Calibration Curves Computing – CCC Software User manual (for
Contents
1. General The software takes the form of an application program called ccc_Software exe stored together with the following files in the subfolder application CCCSoftware_Licence_Agreement pdf CCCSofware_User_Manual pdf Pearson data for Model la xlsx Pearson data for Model 2a xlsx Pearson data for Model 3a xlsx Flow measurement data for Models b xlsx Pearson data for Model la_ElaborationResults txt Pearson data for Model 2a_ElaborationResults txt Pearson data for Model 3a_ElaborationResults txt Flow measurement data for Models b_ElaborationResults txt Pearson data for Model la_ElaborationResults_plot tif Pearson data for Model 2a_ElaborationResults_plot tif Pearson data for Model 3a_ElaborationResults_plot tif Flow measurement data for Models b_ElaborationResults_plot tif splash png Readme txt The application program may be run in either of two ways Double clicking on the executable file ccc_Software exe in Windows Explorer The main graphical user interface GUI is displayed it may take several seconds for the GUI to be displayed Opening an MS DOS window navigating to the folder containing the program typing the name of the program without the extension exe and pressing Return The main GUI is displayed NEW04 Uncertainty VON sun European Association of National The GUI shown in fig 1 is divided into2. model has been selected and at least one exponent has been chosen the ELABORATION button becomes active The number of selected exponents must be smaller than the number of data On pressing the ELABORATION button the optimization problem is solved according to the chosen statistical model and the selected regression exponents see Appendix A for more details on the optimization being implemented Following solution of the optimization problem the text box below the ELABORATION button displays the estimates of the calibration curve parameters together with their associated standard uncertainties and the normalized chi squared value i e the sum of the squared normalized residuals divided by the number of degrees of freedom Moreover the resulting regression curve is added to the scatter plot of the data and the expanded uncertainty bars corresponding to a coverage factor k 2 associated with the fitted y values are visualized 5 6 SELECT OUTPUT FILE Once the elaboration is completed the save button under the SELECT OUTPUT FILE box becomes active and in the box itself a name is automatically suggested for the output file Filename_ElaborationResults txt where Filename is the name of the Excel input data file On pressing the Save button the software displays the browsing window Select the Output data file For OLS and WLS models fitted y values are calculated for the data x values whereas f
3. repetitions and the Var_x and Var_y worksheets must be empty From the repeated measurement values the software evaluates the uncertainty or the covariance matrices as follows OLS Model 1b the common standard deviation of the y data is estimated by means of the standard error of the regression 1 e the square root of the sum of the squared residuals divided by the number of degrees of freedom that is the number n of data minus the number p of model parameters WLS Model 2b the covariance matrix associated with the y data is constructed as a diagonal matrix having on its diagonal the sample variance of each subgroup of data repeated a number of times equal to the number of the measurement repetitions This construction means that all the data pertaining to the same subgroup are considered as uncorrelated and having the same uncertainty and that data pertaining to different subgroups are uncorrelated WTLS Model 3b both the covariance matrices associated with the x and y values are constructed as described for WLS Model 2b above it is also assumed that there is no correlation between the x and y values Depending on the information available about the uncertainty or the covariance matrices associated with the data only those models which make sense with the input data are selectable this fact is graphically evidenced by disabling the check boxes corresponding to not appropriate models 5 4 SELECT REGRESSION EXPONEN
4. the upper right cross of the GUI window 5 8 HELP Clicking on Help in the main menu of the GUI and then clicking on About CCC Software causes a message box displaying information about the software to be displayed Click on OK to close this message box Clicking on Help in the main menu of the GUI and then clicking on Help Documentation causes the User manual this document to be opened using the default program for viewing PDF files Clicking on Help in the main menu of the GUI and then clicking on Licence Agreement causes the Licence Agreement to be opened using the default program for viewing PDF files 6 Acknowledgment The authors are grateful to NPL colleagues for assisting with validating and testing Release 1 1 of the software and for fruitful discussions on a preliminary version thereof The authors wish also to thank all the colleagues of the NEW04 Project for very useful comments and feedback on a previous Release 1 2 of the software The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union 7 References 1 Gertjan Kok Adriaan van der Veen Peter Harris Ian Smith Statistical model and prior knowledge for the determination of calibration curves of flow Deliverable D1 1 4 of WP 1 report of the EMRP joint research project NEW04 Novel mathematical and statistical approaches to uncertainty evaluation 2 http ww
