Engineering Statistics Handbook Episode 3 Part 7 pptx

2. Measurement Process Characterization 2.3. Calibration 2.3.3. Calibration designs 2.3.3.2. General 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) = + - 2.3.3.2.1. General matrix solutions to calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3321.htm (1 of 5) [5/1/2006 10:11:41 AM] 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 s 2 , 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 A -1 matrix shown above. The structure of the individual estimates is contained in the QX' matrix; i.e. the estimate for the ith item can be computed from XQ and Y by 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 The first two columns represent the two NIST kilograms while the third column represents the customers kilogram (i.e., the kilogram being calibrated). The measurements obtained, i.e., the Y matrix, are 2.3.3.2.1. General matrix solutions to calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3321.htm (2 of 5) [5/1/2006 10:11:41 AM] The measurements are the differences between two measurements, as specified by the design matrix, measured in grams. That is, Y(1) is the difference in measurement between NIST kilogram one and NIST kilogram two, Y(2) is the difference in measurement between NIST kilogram one and the customer kilogram, and Y(3) is the difference in measurement between NIST kilogram two and the customer kilogram. The value of the reference standard, R * , is 0.82329. Then If there are three weights with known values for weights one and two, then r = [ 1 1 0 ] Thus and so From A -1 , we have We then compute QX' We then compute B = QX'Y + h'R * This yields the following least-squares coefficient estimates: 2.3.3.2.1. General matrix solutions to calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3321.htm (3 of 5) [5/1/2006 10:11:41 AM] Standard deviations of estimates The standard deviation for the ith item is: where The process standard deviation, which is a measure of the overall precision of the (NIST) mass calibrarion process, is the residual standard deviation from the design, and s days is the standard deviation for days, which can only be estimated from check standard measurements. Example We continue the example started above. Since n = 3 and m = 3, the formula reduces to: Substituting the values shown above for X, Y, and Q results in and Y'(I - XQX')Y = 0.0000083333 Finally, taking the square root gives s 1 = 0.002887 The next step is to compute the standard deviation of item 3 (the customers kilogram), that is s item 3 . We start by substitituting the values for X and Q and computing D Next, we substitute = [0 0 1] and = 0.02111 2 (this value is taken from a check standard and not computed from the values given in this example). 2.3.3.2.1. General matrix solutions to calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3321.htm (4 of 5) [5/1/2006 10:11:41 AM] We obtain the following computations and and 2.3.3.2.1. General matrix solutions to calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3321.htm (5 of 5) [5/1/2006 10:11:41 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.3. What are calibration designs? 2.3.3.3. Uncertainties of calibrated values 2.3.3.3.1.Type A evaluations for calibration designs Change over time Type A evaluations for calibration processes must take into account changes in the measurement process that occur over time. Historically, uncertainties considered only instrument imprecision Historically, computations of uncertainties for calibrated values have treated the precision of the comparator instrument as the primary source of random uncertainty in the result. However, as the precision of instrumentation has improved, effects of other sources of variability have begun to show themselves in measurement processes. This is not universally true, but for many processes, instrument imprecision (short-term variability) cannot explain all the variation in the process. Effects of environmental changes Effects of humidity, temperature, and other environmental conditions which cannot be closely controlled or corrected must be considered. These tend to exhibit themselves over time, say, as between-day effects. The discussion of between-day (level-2) effects relating to gauge studies carries over to the calibration setting, but the computations are not as straightforward. Assumptions which are specific to this section The computations in this section depend on specific assumptions: Short-term effects associated with instrument response come from a single distribution ● vary randomly from measurement to measurement within a design. ● 1. Day-to-day effects come from a single distribution ● vary from artifact to artifact but remain constant for a single calibration ● vary from calibration to calibration● 2. 