Engineering Statistics Handbook Episode 3 Part 14 ppt

2.3.6. Instrument calibration over a regime http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc36.htm (3 of 3) [5/1/2006 10:12:23 AM] Special case of linear model - no calibration required An instrument requires no calibration if a=0 and b=1 i.e., if measurements on the reference standards agree with their known values given an allowance for measurement error, the instrument is already calibrated. Guidance on collecting data, estimating and testing the coefficients is given on other pages. Advantages of the linear model The linear model ISO 11095 is widely applied to instrument calibration because it has several advantages over more complicated models. Computation of coefficients and standard deviations is easy. ● Correction for bias is easy.● There is often a theoretical basis for the model.● The analysis of uncertainty is tractable.● Warning on excluding the intercept term from the model It is often tempting to exclude the intercept, a, from the model because a zero stimulus on the x-axis should lead to a zero response on the y-axis. However, the correct procedure is to fit the full model and test for the significance of the intercept term. Quadratic model and higher order polynomials Responses of instruments or measurement systems which cannot be linearized, and for which no theoretical model exists, can sometimes be described by a quadratic model (or higher-order polynomial). An example is a load cell where force exerted on the cell is a non-linear function of load. Disadvantages of quadratic models Disadvantages of quadratic and higher-order polynomials are: They may require more reference standards to capture the region of curvature. ● There is rarely a theoretical justification; however, the adequacy of the model can be tested statistically. ● The correction for bias is more complicated than for the linear model. ● The uncertainty analysis is difficult.● 2.3.6.1. Models for instrument calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc361.htm (2 of 4) [5/1/2006 10:12:24 AM] Warning A plot of the data, although always recommended, is not sufficient for identifying the correct model for the calibration curve. Instrument responses may not appear non-linear over a large interval. If the response and the known values are in the same units, differences from the known values should be plotted versus the known values. Power model treated as a linear model The power model is appropriate when the measurement error is proportional to the response rather than being additive. It is frequently used for calibrating instruments that measure dosage levels of irradiated materials. The power model is a special case of a non-linear model that can be linearized by a natural logarithm transformation to so that the model to be fit to the data is of the familiar linear form where W, Z and e are the transforms of the variables, Y, X and the measurement error, respectively, and a' is the natural logarithm of a. Non-linear models and their limitations Instruments whose responses are not linear in the coefficients can sometimes be described by non-linear models. In some cases, there are theoretical foundations for the models; in other cases, the models are developed by trial and error. Two classes of non-linear functions that have been shown to have practical value as calibration functions are: Exponential1. Rational2. Non-linear models are an important class of calibration models, but they have several significant limitations. The model itself may be difficult to ascertain and verify. ● There can be severe computational difficulties in estimating the coefficients. ● Correction for bias cannot be applied algebraically and can only be approximated by interpolation. ● Uncertainty analysis is very difficult.● 2.3.6.1. Models for instrument calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc361.htm (3 of 4) [5/1/2006 10:12:24 AM] Example of an exponential function An exponential function is shown in the equation below. Instruments for measuring the ultrasonic response of reference standards with various levels of defects (holes) that are submerged in a fluid are described by this function. Example of a rational function A rational function is shown in the equation below. Scanning electron microscope measurements of line widths on semiconductors are described by this function (Kirby). 