5. 41885397 Covariance matrix associated 2 82789527 0 9809788842 0 7409801278 0 6600553358 28 34845408 11 33938854 with fitted values y 0 9809788842 39 03355671 2 341229847 1 773198798 0 0 0 4166666667 0 0 0 2 916666667 0 7409801278 2 341229847 135 8204274 118 775383 0 EURA European Association of National 0 4166666667 2 916666667 0 6600553358 1 773198798 118 775383 153 1013684 The results in the output file are reported up to ten significant digits in order to provide the User with all the available information in case it should be used for subsequent evaluations However for readability reasons the results shown in the GUI are rounded in the following way uncertainties associated with the parameter estimates are reported to the third significant digit and relevant estimates are reported to the corresponding decimal digit The validation process on OLS and WLS models pointed out an agreement of 8 to 10 digits between the results produced by the CCC software and those produced by an independent software working in higher than normal precision arithmetic NEW04 Uncertainty ESA SUR ARIE European Association of National 5 7 EXIT Pressing the EXIT button of the GUI terminates the program and closes the GUI Alternatively the User may click on File in the main menu of the GUI and then click on Exit or click on
6. B Compiler Runtime MCR installed on the system you will read a message that you do not have to install MCR If you receive this message skip to step 9 If you do not already have the correct version of the MCR installed on your system the application installer will automatically download the MCR and install it along with the deployed CCC Software application Click Next to advance to the License Agreement page If asked about creating the destination folder click Yes Read the license agreement and check Yes to accept it Click Next to advance to Confirmation page Click Install Click Finish In the destination folder selected for the installation the following subfolders will be created appdata application sys uninstall To uninstall the CCC Software NEW04 Uncertainty EURA N European Association of National In the subfolder uninstall bin win64 double click on the executable file uninstall exe Select the products you want to uninstall and press the Uninstall button There may be some remaining files in the destination folder selected for the installation you can manually delete them To uninstall also the MCR Double click on the executable file uninstall exe available within the folder MATLAB MATLAB Compiler Runtime v83 uninstall bin win64 There may be some remaining files in the folder MATLAB MATLAB Compiler Runtime v83 you can manually delete them 5 Using the software 5 1
7. MR ls jointly funded by the EMRP participating countries within EURAMET and the European Union 4 apy Figure 1 Main GUI of the CCC Software 5 2 SELECT INPUT EXCEL FILE Pressing the BROWSE button allows the User to select a Microsoft Excel workbook i e a xls or a xlsx file On pressing the BRowsE button the software opens a browsing window The default NEW04 Uncertainty ESA saunaen European Association of National directory proposed by the software is the directory where the main MATLAB script is located but the User can browse other directories by the usual commands of the browsing window The input data file should have the three worksheets titled and filled in as described in the following Worksheet Data the first column contains the x values and the second column contains the corresponding y values The first line of both columns contains the corresponding headers which will be displayed as axis legends of the data plot If repeated measurements are made they must have the same number of repetitions each group of values being separated from the following by a blank cell Worksheet Var_x contains the covariance matrix associated with the x data If worksheet Data contains x and y values divided into groups worksheet Var_x must be empty Worksheet Var_y contains the covariance matrix associated with the y data it can be provided in the form of a single value or a posi
