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6.
Process or Product Monitoring and Control
6.5. Tutorials 6.5.4. Elements of Multivariate Analysis
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| Hotelling's \(T^2\) distribution | A multivariate method that is the multivariate counterpart of Student's \(t\) and which also forms the basis for certain multivariate control charts is based on Hotelling's \(T^2\) distribution, which was introduced by Hotelling (1947). | ||
| Univariate \(t\)-test for mean | Recall, from Section 1.3.5.2, $$ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} $$ has a \(t\) distribution provided that \(X\) is normally distributed, and can be used as long as \(X\) doesn't differ greatly from a normal distribution. If we wanted to test the hypothesis that \(\mu = \mu_0\), we would then have $$ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} $$ so that $$ \begin{eqnarray} t^2 & = & \frac{(\bar{x} - \mu_0)^2}{s^2 / n} \\ & & \\ & = & n (\bar{x} - \mu_0)(s^2)^{-1} (\bar{x} - \mu_0) \, . \end{eqnarray} $$ | ||
| Generalize to \(p\) variables | When \(T^2\) is generalized to \(p\) variables it becomes $$ T^2 = n (\bar{{\bf x}} - {\bf \mu}_0) {\bf S}^{-1} (\bar{{\bf x}} - {\bf \mu}_0) \, , $$ with $$ \bar{{\bf x}} = \left[ \begin{array}{c} \bar{x}_1 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_p \end{array} \right] \,\,\,\,\,\,\,\,\,\,\,\, {\bf \mu}_0 = \left[ \begin{array}{c} \mu_1^0 \\ \mu_2^0 \\ \vdots \\ \mu_p^0 \end{array} \right] \, . $$ \({\bf S}^{-1}\) is the inverse of the sample variance-covariance matrix, \({\bf S}\), and \(n\) is the sample size upon which each \(\bar{x}_i, \, i=1, \, 2, \, \ldots, \, p\), is based. (The diagonal elements of \({\bf S}\) are the variances and the off-diagonal elements are the covariances for the \(p\) variables. This is discussed further in Section 6.5.4.3.1.) | ||
| Distribution of \(T^2\) | It is well known that when \(\mu = \mu_0\) $$ T^2 \sim \frac{p(n-1)}{n-p} F_{(p, \, n-p)} \, , $$ with \(F_{(p, \, n-p)}\) representing the F distribution with \(p\) degrees of freedom for the numerator and \(n-p\) for the denominator. Thus, if \(\mu\) were specified to be \(\mu_0\), this could be tested by taking a single \(p\)-variate sample of size \(n\), then computing \(T^2\) and comparing it with $$ \frac{p(n-1)}{n-p} F_{\alpha \, (p, \, n-p)} $$ for a suitably chosen \(\alpha\). | ||
| Result does not apply directly to multivariate Shewhart-type charts | Although this result applies to hypothesis testing, it does not apply directly to multivariate Shewhart-type charts (for which there is no \(\mu_0\), although the result might be used as an approximation when a large sample is used and data are in subgroups, with the upper control limit (UCL) of a chart based on the approximation. | ||
| Three-sigma limits from univariate control chart | When a univariate control chart is used for Phase I (analysis of historical data), and subsequently for Phase II (real-time process monitoring), the general form of the control limits is the same for each phase, although this need not be the case. Specifically, three-sigma limits are used in the univariate case, which skirts the relevant distribution theory for each Phase. | ||
| Selection of different control limit forms for each Phase | Three-sigma units are generally not used with multivariate charts, however, which makes the selection of different control limit forms for each Phase (based on the relevant distribution theory), a natural choice. | ||
