data collection and analysis
DESCRIPTION
Data collection and analysis. Jørn Vatn NTNU. Objectives data collection and analysis. Collection and analysis of safety and reliability data is an important element of safety management and continuous improvement - PowerPoint PPT PresentationTRANSCRIPT
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Objectives data collection and analysis
Collection and analysis of safety and reliability data is an important element of safety management and continuous improvement
There are several aspects of utilizing experience data and we will in the following focus on
1. Learning from experience
2. Identification of common problems “Top ten”-lists (visualized by Pareto diagrams)
3. A basis for estimation of reliability parameters MTTF, MDT, aging parameters
3
Collection of data
We differentiate between Accident and incident reporting systems
These data is event-based, i.e. we report into the system only when critical events occur
Examples of such system is Synergy, and Tripod Delta Databases with the aim of estimating reliability parameters
These databases contains system description, failure events, and maintenance activities
The Offshore Reliability Data (OREDA) is one such database Such databases will be denoted RAMS databases in the following
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RAMS data: Boundary description
A clear boundary description is imperative for collecting, merging and analyzing RAMS data from different industries, plants or sources The merging and analysis
will otherwise be based on incompatible data.
For each equipment class a boundary must be defined. The boundary defines what RAMS data are to be collected
INLET GAS CONDITIONING
STARTINGSYSTEM
DRIVERPOWER
TRANSMISSION
COMPRESSOR UNIT
1 st 2nd STAGE STAGE
INTERSTAGECONDITIONING
Recyclevalve
AFTERCOOLER
LUBRICATIONSYSTEM
CONTROL AND MONITORING
SHAFT SEALSYSTEM
MISCEL-LANEOUS
Inlet valve
Outlet valve
Coolant Power Remoteinstr.
Power Coolant
Boundary
5
RAMS Data:Equipment hierarchy
The highest level is the equipment unit class
The number of levels for subdivision will depend on the complexity of the equipment unit and the use of the data
Turnout n
Turnout 3
Turnout i
Turnout 2
Turnout 1
S w itchm echan ism
E lectrica l m otor
Hardw areclassification
Boundaryclassification
Equ
ipm
ent
cla
ssE
quip
men
tu
nit
Sub
un
itM
ain
tain
able
item
Bou
nd
ary
leve
lS
ub-b
oun
da
ryle
vel
Mai
nta
inab
leite
m le
vel
-
(Turnout containsseveral subunits)
(Switch m echanism containsseveral Maintainable item s)
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RAMS database structure
Inventory ..
Inventory 2
Inventory 1
Failure ..
Failure 2
Failure 1
M aintenance 3
M aintenance 2
M aintenance 1
M aintenance ...
S tate in form ation
8
Equipment data
Identification data; e.g. equipment location classification installation data equipment unit data;
Design data; e.g. manufacturer’s data design characteristics;
Application data; e.g. operation, environment
9
Equipment data (Adapted from ISO 14224) Main catego ries
Sub-categories Data
Identification Equipment location - Equipment tag number (*)
Classification - Equipment unit class e.g. (*)- Equipment type (see Annex A) (*)- - Application (see Annex A)(*)
Installation data CountryLine (from A to B)Type of line e.g. double track, high speed lineType of track e.g. main track
Equipment unit data - Equipment unit description (nomenclature)- Unique number e.g. serial number- Subunit redundancy e.g. no of redundant subunits
Design Manufacturer’s data - Manufacturer’s name (*)- Manufacturer’s model designation (*)
Design characteristics
- Relevant for each equipment class e.g. turnout radius, current feeder voltage, see Annex A (*)
Cost data
Application Operation(normal use)
- Mode while in the operating state, e.g. continuous running, standby, normally closed/open, intermittent
- Date the equipment unit was installed or date of production start-up- Surveillance period (calendar time)(*)- The accumulated operating time during the surveillance period - Number of demands during the surveillance period as applicable - Operating parameters as relevant for each equipment class e.g. number of trains passing per
hour, see Annex A
Environmental factors External environment (severe, moderate, benign)a
Remarks Additional information - Additional information in free text as applicable
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Failure data
identification data, failure record and equipment location; failure data for characterizing a failure, e.g. failure date,
maintainable items failed, severity class, failure mode, failure cause, method of observation
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Failure data (From ISO 14224) Cate gory Data Description
Iden tifica tion Failure record (*) Unique failure identification
Equipment location (*) Tag number
Failure date (*) Date the failure was detected (year/month/day)
Failure mode (*) At equipment unit level as well as at maintainable item level)
Impact of failure on operation
Detailed list exist
Failure data Severity class (*) Effect on equipment unit function: critical failure, non-critical failure
Failure descriptor The descriptor of the failure (see Table 18)
Failure cause The cause of the failure (see Table 19)
Subunit failed Name of subunit that failed (see examples in Annex A)
Maintainable Item(s) failed
Specify the failed maintainable item(s) (see examples in Annex A)
Method of observation How the failure was detected (see Table 20)
Re marks Additional information Give more details, if available, on the circumstances leading to the failure, additional information on failure cause etc.
