How MRT Predictive Maintenance Works Using Mathematics: Finding a Fault Before It Becomes a Disruption

A railway component does not usually move from perfectly healthy to failed in one magical instant.

Many failures are preceded by evidence.

A temperature begins drifting upward. A vibration signature changes. A door motor needs slightly longer to complete the same movement. Electrical current becomes less stable. A track measurement slowly departs from its previous pattern. A component begins producing more unusual events than comparable components of the same age and duty.

None of those observations automatically means:

FAILURE TOMORROW

The evidence is noisy. Operating conditions change. Sensors themselves are imperfect. A heavily used component can look different from a lightly used one without being unhealthy. A strange reading can be a genuine warning, a harmless operating condition or simply bad data.

This is why predictive maintenance is not merely:

put sensors everywhere
→ let AI predict failures

It is a mathematical decision system.

Predictive maintenance asks whether the evidence of degradation is strong enough, early enough and reliable enough to justify maintenance before the asset becomes a service problem.

Singapore is moving directly into this territory. In February 2026, the Land Transport Authority, SMRT and SBS Transit announced that Singapore would progressively introduce a more standardised, network-wide condition-monitoring approach across rail lines, alongside an integrated asset-data framework and workforce development in condition monitoring and predictive maintenance.

SMRT has also begun progressively deploying JARVIS, a data and AI platform integrating maintenance, condition-monitoring and operational information. SMRT reported in August 2026 that Phase 1 had launched in January 2026 across the North-South and East-West Lines with 11 analytics modules, and that the system had already helped initiate more than 500 maintenance inspections in response to early signs of component degradation.

New Singapore trains increasingly arrive with this logic built in. LTA says the new Circle Line trains entering service from April 2026 include condition-monitoring systems that gather equipment-health data for predictive maintenance. Similar systems already exist on newer North East Line trains, where Automatic Track Inspection equipment on selected trains uses cameras and sensors to monitor track condition during normal operation.

This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then explore MRT network resilience and the earlier articles linked at the end.

Now we ask a different question:

How can mathematics detect a railway asset becoming unhealthy before the passenger notices anything at all?

The RFE — Why Does Predictive Maintenance Exist?

The weakest possible objective is:

replace everything before it fails

That would certainly reduce some failures.

It would also waste enormous amounts of usable asset life, labour, engineering hours and spare parts.

The opposite extreme is:

keep everything running until it breaks

That extracts maximum component life.

It also transfers risk into passenger service, emergency response and potentially safety.

The RFE sits between these extremes:

observe asset health
→ identify meaningful degradation early
→ estimate risk and remaining useful life
→ intervene at the right time
→ avoid unnecessary maintenance
→ prevent avoidable service disruption

Predictive maintenance therefore exists to make maintenance more timely.

Not earlier than necessary.

Not later than safe.

The best maintenance time is not “as soon as possible”. It is the earliest justified intervention before unacceptable failure risk.

Prompt 1 — What Is the Difference Between Preventive, Condition-Based and Predictive Maintenance?

These approaches are related but mathematically different.

Reactive maintenance

The asset is repaired after failure.

failure occurs
→ detect failure
→ repair

This can be reasonable for non-critical items whose failure has little consequence.

It is poor strategy for assets whose failures can interrupt train service or create safety consequences.

Preventive maintenance

Maintenance is performed according to age, usage or calendar interval.

every T days
or
every N operating cycles
→ inspect or replace

A simple age-replacement policy is:

replace at age τ
unless failure occurs first

This is easy to plan and does not require sophisticated sensing.

But components of the same age may degrade differently because their duty, environment and manufacturing variation differ.

Condition-based maintenance

Maintenance depends on measured health rather than only age.

Let health indicator H(t) be measured over time.

A simple condition rule is:

if H(t) crosses threshold θ
→ inspect or maintain

This asks:

What condition is the component in now?

Predictive maintenance

Predictive maintenance adds a future question:

Given what we observe now,
what is likely to happen next?

It may estimate:

• probability of failure within a time horizon,
• remaining useful life,
• future health state,
• or probability that an anomaly develops into a service-affecting fault.

So:

condition-based maintenance
= respond to current health

predictive maintenance
= forecast future health and act before risk becomes unacceptable

Renewal is a different scale again

Some assets should not be maintained indefinitely.

Entire power, signalling or train systems eventually require renewal.

