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SLIDE 1

Deriving Prognostic Continuous Time Bayesian Networks from D-matrices

Logan Perreault, Monica Thornton, Shane Strasser, and John W. Sheppard

Montana State University

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

Markov Processes

P = (p1, p2, · · · , pn) Q =        x1 x2 · · · xn x1 q1 q12 · · · q1n x2 q21 q2 · · · q2n . . . . . . . . . ... . . . xn qn1 qn2 · · · qn        F(t) = 1 − e−qijt

Deriving CTBNs from D-Matrices Perreault, et al. 6 / 18

SLIDE 7

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: PX
Conditional Intensity Matrices (CIMs): QX|Pa(X)

Features

Disallows simultaneous transitions
Mitigates exponential blowup

Deriving CTBNs from D-Matrices Perreault, et al. 7 / 18

SLIDE 8

CTBN Structure And D-Matrices

D = T1 T2 F1 1 F2 1 1

F1

F2 T1 T2 A D-matrix D and associated network

Deriving CTBNs from D-Matrices Perreault, et al. 8 / 18

SLIDE 9

CTBN Parameters

F1 F2 T1 T2

Q{T1|f 1

1 ,f 1 2 } =

  t1

1

t2

1

t1

1

−q1

1

q1

1

t2

1

q2

1

−q2

1

  Q{T1|f 2

1 ,f 1 2 } =

  t1

1

t2

1

t1

1

−q1

3

q1

3

t2

1

q2

3

−q2

3

  Q{T1|f 1

1 ,f 2 2 } =

  t1

1

t2

1

t1

1

−q1

2

q1

2

t2

1

q2

2

−q2

2

  Q{T1|f 2

1 ,f 2 2 } =

  t1

1

t2

1

t1

1

−q1

4

q1

4

t2

1

q2

4

−q2

4

 

Deriving CTBNs from D-Matrices Perreault, et al. 9 / 18

SLIDE 10

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 QFi = f0

i

f1

i

f0

i

−λi λi f1

i

µi −µi

Deriving CTBNs from D-Matrices

Perreault, et al. 10 / 18

SLIDE 11

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

SLIDE 12

Parameterizing Test Nodes

Ui,1 . . . Ui,j . . . Ui,n ui,1 ui,j ui,n u′

i,1

u′

i,j

u′

i,n

Vi vi v′

i

Line Failure Model

Deriving CTBNs from D-Matrices Perreault, et al. 12 / 18

SLIDE 13

Parameterizing Test Nodes

General Case

Formula to account for the Vi failure function

P(v′

i = 0|ui) = (NDi)P(vi = 1|ui) + (1 − FAi)P(vi = 0|ui)

Remaining terms can be calculated using

P(vi = 0|ui) =

{ui,j∈ui}

P(u′

i,j|ui,j)

CIM for test node with defined in terms of state likelihoods

QTi|Fi =

t0

i

t1

i

t0

i

−P(Ti = 0|Fi)−1 P(Ti = 0|Fi)−1 t1

i

P(Ti = 1|Fi)−1 −P(Ti = 1|Fi)−1

Deriving CTBNs from D-Matrices

Perreault, et al. 13 / 18

SLIDE 14

Parameterizing Test Nodes

Special Case

All required values may not be available for the system
Alternative is to assume no Ui,j line failure
CIMs are as follows

Q{Ti|(∧F ∈FiF=0)} =

t0

i

t1

i

t0

i

−(1 − FAi)−1 (1 − FAi)−1 t1

i

(FAi)−1 −(FAi)−1

Q{Ti|(∨F ∈FiF=1)} =
t0

i

t1

i

t0

i

−(NDi)−1 (NDi)−1 t1

i

(1 − NDi)−1 −(1 − NDi)−1

Deriving CTBNs from D-Matrices

Perreault, et al. 14 / 18

SLIDE 15

Experiments

Probability of T1 through time in the constructed and parameterized synthetic network.

Deriving CTBNs from D-Matrices Perreault, et al. 15 / 18

SLIDE 16

Experiments

Probability of F2 through time in the synthetic network where evidence is applied for T3 and T4 at different points in time

Deriving CTBNs from D-Matrices Perreault, et al. 16 / 18

SLIDE 17

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

SLIDE 18

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