Industrial Data Analytics for Diagnosis and Prognosis. Yong Chen
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which can be interpreted as the standardized distance between the sample mean X̄ and μ0. The distance is standardized by S/n, which is equal to the sample covariance matrix of X̄. When the standardized distance between X̄ and μ0 is beyond the critical value given in the right-hand side of (3.22), the true mean is not likely equal to be μ0 and we reject H0.
The concept of univariate confidence interval can be extended to multivariate confidence region. For p-dimensional normal distribution, the 100(1 − α)% confidence region for μ is defined as
It is clear that the confidence region for μ is an ellipsoid centered at x̄. Similar to the univariate case, the null hypothesis H0 :μ = μ0 is not rejected at level α if and only if μ0 is in the 100(1 − α)% confidence region for μ.
The T2-statistic can also be derived as the likelihood ratio test of the hypotheses in (3.20). The likelihood ratio test is a general principle of constructing statistical test procedures and having several optimal properties for reasonably large samples. The detailed study of the likelihood ratio test theory is beyond the scope of this book.
Substituting the MLE of μ and Σ in (3.16) and (3.17), respectively, into the likelihood function in (3.13), it is easy to see
where
It can be seen that
The likelihood ratio test statistic is the ratio of the maximum likelihood over the subset of the parameter space specified by H0 and the maximum likelihood over the entire parameter space. Specifically, the likelihood ratio test statistic of H0 : μ = μ0 is
The test based on the T2-statistic in (3.21) and the likelihood ratio test is equivalent because it can be shown that
Example 3.2: Hot rolling is among the key steel-making processes that convert cast or semi-finished steel into finished products. A typical hot rolling process usually includes a melting division and a rolling division. The melting division is a continuous casting process that melts scrapped metals and solidifies the molten steel into semi-finished steel billet; the rolling division will further squeeze the steel billet by a sequence of stands in the hot rolling process. Each stand is composed of several rolls. The side_temp_defect data set contains the side temperature measurements on 139 defective steel billets at Stand 5 of a hot rolling process where the side temperatures are measured at 79 equally spaced locations spread along the stand. In this example, we focus on the three measurements at locations 2, 40, and 78, which correspond to locations close to the middle and the two ends of the stands. The nominal mean temperature values at the three locations are 1926, 1851, and 1872, which are obtained based on a large sample of billets without defects. We want to check if the defective billets have significantly different mean side temperature from the nominal values. We can, therefore, test the hypothesis
The following R codes calculate the sample mean, sample covariance matrix, and the T2-statistic for the three side temperature measurements.
side.temp.defect <- read.csv("side_temp_defect.csv",
header = F) X <- side.temp.defect[, c(2, 40, 78)] mu0 <- c(1926, 1851, 1872) x.bar <- apply(X, 2, mean) # sample mean S <- cov(X) # sample var-cov matrix n <- nrow(X) p <- ncol(X) alpha = 0.05 T2 <- n*t(x.bar-mu0)%*%solve(S)%*%(x.bar -mu0) F0 <- (n-1)*p/(n-p)*qf(1-alpha, p, n-p) p.value <- 1 - pf((n-p)/((n-1)*p)*T2, p, n-p)
Using the above R codes, the sample mean and sample covariance matrix are obtained as