Engineering Statistics Handbook Episode 8 Part 15 doc

Inverting a matrix The matrix analog of division involves an operation called inverting a matrix. Only square matrices can be inverted. Inversion is a tedious numerical procedure and it is best performed by computers. There are many ways to invert a matrix, but ultimately whichever method is selected by a program is immaterial. If you wish to try one method by hand, a very popular numerical method is the Gauss-Jordan method. Identity matrix To augment the notion of the inverse of a matrix, A -1 (A inverse) we notice the following relation A -1 A = A A -1 = I I is a matrix of form I is called the identity matrix and is a special case of a diagonal matrix. Any matrix that has zeros in all of the off-diagonal positions is a diagonal matrix. 6.5.3.1. Numerical Examples http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc531.htm (3 of 3) [5/1/2006 10:35:35 AM] D is the sample variance-covariance matrix for observations of a multivariate vector of p elements. The determinant of D, in this case, is sometimes called the generalized variance. Characteristic equation In addition to a determinant and possibly an inverse, every square matrix has associated with it a characteristic equation. The characteristic equation of a matrix is formed by subtracting some particular value, usually denoted by the greek letter (lambda), from each diagonal element of the matrix, such that the determinant of the resulting matrix is equal to zero. For example, the characteristic equation of a second order (2 x 2) matrix A may be written as Definition of the characteristic equation for 2x2 matrix Eigenvalues of a matrix For a matrix of order p, there may be as many as p different values for that will satisfy the equation. These different values are called the eigenvalues of the matrix. Eigenvectors of a matrix Associated with each eigenvalue is a vector, v, called the eigenvector. The eigenvector satisfies the equation Av = v Eigenstructure of a matrix If the complete set of eigenvalues is arranged in the diagonal positions of a diagonal matrix V, the following relationship holds AV = VL This equation specifies the complete eigenstructure of A. Eigenstructures and the associated theory figure heavily in multivariate procedures and the numerical evaluation of L and V is a central computing problem. 6.5.3.2. Determinant and Eigenstructure http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc532.htm (2 of 2) [5/1/2006 10:35:36 AM] By convention, rows typically represent observations and columns represent variables In this case the number of rows, (n = 5), is the number of observations, and the number of columns, (p = 3), is the number of variables that are measured. The rectangular array is an assembly of n row vectors of length p. This array is called a matrix, or, more specifically, a n by p matrix. Its name is X. The names of matrices are usually written in bold, uppercase letters, as in Section 6.5.3. We could just as well have written X as a p (variables) by n (measurements) matrix as follows: Definition of Transpose A matrix with rows and columns exchanged in this manner is called the transpose of the original matrix. 6.5.4. Elements of Multivariate Analysis http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc54.htm (2 of 2) [5/1/2006 10:35:36 AM] Mean vector and variance- covariance matrix for sample data matrix The results are: where the mean vector contains the arithmetic averages of the three variables and the (unbiased) variance-covariance matrix S is calculated by where n = 5 for this example. Thus, 0.025 is the variance of the length variable, 0.0075 is the covariance between the length and the width variables, 0.00175 is the covariance between the length and the height variables, 0.007 is the variance of the width variable, 0.00135 is the covariance between the width and height variables and .00043 is the variance of the height variable. Centroid, dispersion matix The mean vector is often referred to as the centroid and the variance-covariance matrix as the dispersion or dispersion matrix. Also, the terms variance-covariance matrix and covariance matrix are used interchangeably. 6.5.4.1. Mean Vector and Covariance Matrix http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc541.htm (2 of 2) [5/1/2006 10:35:37 AM] Bivariate normal distribution When p = 2, X = (X 1 ,X 2 ) has the bivariate normal distribution with a two-dimensional vector of means, m = (m 1 ,m 2 ) and covariance matrix The correlation between the two random variables is given by 6.5.4.2. The Multivariate Normal Distribution http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm (2 of 2) [5/1/2006 10:35:37 AM] diagonal elements of 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 = 0 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 were specified to be 0 , this could be tested by taking a single p-variate sample of size n, then computing T 2 and comparing it with for a suitably chosen . 