Bayesian Diagnostic Method

8/8/2019 Bayesian Diagnostic Method

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Title:

Bayesian Diagnostic Method

Abstract:

A diagnostic method that uses fault isolation logic and conditional probability todynamically optimize the diagnostic steps required to isolate a cause. Fault isolationlogic processes one or more diagnostic observations to yield a minimized list of possiblecauses and diagnostic steps. Statistical feedback provides the conditional probabilitiesused to calculate the likelihood of each possible cause and subsequently the likelihoodthat a diagnostic step will isolate a possible cause. The calculated likelihood is used toprioritize the possible causes and diagnostic steps. A diagnostic observation is anysymptom, test result, consequence or effect that is observable because of a cause. A cause

is a fault, malfunction, breakdown, or glitch that produces an observable effect. Adiagnostic step is a procedure for detecting or recording one or more diagnosticobservations.

Ford Motor Company Confidential1



INTRODUCTION

Diagnosing automotive problems is not a trivial task. Several electronic control

modules are used to manage and coordinate the operation of systems such as the engine,

transmission, antilock brakes, traction control, anti-theft, climate control, power

doors/windows and others. These systems are constantly interacting and monitoring a

variety of sensors and switches to control solenoids, relays, spark timing, fuel injectors,

fans, pumps, clutches, lights, and more. Several diagnostic methods have been patented

and applied to this problem. This proposed method uses statistical inference to decide the

best diagnostic course of action.

THE DIAGNOSTIC METHOD – IN THEORY

The goal of any diagnostic method is to isolate the cause of an observed effect. In

this section a general-purpose fault isolation methodology is derived using Baye's rule

and conditional probabilities.

Real Causes

Definition 1. Real Cause; an actual possible fault, failure or malfunction of asystem, sub-system or component.

Let Φ=Φ nφ φ φ ,...,, 21 represent the set of Φn real possible causes within a,

system, sub-system or component being diagnosed. When defining a set of causes, it

may be impossible to predict all of the real possible causes. The following concept is

introduced to deal with this reality.




Defined Causes

Definition 2. Defined Cause ; an anticipated fault, failure or malfunction of acomponent, system or sub-system.

Let C ncccC ,...,, 21= represent the set of C n defined causes. Let )( jcΦ

represent the set of real causes that comprise jc . Note that it is possible to define a set

of anticipated causes that does not cover all of the real possible causes or

)(...)()( 21 C nccc Φ∪∪Φ∪Φ⊃Φ .

Example 1. A spark plug ignition circuit consisting of a wire and a spark plug has five real possible causes;

{ },,,,, 54321 φ φ φ φ φ =Φwhere φ 1 represents the wire being shorted to ground,

φ 2 represents the wire being open circuit,

3 represents the wire being shorted to a neighboring ignition wire,φ 4 represents the spark plug’s gap being too large,

and φ 5 represents the spark plug’s gap being too narrow.Someone who is not aware of 3 might define the following anticipated faults;

{ }21 , ccC = ,

where { }211 ,)( φ φ =Φ c represents a wire problem

and { }542 ,)( φ φ =Φ c represents a spark plug problem.

Diagnostic Step

Definition 3. Diagnostic Step; a defined procedure, test, operation or question.

Let { }S s s s n S = 1 2, , ... , represent the set of S n defined diagnostic steps. The role of

the diagnostician is to select a course of action from this set. Each step is defined to

observe any number of possible effects.

Defined Effects

Definition 4. Defined Effect ; an anticipated symptom or test result.




LetS n

k k E E

1=

= represent the set of defined effects wherek E knk k k eee E ,...,, 21= is

the set of k E n effects defined for diagnostic step k s and )( kl eΦ is the subset of real

causes observed by k e .

Cause/Effect Relationship

The cause/effect relationship between defined effects and defined causes is bi-

directional. It can be described either from the perspective of a cause or from the

perspective of an effect. Let )( jc E represent the subset of defined effects of jc .

Conversely, let )( kl eC represent the subset of defined causes of kl e .

Example 2. An engine with three anticipated causes {ignition, fuel, and charging} exhibitstwo possible symptom effects {“no crank” and “no start”}. There are three diagnosticsteps available for this system. An ignition test step has one effect that observes theignition subsystem cause. A fuel test step has one effect that observes the fuel subsystemcause. A comprehensive self-test step has one effect for each subsystem cause. Thethree steps also have a pass effect if no causes are observed. The following table lists thedefined diagnostic steps for this example.