2.3.3.3.1. Type A evaluations for calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3331.htm (1 of 3) [5/1/2006 10:11:42 AM] These assumptions have proved useful but may need to be expanded in the future These assumptions have proved useful for characterizing high precision measurement processes, but more complicated models may eventually be needed which take the relative magnitudes of the test items into account. For example, in mass calibration, a 100 g weight can be compared with a summation of 50g, 30g and 20 g weights in a single measurement. A sophisticated model might consider the size of the effect as relative to the nominal masses or volumes. Example of the two models for a design for calibrating test item using 1 reference standard To contrast the simple model with the more complicated model, a measurement of the difference between X, the test item, with unknown and yet to be determined value, X*, and a reference standard, R, with known value, R*, and the reverse measurement are shown below. Model (1) takes into account only instrument imprecision so that: (1) with the error terms random errors that come from the imprecision of the measuring instrument. Model (2) allows for both instrument imprecision and level-2 effects such that: (2) where the delta terms explain small changes in the values of the artifacts that occur over time. For both models, the value of the test item is estimated as 2.3.3.3.1. Type A evaluations for calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3331.htm (2 of 3) [5/1/2006 10:11:42 AM] Standard deviations from both models For model (l), the standard deviation of the test item is For model (2), the standard deviation of the test item is . Note on relative contributions of both components to uncertainty In both cases, is the repeatability standard deviation that describes the precision of the instrument and is the level-2 standard deviation that describes day-to-day changes. One thing to notice in the standard deviation for the test item is the contribution of relative to the total uncertainty. If is large relative to , or dominates, the uncertainty will not be appreciably reduced by adding measurements to the calibration design. 2.3.3.3.1. Type A evaluations for calibration designs http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3331.htm (3 of 3) [5/1/2006 10:11:42 AM] Level-2 standard deviation is estimated from check standard measurements The level-2 standard deviation cannot be estimated from the data of the calibration design. It cannot generally be estimated from repeated designs involving the test items. The best mechanism for capturing the day-to-day effects is a check standard, which is treated as a test item and included in each calibration design. Values of the check standard, estimated over time from the calibration design, are used to estimate the standard deviation. Assumptions The check standard value must be stable over time, and the measurements must be in statistical control for this procedure to be valid. For this purpose, it is necessary to keep a historical record of values for a given check standard, and these values should be kept by instrument and by design. Computation of level-2 standard deviation Given K historical check standard values, the standard deviation of the check standard values is computed as where with degrees of freedom v = K - 1. 2.3.3.3.2. Repeatability and level-2 standard deviations http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3332.htm (2 of 2) [5/1/2006 10:11:43 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.3. What are calibration designs? 2.3.3.3. Uncertainties of calibrated values 2.3.3.3.4.Calculation of standard deviations for 1,1,1,1 design Design with 2 reference standards and 2 test items An example is shown below for a 1,1,1,1 design for two reference standards, R 1 and R 2 , and two test items, X 1 and X 2 , and six difference measurements. The restraint, R*, is the sum of values of the two reference standards, and the check standard, which is independent of the restraint, is the difference between the values of the reference standards. The design and its solution are reproduced below. Check standard is the difference between the 2 reference standards OBSERVATIONS 1 1 1 1 Y(1) + - Y(2) + - Y(3) + - Y(4) + - Y(5) + - Y(6) + - RESTRAINT + + CHECK STANDARD + - DEGREES OF FREEDOM = 3 SOLUTION MATRIX 2.3.3.3.4. Calculation of standard deviations for 1,1,1,1 design http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3334.htm (1 of 3) [5/1/2006 10:11:43 AM] [...]