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Failure causes (Failure descriptors, From ISO 14224) No. Notation Description
1.0 Mechanical failure- general
A failure related to some mechanical defect, but where no further details are known
1.1 Leakage External and internal leakages, either liquids or gases. If the failure mode at equipment unit level is leakage, a more causal oriented failure descriptor should be used wherever possible
1.2 Vibration Abnormal vibration. If the failure mode at equipment level is vibration, a more causal oriented failure descriptor should be used wherever possible
1.3 Clearance/ alignment failure
Failure caused by faulty clearance or alignment
1.4 Deformation Distortion, bending, buckling, denting, yielding, shrinking, etc.
1.5 Looseness Disconnection, loose items
1.6 Sticking Sticking, seizure, jamming due to reasons other than deformation or clearance/alignment failures
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Failure causes, cont
No. Notation Description
2.0 Material failure- general
A failure related to a material defect, but no further details known
2.1 Cavitation Relevant for equipment such as pumps and valves
2.2 Corrosion All types of corrosion, both wet (electrochemical) and dry (chemical)
2.3 Erosion Erosive wear
2.4 Wear Abrasive and adhesive wear, e.g. scoring, galling, scuffing, fretting, etc.
2.5 Breakage Fracture, breach, crack
2.6 Fatigue If the cause of breakage can be traced to fatigue, this code should be used
2.7 Overheating Material damage due to overheating/burning
2.8 Burst Item burst, blown, exploded, imploded, etc.
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Failure causes, cont
No. Notation Description
3.0 Instrument failure – general
Failure related to instrumentation, but no details known
3.1 Control failure
3.2 No signal/i-ndication/ alarm
No signal/indication/alarm when expected
3.3 Faulty signal-/indication/ alarm
Signal/indication/alarm is wrong in relation to actual process. Could be spurious, intermittent, oscillating, arbitrary
3.4 Out of adjustment Calibration error, parameter drift
3.5 Software failure Faulty or no control/monitoring/operation due to software failure
3.6 Common mode failure
Several instrument items failed simultaneously, e.g. redundant fire and gas detectors
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Failure causes, cont
No. Notation Description
4.0 Electrical failure- general
Failures related to the supply and transmission of electrical power, but where no further details are known
4.1 Short circuiting Short circuit
4.2 Open circuit Disconnection, interruption, broken wire/cable
4.3 No power/ voltage Missing or insufficient electrical power supply
4.4 Faulty power/voltage
Faulty electrical power supply, e.g. over voltage
4.5 Earth/isolation fault Earth fault, low electrical resistance
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Failure causes, cont
No. Notation Description
5.0 External influence – general
The failure where caused by some external events or substances outside boundary, but no further details are known
5.1 Blockage/plugged Flow restricted/blocked due to fouling, contamination, icing, etc.
5.2 Contamination Contaminated fluid/gas/surface e.g. lubrication oil contaminated, gas detector head contaminated
5.3 Miscellaneous external influences
Foreign objects, impacts, environmental, influence from neighbouring systems
6.0 Miscellaneous – generala
Descriptors that do not fall into one of the categories listed above.
6.1 Unknown No information available related to the failure descriptor.
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Maintenance data
Maintenance is carried out To correct a failure (corrective maintenance); As a planned and normally periodic action to prevent failure from
occurring (preventive maintenance).