Anticipated Causes C {Ignition, Fuel, Charging}Ignition 1c =)( 1c E {Charging}Fuel 2c =)( 2c E {Charging}Charging 3c =)( 3c E {Charging}Effects

Step 1: Symptom entry 1 E {No crank, No start} No crank 11e =)( 11eC {Charging} No start 12e =)( 12eC {Ignition, Fuel, Charging} Step 2: Ignition test 2 E {Pass, Fail}Pass 21e =)( 21eC ∅

Fail 22e =)( 22eC {Ignition} Step 3: Fuel test 3 E {Pass, Fail}Pass 31e =)( 31eC ∅

Fail 32e =)( 32eC {Fuel} Step 4: Self-test 4 E {Code 1, Code 2, Code 3, Code 4}Code 1 41e =)( 41eC ∅




Code 2 42e =)( 42eC {Ignition}Code 3 43e =)( 43eC {Fuel}Code 4 44e =)( 44eC {Charging}

For ideal sets of defined causes and effects

)(),()( kl j jkl eC cce ∈∀Φ=Φ (1)

and )(),()( jkl jkl c E ece ∈∀Φ=Φ . (2)

Poorly defined causes and effects will give rise to either a misdiagnosis or a false

alarm. Table I illustrates these situations.

Table I . Misdiagnosis & False Alarms

For an effect k e defined to observe jcand a given real cause .

Defined effectobserves real cause

Defined effect doesnot observe real

cause)( k eΦ∈φ )( k eΦ∉φ

Defined cause includes realcause

)( jcΦ∈φ Ideal Misdiagnosis

Defined cause does notinclude real cause

)( jcΦ∉φ False alarm Ideal

Example 3. Given the causes from example 1, define the following diagnostic effects;{ }2111 , ee E =

where 11e is a wire test result that observes { }111 )( ceC = and 21e is a spark plug test

result that observes { }221 )( ceC = . If in reality 11e is poorly defined such that 11e actually detects

{ }3111 ,)( φ φ =Φ e ,then observing this effect will give rise to misdiagnosis and false alarms. For example, if φ 2 is present 11e cannot observe it, thus failing to diagnose 1c as expected. Also, if

3 is present 11e will observe it, thus falsely implicating 1c which is not defined toinclude this cause.

Observations

Definition 5. Observation; the manifestation of a defined effect due to a set of real possible causes.




Let ),( jkl ceO represent the observation of the defined effect k e caused by jc .

The real causes in jc that manifested in kl e is determined by the intersection

)()(),( jkl jkl ceceO Φ∩Φ= . (3)

Calculating Cause Probabilities

The extent to which a defined cause jc has been detected is a culmination of all the

observations caused by jc . If the defined effects were ideal, the set )( jcΦ could be

described as the union of all the observations caused by jc denoted

)(),()(

jkl c E e jkl j ceOc

∈∀=Φ , (4)

or ( ))(

)()()( jkl c E e

kl j j ecc∈∀

Φ∩Φ=Φ. (5)

Assumption 1. Defined causes and effects are not ideal or)}()({)( kl j j ecc Φ∩Φ⊃Φ .

Since the defined causes and effects may not be ideal,

( ))(

)()()( jkl c E e

kl j j ecc∈∀

Φ∩Φ⊃Φ

. (6)

Figure 1 shows two non-ideal effects defined to detect ic .




Φ

Φ (c1)

Φ (e21)

Φ

Φ Φ

Φ (c1) ⊃ (Φ (c1)∩ Φ (e21))

Φ (c1) ⊃ (Φ (c1) ∩ Φ (e21)) Φ (c1) ⊃ (Φ (c1)∩ Φ (e11)) ∪ (Φ (c1)∩ Φ (e21))

(a) (b)

(c) (d)

Φ (e11)

Φ (e11)

Φ (e11)

Φ (e21) Φ (e21)

Φ (c1) Φ (c1)

Φ (c1)

Figure 1. Targeted cause 1c (a) Detected by effect 11e (b) detected by effect 21e (c)detected by both effects.

The customary functional notation )( a P will be used to represent the probability.

