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Tensor Contribution Decomposition
Move from where the mismatch occurred to what contributed most strongly.
Continue →Tensor SPC Tool
Tensor Residual Localization
Residual localization identifies where reconstruction mismatch is concentrated.
Detection tells us something changed. Localization shows where and how the structured process behavior changed across sensors and time.
Purpose
Move from anomaly detection to structural diagnosis.
A Tensor SPC alarm should not stop at a single Q statistic. Residual localization decomposes the reconstruction error so the user can see which sensor-time region drove the alarm.
Example finding: Structural anomaly detected. The dominant residual contribution occurs in the vibration and surface-finish response region between time steps 55–85.
Core calculation
𝓔 = 𝓧 − 𝓧^
The residual tensor compares the observed tensor with the model reconstruction.
E2s,t = (Xs,t − X^s,t)2
Residual energy is then visualized by sensor and time to locate the structural mismatch.
Theoretical background
What residual localization adds.
Tensor SPC first determines whether structured behavior is abnormal through the Tensor Q statistic. Tensor Residual Localization decomposes that same reconstruction error across the sensor-time field so the user can see where the structural mismatch is concentrated.
Tensor Q
Is the total reconstruction error large enough to indicate abnormal structured behavior?
Residual Localization
Where did the reconstruction error occur across sensors and time?
Manufacturing example
Same process, clearer diagnosis.
This demonstration uses a structured manufacturing process with several related features. The examples show both normal structured behavior and a review condition. The review case is intentionally subtle: individual signals remain plausible, but the relationship between vibration and surface finish changes during the middle of the run.
Example 1: Normal structured behavior
Residuals remain low and distributed across the sensor-time field.
Tensor Q4.2
Q Limit7.9
StatusNo Review Indicated
Normal residual energy heatmap
low reconstruction mismatchFeed Rate
Spindle Load
Tool Temperature
Vibration
Surface Finish
Coolant Flow
0102030405060708090100110
Time Step
low residual elevated residual high residual
In the normal example, residual energy remains low and scattered. The observed sensor-time behavior is consistent with the learned baseline structure, so no review is indicated.
Normal interpretation: The observation is consistent with the learned structured baseline. No concentrated residual region is present, and Tensor Q remains below the review limit.
Example 2: Structural deviation requiring review
Residual energy becomes concentrated in the vibration and surface-finish response region.
Tensor Q18.7
Q Limit7.9
StatusReview Needed
Residual energy heatmap
sensor × time localizationFeed Rate
Spindle Load
Tool Temperature
Vibration
Surface Finish
Coolant Flow
0102030405060708090100110
Time Step
low residual elevated residual high residual
The heatmap is not raw process data. It shows where the tensor reconstruction error is concentrated. Higher residual intensity indicates a larger mismatch between observed behavior and the reconstructed structured behavior.
Top contributing sensors
share of residual energyTime contribution profile
when the anomaly emergedObserved vs reconstructed response
top contributing signalThe tensor model reconstructs the expected structured response from learned sensor-time relationships. The observed response diverges during the same region highlighted by the residual map.
Interpretation
The tensor model detected a localized structural deviation. The largest mismatch occurred in the vibration and surface-finish response during the middle portion of the observation. Individual process values may still appear plausible, but the coordinated behavior no longer matches the learned sensor-time structure.
Practical meaning: Tensor Residual Localization extends the alarm from “something changed” to “this sensor-time region drove the change.”
How to use the result
Localization supports investigation, not automatic conclusion.
The localization output should guide where to investigate. It does not replace engineering review or traditional SPC; it adds a structured diagnostic layer.
| Output | Question answered | Use |
|---|---|---|
| Tensor Q | Is structured behavior abnormal? | Screen for structural process change. |
| Residual heatmap | Where is the mismatch concentrated? | Identify the sensor-time region that drove the alarm. |
| Sensor contributions | Which signals contributed most? | Prioritize investigation. |
| Observed vs reconstructed | How did actual behavior differ from expected behavior? | Explain the anomaly in process terms. |
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