8. NEW04 Uncertainty ESA SUR ahager European Association of National Calibration Curves Computing CCC Software User manual for Release 1 3 Abstract This document is a User manual for the software Calibration Curves Computing CCC Software Release 1 3 which is a software developed in MATLAB ambient for the evaluation of instrument calibration curves specifically aimed at flow meter calibration curves nonetheless applicable to calibration problems from any other metrological field and developed at the Istituto Nazionale di Ricerca Metrologica INRiM The software constitutes the main component of deliverable D1 3 5 Software for calibration problems developed and tested of the EMRP project Novel mathematical and statistical approaches to uncertainty evaluation EMRP NEW04 The software may be applied to pairs x y of measurement values where x is the independent explanatory variable and y the dependent explained variable If uncertainties associated with the data are available they can be provided as inputs to the software otherwise the software is able to evaluate them limited to Type A contribution on the basis of repeated measurements The regression models which can be addressed by the software are fractional polynomial curves In order to determine estimates of the parameters of the curve and evaluate the associated uncertainty and the covariance matrix the software can perform the following kind o
9. TS A subset of fractional polynomials 3 constitutes the class of curves among which the User can choose the regression model i e positive negative and fractional exponents are allowed logarithmic transformation of the independent variable and repeated powers are not The choice of European Association of National NEW04 Uncertainty ESA SUR ARIE the curve to be fitted to the data is performed by selecting the desired exponents among those in the following list This list derives from the experience developed by personnel performing flow meter calibrations over the years it was observed as reported in 1 that in general the calibration curve of a flow meter can be adequately fitted using a polynomial of degree less than or equal to 5 often negative exponents are required to fit high slope regions in the vicinity of the instrument threshold usually negative exponents 1 and 2 are sufficient but for symmetry it was decided to set 5 as the minimum exponent value Moreover fractional exponents 0 5 and 0 5 were also included Only those exponents that are appropriate for the input data may be selected this fact is graphically evidenced by the software disabling the check boxes corresponding to invalid exponents see Appendix B for restrictions on the exponents As the exponents are ticked the text box below displays the updated expression for the resulting calibration curve 5 5 PROCESS DATA When the statistical
10. a number of parts that allow the User to perform the following operations 1 SELECT INPUT EXCEL FILE the User selects a Microsoft Excel workbook Data is loaded from the workbook and displayed on the GUI 2 SELECT STATISTICAL MODEL the User selects a statistical model from a list of possible options 3 SELECT REGRESSION EXPONENTS the User selects the exponents to be used in the regression model from a list of possible exponents 4 PROCESS DATA the User presses the ELABORATION button SELECT OUTPUT FILE the User saves the results to an output data file 6 EXIT the User quits the program by pressing the EXIT button pi Each step is described in detail in the following subsections E ccc Software Ea File Help CCC SOFTWARE FOR FLOW METER CALIBRATION Release 1 3 June 2015 1 SELECT INPUT EXCEL FILE BROWSE Flow measurement data for Models b xisx 2 SELECT STATISTICAL MODEL 3 SELECT REGRESSION EXPONENTS snow Eee Os 04 Os M2 Ma Aos Mo AoA M2 Da De Os uncertainties uncertainties OLS Model ta E 1b y ax 1 b cx WLS Model 2a Y 2b WTLS Model 3a F 3b 4 PROCESS DATA ja 5 u a 1 43e 01 b 9 9215e 01 u b 7 84e 03 c 5 47e 05 u c 8 04e 05 square n p 0 31407 5 SELECT OUTPUT FILE Flow measurement data for Models b_ElaborationResults txt EMRP EA 6 EXIT European Metrology Research Programme B Programme of EURAMET Le DENERA CA The E
11. a x should be non negative When some are negative only integer exponents must be used some x data are negative When some x data are null only positive exponents must be used some x data are zero Either x or y data have zero mean and they cannot be standardized Models 3a and 3b are not applicable Models 3a and 3b apply a standardization of data which is no possible when data have zero mean Errors and warnings when selecting the regression exponents The number of selected exponents must be smaller than the number of data when selecting the regression exponents the number of the ticked exponents has reached the number of the experimental data Errors and warnings when processing the data Minimization problems the result may be unreliable the software has Errors and warnings when saving the results An output file must be selected the User has pressed the Cancel button and therefore not selected a file All x data should be positive since they are flow values should some x data be negative the calibration is still possible but without selecting fractional exponents