The 2026 Rail Reliability Taskforce specifically recommended prioritising renewal of trains, signalling and power systems, while improving condition monitoring and predictive maintenance for asset management.

The complete maintenance architecture is therefore:

react when appropriate
+
maintain periodically where useful
+
monitor condition continuously where valuable
+
predict degradation where evidence allows
+
renew systems when life-cycle economics and risk justify replacement

Predictive maintenance does not replace preventive maintenance. It helps decide where a fixed schedule should become an evidence-based schedule.

Prompt 2 — What Data Does an MRT Maintenance Model Actually See?

A maintenance model does not see “health” directly.

It sees measurements that are related to health.

Possible signals include:

• temperature,
• vibration,
• electrical current and voltage,
• pressure,
• acoustic signals,
• door movement time,
• motor speed,
• braking response,
• event counts and diagnostic codes,
• camera or image measurements,
• track geometry,
• rail-surface condition,
• and operating context such as train load or number of cycles.

LTA gives several public Singapore examples.

New North East Line trains use condition-monitoring systems that collect data from onboard equipment, while two trains carry Automatic Track Inspection systems with cameras and sensors that monitor track defects such as rail cracks, corrugation and missing fasteners during service.

New Bukit Panjang LRT vehicles monitor key systems including doors, brakes and air-conditioning, while selected vehicles monitor power-rail condition in real time.

The newer NSEWL train fleet has built-in condition monitoring and diagnostics for real-time performance tracking, and the renewed track-circuit system has its own monitoring capability for emerging faults.

The Cross Island Line trains will also include condition monitoring and diagnostics, with selected trains carrying Automated Track Inspection equipment.

Time series

Most condition data arrive as time series.

x1, x2, x3, ... , xt

Each x_t is a measurement at time t.

The model may care about:

• absolute level,
• trend,
• rate of change,
• variance,
• periodicity,
• frequency content,
• correlation with other sensors,
• and departures from comparable assets.

A 2025 systematic review of predictive-maintenance time-series analytics describes common tasks including anomaly detection, failure classification and remaining-useful-life estimation.

Raw signal versus feature

Suppose a vibration sensor collects thousands of readings per second.

The model may not use every raw sample directly.

It can construct features such as:

mean
standard deviation
RMS amplitude
peak value
kurtosis
spectral energy
frequency-band power
trend slope

For observations x₁…x_n, root-mean-square value is:

RMS = √[(1/n)Σxi²]

Standard deviation is:

s = √[(1/(n−1))Σ(xi − x̄)²]

A changing RMS or frequency pattern can reveal a change in vibration behaviour even when individual measurements appear noisy.

Context matters

A door motor drawing more current may be degrading.

Or it may simply be moving a door under a different load or temperature condition.

Therefore a health model often uses conditional expectation:

expected signal
=
f(asset type,
age,
load,
environment,
operating mode)

Anomaly is then measured against the expected condition rather than one universal threshold.

This is one reason the 2026 LTA plan for standardised network-wide condition monitoring matters. LTA says existing systems were introduced at different times by different manufacturers and use different parameters. A common baseline and integrated asset-data framework can make cross-line comparison and benchmarking more coherent.

A sensor reading becomes useful only when the system knows what “normal” should have looked like under the same operating condition.

Prompt 3 — How Does Mathematics Detect an Anomaly?

An anomaly is an observation sufficiently inconsistent with the expected pattern.

The simplest possible model uses a z-score.

If normal measurements have mean μ and standard deviation σ:

z = (x − μ)/σ

A large absolute z-score means the observation is unusual relative to the reference distribution.

But one unusual reading is not necessarily a fault.

Railway monitoring therefore needs persistence and context.

Moving average

A moving average smooths short-term noise:

m(t)
=
(1/k)Σi=0k−1 x(t−i)

A slowly increasing moving average may reveal gradual degradation hidden inside noisy readings.

Exponentially weighted moving average

An EWMA gives more weight to recent observations:

zt = λxt + (1−λ)zt−1

where 0 < λ ≤ 1.

Small λ creates heavy smoothing.

Larger λ responds faster to new changes.

The choice is a trade-off:

too sensitive
→ many false alarms

too smooth
→ slow detection

Residual analysis

Suppose a model predicts measurement x̂_t.

The residual is:

et = xt − x̂t

If the asset is healthy and the model is good, residuals should remain inside an expected distribution.

A persistent residual shift suggests the physical system no longer matches the learned healthy relationship.