For subsets where ba ⊃ , ( ) ( )b P a P ≥ . Therefore,

( ) ( )

Φ∩Φ≥Φ

∈∀ )(

)()()( jkl c E e

kl j j ec P c P . (7)

Computing ( ))( jc P Φ using this function is not practical because probability theory for a

union of sets states that

( ) ( ) ( ) ( ) ( n

n

k jik ji

n

ji ji

n

ii

n

ii A A A P A A A P A A P A P A P ∩∩∩∩−+⋅⋅⋅+∩∩+∩−=

−=<<=<== ∑∑∑ ...1 3211

3211

(8)

This requires knowledge of the intersections between all combinations of observations.

General purpose Bayesian Network Software fails because of this.




Assumption 2. Effects by definition approach complete coverage of theanticipated causes they are intended to detect. In other words

)(,)()()( jkl kl j j c E eecc ∈∀Φ∩Φ≈Φ .

Therefore, the best conservative approximation of )( ic P is associated with the

observation that best includes all of the real causes in jc or

( ) ( )( ))()()( max)(

kl jc E e

j ec P c P jkl

Φ∩Φ≈Φ∈∀

. (9)

Baye's rule for conditional probability states that

P A B P A P B A P B P A B( ) ( ) ( | ) ( ) ( | )∩ = = . (10)

By applying this rule, )( jc P becomes a function of the conditional probabilities of

observing the effects of jc

( ( ) (( )(|)()()( max)(

kl jkl c E e

j ec P e P c P jkl

ΦΦΦ≈Φ∈∀

. (11)

By definition, )( j j cc Φ≡ and )( kl kl ee Φ≡ . Therefore,

( ( ) ((kl jkl c E e j

ec P e P c P jkl

|m ax)(∈∀

≈. (12)

Calculating Effect Probabilities

The extent to which the defined effect kl e is observed is a culmination of all of its

causes. If the defined effect is ideal, the set )( kl eΦ could be described as the union of

all the observations of its causes denoted

)(

),()(kl j eC c

jkl kl ceOe∈∀

=Φ , (13)

or ( ))(

)()()(kl j eC c

kl jkl ece∈∀

Φ∩Φ=Φ. (14)

The probability ( ))( kl e P Φ of observing )( kl eΦ follows




( ) ( )

Φ∩Φ=Φ

∈∀ )(

)()()(kl j eC c

kl jkl ec P e P . (15)

As before, this computation is not practical since it requires knowledge of the

intersections between all the defined effects. De Morgan’s law may be applied as shown

in Figure 2 such that

( ) ( )

Φ∩Φ=Φ

∈∀ )(

)()()(kl j eC c


Φ

c2

e11

Φ Φ Φ (e11) = ( Φ (c1)∩ Φ (e11)) ∩ (Φ (c2)∩Φ (e11))

(a)

(b) (c)

c1

c2

e11 c1

Φ (e11) = ( Φ (c1)∩ Φ (e11)) ∪ (Φ (c2)∩Φ (e11))

c1 e11

c2

Figure 2 . Effect 11e (a) targeting causes 1c and 2c (b) represented as a union of observedcauses (c) represented using De Morgan's law.

A reasonable assumption can be made to simplify the probability of these intersections.

Assumption 3. Observations are independent. In other words

( ) ( )( ) ( )( ) ( )( ) jil k ec P ec P ecec P kl ikl ikl ikl i ,,,,)()()()()()()()( ∀Φ∩ΦΦ∩Φ≈Φ∩Φ∩Φ∩Φ

where ji ≠ .

Therefore, ( ))( kl e P Φ is estimated from




( ) ( )( )∏∈∀

Φ∩Φ−−≈Φ)(

)()(11)(kl j eC c


When expressed in terms of conditional probabilities

( ) ( ) ( )( )∏∈∀ΦΦΦ−−≈Φ

)()(|)()(11)(

kl j eC c jkl jkl ce P c P e P . (18)

Again, by definition, )( j j cc Φ≡ and )( kl kl ee Φ≡ . Therefore,

( )∏∈∀

−−≈)(

)|()(11)(kl j eC c

jkl jkl ce P c P e P . (19)

Calculating the Utility of a Diagnostic Step

The goal of a diagnostic procedure is to isolate the cause of an observed problem

using the least amount of effort. Minimizing this effort requires a measure of the

usefulness or utility of each diagnostic step, which in turn is a function of the utility of

each effect that is observable by that step.

Definition 6. Utility of an Effect; the likelihood that the effect will increase the probability of its causes.

