industrial//process monitoring//multivariate statistical process control//T-squared statistic
The T-squared statistic, also called Hotelling's, is a monitoring statistic that measures how far a sample lies from the centre of normal operation inside the plane of normal correlations, in units of the normal spread along each direction; it is used to flag a plant running at an unusual but internally coherent point. A record production day, a heavy load or an unusual feedstock push the plant far along its normal modes while every relation between variables holds, and that shows here.
The T-squared statistic, also called Hotelling's, is a monitoring statistic that measures how far a sample lies from the centre of normal operation inside the plane of normal correlations, in units of the normal spread along each direction; it is used to flag a plant running at an unusual but internally coherent point. A record production day, a heavy load or an unusual feedstock push the plant far along its normal modes while every relation between variables holds, and that shows here.
With the sample's coordinates t=VkTxt=V_k^{\mathsf T}xt=VkTx in the kkk principal directions fitted on normal data, and λi\lambda_iλi the variance of normal operation along each, the statistic is
T2=∑i=1kti2λi.T^2=\sum_{i=1}^{k}\frac{t_i^2}{\lambda_i}.T2=i=1∑kλiti2.
Each coordinate is divided by its own spread before squaring, so a step along a direction where the plant normally varies a lot counts for little and a step along a quiet direction counts for much. It is the Mahalanobis distance restricted to the normal plane; on all ppp variables at once it would be the plain Mahalanobis distance, whose inverse covariance becomes unstable when forty correlated sensors carry only a handful of independent directions.
A high T-squared says unusual, not broken.
It can be a fault that pushes the plant along its normal modes (a fouled exchanger raising every temperature together), or simply a good day. Whether the relations themselves broke is the job of the Q statistic, and the two are read together.
Its control limit comes from an F distribution under Gaussian normal data, or more robustly from the 99th percentile of the statistic on held-out normal operation. With non-Gaussian data or several modes mixed in the training weeks, the F limit misleads and the percentile is safer.
It is the multivariate heir of the single-variable control chart: one number per sample, a line it should not cross, and an operator who sees at a glance whether the plant is inside its normal envelope.
It inherits the choice of kkk. Keep too many components and the noisy small ones, divided by tiny λi\lambda_iλi, inflate it with false alarms; keep too few and real modes of variation move into Q instead.
Both statistics run on every sample in multivariate statistical process control.