Multivariate anomaly detection

One signal can look normal while the combination is abnormal.

Suppose health vector is:

x = [temperature, current, vibration, cycle time]

A Mahalanobis distance measures how unusual the vector is relative to normal covariance:

D²
=
(x − μ)ᵀ Σ−1 (x − μ)

This accounts for correlations among variables.

For example, high current may be normal when cycle time is short, but suspicious when combined with slower movement and higher temperature.

Machine learning

More advanced methods can learn normal or faulty patterns using:

• classification models,
• random forests,
• gradient boosting,
• support vector machines,
• autoencoders,
• recurrent neural networks,
• transformers and other time-series models.

A 2026 survey of railway anomaly detection notes that modern intelligent monitoring is applied across rolling-stock systems, doors, HVAC, traction equipment, track and other infrastructure.

But model complexity does not remove the underlying question.

How different is this observation
from what healthy operation would reasonably produce?

That is anomaly detection in one sentence.

An anomaly detector does not detect failure. It detects evidence that deserves a better question.

Prompt 4 — How Does Mathematics Turn Age into Failure Probability?

Not all components fail at a constant rate.

Some have early-life defects.

Some fail randomly.

Some become increasingly likely to fail as wear accumulates.

Reliability mathematics describes these patterns.

Reliability function

Let random variable T be time to failure.

Reliability is:

R(t) = P(T > t)

Failure probability by time t is:

F(t) = 1 − R(t)

Hazard rate

Hazard rate asks:

Given that the component has survived until time t,
how quickly is its instantaneous failure risk changing now?

Mathematically:

h(t) = f(t)/R(t)

where f(t) is the failure-time probability density.

Weibull model

The Weibull distribution is widely used in reliability engineering because its hazard can take several shapes.

R(t)
=
exp[−(t/η)β]

where:

• η is a scale parameter,
• β is a shape parameter.

Hazard is:

h(t)
=
(β/η)(t/η)β−1

If β < 1, hazard decreases with age.

If β = 1, hazard is constant and the model becomes exponential.

If β > 1, hazard increases with age, which can model wear-out behaviour.

Condition changes the age model

Suppose two components are both five years old.

Component A has stable vibration and temperature.

Component B has a persistent upward temperature trend and abnormal vibration residuals.

A purely age-based model treats them similarly.

A predictive model updates risk using evidence.

Bayes’ theorem provides the conceptual structure:

P(Fault | Data)
=
P(Data | Fault)P(Fault)
/
P(Data)

The age-based failure probability becomes the prior.

Condition-monitoring evidence updates the posterior probability.

age and history
→ prior risk
+
sensor evidence
→ updated risk

This is the mathematical heart of evidence-based maintenance.

Age tells us what tends to happen. Condition tells us whether this particular asset is beginning to behave differently.

Prompt 5 — What Is Remaining Useful Life?

Remaining Useful Life, usually abbreviated RUL, asks:

How much usable life remains
before the asset reaches a defined unacceptable condition?

If failure time is T_f and current time is t:

RUL = Tf − t

The difficulty is that T_f is not known.

It must be estimated.

A degradation model

Suppose health indicator H increases as degradation accumulates.

Failure threshold is H_crit.

A simple linear model is:

H(t) = H0 + kt

Predicted threshold time is:

tf
=
(Hcrit − H0)/k

At current time t:

RUL = tf − t

Real degradation is rarely perfectly linear.

A nonlinear model might be:

H(t) = a + bt + ct²

or a stochastic process with random increments.

Model-based, data-driven and hybrid RUL

A 2025 review of more than 3,000 RUL papers groups methods broadly into:

• model-based methods using physics or degradation equations,
• data-driven methods learning from historical observations,
• hybrid methods combining both.

Physics-based models can be interpretable but difficult when system behaviour is complex.

Data-driven models can capture complex patterns but may struggle when operating conditions differ from the training data.

Hybrid models aim to use both known engineering structure and learned residual behaviour.

RUL must include uncertainty

A model should not simply say:

RUL = 17 days

as though the future were exact.

A better output is a probability distribution:

RUL ~ distribution

with:

expected RUL
confidence interval
probability RUL < maintenance horizon

For example:

median RUL = 18 days
90% interval = 8 to 31 days
P(failure within 14 days) = 0.32

These numbers are fictional.

The structure is the important part.