The utility can also be described as a product of observing an effect and the

incremental increase the effect will have on the probability of its causes. The probability

of observing an effect ( )kl e P is determined using equation (19). The incremental

increase in the probability of its causes, denoted jec P

mn

∆ , is the difference in the

outcome of equation (12) if kl e is observed or

( ) ( )( )( ) ( )( )( )mn jmnc E e

mn jmne P c E e

jeec P e P ec P e P c P

jmnkl jmnkl

|| m axm ax)(1)(),( ∈∀=∈∀

−≈∆ .(20)

Therefore, the utility of an effect denoted ( )kl eU is calculated from the likelihood of

incremental increases in all of its causes or




( ) ( ) ( )∏∈∀

∆⋅−−=

)(

11kl j

kl eC c j

ekl kl c P e P eU . (21)

It follows that the utility of a diagnostic step ( )k sU is calculated from the utilities of

its observable effects or

( )∏=

−−≈k sn

l kl k eU sU

1

)(11)( . (22)

THE DIAGNOSTIC METHOD – IN PRACTICE

Step 1: Initialization of Conditional Probabilities

The conditional probabilities used by this method are calculated from a tally of the

isolated causes and the observed effects captured in a data warehouse filled with

conclusive diagnostic sessions. This data warehouse houses the collective diagnostic

experience of all the users of this system.

Definition 7. Conclusive Diagnostic Session; The isolated anticipated fault andthe accompanying list of symptoms and test results that helped isolate it.

Definition 8. Tally; A counter used to track the number of occurrences of acause, an effect or a cause|effect pair.

Tally the Causes, Effects, and Cause|Effect Pairs

Let id represent a conclusive diagnostic session wherensObservatiod i . Contains the observed effectsuses IsolatedCad i . contains the isolated causes

Loop over each conclusive diagnostic session nsObservatiod i .

Loop over each isolated cause jc in uses IsolatedCad i .

Increment the tally of jc

Increment the tally of all C End LoopLoop over each isolated cause jc in uses IsolatedCad i .

Loop over each effect kl e of jc

If kl e was observed in nsObservatiod i . Or the step k s housing kl e was not performed in id




; the effect must be assumed to be ideal if there was; no attempt made to observe it

Increment the tally of kl j ec

Increment the tally of kl eEnd If

End LoopEnd Loop

End Loop

This step only needs to be done once. Incremental updates to the initial values can be

performed on a regular basis to absorb new information as more diagnostic sessions are

completed and added to the warehouse.

Step 2: Compute Initial Cause|Effect Conditional Probabilities

The tallies compiled in step 1 are used to compute initial cause|effect conditional

probabilities.

Compute conditional Cause|Effect probabilities

Loop over each cause|effect pair kl j ec

Compute )(/)|()|( kl kl jkl j eTallyecTallyec P =

Compute )(/)|()|( jkl j jkl cTallyecTallyce P =

End Loop

This step only needs to be done once or as needed when tallies are updated in step 1.

Step 3: Compute Initial Cause Probabilities

The tallies compiled in step 1 are also used to compute initial cause probabilities.

Compute initial Cause probabilities

Loop over each cause jcCompute )(/)()( C TallycTallyc P j j =

End Loop

This step also needs to be done only once or as needed when tallies are updated. At

this point the system is prepared to handles diagnostic sessions.




Step 4: Start a Diagnostic Session

To start a diagnostic session the user must provide one observed effect selected from

any diagnostic step for the system being diagnosed. If more than one effect is submitted,

they must be processed one at a time to manage the possibility of multiple faults.

Remember, an observed effect could be a symptom or test result. The observed effect is

added to the newly created diagnostic session for processing.

Add Observed Effect to the Diagnostic Session

Start a new diagnostic session id where

id is the diagnostic sessionnsObservatiod i . contains the observed effects

Effectsd i . contains the working set of effectsCausesd i . contains the working set of causesComponentsd i . contains the working set of componentsStepsd i . contains the working set of Steps

Add the initial observed effect kl e to nsObservatiod i .

Step 5: Generate a Working Set of Causes and Effects

To optimize the performance of this diagnostic methodology, a working subset of

causes and effects are extracted from given initial observation. Only the direct causes of

the initial observed effect are significant as well as all of the effects of these direct

causes. This list of direct causes and the list of their effects form the working sets of

causes and effects processed by this algorithm.