2.3.6.1. Models for instrument calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc361.htm (4 of 4) [5/1/2006 10:12:24 AM] 2.3.6.2. Data collection http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc362.htm (2 of 2) [5/1/2006 10:12:24 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.6. Instrument calibration over a regime 2.3.6.4.What can go wrong with the calibration procedure Calibration procedure may fail to eliminate bias There are several circumstances where the calibration curve will not reduce or eliminate bias as intended. Some are discussed on this page. A critical exploratory analysis of the calibration data should expose such problems. Lack of precision Poor instrument precision or unsuspected day-to-day effects may result in standard deviations that are large enough to jeopardize the calibration. There is nothing intrinsic to the calibration procedure that will improve precision, and the best strategy, before committing to a particular instrument, is to estimate the instrument's precision in the environment of interest to decide if it is good enough for the precision required. Outliers in the calibration data Outliers in the calibration data can seriously distort the calibration curve, particularly if they lie near one of the endpoints of the calibration interval. Isolated outliers (single points) should be deleted from the calibration data. ● An entire day's results which are inconsistent with the other data should be examined and rectified before proceeding with the analysis. ● 2.3.6.4. What can go wrong with the calibration procedure http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc364.htm (1 of 2) [5/1/2006 10:12:24 AM] Systematic differences among operators It is possible for different operators to produce measurements with biases that differ in sign and magnitude. This is not usually a problem for automated instrumentation, but for instruments that depend on line of sight, results may differ significantly by operator. To diagnose this problem, measurements by different operators on the same artifacts are plotted and compared. Small differences among operators can be accepted as part of the imprecision of the measurement process, but large systematic differences among operators require resolution. Possible solutions are to retrain the operators or maintain separate calibration curves by operator. Lack of system control The calibration procedure, once established, relies on the instrument continuing to respond in the same way over time. If the system drifts or takes unpredictable excursions, the calibrated values may not be properly corrected for bias, and depending on the direction of change, the calibration may further degrade the accuracy of the measurements. To assure that future measurements are properly corrected for bias, the calibration procedure should be coupled with a statistical control procedure for the instrument. Example of differences among repetitions in the calibration data An important point, but one that is rarely considered, is that there can be differences in responses from repetition to repetition that will invalidate the analysis. A plot of the aggregate of the calibration data may not identify changes in the instrument response from day-to-day. What is needed is a plot of the fine structure of the data that exposes any day to day differences in the calibration data. Warning - calibration can fail because of day-to-day changes A straight-line fit to the aggregate data will produce a 'calibration curve'. However, if straight lines fit separately to each day's measurements show very disparate responses, the instrument, at best, will require calibration on a daily basis and, at worst, may be sufficiently lacking in control to be usable. 2.3.6.4. What can go wrong with the calibration procedure http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc364.htm (2 of 2) [5/1/2006 10:12:24 AM] This plot shows the differences between each measurement and the corresponding reference value. Because days are not identified, the plot gives no indication of problems in the control of the imaging system from from day to day. REFERENCE VALUES (µm) This plot, with linear calibration lines fit to each day's measurements individually, shows how the response of the imaging system changes dramatically from day to day. Notice that the slope of the calibration line goes from positive on day 1 to 2.3.6.4.1. Example of day-to-day changes in calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3641.htm (2 of 3) [5/1/2006 10:12:25 AM] negative on day 3. REFERENCE VALUES (µm) Interpretation of calibration findings Given the lack of control for this measurement process, any calibration procedure built on the average of the calibration data will fail to properly correct the system on some days and invalidate resulting measurements. There is no good solution to this problem except daily calibration. 2.3.6.4.1. Example of day-to-day changes in calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3641.htm (3 of 3) [5/1/2006 10:12:25 AM] [...]