbalance Aggregation Reliability Information: Prior knowledge Reliability test data Absent information M. Li 13
State of the Art Methodology System Reliability Modeling Summary Parametric Semi-parametric/non- methods parametric methods Ramamoorty [5] , Camarda et al . [6] , Cui Klein and Moeschberger [13] , Meeker Multi-level No et al . [7] , Hoyland and Rausand [8] , and Escobar (Chapter 3) [14] , Ibrahim information Coit [9] , Jin and Coit [10] , Martz and et al . [15] aggregation Walker [11] , Hamada et al . [12] , etc. Martz et al . [16] , Martz and Walker [17] , Yes To be presented Hulting and Robinson [18] • Features of the proposed model: • Failure-time data with covariates and censoring • Semi-parametric modeling • Information aggregation from lower levels M. Li 14
Overview of the Proposed Work Upward Recursively Posterior of X ( l- 1,1) Step 1 Combined Prior of X ( l- 1,1) Data of X ( l- 1,1) Step 4 X ( 1,1) l Native Prior of X ( l- 1,1) Induced Prior of X ( l- 1,1) Step 3 Aggregated posterior of X ( l- 1,1) Step 2 Posterior of X ( l ,2) Posterior of X ( l ,1) X X ( ,2) l ( ,1) l Step 1 Step 1 Data of X ( l ,1) Prior of X ( l ,1) Data of X ( l ,2) Prior of X ( l ,2) M. Li 15
Modeling of Individual Element Unique Proportion Hazard Model (PHM): Reliability: ( ) ( ) R t H t X ( , ) ( , ) l k l k ( , ) l k β T b ( ) ( )exp H t H t u Covariate ( , ) ( , ) ( , ) ( , ) l k l k l k l k Covariate Cumulative Baseline cumulative coefficient hazard function hazard function Gamma Process Prior: Multivariate Normal Prior: ( ) ( ( ) , ) β Σ Z t G c t c ( ) ( , ) N 0 0 ( , ) ( , ) ( , ) l k l k l k l l l Confidence Mean function Mean vector Covariance matrix parameter Baseline cumulative hazard increments: Carry information b ~ ( ( ( ) ( ) ), ) H Gamma c s s c , ( , ) 0 1 0 j l k j j for aggregation M. Li 16
Aggregation Procedure: Step 1 • Step 1 – Compute the posterior (lower-level element): Joint Posterior Likelihoods Joint Priors: β β β b b b ( , | ) ( , | ) ( ) ( ) p H L H H ( , ) ,( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) l k j l k l k l k j l k l k l k j l k Prior knowledge Integration of Failure-time data with data and prior covariates and censoring X t 5 t 1 t 3 t 4 t 2 Example: ( 1,1) l s 1 s 2 s 3 0 t n : the actual failure time stamp of test unit n Step 1 Bayesian PHM integrates the reliability prior X X ( ,1) l ( ,2) l knowledge and failure data M. Li 17
Aggregation Procedure: Failure Relationship • Failure relationship between two levels: General Relationships Q Reliability functions: ( ) ( ) , R t f R t ( 1, ) ( , ) ( 1, ) l k l l k Baseline cumulative Q A b b ( ), H g H hazard increments: ,( 1, ) ,( , ) ( 1, ) j l k j l l k Example: Series configuration X ( 1,1) l A ( ) ( ) ( ) R t R t R t Aggregated ( 1,1) ( ,1) ( ,2) l l l Step 2 A b b b ( ) ( ) ( ) H t H t H t X X j l ( 1,1) j l ( ,1) j l ( ,2) ( ,1) l ( ,2) l b Information is aggregated through based on failure H j relationships M. Li 18
Aggregation Procedure: Steps 2-3 • Step 2 - Aggregate the posterior: Aggregated posterior: A b b ( ), 1,2 H g H ,( 1,1) ,( , ) j l j l I b Induced prior: H ,( 1,1) j l • Step 3 - Approximate the Step 3 induced prior: A b Aggregated posterior: H j l ,( 1,1) Induced prior: I b A b H H ,( 1,1) ,( 1,1) j l j l Step 2 X Gamma I b I I ( 1,1) l ~ ( , ) H Posterior: Posterior: ,( 1,1) ,( 1,1) ,( 1,1) j l j l j l b b H H ,( ,1) j l ,( ,2) j l (Validate by K-S goodness fitness test) X X ( ,2) l ( ,1) l M. Li 19
Aggregation Procedure: Step 4 • Step 4 – Combine the native prior and the induced prior: Combined prior: C b C C ~ ( , ) H Gamma ,( 1,1) ,( 1,1) ,( 1,1) j l j l j l C I N (1 ) Weighting factor: 0 1 w w w ,( 1,1) ,( 1,1) ,( 1,1) j l j l j l C I N (1 ) w w b ,( 1,1) ,( 1,1) ,( 1,1) H j l j l j l Posterior: j l ,( 1,1) Step 1 w : balance native prior and induced prior Ω Failure data: l ( 1,1) • Step 1 – Compute the posterior C b (higher-level element): Combined prior: H ,( 1,1) j l Step 4 Similar Bayesian inference Native prior: Induced prior: Gamma C b C C ~ ( , ) H ,( 1,1) ,( 1,1) ,( 1,1) N b I b j l j l j l H H j l ,( 1,1) j l ,( 1,1) X ( 1,1) l M. Li 20
Information Aggregation: Procedure Review • Recursive • Flexible • Generic Aggregate Upward b Posterior: H Recursively ,( 1,1) j l Step 1 Ω Failure data: Combined prior: C b H l ( 1,1) j l ,( 1,1) Step 4 X Induced prior: I b Native prior: N b ( 1,1) H H l ,( 1,1) j l ,( 1,1) j l Step 3 Level l -1 Aggregated posterior: A b H ,( 1,1) j l Step 2 X Level l X Posterior: A b Posterior: A b H H ( ,2) l ( ,1) l ,( ,2) ,( ,1) j l j l Step 1 Ω prior: C b Ω Failure data: prior: C b H Failure data: H ( ,1) l ,( ,1) j l ( ,2) l j l ,( ,2) M. Li 21
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