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Maintenance data (From ISO 14224)
Category Data Description
Identifi cation Maintenance record (*) Unique maintenance identification
Equipment location (*) Tag number
Failure record (*) Corresponding failure identification (corrective maintenance only)
Date of maintenance (*) Date when maintenance action was undertaken
Maintenance category Corrective maintenance or preventive maintenance
Maintenance activity Description of maintenance activity (see Table 21)
Impact of maintenance on operation
Zero, partial or total, (safety consequences may also be included)
Mainten ance data Subunit maintained Name of subunit maintained (see Annex A)NOTE - For corrective maintenance, the subunit maintained will normally be identical with the one specified on the failure event report
Maintainable item(s) maintained Specify the maintainable item(s) that were maintained (see Annex A)
Spare parts Spare parts required to restore the itemCost of spare parts, or links to a cost structure database..
Maintenance resources
Maintenance man-hours, per discipline
Maintenance man-hours per discipline (mechanical, electrical, instrument, others)
Maintenance man-hours, total Total maintenance man-hours.
Maintenance time Active maintenance time Time duration for active maintenance work on the equipment
Down time The time interval during which an item is in a down state
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State information
State information (condition monitoring information) may be collected in the following manners: Readings and measurements during maintenance Observations during normal operation Continuous measurements by use of sensor technology
20
State information, discrete readings
Category Data Description
Identifi cation State information record Unique state information identification
Equipment location Tag number
Maintenance record Corresponding maintenance identification, i.e. an observation is recorded either related to corrective or preventive maintenance
Failure record Corresponding failure identification (if no maintenance is performed in relation to the failure)
Date of observation Date when state information was read
State information
Type of measurement What measurement is obtained? For example a distance measure,
Value What are the readings of the measurement?
Remarks Additional information Give more details
If the readings are taken during normal operation, there will not be a corresponding maintenance or failure record. In this case the state information is linked directly to the inventory record
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State information, continuous readings
Category Data Description
Identifi cation State information record Unique state information identification
Equipment location Tag number
Type of measurement What measurement is obtained? For example a distance measure,
Sampling frequency What is the sampling frequency?
State information
Sensor What type of sensor is used
Data compression principle
How is data compressed, e.g. Fast Fourier Transform
Remarks Additional information Give more details
State information is linked directly to the inventory record for continuous readings
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Data analysis
Graphical techniques Histogram Bar charts Pareto diagrams Visualization of trends
Parametric models Estimation of constant failure rate Estimation of increasing hazard rate Estimation of global trends (over the system lifecycle)
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Pareto diagram (“Top ten”, components)
Contr
ol- and
Signa
ling Syste
m Overh
ead
line
Other
reasons -
Infras
tructu
re Break
of rai
lIrreg
ularties
by
cons
tructi
on/ r
epair
work
Telecom
m-
unica
tion
0 %
5 %
10 %
15 %
20 %
25 %
30 %
Co
ntr
ibu
tio
n t
o d
elay
tim
e [%
]
24
Presenting raw data and rates from accident and incident reporting systems
When presenting a “snapshot” of the indicators we often compare with targets value Colour codes may be used
For “occurrences” we just plot the raw data For frequencies we need to establish the “exposure”
Number of working hours in the period Number of critical work operations
25
Cross-tabulation
To see the effect of explanatory variables we could plot the number of occurrence or frequencies as a function of one or two explanatory variables
we get an indication whether the risk is unexpected high among certain groups of workers, during specific work operations, in special periods etc
26
Example of cross tabulation (dummy figures)
Onshore Offshore
ONGC employees 4 per 106 hrs 7.1 per 106 hrs
Contractors 3 per 106 hrs 2.4 per 106 hrs
Sub-contractors 8.2 per 106 hrs 12 per 106 hrs
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Root cause analysis
The objective is to present the contributing factors to the HSE indicators
Occurrences and/or frequencies are plotted against the causation codes, see next slide
Challenges How to treat more than one causation code? Causation codes are organised in a structure
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Causation codes in an MTO structuring
Triggering factors Underlying causes
Work organisation Work supervision Change routines Communication Working environment Requirements/procedures/guidelines
Management of company/entity Deficient safety culture Poor quality of established systems
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HMS avvik fordelt pr. utløsende årsakerKEP 2005
0
50
100
150
200
250
300
2003 15 4 11 11 2 8 8 8 1 6 7 5 1 4
2004 262 251 124 221 163 168 169 135 97 48 79 74 48 37
2005 201 81 178 69 78 54 33 62 106 126 68 70 81 68
Iverks ikke tilstr. sikr. av arb.pl.
Unnlot å inform./varsle/komm.
Uryd. arb. pl./mangl.
renhold
Brukte utst./verkt. på feil m.
