Hotelling’s statistic

Multivariate monitoring charts based on Hotelling s statistic (T ) and squared prediction errors SPEx and SPEy) are constructed using the PLS models. Hotelling s statistic for a new independent t vector is [298]... 

The sum of normalized squared scores, known as the Hotelling s statistic, is a measure of the variation in each sample within the PCA model. The T statistic is defined as ... 

FIGURE I Hotelling s plot for a given process. The horizontal line corresponds to the critical threshold of the Hotelling s statistic. The points circled in red correspond to samples that are outside the acceptable limits for the process. See Figures 2 and 3 for contribution plots of samples marked Fault I and Fault 2. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this chapter.)... 

Suppose that it is desired to use SPC techniques to monitor p variables, which are correlated and normally distributed. Let x denote the column vector of these p variables, x = col x, X2,..., Jc ]. At each samphng instant, a subgroup of n measurements is made for each variable. The subgroup sample means for the Ath sampling instant can be expressed as a column vector x(k) = col k), X2 k), Xp k) Multivariate control charts are traditionally based on Hotelling s statistic (Montgomery, 2009). 

Consider the wastewater treatment problem of Examples 21.5 and 21.6 and five new pairs of measurements shown below. Calculate the value of Hotelling s statistic for each pair using the information for Example 21.6, and plot the data on a 7 chart. Based on the number of chart violations for the new data, does it appear that the current process behavior is normal or abnormal ... 

Plots of leverage or Hotellings (Chapter 3 and Section 5) are useful for detecting samples that are far from the centre within the space described by the model. The Hotelling s statistics critical limit is based on an F-test [33] whereas the critical limit for Leverage is based on one ad hoc rule [2]. 

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 from the models can be used to monitor the performance of the controller. 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 7 provides an indication of where the process is operating relative to the conditions used to develop the PLS model, while the Q statistic is a measure of the variability of a sample s response relative to the model. Thus the use of a multivariate model (PCA or PLS) 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 ... 

Figure 4.24 Hotelling s T image for the second score image of the negative of the bacteria/butter data. The color bar indicates the values of the colors in the image. The numbers adjacent to the arrows on the color bar are the confidence intervals for the statistical test.

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). 

---