JPM turns resistance measurements into an individualized, continuously updated risk estimate. It models each cell’s internal-resistance trajectory, separates common degradation behavior from cell-specific variation, and then uses the cell-specific trajectory as a time-dependent input to a Cox proportional hazards model. A Bayesian update recalculates that cell’s survival function and failure probability as new testing data arrives—without refitting the population-level model at every time step.
Core takeaway: JPM connects how a cell is degrading to how likely it is to fail. Internal resistance provides the evolving health signal, while the survival model converts that signal into an updated, cell-specific failure probability.
How JPM Connects Condition Monitoring to Failure Risk
The two data streams
JPM combines two fundamentally different types of information:
- Time-to-event data: when cells fail, or whether testing ends before failure occurs.
- Condition-monitoring data: continuous or repeated measurements such as internal resistance over cycling time.
Time-to-event data describes the failure behavior of the population. Internal resistance describes the evolving health state of each individual cell.
Internal resistance as the health signal
Internal resistance is typically calculated from dynamic voltage and current changes:
[ R = \frac{\Delta V}{\Delta I} ]
The current must vary between the measurement points. Under perfectly constant current, (\Delta I) is zero, so this calculation is not meaningful.
Resistance is useful because its trajectory often reflects degradation. As a cell ages, increasing resistance can indicate declining conductivity, loss of active material, or other internal changes associated with reduced performance.
How JPM Models Individual Cell Degradation
Population trends and cell-specific behavior
JPM commonly models the condition-monitoring signal with a linear mixed-effects model. A representative resistance trajectory can be written as:
[ CM_i(t) = b_{i0} + b_{i1}t + b_{i2}t^2 + e_i ]
Here, (CM_i(t)) is the measured condition signal for cell (i) at time (t).
The fixed effects describe the average resistance trend across the cell population. The random-effects vector (b_i) captures how a particular cell differs from that average, including its initial resistance and its individual degradation rate or curvature.
Why random effects matter
Two cells can have similar average behavior but different individual trajectories. One may begin with higher resistance, while another may degrade more rapidly.
The random-effects structure allows JPM to represent this cell-to-cell variability rather than forcing every cell to follow one common degradation curve.
Measurement noise is separated from degradation
The term (e_i) represents measurement error or short-term variation. Separating noise from the underlying trajectory helps prevent a single irregular resistance measurement from being interpreted as a genuine change in cell health.
How the Resistance Model Produces Failure Probabilities
Resistance becomes a time-dependent hazard input
The modeled resistance trajectory is embedded in a Cox proportional hazards model. In general form, the hazard for cell (i) at time (t) can be represented as:
[ h_i(t) = h_0(t)\exp{\beta , CM_i(t)} ]
where:
- (h_i(t)) is the cell’s instantaneous failure hazard,
- (h_0(t)) is the baseline hazard for the population,
- (CM_i(t)) is the cell’s current modeled condition signal,
- (\beta) measures how the condition signal affects failure risk.
If higher resistance is associated with failure, a rising modeled resistance value increases the estimated hazard when (\beta) is positive.
From hazard to survival
The hazard is converted into a survival function, which expresses the probability that the cell remains operational beyond a future time:
[ S_i(t) = P(T_i > t) ]
The corresponding failure probability by time (t) is:
[ P(T_i \leq t) = 1 - S_i(t) ]
This enables JPM to estimate not only whether a cell is at elevated risk, but also its probability of surviving to a specified future cycle or test time.
How JPM Updates Predictions During Testing
Bayesian updating of the individual cell
At the beginning of testing, a cell’s individual degradation parameters are uncertain. JPM starts with population information about the random effects and combines it with that cell’s observed resistance history.
As new resistance measurements arrive, Bayesian updating revises the posterior distribution of the cell’s random-effects vector (b_i).
This progressively improves the estimate of the cell’s actual degradation trajectory.
Real-time prediction without refitting the whole model
The key operational advantage is that JPM does not need to re-estimate all fixed effects, baseline hazard parameters, and population distributions after every new measurement.