Mangelfull skilting/ avskj.
Arb.pl. var/ble
ikke tilr.
Brukte ikke korr.
pers. verne
Feilpl. gjen- stander
Mangelf. kv.kontr./ verif. av
Løse gjen- stander
Brudd på trafikk- regler
Oppf. ikke sign./tegn/
skilt
Mangelf. verneutstyr
Feil el. svikt i
utst/tekn.
HSE deviations per triggering factor
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Trend curves, three alternatives
1. Plot number of occurrences as a function of time (histogram)
2. Plot frequencies (number/exposure) as a function of time
3. Plot both number and exposure as a function of time in the same diagram
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HMS avvik fordelt pr. kvartalKEP 2005
0
100
200
300
400
500
600
700
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
Antall registreringer 2 2 22 42 67 334 635 646 518 451 216
Arbeidstimer 270743 127600 182706 297263 523307 774040 694858 850900 769685 402764 279959
2003 1 2003 2 2003 3 2003 4 2004 1 2004 2 2004 3 2004 4 2005 1 2005 2 2005 3
Exposure (hours worked)
Incidents
Quarterly HSE deviation
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Challenges
Difficult to see trends due to the stochastic nature of the number of events
As an alternative, plot cumulative number of events as a function of time (adjusted for exposure)
Convex plot indicates increasing risk level Concave plot indicates an improving situation The following example is based on the previous plot
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0
0.001
0.002
0.003
0.004
0.005
0.006
2003
-1
2003
-2
2003
-3
2003
-4
2004
-1
2004
-2
2004
-3
2004
-4
2005
-1
2005
-2
2005
-3
Cumulative number of deviations
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Interpretation of cumulative plot
A convex plot indicates an increasing frequency of incidents ()
A concave plot indicates improvement ()
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Note
Cross-tabulation and trend curves are used to focus on safety problems, but do not indicate improvement measures
Root cause analysis identifies significant causes behind the undesired events/accidents cue on measures
Risk reducing measure should be based on an understanding of That the measure is directed against one or more failure causes (causation code) That the measure is effective in terms of e.g., cost That no negative effects of the measure is anticipated
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Parameter estimation /bathtub curve
Time
Fai
lure
rat
e
• The bathtub curve is a basis for reliability modelling, but
• There are two such curves• The hazard rate for ”local time”• The failure intensity for ”global time”
• Combining the two:
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Performance loss
Local time
Fai
lure
inte
nsity
/P
erfo
rman
ce lo
ss
Local time Local time
Global (system) time
1
23
4
38
Plotting techniques, lifetime data (local bath tubcurve)
Several plots exists to visualize characteristics of lifetime data TTT-plot Kaplan-Meier plot Hazard plot
All these plots assume Failure times are identical,
and independent distributes I.e. no change over
system lifetime
Examples of how life timesare generated are shown to the right
1
2
3
4
5
6
7
T 1
T 2
T 3
T 4
T 5*
T 6
T 7
t=0 End
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TTT- Total Time on Test plot
Let T1,T2,T3,..,Tn be the recorded lifetimes
Let T(1),T(2),T(3),.. be ordered lifetimes, i.e. T(1) T(2)T(3).. Define the total test on time at time t by
where i is such that T(i) t < T(i+1)
The TTT-plot is obtained by plotting for i = 1,..,n:
ti-n+T = tTTT j
i
1=j
)()( )(
)(
)(
)(
)(
TTTTTTTT
,n
i
n
i
40
Examplei T(i)
ST(i)ST(i)+(n-i)T(i) i/n TTT Transform
0 0 0 0 0 0 00.02 0.310582
1 6000.00 6000.00 66000 0.09 0.35 0.512 8000.00 14000.00 86000 0.18 0.46 0.633 12000.00 26000.00 122000 0.27 0.65 0.714 14000.00 40000.00 138000 0.36 0.74 0.785 16000.00 56000.00 152000 0.45 0.81 0.836 18000.00 74000.00 164000 0.55 0.88 0.877 19000.00 93000.00 169000 0.64 0.90 0.918 20000.00 113000.00 173000 0.73 0.93 0.949 23000.00 136000.00 182000 0.82 0.97 0.96
10 24000.00 160000.00 184000 0.91 0.98 0.9811 27000.00 187000.00 187000 1.00 1.00 1.00
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Interpretation
A plot around the diagonal indicates a constant hazard rate, i.e. failure times can be considered exponentially distributed.