... STANDARD DEVIATIONS WT FACTOR K1 1 1 1 1 1 0 .35 36 + 1 0 .35 36 + 1 0.6124 + 1 0.6124 + 0 0 .70 71 + - FACTORS FOR LEVEL-2 STANDARD DEVIATIONS WT FACTOR K2 1 1 1 1 1 0 .70 71 + 1 0 .70 71 + 1 1.22 47 + http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 4.htm (2 of 3) [5/1/2006 10:11: 43 AM] 2 .3. 3 .3. 4 Calculation of standard deviations for 1,1,1,1 design 1 0 1.22 47 1.4141 + + - The first table shows factors... standard deviations using the unifying equation For this example, and http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 4.htm (3 of 3) [5/1/2006 10:11: 43 AM] 2 .3. 3 .3. 5 Type B uncertainty http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 5.htm (2 of 2) [5/1/2006 10:11:44 AM] 2 .3. 3 .3. 6 Expanded uncertainties Degrees of freedom using the Welch-Satterthwaite approximation Therefore, the... sets 1 1,1,1 design 2 1,1,1,1 design 3 1,1,1,1,1 design 4 1,1,1,1,1,1 design 5 2,1,1,1 design 6 2,2,1,1,1 design 7 2,2,2,1,1 design 8 5,2,2,1,1,1 design 9 5,2,2,1,1,1,1 design 10 5 ,3, 2,1,1,1 design 11 5 ,3, 2,1,1,1,1 design 12 5 ,3, 2,2,1,1,1 design 13 5,4,4 ,3, 2,2,1,1 design 14 5,5,2,2,1,1,1,1 design http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc341.htm (3 of 4) [5/1/2006 10:11:45 AM] ... http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc341.htm (2 of 4) [5/1/2006 10:11:45 AM] 2 .3. 4.1 Mass weights 2nd series using 5 ,3, 2,1,1,1 design The second series is a 5 ,3, 2,1,1,1 design where the restraint over the 500g, 30 0g and 200g weights comes from the value assigned to the summation in the first series; i.e., The weights assigned values by this series are: q 500g, 30 0g, 200 g and 100g test...2 .3. 3 .3. 4 Calculation of standard deviations for 1,1,1,1 design DIVISOR OBSERVATIONS Y(1) Y(2) Y (3) Y(4) Y(5) Y(6) R* Explanation of solution matrix Factors for computing contributions of repeatability and level-2 standard deviations to uncertainty = 8 1 1 1 1 2 1 1 -1 -1 0 4 -2 -1 -1 1 1 0 4 0 -3 -1 -3 -1 2 4 0 -1 -3 -1 -3 -2 4 The solution matrix gives values... items For example, Sum 1 1 1 2 Factor 0.0000 0.8166 0.8166 1.4142 1 + implies that the standard deviations for the estimates are: http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc34.htm (3 of 3) [5/1/2006 10:11:45 AM] 1 1 + + + + 2 .3. 4.1 Mass weights First series using 1,1,1,1 design The calibrations start with a comparison of the one kilogram test weight with the reference kilograms (see the... uncertainty Notice that the standard deviation of the restraint drops out of the calculation because of an infinite degrees of freedom http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 6.htm (2 of 2) [5/1/2006 10:11:44 AM] 2 .3. 4 Catalog of calibration designs Information: Design Given q q Solution Factors for computing standard deviations Convention for showing the measurement sequence n =... shown below: 1 Y(1) = + Y(2) = + Y (3) = + Solution matrix Example and interpretation 1 - 1 - - The cross-product of the column of difference measurements and R* with a column from the solution matrix, divided by the named divisor, gives the value for an individual item For example, Solution matrix Divisor = 3 Y(1) Y(2) Y (3) R* 1 0 0 0 +3 1 -2 -1 +1 +3 1 -1 -2 -1 +3 implies that estimates for the restraint... Divisor = 3 Y(1) Y(2) Y (3) R* 1 0 0 0 +3 1 -2 -1 +1 +3 1 -1 -2 -1 +3 implies that estimates for the restraint and the two test items are: http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc34.htm (2 of 3) [5/1/2006 10:11:45 AM] 2 .3. 4 Catalog of calibration designs Interpretation of table of factors The factors in this table provide information on precision The repeatability standard deviation, ,... summation in the first series; i.e., The weights assigned values by this series are: q 500g, 30 0g, 200 g and 100g test weights q 100 g check standard (2nd 100g weight in the design) q Summation of the 50g, 30 g, 20g weights Other starting points The calibration sequence can also start with a 1,1,1 design This design has the disadvantage that it does not have provision for a check standard Better choice of . 1 1 1 1 1 0 .35 36 + 1 0 .35 36 + 1 0.6124 + 1 0.6124 + 0 0 .70 71 + - FACTORS FOR LEVEL-2 STANDARD DEVIATIONS WT FACTOR K 2 1 1 1 1 1 0 .70 71 + 1 0 .70 71 + 1 1.22 47 + 2 .3. 3 .3. 4. Calculation. example, and ● 2 .3. 3 .3. 4. Calculation of standard deviations for 1,1,1,1 design http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 4.htm (3 of 3) [5/1/2006 10:11: 43 AM] 2 .3. 3 .3. 5. Type B uncertainty http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 5.htm. design http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc 333 4.htm (1 of 3) [5/1/2006 10:11: 43 AM] DIVISOR = 8 OBSERVATIONS 1 1 1 1 Y(1) 2 -2 0 0 Y(2) 1 -1 -3 -1 Y (3) 1 -1 -1 -3 Y(4) -1 1 -3 -1 Y(5) -1 1 -1 -3 Y(6)