A 2025 review in Mechanical Systems and Signal Processing argues that predictive-maintenance prognostics should be evaluated not only for accuracy but also for uncertainty, robustness, interpretability and feasibility.

That is exactly right for a railway.

A precise-looking prediction with unknown uncertainty can be more dangerous than a wider but honest estimate.

The maintenance planner does not need a confident number. The planner needs an honest estimate of what could happen and how sure the model is.

Prompt 6 — How Does the System Decide Whether an Alarm Is Worth Acting On?

A prediction is useful only if it improves decisions.

Suppose an anomaly detector produces an alarm score S.

A simple decision rule is:

if S ≥ θ
→ inspect asset

Threshold θ creates a trade-off.

False positive

The system alarms, but the component is healthy.

Cost:

• unnecessary inspection,
• engineering labour,
• possible train withdrawal,
• alarm fatigue,
• and wasted parts if replacement is too aggressive.

False negative

The system remains quiet, but a genuine fault is developing.

Cost can be far larger:

• unscheduled breakdown,
• service delay,
• secondary damage,
• emergency maintenance,
• or safety consequence depending on the asset.

Therefore the threshold should reflect consequence.

Confusion matrix

                 Actual Fault   Actual Healthy
Alarm                  TP             FP
No Alarm               FN             TN

Sensitivity or recall is:

Recall = TP/(TP + FN)

Precision is:

Precision = TP/(TP + FP)

A highly sensitive detector catches most faults but may create many false alarms.

A very conservative detector produces fewer false alarms but may miss early failures.

Expected decision cost

Let:

CFP = cost of false positive
CFN = cost of false negative
CTP = cost of justified inspection or maintenance

Expected cost can be written:

E[C]
=
CFP P(FP)
+ CFN P(FN)
+ CTP P(TP)

The optimal threshold minimises expected whole-system cost while satisfying safety rules.

For high-consequence assets, CFN can dominate.

For low-consequence assets, excessively frequent false alarms may dominate maintenance resources.

Bayesian maintenance decision

Suppose current evidence gives failure probability p during the next maintenance horizon.

If preventive maintenance now costs C_M and failure costs C_F:

Maintain now when:

CM
<
p CF
+ expected future maintenance cost

Again, real railway safety-critical decisions cannot be reduced to a financial inequality alone.

Mandatory safety limits and engineering rules come first.

But for decisions within the permitted maintenance envelope, expected-cost reasoning helps allocate scarce engineering resources.

The maintenance threshold is not a magic property of the sensor. It is a decision boundary built from risk, consequence and available action.

Prompt 7 — How Does Predictive Maintenance Decide When to Inspect, Replace or Wait?

Once an asset’s health is estimated, maintenance still has to be scheduled.

This is an operations-research problem.

The railway has limited:

• engineering hours,
• maintenance staff,
• depot space,
• replacement parts,
• specialist tools,
• and opportunities to withdraw trains without harming service.

Inspection interval

Suppose inspections happen every Δt days.

Shorter interval:

more inspections
→ earlier detection
→ higher maintenance cost

Longer interval:

fewer inspections
→ lower routine cost
→ larger window for undetected degradation

If degradation can be monitored continuously, the asset may move away from fixed interval inspection towards condition-triggered inspection.

Maintenance priority score

Suppose asset i has:

• probability of failure p_i,
• consequence severity c_i,
• uncertainty u_i,
• maintenance duration m_i,
• remaining useful life estimate r_i.

An illustrative priority index might be:

Priorityi
=
(pi ci)/(ri + ε)
× ui

This is not an operator formula.

It illustrates why probability alone is not enough.

A component with modest failure probability but severe consequences can deserve greater attention than a component with high failure probability and trivial consequence.

Maintenance-window optimisation

Let binary variable x_ij equal 1 when asset i is maintained during engineering window j.

xij ∈ {0,1}

A simplified objective is:

minimise
Σi,j xij Cmaintenance,i,j
+
Σi expected failure risk if deferred

subject to:

available staff
available engineering time
parts availability
train availability
required service level
safety constraints

This is where predictive maintenance becomes fleet and timetable planning.

Spare-parts forecasting

The Rail Reliability Taskforce’s 2026 recommendations explicitly include improving spares forecasting using more comprehensive data and analysis.

Suppose component failures over a period follow count random variable N.

If expected failures are λ and a simple Poisson approximation is reasonable:

P(N=k)
=
e−λ λk/k!

