Can the components predict the decline?

The question

A lithium-ion battery (NASA cell B0005) was charged and discharged 168 times, and its capacity, the charge it can still deliver, was measured after every cycle. If a computer splits the first 118 measurements into a slow trend and faster wiggles, forecasts each part and adds the forecasts up, does it predict the last 50 cycles better than forecasting the curve directly?

The answer

No, not on this battery. Over the last 50 cycles the split-and-add forecast (CEEMDAN + LSTM) has a typical error (RMSE) of 0.052 Ah, 2.9 times that of an LSTM trained on the capacity curve directly (0.018 Ah). It does beat a straight-line trend (0.075 Ah). Compare the shapes of the forecasts, not just their average errors.

Split → train separately → add

Both methods see only the first 118 measured cycles. CEEMDAN, a signal-decomposition method, splits that capacity curve into components called IMFs: three small, fast wiggles (IMF 1–3) and one slow downward trend (IMF 4). Added together, they give back the original curve.

Original capacity · 118 cycles

Train a separate model for each component

16 consecutive values→ own LSTM →next value

Each component gets its own LSTM, a neural network that reads a sequence of values. It learns from a sliding window: cycles 1–16 predict cycle 17, cycles 2–17 predict cycle 18, and so on, giving 102 training examples per component.

Each component is standardized (shifted and scaled) using its training values only. Each model predicts the change from the last value, adds it to that value, and is trained against the actual next value for 300 epochs (passes over the training examples).

Solid: training historyDashed + shaded: forecast

IMF 1 LSTM 1-0.039 to 0.038 Ah

IMF 2 LSTM 2-0.025 to 0.028 Ah

IMF 3 LSTM 3-0.024 to 0.024 Ah

IMF 4 LSTM 41.357 to 1.833 Ah

Cycle 1118168

Σ Add the component forecasts

Convert each component forecast back to Ah and add them cycle by cycle. The sum is the CEEMDAN + LSTM forecast in “Then, compare their forecasts”.

Direct route

Original capacity curve → one LSTM (the raw LSTM) → forecast

How to read the rows: each row shows one component from cycle 1 to 168. The solid line is its training history; after the dashed boundary at cycle 118, the dashed line in the shaded area is that component’s forecast for the 50 later cycles. Each row has its own vertical scale, printed beside its name. Every forecast step uses the model’s own earlier predictions as input, never later measurements. CEEMDAN also leaves a residue (what remains after the four components); here it is zero to rounding error, so it is carried forward without a model.

Then, compare their forecasts

Learned forecast Compare with

Measured capacityLearned forecastComparison forecast

Reveal held-out measurementsThrough cycle 168

Typical error over the 50 held-out cycles · RMSE, Ah

Same held-out cycles and error measure

How to read this chart: the Measured capacity line is the battery’s real capacity; the shaded area after the dashed line at cycle 118 holds the 50 cycles no model was trained on. Both forecasts were made once from the first 118 cycles, each step using the model’s own earlier predictions, never later measurements. Use the reveal button (Replay the reveal) or drag the slider to uncover the measured values one cycle at a time; the forecasts do not change.

Scores and training settings

Method / protocol	RMSE (Ah)	MAE (Ah)
Raw LSTM · one step, measured history	0.009832	0.006400
Raw LSTM · recursive	0.017799	0.014947
CEEMDAN + LSTM · recursive	0.051508	0.047141
Persistence	0.076269	0.067619
Linear trend	0.074839	0.065231

“One step” gives the raw LSTM the 16 true previous measurements at every step, so its errors cannot build up; “recursive” feeds each prediction back in as input, as in the chart above. Data: NASA battery B0005; see the Dataset page.

.forecast-stage{display:none}JavaScript enables the forecast menus and the reveal. Saved comparison: black measured capacity, blue CEEMDAN + LSTM, orange linear trend; the dashed line marks the end of training (cycle 118).Cycle 1118 · training ends168