Deriving Prognostic Continuous Time Bayesian Networks from D-matrices Logan Perreault, Monica Thornton, Shane Strasser, and John W. Sheppard Montana State University
Motivation Diagnosis and Prognosis with ATS Given: A UUT to be tested by an intelligent TPS 1 Develop/derive models from system design data 2 Perform tests according to test program 3 Manage test uncertainty in diagnostic process 4 Determine health state using diagnostic models 5 Track and predict failures through time Deriving CTBNs from D-Matrices Perreault, et al. 2 / 18
Overview Approach • Probabilistic graphical models provide a method for performing diagnostics and prognostics in complex systems • CTBNs are probabilistic graphical models that allow the user to track the state of the system through time • Building the model directly requires a significant amount of data on the unit under test (UUT) • D-matrices and reliability information can be used to construct a CTBN Deriving CTBNs from D-Matrices Perreault, et al. 3 / 18
D-Matrices Definition • D-matrix is an adjacency matrix that explicitly represents relationships between tests and faults • Columns correspond to tests and rows correspond to potential failures observed by tests Features • Useful in diagnostic contexts • Logical relationships between tests and faults are captured, but probabilistic information is not • D-matrix usually does not provide information on test-to-test relationships Deriving CTBNs from D-Matrices Perreault, et al. 4 / 18
Continuous Time Markov Processes Definition • Continuous time Markov process (CTMP): describes a set of discrete state variables that evolve in continuous time • Two components: ◦ Initial Distribution: P ◦ Intensity Matrix: Q Features • Generally utilizes exponential distributions • Satisfies the Markov assumption (memoryless) • Exponential in the number of variables Deriving CTBNs from D-Matrices Perreault, et al. 5 / 18
Markov Processes P = ( p 1 , p 2 , · · · , p n ) x 1 x 2 · · · x n x 1 q 1 q 12 · · · q 1 n · · · x 2 q 21 q 2 q 2 n Q = . . . . ... . . . . . . . . x n q n 1 q n 2 · · · q n F ( t ) = 1 − e − q ij t Deriving CTBNs from D-Matrices Perreault, et al. 6 / 18
Continuous Time Bayesian Networks Definition • Continuous time Bayesian network (CTBN): factors CTMPs by taking advantage of conditional independencies in system • Graph structure G • Parameters (for each node) ◦ Initial Distributions: P X ◦ Conditional Intensity Matrices (CIMs): Q X | Pa( X ) Features • Disallows simultaneous transitions • Mitigates exponential blowup Deriving CTBNs from D-Matrices Perreault, et al. 7 / 18
CTBN Structure And D-Matrices F 1 F 2 � T 1 T 2 � F 1 1 0 D = T 1 T 2 F 2 1 1 A D-matrix D and associated network Deriving CTBNs from D-Matrices Perreault, et al. 8 / 18
CTBN Parameters F 1 F 2 T 1 T 2 t 1 t 2 t 1 t 2 1 1 1 1 t 1 − q 1 q 1 t 1 − q 1 q 1 Q { T 1 | f 1 2 } = Q { T 1 | f 1 2 } = 1 1 1 1 2 2 1 ,f 1 1 ,f 2 t 2 q 2 − q 2 t 2 q 2 − q 2 1 1 1 1 2 2 t 1 t 2 t 1 t 2 1 1 1 1 t 1 − q 1 q 1 t 1 − q 1 q 1 Q { T 1 | f 2 2 } = Q { T 1 | f 2 2 } = 1 3 3 1 4 4 1 ,f 1 1 ,f 2 t 2 q 2 − q 2 t 2 q 2 − q 2 1 3 3 1 4 4 Deriving CTBNs from D-Matrices Perreault, et al. 9 / 18
Parameterizing the CTBN Fault Nodes • By construction, fault nodes have no parents and hence a single unconditional intensity matrix • Failure rate λ indicates rate at which a fault will occur given that the given fault currently does not exist • Repair rate µ indicates rate at which a component will transition back to having no failure � f 0 f 1 i i � f 0 − λ i λ i i Q F i = f 1 µ i − µ i i Deriving CTBNs from D-Matrices Perreault, et al. 10 / 18
Parameterizing the CTBN Test Nodes • Goal is to define a transition distribution for each test node given the faults it monitors • CIMs for test nodes are parameterized in terms of non-detect and false alarm rates ◦ False alarm: an indication of a fault where no fault exists ◦ Non-detect: an indication of no fault where a fault exists • For each type of false indication, there can be two potential sources of failure ◦ Failure specific to each fault ◦ Failure due to a malfunction with the test Deriving CTBNs from D-Matrices Perreault, et al. 11 / 18
Parameterizing Test Nodes u ′ u i, 1 i, 1 U i, 1 . . . u ′ v ′ u i,j v i i,j i U i,j V i . . . u ′ u i,n i,n U i,n Line Failure Model Deriving CTBNs from D-Matrices Perreault, et al. 12 / 18
Parameterizing Test Nodes General Case • Formula to account for the V i failure function P ( v ′ i = 0 | u i ) = ( ND i ) P ( v i = 1 | u i ) + (1 − FA i ) P ( v i = 0 | u i ) • Remaining terms can be calculated using � P ( u ′ P ( v i = 0 | u i ) = i,j | u i,j ) { u i,j ∈ u i } • CIM for test node with defined in terms of state likelihoods t 0 t 1 i i � � t 0 − P ( T i = 0 | F i ) − 1 P ( T i = 0 | F i ) − 1 i Q T i | F i = t 1 P ( T i = 1 | F i ) − 1 − P ( T i = 1 | F i ) − 1 i Deriving CTBNs from D-Matrices Perreault, et al. 13 / 18
Parameterizing Test Nodes Special Case • All required values may not be available for the system • Alternative is to assume no U i,j line failure • CIMs are as follows t 0 t 1 i i � � t 0 − (1 − FA i ) − 1 (1 − FA i ) − 1 i Q { T i | ( ∧ F ∈ F i F =0) } = t 1 ( FA i ) − 1 − ( FA i ) − 1 i t 0 t 1 i i � � t 0 − ( ND i ) − 1 ( ND i ) − 1 i Q { T i | ( ∨ F ∈ F i F =1) } = t 1 (1 − ND i ) − 1 − (1 − ND i ) − 1 i Deriving CTBNs from D-Matrices Perreault, et al. 14 / 18
Experiments Probability of T 1 through time in the constructed and parameterized synthetic network. Deriving CTBNs from D-Matrices Perreault, et al. 15 / 18
Experiments Probability of F 2 through time in the synthetic network where evidence is applied for T 3 and T 4 at different points in time Deriving CTBNs from D-Matrices Perreault, et al. 16 / 18
Conclusion Contributions • Structure for bipartite CTBN consisting of fault and test nodes can be constructed directly from D-matrix • Fault nodes in network can be parameterized using information about mean time between failures and mean time to repair • Test nodes can be parameterized using the non-detect and false alarm rates associated with each fault-test relationship and for the test alone • Correctness and effectiveness of our approach demonstrated with three experiments Deriving CTBNs from D-Matrices Perreault, et al. 17 / 18
Future Work Future Work • Inter-test relationships ◦ Logical closure ◦ Transitive reduction • Generating CTBNs from fault-trees ◦ Produces network with fault and hazard nodes ◦ Fault nodes are the same as D-matrix network ◦ Define and parameterize nodes representing logic gates Deriving CTBNs from D-Matrices Perreault, et al. 18 / 18
Recommend
More recommend