12. correlation between x and y values Release 1 3 generalizes the treatment of model 2a described in 1 in which the y values are uncorrelated European Association of National NEW04 Uncertainty ESA curar Models a assume that uncertainty or covariance information associated with the measurement data are known and supplied by the User as described below OLS Model la a single value is provided as the common squared standard uncertainty associated with the y values in any of the cells within the Var_y worksheet WLS Model 2a a square n x n positive definite matrix not necessarily diagonal is provided in any place within the Var_y worksheet as the covariance matrix associated with the y values WTLS Model 3a a square n x n positive definite matrix not necessarily diagonal with n equal to data size is provided in any place within the Var_x worksheet as the covariance matrix associated with the x values and a square n x n positive definite matrix not necessarily diagonal is provided in any place within the Var_y worksheet as the covariance matrix associated with the y values Models b assume that no uncertainty or covariance information associated with the measurement data is available In this case the measurement data within the Data worksheet must be organized into subgroups of repeated measurement values separated by blank spaces each subgroup having the same number of
13. epts the terms of the agreement Calibration Curves Computing CCC Software Software User Licence Agreement This User Licence Agreement of the Calibration Curves Computing CCC Software is made and is effective on the first day the User installs the Software by and between the Istituto Nazionale di Ricerca Metrologica hereinafter referred to as INRiM the Italian National Metrological Institute NMI and you the User either acting on behalf of your organisation or yourself It is hereby agreed that the User will be permitted to use INRiM s CCC Software hereinafter referred to as the Software free of charge for the express purpose of their own use in accordance with the following terms and conditions of this Agreement Copyright The Software is protected by copyright laws and international copyright treaties as well as other intellectual property laws and treaties Copyright ownership of the Software is vested in the INRiM All title logo or parts of the Software including but not limited to any images text sample codes or examples etc are copyrighted by the INRiM Performance of the Software The Software is provided by INRiM as is without warranty of any kind INRiM disclaims all implied warranties including without limitation any implied warranties of merchantability or of fitness for a particular purpose INRiM will make any reasonable effort to resolve any reported problems found in runnin
14. es When such repetitions occur only Models 1b and 2b are allowed one or more groups of data x contain the same repeated values Some groups of x or y data contain the same repeated values When such repetitions occur only Model 1b is allowed one or more groups of data y and or data x contain the same repeated values When x and y data are not divided into groups at least the covariance matrix Vy must be given in the worksheet Var_y no covariance matrix is provided although data are not divided into groups The dimensions of the covariance matrix Vy must agree with the size of y data matrix dimension do not agree with data The dimensions of the covariance matrix Vx must agree with the size of x data matrix dimension do not agree with data The squared standard uncertainty associated with y data must be positive a non positive value has been provided as the squared standard uncertainty of y data Covariance matrix Vy must be positive definite the covariance matrix associated with y values are not positive definite NEW04 Uncertainty EURA European Association of National Both covariance matrices must be positive definite either or both of the covariance matrices are not positive definite Covariance matrices Vx and Vy must contain only numeric values some not a number NaN value is present within the covariance matrices All input dat
15. f regression procedures corresponding to statistical models la 1b 2a 2b 3a and 3b respectively described in deliverable D1 1 4 of the NEW04 project Ordinary least squares regression OLS for models la and 1b Weighted least squares regression WLS for models 2a and 2b Weighted total least squares regression WTLS for models 3a and 3b European Association of National NEW04 Uncertainty ESA sua Table of contents oP Pp B BACKETOUNG NN 3 Introduction 3 Licence Aprecio 4 Installing and Uninstalling the software oooooococccncnononononnnnnnnonononononnnnnnnnnnnonenonnnnnnnnnnnnnnnnnnnnnnnnnnnnennnnnancnnnns 5 Usine TIE SO MW RNE cia 6 O 6 5 2 SELECT INPUT EXCEL FILE coccion eri ada asi pad is 7 5 3 SELECT STATISTICAL MODEL cocoa mf A tania tadas 8 5 4 SELECT REGRESSION EXPONENTS cccceecceeseeesceeseeceaeceaeceaecaeceaeeeeeeeeeeeeeeseaeeeaeesaeeeaaessaeenaeeeaeeeaeees 9 Bibs PROCESS DATA adi dt tios 10 56 A scssccecsccaceescgevzen scot ddes szcisaessecdsicacacagen sank edacuvacadnta isackea agen ee aeaoe aeaa EOAR 10 57e NN 13 Dies MEP a datado 13 AckNoOWIe dE MEN tii asa 13 RefErENCES siiis aa a 13 Statistical Models rica at 14 A 1 Nota on SREE OTE ONI TO E O OTOOTO A AOOO 14 A 2 MOHE Siei ad A A R A 14 A 3 A 15 AA WHS FRIAS RN 15 A5 WLS A 15 Mess Suma a AA Ad dai 17 NEW04 Uncertainty ESA SUR ARIE European Association of National 1 Background The Calibration Curves Comp
