Control Charts

Multivariate Control Chart (Hotelling T²)

Hotelling T² for individual observations with Phase II UCL from the F distribution.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Control Charts > Multivariate Charts > T²

Description

Computes T²ᵢ = (xᵢ − x̄)′ S⁻¹ (xᵢ − x̄) for every observation. The Phase II individuals UCL is UCL = (p(m+1)(m−1))/(m²−mp) · F_{α, p, m−p}, with p = number of variables, m = baseline sample size, α = false-alarm probability (typically 0.0027).

Formulas

T² statistic

T_{i}^{2} = (x_{i} - \overset{ˉ}{x})^{'} S^{- 1} (x_{i} - \overset{ˉ}{x})

Squared Mahalanobis distance

UCL Phase II

UCL = \frac{p ( m + 1 ) ( m - 1 )}{m ^{2} - m p} F_{α, p, m - p}

Individual observations

Assumptions

Observations multivariate normal
Baseline m > p, otherwise covariance singular

Limitations

Highly correlated variables → covariance unstable or singular
A T² alarm does not say which variable is responsible

References

Hotelling, H. (1947), Multivariate Quality Control
Montgomery, Introduction to Statistical Quality Control, 8th Ed.