... 0.60086 0.80 130 0.80122 0.80127 1.001 73 1.00164 1.001 73 1.20227 1.20218 1.20227 1.40282 1.40278 1.40279 1.6 034 4 1.6 033 9 1.6 034 1 1.80412 1.80409 1.80411 2.00485 2.00481 2.004 83 http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3651.htm (1 of 2) [5/1/2006 10:12:25 AM] 2 .3. 6.5.1 Data on load cell #32 066 21 21 21 2.10526 2.10524 2.10524 http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3651.htm... http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc365.htm (3 of 3) [5/1/2006 10:12:25 AM] 2 .3. 6.5.1 Data on load cell #32 066 2 Measurement Process Characterization 2 .3 Calibration 2 .3. 6 Instrument calibration over a regime 2 .3. 6.5 Data analysis and model validation 2 .3. 6.5.1 Data on load cell #32 066 Three repetitions on a load cell at eleven known loads X 2 2 2 4 4 4 6 6 6 8 8 8 10 10 10 12 12 12 14 14 14 16 16 16... F DISTRIBUTION WITH 8 AND 22 DEGREES OF FREEDOM COEFFICIENT ESTIMATES 1 a -0.1 839 80E-04 ST DEV T VALUE (0.2450E-04) -0.75 2 b 0.100102 (0.4 838 E-05) 3 c 0.7 031 86E-05 (0.2013E-06) 0.21E+05 35 RESIDUAL STANDARD DEVIATION = 0.000 037 635 3 RESIDUAL DEGREES OF FREEDOM = 30 Note: The T-VALUE for a coefficient in the table above is the estimate of the coefficient divided by its standard deviation The F-ratio... http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc366.htm (3 of 3) [5/1/2006 10:12:26 AM] 2 .3. 6.7 Uncertainties of calibrated values propagation of error for the linewidth calibration data are also illustrated An example of the derivation of propagation of error type A uncertainties for calibrated values from a quadratic calibration curve for loadcells is discussed on the next page http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc367.htm... to the size of the reference standards requires a root The correct root (+ or -) can usually be identified from practical considerations http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc366.htm (2 of 3) [5/1/2006 10:12:26 AM] 2 .3. 6.6 Calibration of future measurements Power curve The inverse of the calibration curve for the power model gives the calibrated value where b and the natural logarithm... significant at the 5% level Other software may report in other ways; therefore, it is necessary to check the interpretation for each package http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc365.htm (2 of 3) [5/1/2006 10:12:25 AM] 2 .3. 6.5 Data analysis and model validation The t-values are used to test the significance of individual coefficients The t-values can be compared with critical values from...2 .3. 6.5 Data analysis and model validation Run software macro F-ratio for judging the adequacy of the model Coefficients and their standard deviations and associated t values read loadcell.dat x y quadratic fit y x return the following output: LACK OF FIT F-RATIO = 0 .34 82 = THE 6 .34 45% POINT OF THE F DISTRIBUTION WITH 8 AND 22 DEGREES OF FREEDOM COEFFICIENT ESTIMATES 1 a -0.1 839 80E-04 ST... F-ratio = 0 .34 82 with v1 = 8 and v2 = 20 degrees of freedom The critical value of F(0.05, 8, 20) = 2.45 indicates that the quadratic function is sufficient for describing the data A fact to keep in mind is that an F-ratio < 1 does not need to be checked against a critical value; it always indicates a good fit to the data Note: Dataplot reports a probability associated with the F-ratio (6 .33 4%), where... derivation of propagation of error type A uncertainties for calibrated values from a quadratic calibration curve for loadcells is discussed on the next page http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc367.htm (2 of 2) [5/1/2006 10:12:26 AM] ... http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3651.htm (1 of 2) [5/1/2006 10:12:25 AM] 2 .3. 6.5.1 Data on load cell #32 066 21 21 21 2.10526 2.10524 2.10524 http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3651.htm (2 of 2) [5/1/2006 10:12:25 AM] 2 .3. 6.6 Calibration of future measurements Linear calibration line The inverse of the calibration line for the linear model gives the calibrated value Tests for the intercept and slope of calibration curve . VALUE 1 a -0.1 839 80E-04 (0.2450E-04) -0.75 2 b 0.100102 (0.4 838 E-05) 0.21E+05 3 c 0.7 031 86E-05 (0.2013E-06) 35 . RESIDUAL STANDARD DEVIATION = 0.000 037 635 3 RESIDUAL DEGREES OF FREEDOM = 30 Note: The. 0.80 130 8. 0.80122 8. 0.80127 10. 1.001 73 10. 1.00164 10. 1.001 73 12. 1.20227 12. 1.20218 12. 1.20227 14. 1.40282 14. 1.40278 14. 1.40279 16. 1.6 034 4 16. 1.6 033 9 16. 1.6 034 1 . followed. 2 .3. 6.5. Data analysis and model validation http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc365.htm (3 of 3) [5/1/2006 10:12:25 AM] 2. Measurement Process Characterization 2 .3. Calibration 2 .3. 6.