Tensor SPC Method
How Tensor SPC works
Tensor SPC extends traditional process monitoring by preserving the natural multiway structure of process data. Instead of treating a run as one long row of numbers, Tensor SPC keeps the relationship between observations, sensors, and time visible to the model.
Core idea
Tensor SPC monitors structure, not just individual values.
Many process signals are meaningful because of how they move together. A single variable may look acceptable while the coordinated process behavior changes. Tensor SPC is designed to detect those structured changes.
Excellent for individual variables, shifts, spikes, and instability in one signal at a time.
Useful for multivariate behavior, but the sensor-time layout is converted into a long vector.
Preserves observation × sensor × time structure so the model can learn coordinated behavior.
Data representation
Flattening versus preserving structure
The same data can be represented two different ways. The first approach flattens sensor-time data into one long vector. The tensor approach keeps each mode separate, which makes the relationships easier to model and interpret.
Modeling step
The tensor model learns normal structured behavior.
A low-rank tensor model represents the dominant patterns in the data. In practical terms, it learns common sensor relationships, time patterns, and observation-to-observation variation.
Low-rank tensor approximation
𝓧 is the structured data tensor. 𝓖 is the core tensor. The factor matrices represent major patterns by mode: observations, sensors, and time.
Mode interpretation
| Mode | Meaning |
|---|---|
| A | Observation/run behavior |
| B | Sensor relationship patterns |
| C | Time evolution patterns |
Keep observation × sensor × time layout.
Learn dominant low-rank process structure.
Estimate what normal structured behavior should look like.
Flag behavior the model cannot reconstruct well.
Monitoring statistics
Tensor SPC uses both latent scores and residual energy.
Two common monitoring ideas are retained: a score-space statistic for unusual movement within the learned structure, and a residual statistic for behavior outside that structure.
Score-space monitoring
This monitors whether a new observation is unusual within the learned latent tensor structure.
Residual monitoring
This monitors how much structured behavior remains unexplained after reconstruction. A high Q value indicates behavior the normal tensor model could not reproduce.
Visual explanation
Residual maps help show where the structure changed.
Because the tensor keeps sensor and time modes separate, residual energy can be displayed as a sensor-time map. This makes the result more diagnostic than a single alarm number.
Where it fits
Tensor SPC is an extension, not a replacement.
Traditional SPC should remain the first layer for individual process stability. Tensor SPC adds a second layer for structure-aware monitoring when processes involve meaningful relationships across variables and time.
| Monitoring layer | Best at detecting | Typical question answered |
|---|---|---|
| Traditional SPC | Single-variable shifts, spikes, instability, and special causes | Did this signal move outside expected behavior? |
| PCA/MSPC | Overall multivariate variation after flattening | Is this observation unusual in multivariate space? |
| Tensor SPC | Sensor-time relationship changes and structured residual patterns | Did the process stop behaving coherently? |