A concave plot (above the diagonal) indicates an increasing hazard rate (IFR).
A convex plot (under the diagonal) indicates a decreasing hazard rate (DFR).
A plot which fist is convex, and then concave indicates a bathtub like hazard rate
A plot which first is concave, and then convex indicates heterogeneity in the data, see Vatn (1996).
43
Exercise
Assume that the following failure data for one component type has been recorded (in months)
8,9,7,6,12,18,14,6,9,11,24 Construct the TTT plot What would you say about the hazard rate?
44
The Nelson Aalen plot for global trend over the system lifetime
The Nelson-Aalen plot shows the cumulative number of failures on the Y-axis, and the X-axis represents the time
A convex plot indicates a deteriorating system, whereas a concave plot indicates an improving system
The idea behind the Nelson-Aalen plot is to plot the cumulative number of failures against time
Actually we plot W(t) which is the expected cumulative numbers of failures in a time interval
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Nelson Aalen procedure
When estimating W(t) we need failure data from one or more processes (systems)
Each process (system) is observed in a time interval (ai,bi] and tij denotes failure time j in process i (global or calendar time)
To construct Nelson Aalen plot the following algorithm could be used Group all the tij’s and sort them, and denote the result tk, k = 1,2,…..
For each k, let Ok denote the number of processes that are under observation just before time tk
Let Let, k = 1,2,… Plot
0ˆ0 W
kkk OWW /1ˆˆ1
)ˆ,( kk Wt
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Example of Nelson Aalen plot
ai bi tij
0 50 7, 20, 35, 44
20 60 26, 33, 41, 48, 57
40 100 50, 60, 69, 83, 88, 92, 99
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90 100
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Parameter estimation
Constant hazard rate, homogeneous sample Constant hazard rate, non-homogeneous sample Increasing hazard rate
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Constant failure rate homogeneous sample
In this situation we only need the following information t = aggregated time in service n = the total number of observed failures in the period
An estimate for the failure rate is given by
t
n
service in timeAggregated
failures ofNumber ̂
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Multi-Sample Problems
In many cases we do not have a homogeneous sample of data
The aggregated data for an item may come from different installations with different operational and environmental conditions, or we may wish to present an “average” failure rate estimate for slightly different items
In these situations we may decide to merge several more or less homogeneous samples, into what we call a multi-sample
The various samples may have different failure rates, and different amounts of data
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Illustration, multi-sample dataSample
1
2
3
k
Total
Failure rate(failures per 104 hours)1 2 53 10984 6 7 11 12
Uncertainty limits
51
Estimation principles, multi-sample
The OREDA-estimator used in the OREDA data handbook is based on the following assumptions: We have k different samples. A sample may e.g., correspond to a
platform, and we may have data from similar items used on k different platforms.
In sample no. i we have observed ni failures during a total time in service ti, for i =1,2,…, k.
Sample no. i has a constant failure rate i, for i =1,2,…, k. Due to different operational and environmental conditions, the
failure rate i may vary between the samples. This variation is described by a probability density function, say
() The OREDA handbook presents expectation and standard
deviation in the estimated distribution of ()
52
Increasing hazard rate The estimation of parameters e.g., the Weibull distribution
requires Maximum Likelihood procedures. Let be the parameter vector of interest, for example = [,] if
the Weibull distribution is considered Let tj denote the observed life times, both censored and real life
times The likelihood function is now given by
where CL, U and CR are the set of left-censored (start of observation not known, uncensored (real lifetimes) and right-censored life times (failure time not known)
The estimator is the value of that maximizes L(;t)
RL Cj
jUj
jCj
j tRtftFL );();();();( θθθtθ
53
How to do it?
Usually we use numerical methods to maximize the likelihood function
Such a procedure is implemented in the TTTPlot.xls file
For the example data we get
a3.05630302
l 0.000052
54
Exercise
Assume that the following failure data for one component type has been recorded (in months)
8,9,7,6,12,18,14,6,9,11,24 Find the ML estimators for the parameters in the Weibull
distribution Compare the parametric plot (Weibull) with the TTT plot,
and judge how well the data fit the Weibull distribution
55
Likelihood function, Weibull model
Assume t(1), t(2),…t(n) are ordered failure times, and I(1), I(2),…I(n)
are indicators such that I(i) = 1 if failure time (i) is a failure time, and 0 if it is a censoring life time.
