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Tensor SPC Research
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Why Tensor SPC Can Find What Ordinary Charts Miss
This example starts with a simple system: light rises, temperature follows, and ventilation should respond. The fault is not a single extreme sensor value. The fault is that the expected relationship between sensors breaks.
The core idea
The anomaly is in the interaction, not just the value
1
Normal behavior
Light increases. Temperature increases shortly after. Ventilation responds to remove heat.
Light ↑Temperature ↑Ventilation ↑
2
Structural fault
Light and temperature behave normally, but ventilation does not respond as expected.
Light ↑Temperature ↑Ventilation —
3
Ordinary SPC view
Individual values can remain inside limits, so a one-sensor-at-a-time chart may look acceptable.
No obvious single-sensor alarm
4
Tensor SPC view
The sensor × time structure is no longer coherent, so residual energy and Q increase.
Relationship breakdown detected
Controls
Change the relationship fault
The slider controls how strongly the ventilation response is suppressed during the highlighted time window. The sensor values can still look reasonable, but the expected light → temperature → ventilation relationship weakens.
Loading Python runtime…
Result
What changed?
Healthy Q—
Faulted Q—
Increase—
Run the example to compute residual energy.
Interpretation will appear after the first run.
Step 1 and 2
First look at the sensor relationship
This plot is intentionally simple. It shows why the example matters before any tensor terminology is introduced. In the faulted case, ventilation fails to follow the normal response pattern during the highlighted window.
Relationship plot will appear after Python loads.
Step 3
Why ordinary charts can miss it
The values are not necessarily extreme. The issue is that one sensor no longer behaves correctly relative to the others. A chart focused only on individual limits may not show a clear violation.
Classical view will appear after Python loads.
Step 4
What Tensor SPC adds
Tensor SPC compares the observation to the learned sensor × time structure. When the relationship breaks, the model leaves unexplained energy behind. That unexplained energy is the residual, and the total residual energy is summarized by Q.
Q contribution plot will appear after Python loads.
Residual Map
Where the model struggles to reconstruct the observation
The residual map is not a raw data plot. It shows squared reconstruction error after the learned structure is removed. Brighter cells mean the tensor model had more difficulty explaining that sensor at that time.
Residual map will appear after Python loads.
How to read it
The highlighted window is the known fault location
- Raw values: shows the sensor readings themselves.
- Residual energy: shows what the learned sensor × time structure failed to explain.
- Highlighted window: the known time window where the ventilation response was suppressed.
- Q statistic: the total residual energy across the observation.
Main takeaway: Tensor SPC is useful when the problem is not a single high or low value, but a breakdown in how variables move together over time.
Next step
From example to application
A production app can extend this idea with file uploads, rank selection, T²-Q monitoring charts, residual heatmaps, and Excel/PDF reporting. The browser example is intended to make the reason for Tensor SPC visible before introducing the full research workflow.
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