Method Guide

What is Tensor-Structured SPC?

Tensor SPC is a structure-aware extension of multivariate process monitoring. It is designed for data that naturally have multiple modes, such as sensor × time × condition, rather than data that should be flattened into one long vector.

The problem with flattening

Traditional PCA/MSPC often converts structured observations into a vector. That can work, but it hides relationships among sensors, time, stages, locations, or operating conditions. For manufacturing and test data, those relationships often carry the process signal.

Tensor SPC keeps the observation in its natural form. A process signature can remain a matrix or higher-order tensor while the model learns dominant structure across each mode.

A tensor observation can preserve several modes at once: variables, time points, and conditions.

Core idea

Instead of modeling a flattened vector, Tensor SPC uses multilinear decomposition, such as Tucker/HOSVD, to represent the dominant low-rank structure of the process. New observations are compared to this learned structure using complementary monitoring statistics.

Tucker decomposition

Separates variation across modes and produces a low-dimensional representation of normal structured behavior.

Tensor T²

Measures unusual behavior inside the modeled score space. It is read as “T-squared,” following Hotelling-style notation.

Tensor Q

Measures residual energy outside the modeled tensor subspace. It is useful for localized or unmodeled anomalies.

T² and Q provide complementary views: modeled-structure variation and residual variation.

How the statistics are interpreted

High T², low Q: the sample follows the modeled structure but is unusual within that structure.
Low T², high Q: the sample contains variation that the model does not explain well.
High T² and high Q: the sample is unusual both inside and outside the modeled subspace.

This distinction is why Tensor SPC can support both detection and interpretation.

Plain-language summary

Tensor SPC keeps the shape of the data intact.

It monitors whether a new structured observation behaves like the learned normal structure. When something changes, T² and Q help separate whether the change is part of the modeled pattern or residual structure outside the model.

Continue to worked example