16. g the Software but INRiM does not guarantee to fix any problems in the Software or to provide any updates or improvements by a particular date Other Restrictions The User will not sell rent lease supply publish sublicense or distribute the Software either in whole or in part or use it for commercial purposes without INRiM prior written permission The User will not alter or adapt or edit the Software No Liability for Consequential Damages To the maximum extent permitted by applicable law in no event shall INRiM be liable for any damages whatsoever including without limitation damages for loss of business profits business interruption loss of business information or other pecuniary loss arising out of the use or inability to use the Software The entire risk arising out of the use or performance of the Software and documentation remains with the User The User hereby understands that any use of INRiM CCC Software including small or unlimited use will be entirely subject to the terms of this Licence Agreement The User also understands that they will NOT make use of or be permitted to gain access to INRIM CCC Software if they are unable to accept and or comply to the terms of this Licence Agreement unless a written agreement is obtained from the INRiM INRiM Istituto Nazionale di Ricerca Metrologica Strada delle Cacce 91 10135 Torino Italy Telephone 39 O11 3919 1 switchboard Fax 39 O11 346384 Email INRiM INRiM
17. it VAT Number IT09261710017 NEW04 Uncertainty ESA SUR ARIE European Association of National 4 Installing and uninstalling the software NOTE The software is intended to run on personal computers with a 64 bit Microsoft Windows operating system To run the software the User must first install MATLAB s Compiler Runtime MCR libraries version 8 3 R2014a 2 see Section 4 The MCR installer file is not included in the CCC Software distribution but it can be automatically downloaded from the internet during the installation process itself see the following point 6 Information on how to obtain the MCR installer file is also given in the file README txt included in the software distribution The User must accept the terms of the MCR Library License as part of the installation of the MCR libraries To install the software undertake the following steps l 8 9 10 11 Extract the application installer CCC_Software_Installer_w64 exe from the folder CCC_Software_for_redistribution zip and run it If you connect to the internet using a proxy server enter the server s settings a Click Connection Settings b Enter the proxy server settings in the provided window c Click OK Click Next to advance to the Installation Options page Click Next to advance to the Required Software page If asked about creating the destination folder click Yes If you already have the correct version of the MATLA
18. mat as Filename_ElaborationResults_plot tif in the same directory where the output file is saved In the following an example output file is reported EMRP NEWO4 CCC Calibration Software Release 1 3 programming language MATLAB R2013a Program identifier EMRP NEWO4 CCC Calibration Software Release 1 3 Authors A Malengo F Pennecchi P Spazzini Istituto Nazionale di Ricerca Metrologica INRIM Italy Release Date June 30 2015 Developed in the framework of WP1 EMRP Project NEWO4 Project financed by EURAMET Data identifier Ordinary weighted and weighted total least squares method for fitting fractional polynomial curves to given data x y Ux and Uy according to models la 1b 2a 2b 3a and 3b defined in Deliverable 1 1 4 Input data from File C Few data xlsx Experimental data x and y 10 1 38 42 a fs Oe o 3 Total experimental points no 4 NEW04 Uncertainty Known covariance matrix associated with x 0 4166666667 0 0 0 0 0 4166666667 0 0 Known covariance matrix associated with y 2 916666667 0 0 0 Regression Model Model 3a 0 2 916666667 0 0 Fitted polynomial curve y a bx REGRESSION RESULTS a 6 4281692 u a 9 370249411 b 12 47127207 u b 3 367400858 Chi square n p 0 58342 Covariance matrix associated with parameter estimates 87 80157402 28 34845408 Fitted values y 9 8295830373 9 1412094040 38 2350197603 41 79
19. matrix The software can perform the following kinds of regression Ordinary least squares regression OLS Weighted least squares regression WLS Weighted total least squares regression WTLS The software relies on statistical models la 1b 2a 2b 3a and 3b described in deliverable D1 1 4 Section 3 provides information about the Licence Agreement Section 4 tells how to install and uninstall the software Section 5 describes how to use the software the software graphical user interface is described Section 5 1 information on the input data file format is given Section 5 2 guidance on the criteria for choosing the statistical model appropriate for the available measurement data is provided Section 5 3 the list of possible regression exponents is given Section 4 4 information on how processing the data Section 5 5 and saving the results Section 5 6 is also provided Appendix A informs on the implementation of the different least squares methods corresponding to the statistical models described in Section 5 3 Appendix B lists all the possible messages of warnings and errors European Association of National NEW04 Uncertainty ESA sua 3 Licence Agreement Release 1 3 of the CCC Software is provided with a Software User Licence Agreement which is fully reported in the following and the use of the software is subject to the terms laid out in that agreement By installing and running the software the User acc