The inventory planner can choose stock S such that:

P(N ≤ S) ≥ service target

Real spares planning must consider lead times, part obsolescence, common-cause failures and non-Poisson demand.

But the probability structure explains why maintenance analytics and supply-chain resilience belong together.

Predicting a failure is not useful if the required replacement part will not exist when the maintenance window arrives.

Grouping maintenance

Two assets may each be slightly early for maintenance.

Maintaining them together can reduce mobilisation and train-withdrawal cost.

This creates an economic dependence:

maintain A now alone
+
maintain B later alone

versus

maintain A and B together now

The second option replaces some useful life early but may reduce total engineering cost and service impact.

Predictive maintenance does not merely predict when something may fail. It helps place maintenance inside the least harmful available moment.

Prompt 8 — What Does a Whole-Railway Predictive Maintenance System Look Like?

At railway scale, the problem is no longer one sensor and one asset.

There may be thousands of monitored components producing millions of observations.

The system therefore needs several layers.

physical asset
→ sensor
→ data quality checks
→ health features
→ anomaly model
→ degradation model
→ failure/RUL estimate
→ uncertainty
→ maintenance priority
→ human engineering review
→ work order
→ inspection result
→ model update

Data quality first

A bad sensor can create a convincing false fault.

So before health prediction, the system should ask:

Is the sensor alive?
Is the timestamp correct?
Is the reading physically possible?
Are data missing?
Has calibration drifted?

Let data-quality indicator q_t lie between 0 and 1.

A health estimate can be down-weighted when data quality is weak:

effective evidence
=
qt × model evidence

The exact implementation varies, but the principle is essential.

Uncertain data should not silently become certain diagnosis.

Cross-asset benchmarking

Suppose 500 similar door motors operate across the network.

Instead of asking whether one motor crossed an absolute threshold, the model can ask where it sits relative to peers of similar:

• age,
• duty cycle,
• environment,
• maintenance history,
• and operating line.

A peer-normalised score is:

zi
=
(Hi − μpeer)/σpeer

This can identify assets behaving unusually even when all remain inside a broad design limit.

LTA’s planned integrated data framework and standardised condition-monitoring baseline can make this kind of cross-line benchmarking more meaningful because asset data will be more consistent.

Human review

A maintenance model should not be treated as an oracle.

Engineers understand:

• physical failure modes,
• recent work done on the asset,
• operating context,
• known sensor limitations,
• and whether a recommended intervention is physically sensible.

SMRT’s JARVIS description reflects this human-machine partnership: predictive analytics and visualisation help engineers identify issues earlier, prioritise work and make faster decisions.

The correct architecture is therefore not:

AI detects anomaly
→ AI orders repair

It is closer to:

data finds evidence
→ model estimates significance
→ system ranks risk
→ engineer investigates
→ authorised maintenance action
→ observed result returns to the model

Continual learning

Railway assets and operating conditions change.

New trains enter service.

Components are renewed.

Passenger demand changes.

Maintenance itself changes the data distribution.

A model trained on yesterday can therefore degrade.

This is data drift or concept drift.

Let model error over time be:

E(t) = performance error at time t

If E(t) rises beyond a monitoring threshold, retraining or recalibration is needed.

Research on continual learning for predictive maintenance specifically addresses changing condition-monitoring data streams and non-stationary operating environments.

The predictive-maintenance model itself therefore needs maintenance.

A system that predicts ageing assets but assumes its own model never ages has missed the lesson.

A Complete Fictional Predictive-Maintenance Example

Let us build an educational example around a fictional train-door drive unit.

This example does not use actual Singapore MRT alarm limits, component specifications or maintenance thresholds.

Step 1 — Observe the signals

For each door cycle, the system records:

closing time T
peak motor current I
motor temperature θ
vibration RMS V

During healthy operation, peer-normalised means are approximately:

T = 1.00
I = 1.00
θ = 1.00
V = 1.00

where each value has been scaled relative to the healthy baseline.

Step 2 — Observe gradual drift

Day 1:  T 1.01, I 1.00, θ 1.01, V 1.00
Day 5:  T 1.03, I 1.02, θ 1.03, V 1.02
Day 10: T 1.07, I 1.05, θ 1.08, V 1.05
Day 15: T 1.12, I 1.09, θ 1.14, V 1.10

No single signal is catastrophic.

The combination is moving in one direction.

