Window influence on CPD score

The idea

A good model turns change into a number

A solid model of the plant's nominal behavior renders its prediction error — the residual — into an interpretable metric of system change: small and noise-like while the plant behaves, and rising once it departs from what the model knows.

Conventionally the residual is read through two sliding windows: a recent test window to watch for the change, and an older reference window to compensate for measurement noise. The score is their ratio:

score = max( D_test / D_ref − 1 , 0 )

where D is the mean residual over a window. It sits near zero while both windows see the same nominal behavior, and rises as the test window slides into a change the reference hasn't reached yet — that rise is the detection.

Try it

Easy to set, easy to read

The score's whole character comes from how the two windows are sized and spaced. Here's what each of the three knobs does — drag them and watch the figure:

test window

The fast, recent view — the detection delay ≈ this length.

shorter · peaks sooner, noisierlonger · peaks later, steadier

reference window

The local baseline — fast wiggles that aren't a real change sit in both windows and cancel.

shorter · fast-adapting, hugs recent normallonger · a steady anchor

lag

The gap between test and reference — it sets how long the alarm stays raised once a change hits.

shorter · fires, then quickly goes silentlonger · the alarm is held up longer

The trade-off

Bound to real value

A threshold turns the score into a yes/no alarm. Set it too low and you cry wolf — a high FAR; too high and you miss the event — a high MAR.

threshold

Don't like setting magic constants? Us neither! The score is a relative increase — the test error as a fraction of the reference (0.25 = 25% above baseline) — so it is dimensionless and the same threshold transfers across signals of any scale.
Not convinced? Check the sweep view over all reasonable thresholds.

sweep view

The threshold-free summary is the AUC in the table: higher is better, and it doesn't depend on where you set the bar.

Make it harder

Real changes aren't clean steps

Slow drifts and noisy sensors are what actually break detectors. Make the transition gradual and turn the noise up — the delay grows and the AUC falls. But the detection

change type

transition — sharp ↔ slow

noise

The real thing

A two-tank plant, six ways to identify it

So far so good? Now comes the difficult part! The whole idea holds only if we can identify a model that tracks the system well. Here the residual come from a real nonlinear two-tank simulation whose outflow slowly fouls.

The model is one of six DMD identifiers — batch vs online, input-blind vs control-aware, full vs rank-truncated. Switch them on and compare: which fires first, with fewest false alarms?

The upper panels then show each tank's level in metres (grey = plant, coloured dashed = each model's prediction) and the input flow. The residual is the plant–model mismatch — the gap between the grey and dashed lines — so it is in metres too; where a model's prediction peels off the plant, its residual climbs.

Go deeper

Reproduce & explore

Export the exact figure or the underlying numbers.

Export figure & data

Glossary. FAR = false-alarm rate · MAR/s = missed per sample · MAR/e = missed per event · AUC = ranking quality · delay = samples to detection. DMD variants: c = control-aware, t = rank-truncated, o/online = streaming/adaptive.