The probability density function is given by
The survival function is given by
The log likelihood function is given by
56
Exercise
Assume that the following failure data for one component type has been recorded (in months)
8,9,7,6,12,18,14,18*,6,9,11,24,30*,28* Find the ML estimators for the parameters in the Weibull
distribution where failure times with a star (*) represent censoring failure times
57
Kaplan Meier estimator
The standard TTT plot assumes that we do not have censoring failure times
The Kaplan Meier estimator and corresponding plot may be used for censoring life times
Let t(1), t(2),…t(n) be ordered failure times with corresponding indicator variable to indicate the real failure times
Let n(i) be the number of components “at risk” just prior to t(i) and s(i) the number of “deaths” at that time
The Kaplan Meier estimator is given by:
58
Exercise
Assume that the following failure data for one component type has been recorded (in months)
8,9,7,6,12,18,14,18*,6,9,11,24,30*,28* Construct the Kaplan Meier plot, and insert a Weibull
distribution overlay curve with the parameters estimated
59
Estimation in NHPP
The Non-Homogeneous Poison Process (NHPP) is a model defined by: A system is put into service at time t = 0. If the system fails, a repair is conducted and the system is put into
service after a time that could be neglected The repair action set the system back to a state as good as it was
immediately prior to the failure, i.e. a minimal repair.
The important parameter is w(t) = ROCOF = Probability of failure in (t, t + t) divided by t
60
For the NHPP we have The rate of occurrence of failures, ROCOF = w(t) is
generally not constant. The number of failures in an interval (a,b) is Poisson
distributed with parameter The mean number of failures in an interval (a,b) is The cumulative number of failures up to time t is
61
Properties for selected NHPP models
Property Model Power law model
Linear model
Log-linear model
ROCOF = w(t) t-1 (1+t) e+t
W(t) t (t+t2/2) (e+t - e)/System improves for < 1 < 0 < 0
System deteriorates for > 1 > 0 > 0
Average failure rate when replaced at time
-1 (1+/2) (e+ - e)/()
62
Estimation in NHPP
A NHPP observed over a period 0 a < b We have observed n failure times t1, t2,…, tn sorted in time
The likelihood function, say L(,t), is now the probability that we have observed the actual failure times, i.e. t = [t1, t2,…, tn] as a function of
Consider small time intervals around the observed failure times and let ti be such a small time interval following ti
The likelihood function
64
Bayesian estimation
In some situations we may have tacit knowledge in terms of expert knowledge
Experts are typically experienced people in the project organisation
By an elicitation procedure we may get the experts to state their uncertainty distribution regarding parameters of interest
This uncertainty distribution is combined by data to find the final parameter estimates
65
Procedure Specify a prior uncertainty distribution of the reliability
parameter, () Structure reliability data information into a likelihood
function, L(;x) Calculate the posterior uncertainty distribution of the
reliability parameter vector, (x) The posterior is found by (x) L(;x) (), and
the proportionality constant is found by requiring the posterior to integrate to one
The Bayes estimate for the reliability parameter is given by the posterior mean
66
Example: Constant failure rate
= failure rate treated as a random quantity Prior expert distribution from the elicitation procedure:E() = 0.710-6 (failures / hour)SD() = 0.310-6
For mathematical simplicity, a gamma prior is used with parameters and where E = / and Var = / 2 = E/SD2 = (0.710-6)/( 0.310-6)2 = 7.78106 = E = (7.78106) (0.710-6) = 5.44
67
Example, cont
Data:t = total time in service, = 525 600 hours (e.g. 60 detector years)n = 1 = number of failures observed
Constant exponentially distributed failure times the number of failures in a period of length t, N(t), is Poisson distributed with parameter t
The probability of observing n failures is then L(;n,t) = Pr(N(t) = n) ne-t = likelihood
68
Example, cont
The posterior distribution is found by multiplying the prior distribution with the likelihood function(n) L(;n,t) () ne-t -1e- (+ n)-1e-(+t)
(+ n)-1e-(+t) is recognized as a gamma distribution with new parameters ’ =+ n, and ’ = +t
The Bayes estimate is given by the mean in this distribution, i.e.
(MLE: 1.910-6, prior mean = 0.710-6)
66
5.44 1ˆ 0.78 107.78 10 525600
n
t