20. or WTLS models they are calculated for the estimated x values see also Appendix A NEW04 Uncertainty EURA European Association of National which automatically opens the same directory where the main MATLAB script is located However the file name and its location can be modified placing the file in the desired directory and saving it by pressing the Save button of the browsing window If such a file already exists in the selected directory it is possible to overwrite it or to save it with a different name The chosen file name is then displayed within the text box below the Save button of the GUI and the Save button is disabled The output file reports the following information Main information on the software Name of the input data Excel workbook Two column vectors of the experimental x and y data Number of experimental data couples Covariance matrices or single variance value associated with data y and possibly with data x the covariance matrices are provided by the User for models a or are estimated by the software for models b Statistical model Regression curve Parameter estimates and associated standard uncertainty Normalized chi squared value of the regression Covariance matrix associated with the parameter estimates Fitted y values and the associated covariance matrix The plot of the regression curve displayed in the GUI is automatically saved in a tif for
21. th data are used in this section irrespective of whether they have been provided by the User for models a or they have been estimated by the software for models b A 2 Models For OLS and WLS fitting the underlying statistical model is given by y XBre 1 where x Ft xPP X s o 2 Xy 2 Xp Bb NEW04 Uncertainty A sua European Association of National l Y tes For WTLS fitting in which also the x values are subject to uncertainty the model is given by y X Bre 3 x x 0 4 The choice of the fitting procedure depends on the assumptions made about the nature of errors and 6 according to models to 3 described in 1 A 3 OLS fitting When model errors e in 1 are assumed to be i i d as N 0 uy normal distribution with zero mean and standard deviation equal to u for i 1 n then the following formulae are implemented B XTX XTY 5 Vp u XTX 6 7 XB 7 V5 XVpX 8 1 ZEAN Ea E 9 A 4 WLS fitting When model errors e in 1 are assumed to be distributed as N 0 V multivariate normal distribution with zero mean vector and covariance matrix V then the following formulae are implemented B XTV 1X XTV ty 10 Vp XTV 1X 11 X 12 V XVgx 13 t 0 9 V 0 y 14 A 5 WTLS fitting When model errors 6 in 4 and in 3 are assumed to be distributed as 5 N 0 V and e N 0 V respectively then the follo
22. the software is reported Errors and warnings when inputting data An input data file must be selected the User has pressed the cancel button and therefore not selected a file 193 The input data file must be an Excel workbook the extension of the input data file is not xls or xlsx 66 The input data file must contain the worksheets Data Var_x and 29 Var_y at least one of the prescribed worksheets is not present in the input data file 113 The worksheet Data must contain x and y data measurement data are not present in the worksheet Data nm 7 he worksheet Data must contain headers for x and y data headers for x and y data are not present in the worksheet Data 113 The number of data x must be equal to that of data y x and y data have different size Cn he number of groups within data x must be equal to that within data y X and y data are divided into a different number of groups The number of data within each group must be the same groups of data have different dimensions 3 When x and y data are divided into groups the worksheets Var_x and Var_y must be empty covariance matrices are provided although data are grouped 193 The groups within x and y data must be separated by a single empty row groups of data are separated by more than one blank cell Some groups of x data contain th sam repeated valu
