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Multivariate process monitoring with Hotelling T² and SPE
Read correlated sensors through T² and SPE together rather than one statistic, interpret contribution plots, and learn where the method’s assumptions break.
What are Hotelling T2 and SPE in multivariate process monitoring?
Short answer
Hotelling T2 measures how extreme a point is within the normal variation captured by a principal component analysis (PCA) model plane. SPE, also called Q, measures how far the point lies off that plane, which signals that the correlation structure has broken. The two statistics catch different faults, so both are monitored together.
Why monitor variables together
Flow, pressure and motor current usually move together. A chart for each variable checks limits one at a time, so it can miss a combination that is unusual although each variable stays in range.
Principal component analysis (PCA) reduces the normal data to the k directions that carry the most variation, which define the model plane. T2 measures where a point sits in that plane, relative to normal variation: a Mahalanobis-type distance. SPE is the squared distance from the point to the plane. It rises when the point no longer follows the normal correlation pattern. The two respond to different deviations, so both are needed.
Hypothetical two-variable example
Suppose a line has two standardised variables, flow z1 and pressure z2, with normal correlation 0.9. The first principal component points along (1, 1)/√2 with variance 1 + 0.9 = 1.9, so k = 1 gives a model line.
Point A = (2, 2): both values are two standard deviations high, as in normal behaviour. Its score is t = (2 + 2)/√2 ≈ 2.83, so T2 = t²/λ = 8/1.9 ≈ 4.21. The residual is zero, so SPE = 0. The point is extreme along the normal pattern but has not broken it.
Point B = (2, −2): each value is again two standard deviations from its mean, but in opposite directions. Its score is t = 0, so T2 = 0. The residual is (2, −2), so SPE = 2² + (−2)² = 8. Point B sits at the centre of the model line but far from it, so only SPE flags it.
Limits and common mistakes
Both statistics assume roughly linear relationships and clean normal training data. If a fault is hidden in the training window, the model learns it as normal and the alarm never fires. Variables must be standardised first; otherwise a sensor with large numeric units, such as pressure in pascal rather than bar, will dominate the components.
Control limits can come from F-distribution based approximations or from empirical percentiles of the training data, and both rest on the normal-data assumption. Retrain after maintenance or a regime change. When an alarm fires, contribution plots show which variables drove the statistic. That is the start of diagnosis, not the answer.
Follow-up questions
Which should I watch, T2 or SPE?
Both. T2 signals unusual movement within the normal correlation pattern, while SPE signals that the pattern itself has changed, for example when a sensor stops following its partners. Watching only one leaves a whole class of faults undetected.
Can I use a fixed threshold for T2?
A single fixed number is usually not appropriate. The limit depends on the number of variables and components, the number of training samples, and the false-alarm rate you accept. Practitioners derive it from an F-distribution approximation or from percentiles of the normal training data, and recheck it when the process changes.
Sources
- NIST/SEMATECH e-Handbook: 6.5.4.3. Hotelling's T squared ↗
- Principal component analysis (Wikipedia) ↗
Example numbers are hypothetical, not measured field results.
Read the full guide.
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