Instead, the established population model remains in place while the individual cell’s parameters are updated. The revised parameters are then used to calculate an updated survival function and failure probability.
Why predictions improve with more data
Early in life, resistance trajectories from different cells may be very similar. At that stage, there is limited evidence for distinguishing which cells are likely to fail early.
As cycling continues, differences in resistance level and growth rate become clearer. The model can then make more discriminating predictions, often improving classification accuracy and sensitivity.
What the Testing System Must Provide
Dynamic voltage and current measurements
Accurate resistance estimation requires dynamic electrical behavior, such as voltage and current measurements taken at distinct operating points.
A laboratory battery testing system must therefore capture sufficiently resolved voltage and current responses rather than relying only on steady-state constant-current data.
Consistent measurement conditions
Resistance depends on operating conditions, including temperature, state of charge, discharge rate, and measurement timing. These conditions should be controlled or included in the analysis so that changes in the signal are not confused with changes in health.
A sufficiently long monitoring history
JPM benefits from the trajectory of the signal, not merely from one resistance value. Repeated measurements reveal whether resistance is stable, increasing gradually, or accelerating.
That history is what allows the model to distinguish ordinary variation from meaningful degradation.
Understanding the Trade-offs
A fixed degradation curve may be inadequate
Standard JPM formulations often assume a smooth, continuous degradation pattern. Real battery resistance growth may instead exhibit a change point, after which degradation accelerates sharply.
If this behavior is ignored, the model can estimate the resistance trajectory inaccurately and produce poor failure or remaining-life predictions.
Change-point extensions may be necessary
A change-point formulation, such as a JPM-C approach, introduces an unknown point at which the degradation dynamics change. The change point can be estimated from the observed signal, using measures such as the Concordance Correlation Coefficient.
This is more flexible, but it also adds model complexity and requires enough data around the transition to estimate the change reliably.
Resistance is informative but not definitive
Internal resistance is a valuable health indicator, but it is not a universal failure mechanism detector. Resistance can vary with temperature, SOC, test rate, and electrochemical operating regime.
For robust prognostics, resistance may be combined with other signals, such as capacity, voltage response, temperature, or pressure, when those measurements are available and physically relevant.
Early predictions remain uncertain
JPM cannot create information that the test has not yet revealed. When cells have nearly identical early-life signals, individual failure probabilities may remain close to the population estimate.
The model’s uncertainty should therefore be reported alongside the predicted probability, especially during the early testing period.
How to Apply This to Your Project
JPM is most effective when the failure definition, dynamic resistance measurement process, and statistical model are designed as one integrated workflow.
- If your primary focus is real-time cell screening: Use repeated dynamic resistance measurements and update each cell’s random effects as data arrives, allowing the Cox model to rank cells by current failure risk.
- If your primary focus is remaining useful life: Model the full resistance trajectory, test for degradation acceleration, and use a change-point extension if a single smooth curve does not fit the data.
- If your primary focus is population reliability: Use time-to-failure outcomes to estimate the baseline hazard and use resistance histories to explain why individual cells deviate from the population trend.
- If your primary focus is prediction accuracy: Control measurement conditions and consider complementary health signals so that temperature- or SOC-driven resistance changes are not mistaken for irreversible degradation.
By combining population failure behavior with continuously updated cell-specific degradation evidence, JPM converts battery test data into actionable individual risk predictions.
Summary Table:
| Key Element | Description |
|---|---|
| Data Inputs | Time-to-event data (failures) and condition-monitoring data (e.g., internal resistance) |
| Health Signal | Internal resistance (ΔV/ΔI) tracked over cycling time |
| Model Components | Linear mixed-effects model: $CM_i(t) = b_{i0} + b_{i1}t + b_{i2}t^2 + e_i$ |
| Survival Model | Cox proportional hazards: $h_i(t) = h_0(t)\exp{\beta , CM_i(t)}$ |
| Updating | Bayesian updating of cell-specific random effects as new data arrives |
| Output | Updated survival function and failure probability for each cell |
| Key Advantage | Real-time predictions without refitting entire population model |
| Challenges | Potential for change points; resistance not sole failure indicator |
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