Step 3 — Build a health index

Suppose an illustrative health score is:

H
=
0.30(T−1)
+ 0.25(I−1)
+ 0.25(θ−1)
+ 0.20(V−1)

At Day 15:

H
= 0.30(0.12)
+ 0.25(0.09)
+ 0.25(0.14)
+ 0.20(0.10)

= 0.036 + 0.0225 + 0.035 + 0.020
= 0.1135

The exact coefficients are fictional.

They demonstrate how several weak signals can combine into stronger evidence.

Step 4 — Compare with peers

Suppose healthy peer health scores have:

μpeer = 0.020
σpeer = 0.025

Then:

z
= (0.1135 − 0.020)/0.025
= 3.74

The unit is now statistically unusual relative to comparable healthy assets.

Step 5 — Estimate degradation rate

Suppose the health index is approximately linear over the last ten days:

H(t) ≈ 0.025 + 0.006t

where t is days since the start of the trend.

Assume an illustrative intervention boundary:

Hcrit = 0.16

Predicted crossing time is:

tcrit
= (0.16 − 0.025)/0.006
= 22.5 days

If the trend has already run for 15 days:

point-estimate RUL
≈ 22.5 − 15
= 7.5 days

The system should not stop there.

Step 6 — Add uncertainty

Suppose degradation-slope uncertainty produces:

median RUL = 8 days
90% interval = 3 to 17 days

There is meaningful probability of reaching the intervention boundary before the next routine 14-day maintenance opportunity.

The question becomes a decision problem.

Step 7 — Compare maintenance opportunities

Option A:

inspect tonight

Option B:

wait 7 days for planned depot access

Option C:

wait 14 days for routine maintenance

Suppose the model estimates:

P(cross boundary within 1 day) = 0.04
P(cross within 7 days) = 0.42
P(cross within 14 days) = 0.79

These numbers are fictional.

They show why predictive maintenance changes the decision from:

Is the unit failed?

to:

Is waiting until the next maintenance window still justified?

Step 8 — Human inspection

The engineering team inspects the unit.

Suppose the inspection confirms increased mechanical resistance.

Maintenance is performed.

After maintenance:

T returns near baseline
I returns near baseline
θ stabilises
V falls

The inspection result becomes a labelled event for future model learning.

prediction
→ inspection
→ confirmed physical cause
→ maintenance
→ signal recovery
→ model validation

That is the complete predictive-maintenance loop.

The Predictive-Maintenance Deletion Tests

Remove sensors

The system returns towards calendar-based maintenance and manual inspection.

Remove context

Normal high-load behaviour may be misclassified as degradation.

Remove uncertainty

Predictions look more certain than the evidence deserves.

Remove false-alarm cost

The system can flood engineers with low-value alerts.

Remove missed-failure cost

The model can become too conservative and fail to warn early enough.

Remove human engineering review

Statistical anomaly becomes automatic physical diagnosis.

Remove spare parts and engineering windows

The system predicts accurately but cannot act.

Remove post-maintenance feedback

The model never learns whether its alarms were physically correct.

Remove model monitoring

The predictive system can quietly become obsolete as the railway changes.

A predictive-maintenance system is only as strong as its weakest link between measurement and justified action.

The Predictive-Maintenance Paradoxes

Paradox 1 — Better monitoring can initially create more maintenance work

The system begins finding degradation that was previously invisible.

Paradox 2 — Fewer breakdowns can make failure data scarcer

Successful prevention reduces the number of real failures available to train future models.

Paradox 3 — A highly accurate model can still be operationally useless

If it predicts too late, cannot explain itself or requires unavailable data, it cannot support maintenance well.

Paradox 4 — A false alarm can be more valuable than silence

For high-consequence assets, the acceptable balance may deliberately favour sensitivity over convenience.

Paradox 5 — Replacing too early can reduce reliability knowledge

If components are always removed before meaningful degradation develops, the organisation learns less about the true failure process.

Paradox 6 — More data can make a worse model

Poorly labelled, inconsistent or drifting data can overwhelm useful evidence.

Paradox 7 — The AI is not the difficult part

The hardest problem may be standardising asset definitions, collecting reliable data, integrating maintenance history, securing engineering windows and changing workflows so that insights actually become action.

Singapore’s network-wide standardisation effort directly addresses this systems problem.

How Predictive-Maintenance Mathematics Grows from School to Research

Primary Mathematics

• averages,
• rates of change,
• tables and graphs,
• percentages and probability.