Generate a Working Set of Causes and Effects

Loop over each cause jc of initial effect kl e in nsObservatiod i .

Add cause jc to Causesd i . Add the component housing jc to Componentsd i .

Loop over each effect kl e of cause jc

Add effect kl e to Effectsd i .

Add the diagnostic step housing kl e to Stepsd i .




End LoopEnd Loop

Step 6: Update Cause Probabilities given Observed Effects

With each observed effect in a diagnostic session update cause probabilities using the

computed conditional probabilities. Note that for observed effects 1)( =kl e P .

Update Cause Probabilities

Loop over each observed effect kl e in nsObservatiod i .

Loop over each cause jc of kl e

Compute )]|(),(max[)( kl j j j ec P c P c P =

End LoopEnd Loop

This step needs to be done each time a new observation is made.

Step 7: Compute Effect Probabilities

For observed effects, 1)( =kl e P . The probability of observing other effects is

computed from the cause probabilities and initial conditional probabilities computed in

step 2.

Compute Effect Probabilities

Loop over each effect kl e in Effectsd i .

Set 1)( =kl e P

If kl e is not in nsObservatiod i .

Loop over each cause jc of kl e

Compute ))|()(1()()( jkl jkl kl ce P c P e P e P −⋅=End LoopSet )(1)( kl kl e P e P −=

End If End Loop





Step 8: Compute Utility of Diagnostic Steps

The utility of a diagnostic step )( k sU is calculated from the effect probabilities and

the conditional probabilities calculated in the preceding steps.

Compute Diagnostic Step Utilities

Loop over each diagnostic step k s in Stepsd i .

Set 1)( =k sU

Loop over each effect kl e in k s

Set 1)( =kl eU

Loop over each cause ic of effect kl e

Compute )))()|(()(1()()( ikl ikl kl kl c P ec P e P eU eU −⋅−⋅=

End LoopSet )(1)( kl kl eU eU −=

Compute ))(1()()( kl k k eU sU sU −⋅=

End LoopSet )(1)( k k sU sU −=

End Loop


Step 9: Determine if further Observations are Useful

Further observations will be useful only if their utility is greater than zero. Possible

causes will not be isolated any further otherwise.

Determine if Further Observations are Useful

Loop over each diagnostic step k s in Stepsd i .

Loop over each effect kl e in nsObservatiod i .

If 0)( >k sU go to Step 10 to submit further observations

End LoopEnd LoopEnd Diagnostic Session




Step 10: Submit Observation

Diagnostic steps housing the possible effects are presented to the user. When one is

observed, it is added to the current diagnostic session for processing.

Add Observed Effect to the Diagnostic Session

Add observed effect kl e to nsObservatiod i .Go to Step 6 to process further observations

Summary

Figure 3 summarizes the steps involved in the diagnostic process.

Obser vati on Processi ng Obser vati onsI ni t i al i z at i on

Step 1Tal l y the Causes,

Eff ects, and Cause| Eff ect Pai rs

Step 7Comput e Ef f ect Probabi l i ti es

Step 8Comput e Di agnost i c Step

Ut i l i t i es

Step 2Comput e I ni t i al Cause

Probabi l i ti es

Step 4St art Di agnosti c Sessi onwi th I ni ti al Observati on

Step 9 Are furt hur observati ons

useful ?

es

End Di agnost i c Sessi on

I ni t i al Observat i on

o

Step 10Submi t Observati on

Step 5Generate Worki ng Set of

Causes and Ef f ects

End Observat i on

St ar t I n i t i al i z at i on

Step 3Compute I ni t i al

Cause| Eff ect Condi ti onal Probabi l i ti es

Step 6Update Cause

Probabi l i ti es

Figure 3. Summary of the diagnostic process.




DATA MODEL (DATA DEVELOPMENT PROCESS)

Step 1: Establish Content Model

The backbone of all diagnostic and service information mirrors the content of the

vehicle. It is typically organized according to vehicle systems, sub-systems, assemblies,

sub-assemblies and components.

Content Model

Engine

Brakes

Body

Front Disk

Anti-Lock Control

Rear Drum

ABS Module

Front Sensor

Rear Sensor

Instrument Cluster

ABS Warning

Bulb




Step 2: Failure mode and effects analysis (FMEA)

The failure mode and effects analysis process establishes the fault population for the

vehicle contents and it establishes the corresponding observable effects or symptoms.Their relationships are established through cause/effect and parent/child links.

