2.3.3.2.1. General matrix solutions to calibration designs

2. Measurement Process Characterization
2.3. Calibration
2.3.3. What are calibration designs?
2.3.3.2. Solutions to calibration designs

2.3.3.2.1. General matrix solutions to calibration designs

Requirements

Solutions for all designs that are cataloged in this Handbook are included with the designs. Solutions for other designs can be computed from the instructions below given some familiarity with matrices. The matrix manipulations that are required for the calculations are:

transposition (indicated by ')
multiplication
inversion

Notation

n = number of difference measurements
m = number of artifacts
(n - m + 1) = degrees of freedom
X= (nxm) design matrix
r'= (mx1) vector identifying the restraint
= (mx1) vector identifying ith item of interest consisting of a 1 in the ith position and zeros elsewhere
R*= value of the reference standard
Y= (mx1) vector of observed difference measurements

Convention for showing the measurement sequence

The convention for showing the measurement sequence is illustrated with the three measurements that make up a 1,1,1 design for 1 reference standard, 1 check standard, and 1 test item. Nominal values are underlined in the first line .

                 1     1     1
          Y(1) = +     -

          Y(2) = +           -

          Y(3) =       +     -

Matrix algebra for solving a design

The (mxn) design matrix X is constructed by replacing the pluses (+), minues (-) and blanks with the entries 1, -1, and 0 respectively.

The (mxm) matrix of normal equations, X'X, is formed and augmented by the restraint vector to form an (m+1)x(m+1) matrix, A:

Inverse of design matrix

The A matrix is inverted and shown in the form:

where Q is an mxm matrix that, when multiplied by s², yields the usual variance-covariance matrix.

Estimates of values of individual artifacts

The least-squares estimates for the values of the individual artifacts are contained in the (mx1) matrix, B, where

where Q is the upper left element of the Ainv matrix shown above. The structure of the individual estimates is contained in the QX' matrix; i.e. the estimate for the ith item can computed from XQ and Yby

Cross multiplying the ith column of XQ with Y
And adding R*(nominal test)/(nominal restraint)

Clarify with an example

We will clarify the above discussion with an example from the mass calibration process at NIST. In this example, two NIST kilograms are compared with a customer's unknown Kilogram.

The design matrix, X, is