Secondary Mathematics

• standard deviation,
• trend lines,
• quadratic and exponential functions,
• probability,
• classification tables.

Junior College

• probability distributions,
• conditional probability,
• Bayes’ theorem,
• calculus,
• optimisation,
• time-series reasoning.

University and Research

• reliability engineering,
• survival analysis,
• signal processing,
• machine learning,
• Bayesian inference,
• remaining-useful-life prognostics,
• stochastic optimisation,
• inventory theory,
• continual learning and model monitoring.

Again, reality removes the textbook chapter boundaries.

A vibration trace can become a probability distribution.

The probability can become a maintenance priority.

The priority can become a depot schedule.

The depot decision can prevent a passenger delay.

The World Return — When the Inspection Answers the Algorithm

Predictive maintenance is incomplete until the physical world answers.

The algorithm predicts an anomaly.

An engineer inspects the component.

Several outcomes are possible.

Prediction confirmed

model warning
→ physical degradation found
→ maintenance performed

This becomes evidence that the model detected a meaningful precursor.

Prediction not confirmed

model warning
→ no relevant degradation found

Possible explanations include:

• false alarm,
• inspection method missed the degradation,
• sensor issue,
• or operating context not represented in the model.

No warning, later fault

This is especially important.

The system asks:

Was there precursor evidence?
Did the sensors capture it?
Did the model ignore it?
Was the fault genuinely sudden?
Was the asset operating outside known conditions?

Every failure becomes a model audit.

SMRT’s JARVIS example is useful because the system is not described merely as producing predictions. It has already generated maintenance inspections. That closes the distance between analytics and physical engineering action.

The learning loop is:

sensor data
→ anomaly
→ prediction
→ engineering inspection
→ physical finding
→ maintenance
→ post-maintenance signal
→ labelled evidence
→ model revision

The world is the final label.

A predictive model becomes trustworthy by repeatedly surviving contact with the machine it claims to understand.

RFE Return — What Does Good Predictive Maintenance Owe the Passenger?

The passenger does not need a dashboard showing vibration spectra.

They should not need to know the posterior probability of a door-drive anomaly.

The system exists precisely so that most of this work remains invisible.

Its public outcome should be:

fewer avoidable failures
+
fewer emergency repairs
+
more predictable engineering work
+
more efficient use of parts and staff
+
better-informed renewal
+
stronger reliability

But predictive maintenance also owes the passenger restraint.

It should not become a theatre of artificial certainty.

A good system should know:

• what it measured,
• what it inferred,
• how uncertain the inference is,
• which alarms require human confirmation,
• and when the data are not sufficient to make a strong claim.

The RFE is therefore not:

predict everything

It is:

find enough trustworthy evidence early enough that engineers can prevent avoidable failure without wasting the railway on unnecessary intervention.

Conclusion — The Best Fault Is the One the Passenger Never Meets

A traditional passenger experiences maintenance after the fact.

A train breaks.

Service slows.

Engineers respond.

Predictive maintenance tries to move the intervention backwards in time.

failure
← degradation threshold
← abnormal pattern
← subtle signal drift
← normal operation

The closer the system can detect meaningful evidence to the left side of that timeline, the more choices engineers have.

They may be able to:

• inspect during normal engineering hours,
• order a spare before it is urgently required,
• combine maintenance with other work,
• withdraw one train without disturbing peak service,
• and repair the component before it becomes a passenger event.

This is the true value of the mathematics.

The sensor sees a waveform.

The statistician sees a residual.

The reliability engineer sees an updated hazard.

The prognostics model sees a distribution of remaining useful life.

The planner sees a maintenance window.

The stores team sees a spare-parts requirement.

The engineer sees a component that should be inspected.

The passenger sees nothing.

The train simply arrives.

measurement
→ evidence
→ anomaly
→ risk
→ RUL
→ maintenance decision
→ engineering action
→ physical inspection
→ revised evidence

Singapore’s 2026 move towards network-wide condition-monitoring standards and an integrated asset-data framework is therefore larger than an AI programme.

It is an attempt to make the railway better at knowing its own health.

New Circle Line, North East Line, NSEWL, BPLRT and future Cross Island Line rolling stock increasingly contain the sensors and diagnostic systems required for that transition. SMRT’s JARVIS platform adds another layer by connecting condition data, maintenance history and operational evidence into engineering decisions.

The achievement will not be measured by how many predictions an algorithm makes.