Content Model

Engine

Brakes

Body

Front Disk

Anti-Lock Control

Rear Drum

ABS Module

Front Sensor

Rear Sensor

Instrument Cluster

ABS Warning

Bulb

No Power

Bad Pin Front Sensor

Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Open Circuit

Short Circuit

Anti-Lock not functional

Symptoms

ABS Warning Lamp Never On

ABS Warning Lamp Always On

Symptoms

Missing Signal

Missing Signal

Cause/Effect Child/ParentParent/Child

FMEA - faults FMEA - effects FMEA – symptom groups




Step 3: Diagnostic Authoring

Diagnostic tests are then added to isolate the faults.

Content Model

Anti-Lock Control

ABS Module

Front Sensor

Rear Sensor

ABS Warning

Bulb

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Open Circuit

Short Circuit


Symptoms



Symptoms

Missing Signal

Missing Signal

Cause/Effect Child/ParentParent/Child

FMEA Faults FMEA Effects FMEA Symptom Groups

Pass

CODE1 = No Power

CODE2 = Front Sensor

CODE3 = Rear Sensor

CODE0 = Pass

Found Open Circuit

Found Short Circuit

ABS Self-Test

Check Front Sensor

Check Rear Sensor

No Signal

Pass

Found Open Circuit

Found Short Circuit

No Signal

Check Bulb

Burned Out

Diagnostic Test Results Diagnostic Tests




APPLYING THE BAYSIAN ALGORITHM

Start Diagnostic Session with Initial Observation

Any initial symptom or test result could establish a diagnostic session. The cause/effect

and parent/child relationships are used to generate a working set of components, faults,

effects and tests containing only the information necessary to diagnose the problem.

Anti-Lock Control

ABS Module

Front Sensor

Rear Sensor

ABS Warning

Bulb

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Open Circuit

Short Circuit


Symptoms



Symptoms

Missing Signal

Missing Signal

Pass

CODE1 = No Power


CODE3 = Rear Sensor

CODE0 = Pass

Found Open Circuit

Found Short Circuit

ABS Self-Test

Check Front Sensor

Check Rear Sensor

No Signal

Pass

Found Open Circuit

Found Short Circuit

No Signal

Check Bulb

Burned Out

1: First Observation

2: Possible Causes

3: Suspect Components

4: Applicable Test Results 5: Applicable Tests

Diagnostic Session 1: ABS Warning Lamp Always On




Compute Fault Probabilities

Counters on all of the nodes and links are used to calculate conditional probabilities.

The conditional probability of a cause given an effect P(C|E) = (link count)/(effectcount).

The probability of a cause is the maximum conditional probability of the observed

effects.

( ( ) (( kl jkl c E e

j ec P e P c P jkl

|max)(∈∀

≈

ABS Module

Front Sensor

Rear Sensor

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Short Circuit

Symptoms


Symptoms

Pass

CODE3 = Rear Sensor

ABS Self-Test

Check Front Sensor

Check Rear Sensor

Pass

Found Open Circuit

Diagnostic Session E1: Symptom - > ABS Warning Lamp Always On

1

1

1

3

1

1

1

10

1

3

1

1

5

1

1

1

1

1

9

1

1

1

1

3

1

1

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 3/10 = 30%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 =10% P(C|E2) = 1/5 =20%

P(C|E2) = 1/5 = 20%

P(C) = MAX = 20%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 100%

P(C) = MAX = 20%

3

1

1

3

E2: ABS Self-Test -> CODE3 = Rear Sensor

E3: Check Rear Sensor -> Found Open Circuit


CODE1 = No Power


CODE0 = Pass

Found Short Circuit

Found Open Circuit

Found Short Circuit

P(C|E2) = 3/5 = 60% P(C|E3) = 3/3 = 100%




Compute Effect Probabilities

The conditional probability of an effect given a cause P(E|C) = (link count)/(cause

count).The probability of an effect is a function of the probabilities of the causes and the

conditional probabilities of the effect given the causes.

