Hotelling’s T2 statistic

The true values of X and jx are usually estimated from a small sample of size n. When n is very large, the estimates x and S are very good however, n is usually small, and thus the estimates x and S have a lot of uncertainty. In this case it is necessary to make an adjustment for the confidence interval, 100%(1 - a), of the sample mean and scatter matrix by use of Hotelling s T2 statistic. 

Mahalanobis Distances and Probability Densities from Hotelling s T2 Statistic for Test Samples of Sulfamethoxazole Compared with the Sulfamethoxazole Training Set... 

TVT is diagonal and contains the R largest eigenvalues of X X on its diagonal. It is readily seen that the squared Mahalanobis distance is closely related to Hotelling s T2 statistic, the only difference being a scalar correction. This distance is used extensively for outlier detection, e.g. in multivariate statistical process control (see Chapter 10). 

Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

Several statistics that can be used to monitor the performance of the controller have been defined.44 45 Square prediction error (SPE) gives an indication of the quality of the PLS model. If the correlation of all variables remains the same, the SPE value should be low, and indicate that the model is operating within the limits for which it was developed. Hotelling s T2 provides an indication of where the process is operating relative to the conditions used to develop the PLS model. Thus the use of a PLS model within a control system can provide information on the status of the control system. 

The following statistical measures are those most commonly found in software packages. First we mention HOTELLING S T2 for the 2-class case which is based on a generalized distance measure, the MAHALANOBIS distance D2, and from which a / -test can be derived ... 

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