23. tive definite square matrix If worksheet Data contains x and y values divided into groups worksheet Var_y must be empty Once the data is loaded the file name is displayed within the text box below the BROWSE button and the data couples x y are shown as a scatter plot within a figure in the GUI Specific restrictions and requirements relevant to the input Excel file are addressed in Appendix B 5 3 SELECT STATISTICAL MODEL The following regression models are available according to models described in 1 OLS Model la OLS model with known uncertainties OLS Model 1b OLS model with uncertainties evaluated from repeated data WLS Model 2a WLS model with known uncertainties WLS Model 2b WLS model with uncertainties evaluated from repeated data WTLS Model 3a WTLS model with known uncertainties WTLS Model 3b WTLS model with uncertainties evaluated from repeated data OLS models assume that only the y values are subject to uncertainty or equivalently the uncertainties associated with the x values are negligible and that all the y values have the same uncertainty WLS models assume that only the y values are subject to uncertainty but without necessarily being homoscedastic and could also be correlated WTLS models assume that the x values are also subject to uncertainty The x values may be correlated among themselves and the y values may be correlated among themselves but there may be no
24. uting CCC Software constitutes the main component of deliverable D1 3 5 Software for calibration problems developed and tested of the EMRP project Novel mathematical and statistical approaches to uncertainty evaluation EMRP NEW04 Deliverable D1 3 5 is part of the series of deliverables D1 1 4 D1 1 8 D1 1 12 D1 2 4 D1 2 8 D1 3 3 and D1 3 6 dedicated to the statistical issues related to flow meter measurement data and the determination of relevant calibration curves The software relies on the statistical models described in deliverable D1 1 4 1 in particular on models la 1b 2a 2b 3a and 3b Although the software was developed for flow meter calibration it can easily be applied to similar calibration problems from different metrology areas Release 1 3 of the CCC Software was developed in MATLAB R2013a with Optimization Toolbox 6 4 ona personal computer running Microsoft Windows 7 2 Introduction This User manual describes the Calibration Curves Computing CCC Software software for the evaluation of instrument calibration curves The User provides measured data uncertainty and covariance information associated with those data and assigns the mathematical and statistical model from the options available The software determines estimates of the calibration curve parameters and their associated covariance matrix as well as estimates of values on the calibration curve and their associated covariance
25. w mathworks com products compiler mcr index html 3 Royston P and D G Altman 1994 Regression using fractional polynomials of continuous covariates Parsimonious parametric modelling Applied Statistics 43 429 467 4 A Malengo and F Pennecchi A weighted total least squares algorithm for any fitting model with correlated variables Metrologia 2013 50 654 NEW04 Uncertainty ESA SUR ARIE European Association of National A Statistical models In this Appendix information is provided on the implementation of the different least squares methods corresponding to the statistical models described in Section 4 3 A 1 Notation x vector of x values n x 1 y vector of y values n x 1 X design matrix n x p B vector of regression parameters p x 1 E vector of errors in variable y n x 1 vector of errors in variable x n x 1 for WTLS models only Uy common standard uncertainty associated with y values for OLS models V covariance matrix associated with y values n x n for WLS and WTLS models V covariance matrix associated with x values n x n for WTLS models only B vector of parameter estimates p x 1 Va covariance matrix associated with B p x p vector of fitted values n x 1 a vector of fitted X values n x 1 for WTLS models Xi normalized chi squared value of the regression The mathematical symbols mentioned above for uncertainties and covariance matrices associated wi
26. wing formulae are implemented NEW04 Uncertainty ESA sua European Association of National B argminge gy C x V x x X B y Vy X B y 15 Visa HD V H D 16 where H is the Hessian matrix of the cost function 1 e the expression to be minimized in 15 D is the matrix of the mixed second order derivatives of the cost function with respect to parameters x and and with respect to input data x and y and V 0 v 6 v Covariance matrix V is the lower right sub matrix in Vip of dimension p x p Moreover y9 XP 17 V5 J Veg 18 where J is the matrix of the first order derivatives of the regression model XB with respect to both x and B calculated at estimates Bl and MS ROW R 0 0 9 O y 19 For more details on the implemented WTLS fitting procedure see 4 The expression to be minimized is non linear in its parameters B and x hence a numerical solution is found by implementing an algorithm based on the MATLAB function fminunc m available within the MATLAB Optimization Toolbox Slightly different results from those reported in the paper may be obtained since in Release 1 3 of the CCC software the minimization option GradObj of the fminunc m function has not been implemented NEW04 Uncertainty EURA European Association of National B Messages In this section a list of the messages that can be visualized when running
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