( )∏∈∀

−−≈)(

)|()(11)(kl j eC c

jkl jkl ce P c P e P

ABS Module

Front Sensor

Rear Sensor

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Short Circuit

Symptoms


Symptoms

Pass

CODE3 = Rear Sensor

ABS Self-Test

Check Front Sensor

Check Rear Sensor

Pass

Found Open Circuit


1

1

3

1

1

1

10

1

3

1

1

5

1

1

1

1

1

9

1

1

1

1

3

1

1

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 3/10 = 30%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 =10%

P(E|C) = 3/3 =100%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 30%

P(C) = MAX =10%

3

1

1


CODE1 = No Power


CODE0 = Pass

Found Short Circuit

Found Open Circuit

Found Short Circuit

P(E|C) =1/1 = 100%

P(E|C) =1/1 = 100%

1

P(E) =1(1-10%x 100%)(1-30%x100%)(1-10%x100%) = 43%




Compute Effect Utilities

The utility of an effect denoted ( )kl eU is calculated from the likelihood of

incremental increases in all of its causes or

( ) ( ) ( )∏∈∀

∆⋅−−=

)(

11kl j

kl eC c jekl kl c P e P eU .

where the incremental increase in the probability of an effect's causes, denoted

jec P

mn

∆ , is

( ) ( )( )( ) ( )( )( )m n jmnc E e

mn jmne P c E e

je ec P e P ec P e P c P

jmnkl jmnkl || m axm ax )(1)(),( ∈∀=∈∀

−≈∆ .




ABS Module

Front Sensor

Rear Sensor

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Short Circuit

Symptoms


Symptoms

Pass

CODE3 = Rear Sensor

ABS Self-Test

Check Front Sensor

Check Rear Sensor

Pass

Found Open Circuit


1

1

3

1

1

1

10

1

3

1

1

5

1

1

1

1

1

9

1

1

1

1

3

1

1

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 3/10 = 30%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 = 10%

P(C|E1) = 1/10 =10%∆ P(C) = 20% -10% = 10%

P(C) = MAX = 10%

∆ P(C) =20% -10% = 10%

∆ P(C) =60% -30% = 30%

3

1

1


CODE1 = No Power


CODE0 = Pass

Found Short Circuit

Found Open Circuit

Found Short Circuit

P(E) = 43%

1

U(E) =1 - (1-10%x 43%)(1-30%x43%)(1-10%x43%) = 20%

P(C|E2) = 1/5 =20%

P(C|E2) =3/5 =60%

P(C|E2) = 1/5 =20%




Learn from Completed Diagnostic Session - Update Counters

Increment the counter of the isolated fault, the observed effects and all the links between

the observed effects and the isolated fault.

ABS Module

Front Sensor

Rear Sensor

No Power


Bad Pin Back Sensor

Open Circuit

Short Circuit

Open Circuit

Short Circuit

Symptoms


Symptoms

Pass

CODE3 = Rear Sensor

ABS Self-Test

Check Front Sensor

Check Rear Sensor

Pass

Found Open Circuit


1

1

1

3

1

1

1

10

1

3

1

1

5

1

1

1

1

1

9

1

1

1

1

3

1

1

P(C) = MAX = 20%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 10%

P(C) = MAX = 100%

P(C) = MAX = 20%

3

1

1

3

E2: ABS Self-Test -> CODE3 = Rear Sensor

E3: Check Rear Sensor -> Found Open Circuit


CODE1 = No Power


CODE0 = Pass

Found Short Circuit

Found Open Circuit

Found Short Circuit

Isolated Fault = Rear Sensor -> Open Circuit

+1

+1

+1

+1

+1

+1

+1

NEXT STEPS

This method uses statistical inference to decide the best diagnostic course of action.

Another parameter or variable that should be considered when deciding the best

diagnostic course of action is the cost of performing the steps. Each step has an

associated cost, which is typically measured in time. Additionally each step may require




certain conditions or states to be met which also impact the cost. These issues must be

investigated further if one wishes to optimize diagnostic to minimize cost.

REFERENCES

[1] Stonehocker V. T., Computer-based engine diagnostic method , U.S. Patent 5,010,487Mar 23, 1991.

[2] Marko K. A., et al, Diagnostic system using pattern recognition for electronicautomotive control systems, U.S. Patent 5,041,976 Aug 20, 1991.

[3] Forchert T., et al, Method for detecting malfunctions in a motor vehicle, U.S. Patent5,396,422 Mar 7, 1995.

[4] Jensen F. V., An Introduction to Bayesian Networks , Springier-Verlag, 1996.

[5] Netica Application, http://www.norsys.com/ , Norsys Software Corporation, 1999.

http://www.norsys.com/

http://www.norsys.com/

Documents

Bayesian Diagnostic Method