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 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. 6.5.4.3. Hotelling's T squared http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm (2 of 2) [5/1/2006 10:35:38 AM] Estimating the variances and covariances The variances and covariances are similarly averaged over the subgroups. Specifically, the s ij elements of the variance-covariance matrix S are obtained as with s ijl for i j denoting the sample covariance between variables X i and X j for the lth subgroup, and s ij for i = j denotes the sample variance of X i . The variances (= s iil ) for subgroup l and for variables i = 1, 2, , p are computed as . Similarly, the covariances s ijl between variables X i and X j for subgroup l are computed as . Compare T 2 against control values As with an chart (or any other chart), the k subgroups would be tested for control by computing k values of T 2 and comparing each against the UCL. If any value falls above the UCL (there is no lower control limit), the corresponding subgroup would be investigated. Formula for plotted T 2 values Thus, one would plot for the jth subgroup (j = 1, 2, , k), with denoting a vector with p elements that contains the subgroup averages for each of the p characteristics for the jth subgroup. ( is the inverse matrix of the "pooled" variance-covariance matrix, , which is obtained by averaging the subgroup variance-covariance matrices over the k subgroups.) Formula for the upper control limit Each of the k values of given in the equation above would be compared with 6.5.4.3.1. T2 Chart for Subgroup Averages Phase I http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc5431.htm (2 of 3) [5/1/2006 10:35:39 AM] Lower control limits A lower control limit is generally not used in multivariate control chart applications, although some control chart methods do utilize a LCL. Although a small value for might seem desirable, a value that is very small would likely indicate a problem of some type as we would not expect every element of to be virtually equal to every element in . Delete out-of-control points once cause discovered and corrected As with any Phase I control chart procedure, if there are any points that plot above the UCL and can be identified as corresponding to out-of-control conditions that have been corrected, the point(s) should be deleted and the UCL recomputed. The remaining points would then be compared with the new UCL and the process continued as long as necessary, remembering that points should be deleted only if their correspondence with out-of-control conditions can be identified and the cause(s) of the condition(s) were removed. 6.5.4.3.1. T2 Chart for Subgroup Averages Phase I http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc5431.htm (3 of 3) [5/1/2006 10:35:39 AM] Phase II control limits with a denoting the number of the original subgroups that are deleted before computing and . Notice that the equation for the control limits for Phase II given here does not reduce to the equation for the control limits for Phase I when a = 0, nor should we expect it to since the Phase I UCL is used when testing for control of the entire set of subgroups that is used in computing and . 6.5.4.3.2. T2 Chart for Subgroup Averages Phase II http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc5432.htm (2 of 2) [5/1/2006 10:35:42 AM] Delete points if special cause(s) are identified and corrected As in the case when subgroups are used, if any points plot outside these control limits and special cause(s) that were subsequently removed can be identified, the point(s) would be deleted and the control limits recomputed, making the appropriate adjustments on the degrees of freedom, and re-testing the remaining points against the new limits. 6.5.4.3.3. Chart for Individual Observations Phase I http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc5433.htm (2 of 2) [5/1/2006 10:35:43 AM] [...]... of a univariate moving range chart might be considered as an estimator of the variance-covariance matrix for Phase I, although the distribution of the estimator is unknown http://www.itl.nist.gov/div8 98 /handbook/ pmc/section5/pmc5435.htm [5/1/2006 10:35:43 AM] . natural choice. 6.5.4.3. Hotelling's T squared http://www.itl.nist.gov/div8 98 /handbook/ pmc/section5/pmc543.htm (2 of 2) [5/1/2006 10:35: 38 AM] Estimating the variances and covariances The variances and. central computing problem. 6.5.3.2. Determinant and Eigenstructure http://www.itl.nist.gov/div8 98 /handbook/ pmc/section5/pmc532.htm (2 of 2) [5/1/2006 10:35:36 AM] By convention, rows typically represent observations and. the transpose of the original matrix. 6.5.4. Elements of Multivariate Analysis http://www.itl.nist.gov/div8 98 /handbook/ pmc/section5/pmc54.htm (2 of 2) [5/1/2006 10:35:36 AM] Mean vector and variance- covariance matrix