Yong Chen - Industrial Data Analytics for Diagnosis and Prognosis

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 μis defined as

It is clear that the confidence region for μis an ellipsoid centered at x̄. Similar to the univariate case, the null hypothesis H 0: μ= μ 0is not rejected at level α if and only if μ 0is in the 100(1 − α )% confidence region for μ.

The T 2-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 μand Σin ( 3.16) and ( 3.17), respectively, into the likelihood function in ( 3.13), it is easy to see

where is the MLE of Σgiven in ( 3.17). Under the null hypothesis H 0: μ= μ 0, the MLE of Σwith μ= μ 0fixed can be obtained as

It can be seen that is the same as except that X̄is replaced by μ 0.

The likelihood ratio test statistic is the ratio of the maximum likelihood over the subset of the parameter space specified by H 0and the maximum likelihood over the entire parameter space. Specifically, the likelihood ratio test statistic of H 0: μ= μ 0is

(3.23)

The test based on the T 2-statistic in ( 3.21) and the likelihood ratio test is equivalent because it can be shown that

(3.24)

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_defectdata 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 Rcodes calculate the sample mean, sample covariance matrix, and the T 2-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 Rcodes, the sample mean and sample covariance matrix are obtained as

The T 2-statistic is obtained by ( 3.21) as T 2= 19.71. The right-hand side of ( 3.22) at α = 0.05 is obtained as F 0= 8.13. Since the observed value of T 2exceeds the critical value F 0, we reject the null hypothesis H 0and conclude that the mean vector of the three side temperatures of the defective billets is significantly different from the nominal mean vector. In addition, the p -value is 0.0004 < α =0.05, which further confirms that H 0should be rejected.

Let D = { x 1, x 2,…, x n } denote the observed data set. In the maximum likelihood estimation, the distribution parameters are considered as fixed. The estimation errors are obtained by considering the random distribution of possible data sets D . By contrast, in Bayesian inference, we treat the observed data set D as the only data set. The uncertainty in the parameters is characterized through a probability distribution over the parameters.

In this subsection, we focus on Bayesian inference of normal distribution when the mean μis unknown and the covariance matrix Σis assumed as known. The Bayesian inference is based on the Bayes’ theorem. In general, the Bayes’ theorem is about the conditional probability of an event A given that an event B occurs:

Applying Bayes’ theorem for Bayesian inference of μ, we have

(3.25)

where g ( μ) is the prior distribution of μ, which is the distribution before observing the data, and f ( μ| D ) is called as the posterior distribution , which is the distribution after we have observed D . The function f ( D | μ) on the right-hand side of ( 3.25) is the density function for the observed data set D . If it is viewed as a function of the unknown parameter μ, f ( D | μ) is exactly the likelihood function of μ. Therefore the Bayes’ theorem can be stated in words as