Tensor Q Explained

What Tensor Q represents

Tensor Q measures the total structured reconstruction mismatch remaining after the monitored process data is reconstructed using the learned tensor model.

A low Tensor Q value indicates that the observed process behavior closely matches the learned normal tensor structure. A high Tensor Q value indicates that structured behavior exists which the tensor model could not adequately reproduce.

Tensor reconstruction error

Tensor SPC first reconstructs the observed tensor using the learned tensor subspace model. The residual tensor represents the portion of structured behavior left unexplained:

Tensor Q is then computed as the squared Frobenius norm of this residual tensor.

Interpretation

Tensor Q answers a practical engineering question:

Traditional SPC often focuses on individual variable deviations. Tensor Q instead evaluates whether the coordinated multiway structure of the process has changed.

Low Q vs high Q

Q limits and thresholds

Tensor Q values are compared against a learned control threshold or Q limit. This limit represents the expected range of reconstruction error under normal structured behavior.

Since the observed Tensor Q exceeds the learned limit, the process enters a review condition. In practice, Q limits may be estimated from historical baseline behavior, percentile-based thresholds, simulation-based estimation, or statistical confidence approaches.

Detection → localization → contribution