Tensor SPC Tool
Tensor Contribution Decomposition
Residual localization shows where structured mismatch occurred. Contribution decomposition shows which sensors and time regions drove the Tensor Q statistic.
Purpose
Move from location to contribution.
A Tensor SPC alarm becomes more useful when the total residual energy is decomposed into ranked contributions. This helps the user understand which variables, time regions, and sensor-time patterns should be reviewed first.
Core calculation
The Tensor Q statistic is the total residual energy for one observation.
Sensor contribution expresses each feature's share of the total residual energy.
Time contribution identifies when the structured mismatch was strongest.
Theoretical background
What contribution decomposition adds.
Tensor Residual Localization answers where the model reconstruction error is concentrated. Tensor Contribution Decomposition ranks that same residual energy by sensor and time, turning a heatmap into a prioritized investigation path.
Residual Localization
Where did the structured mismatch occur across the sensor-time field?
Contribution Decomposition
Which sensors and time regions contributed most strongly to the Tensor Q result?
Manufacturing example
Normal behavior compared with a review condition.
The examples use the same structured manufacturing process introduced in the residual localization page. The normal example shows low, distributed residual energy. The review example shows concentrated contribution from vibration and surface finish during the middle of the observation.
Example 1: Normal structured behavior
Residual energy is low and broadly distributed. No single sensor or time region dominates the Tensor Q statistic.
Normal contribution map
distributed residual energyTop contributing sensors
normal exampleTime contribution profile
normal exampleInterpretation
The normal example shows low residual energy with no dominant sensor-time contribution. The Tensor Q value remains below the review limit, and the contribution profile does not identify a concentrated structural mismatch.
Example 2: Structural deviation requiring review
Residual energy is concentrated in vibration and surface finish during the middle of the process cycle.
Review contribution map
concentrated residual energyHigher contribution intensity indicates a larger share of the Tensor Q statistic from that sensor-time region.
Top contributing sensors
review exampleTime contribution profile
review exampleInterpretation
The review example shows a concentrated contribution pattern. Surface Finish and Vibration account for most of the Tensor Q statistic, and the time contribution profile peaks during the middle of the cycle. This suggests the review should begin with the coordinated vibration/surface-finish behavior rather than treating the result as a generic process alarm.
Practical meaning: Tensor Contribution Decomposition extends localization by ranking what drove the structural anomaly.
How to use the result
Contribution decomposition prioritizes investigation.
The output should not be treated as an automatic root cause. It is a structured diagnostic guide that helps focus review on the variables and time regions most responsible for the Tensor Q result.
| Output | Question answered | Use |
|---|---|---|
| Tensor Q | Is structured behavior abnormal? | Screen for a structural process change. |
| Residual localization | Where is the mismatch concentrated? | Locate the sensor-time region of concern. |
| Sensor contribution | Which features drove the result? | Prioritize the investigation path. |
| Time contribution | When did the mismatch dominate? | Identify the process window requiring review. |