It will be measured by how many predictions become useful actions, how many false alarms are controlled, how much uncertainty is handled honestly, and how many avoidable failures never reach the passenger.

The best predictive-maintenance event is often the disruption that never becomes news because the mathematics found it while it was still only a change in the data.

Key Equations

RMS = √[(1/n)Σxi²]
Signal magnitude feature

s = √[(1/(n−1))Σ(xi − x̄)²]
Sample standard deviation

z = (x − μ)/σ
Standardised anomaly score

zt = λxt + (1−λ)zt−1
Exponentially weighted moving average

et = xt − x̂t
Prediction residual

D² = (x−μ)ᵀΣ−1(x−μ)
Mahalanobis distance

R(t) = P(Tfailure > t)
Reliability function

F(t) = 1 − R(t)
Cumulative failure probability

h(t) = f(t)/R(t)
Hazard rate

RWeibull(t) = exp[−(t/η)β]
Weibull reliability

P(Fault|Data)
= P(Data|Fault)P(Fault)/P(Data)
Bayesian evidence update

RUL = Tf − t
Remaining useful life

H(t) = H0 + kt
Simple linear degradation model

Recall = TP/(TP+FN)
Fault-detection sensitivity

Precision = TP/(TP+FP)
Alarm precision

E[C] = CFP P(FP) + CFN P(FN) + CTP P(TP)
Illustrative decision cost

P(N=k) = e−λ λk/k!
Poisson spare-demand example

Priorityi = (pici)/(ri+ε) × ui
Illustrative maintenance-priority score

zi = (Hi − μpeer)/σpeer
Peer-normalised health score

e(t) = model performance error
Model-drift monitoring

Reader-safety note: All thresholds, component examples, health scores and numerical maintenance decisions in this article are fictional teaching models. They do not reproduce Singapore MRT alarm thresholds, failure limits, maintenance instructions, asset vulnerability data or restricted engineering procedures. Real rail maintenance uses validated asset-specific limits, authorised engineering judgement and confidential operational data.

Continue the MRT Mathematics Series

• How MRT Timing Works Using Mathematics — the full system pillar.
• How MRT Braking Works Using Mathematics — stopping distance and regenerative braking.
• How MRT Headway Works Using Mathematics — train spacing and throughput.
• How MRT Station Dwell Time Works Using Mathematics — queues, doors and platform flow.
• How MRT Delays Propagate and Recover Using Mathematics — instability, regulation and recovery.
• How an MRT Timetable Is Built Using Mathematics — demand, train cycles and robustness.
• How MRT Energy Use Is Optimised Using Mathematics — traction, coasting and regenerative energy.
• How MRT Passenger Capacity Is Calculated Using Mathematics — vehicle, station and delivered capacity.
• How MRT Routes and Transfers Are Optimised Using Mathematics — graph theory, transfer cost and route choice.
• How MRT Network Resilience Is Measured Using Mathematics — redundancy, cascading overload and recovery.

The next natural leg is How MRT Track Inspection Works Using Mathematics: track geometry, rail wear, sensor fusion, computer vision, sampling, defect probability, measurement error and how inspection trains convert kilometres of rail into a continuous health map.

Sources and Further Reading

• LTA, SMRT and SBS Transit — Rail Reliability Taskforce recommendations, condition monitoring and predictive maintenance
• LTA — Independent Advisory Panel call for comprehensive condition monitoring and predictive maintenance
• SMRT — JARVIS predictive analytics platform and 500+ maintenance inspections
• LTA — New Circle Line trains with condition-monitoring systems
• LTA — NEL condition monitoring and Automatic Track Inspection
• LTA — BPLRT condition monitoring for doors, brakes, air-conditioning and power rail
• LTA — NSEWL train and track-circuit condition monitoring
• LTA — Cross Island Line condition monitoring and automated track inspection
• Mechanical Systems and Signal Processing — Review of data-driven prognostics, uncertainty, robustness and interpretability
• Machine Learning with Applications — Comprehensive review of remaining useful life prediction
• Decision Analytics Journal — Time-series algorithms and analytics in predictive maintenance
• Measurement — Predictive diagnostics for railway axle bearings
• Engineering Applications of Artificial Intelligence — Deep-learning anomaly detection in railway systems
• Computers & Industrial Engineering — Continual learning for failure prediction under non-stationary conditions

Return to the complete MRT Mathematics hub: How MRT Works | It’s Mathematics.