Hotelling T²
Two readings, each in spec, yet impossible together.
Two readings can each sit comfortably in spec and yet be impossible together. Hotelling T-squared is the multivariate control chart that watches the relationship between signals, catching the joint excursion that single-variable charts cannot see.
What it measures
When process variables are correlated, controlling them one at a time misses faults that live in the correlation. Hotelling T-squared collapses many variables into one statistic that respects their relationships:
- The statistic: T² = n(x̄ − μ)ᵀ S⁻¹ (x̄ − μ), the squared distance of a multivariate observation from the process center, scaled by the covariance matrix S. Because S encodes how the variables move together, T² measures distance in the shape of the real process, not in a box of independent limits.
- The control region: for two variables this is a tilted ellipse, not a rectangle. The rectangle is what you get from two independent charts; the ellipse is the truth. A point inside the rectangle but outside the ellipse is in spec on every individual variable and still out of control jointly.
- The limit: T² is compared to an upper control limit derived from the F-distribution, so a single threshold governs the whole multivariate process.
How to read the output
A T² breach says the joint state of the process is abnormal even when no single variable looks wrong, so the first follow-up is decomposition: which variables, and which broken correlation, drove the statistic up. The classic signature is two normally-correlated signals that have decoupled, temperature and pressure that always track together suddenly diverging, each still in its own band. Read T² alongside the univariate charts, not instead of them: the univariate charts localize a single- variable shift, and T² catches the relationship faults they are blind to. A process in multivariate control keeps T² below its limit while the variables move together within the ellipse.
A real use case
A coating process is governed by slurry viscosity and coating-gap setting, which normally move together: a thicker slurry runs with a wider gap to hold areal weight constant. One day the viscosity drifts up while the gap stays put, each still comfortably inside its own control limits, so two univariate charts show nothing. But the pair has left the control ellipse: this viscosity with this gap means the coating is running heavy, a combination that does not occur in a healthy process. The Hotelling T² chart flags the joint excursion, decomposition points at the viscosity-gap decoupling, and the operator corrects the gap before a roll of overweight electrode is made, a fault that single-variable SPC would have shipped.
Common mistakes
- Controlling correlated variables with independent charts. The rectangle of individual limits passes joint states the process never actually produces when healthy.
- Reading a T² breach without decomposing it. The statistic says something is jointly wrong; you still have to find which variables and which correlation broke.
- Estimating the covariance matrix from too little or already-out-of-control data, so the ellipse itself is wrong.
- Dropping the univariate charts. T² catches relationship faults but is less direct for localizing a plain single-variable shift; the two are complementary.
- Applying it to genuinely independent variables, where it adds complexity without benefit. T² earns its keep when the variables are correlated.
Joint control on correlated signals
Niobia computes the Hotelling T-squared statistic across the correlated process variables you monitor, estimates the covariance structure, and charts T² against its F-distribution control limit so a joint excursion is flagged even when every individual variable is in spec. On a breach it surfaces the contribution of each variable so the broken relationship is identifiable, and it runs alongside the univariate X-bar and R charts so single-variable shifts and multivariate relationship faults are both covered. A joint excursion becomes an alert with the contributing variables named.
Frequently asked
Why isn't running a control chart on each variable enough?
Because correlated variables have joint states that are individually in spec but jointly impossible in a healthy process. Independent charts draw a rectangle of limits; the real control region is a tilted ellipse. Points in the rectangle but outside the ellipse are exactly the faults univariate SPC misses.
What do I do when T-squared signals?
Decompose it. The statistic tells you the joint state is abnormal; the contribution of each variable tells you which signals and which broken correlation drove it. That decomposition turns a multivariate alarm into a specific, actionable cause.
How many variables can it handle?
Many: T-squared generalizes to any number of correlated characteristics through the covariance matrix. The practical limits are having enough clean in-control data to estimate that covariance well, and keeping the variable set to ones that are genuinely related.