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/>REGRESSION</link></c></inlineAttr></u></f></p></text></region><region region-id="1321" left="36" top="90.75" width="115.5" height="13.5" align-x="36" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nonlinear Regression </p></text><indexes><phrase>nonlinear regression</phrase><phrase>regression, nonlinear</phrase></indexes></region><region region-id="1320" left="66" top="115.5" width="311.25" height="12.75" align-x="66" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">LeastSquaresFit(vx, vy, F(x,<f family="Symbol" charset="2" size="11"><b><inlineAttr font-style="normal" underline="false" line-through="false"><c val="#800000">b</c></inlineAttr></b></f>),<sp count="2" />guess, conf, [Stdy], [LBUB], [Acc]) </p></text><indexes><phrase>LeastSquareFit</phrase><phrase>functions, LeastSquareFit</phrase></indexes></region><region region-id="2485" left="66" top="133.5" width="65.25" height="12.75" align-x="66" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Minerr(guess)</p></text><indexes><phrase>Minerr</phrase><phrase>functions, Minerr</phrase></indexes></region><region region-id="2490" left="156" top="133.5" width="127.5" height="12.75" align-x="156" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">genfit(vx, vy, guess, F(x,<f family="Symbol" charset="2" size="11"><b><inlineAttr font-style="normal" underline="false" line-through="false"><c val="#800000">b</c></inlineAttr></b></f>))</p></text><indexes><phrase>genfit</phrase><phrase>functions, genfit</phrase></indexes></region><region region-id="2622" left="36" top="164.25" width="309" height="60" align-x="36" align-y="174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Nonlinear regression minimizes the distance of all data points to a curve described by a nonlinear function of multiple parameters. Choosing the function may be informed by the physics of a problem, or it may be somewhat arbitrary to simply match the shape of the data.<sp count="2" /></p></text></region><region region-id="1993" left="36" top="242.25" width="306" height="96" align-x="36" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solvers equipped to handle the problem of minimizing least-squares sums over an arbitrarily complex formula are varied in their approach, but all will require guess values, and derivatives with respect to each of the parameters in the equation. These derivatives can be found numerically by finite differences, or they can be supplied in closed form. For large data sets, analytical expressions for derivatives may provide additional speed and accuracy.</p></text></region><region region-id="1996" left="36" top="355.5" width="194.25" height="12.75" align-x="36" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Sequential Quadratic Programming solver</p></text><indexes><phrase>sequential quadratic programming solver</phrase></indexes></region><region region-id="1997" left="36" top="374.25" width="299.25" height="84" align-x="36" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">LeastSquaresFit</c></inlineAttr></b></f> solver gives the most flexibility for solving nonlinear regression problems. It allows you to enter constraint equations for any of the dependent parameters, lower and upper bounds on parameters, standard deviations on the x values, as well as a confidence limit for the calculation. With so many arguments, it's best to explain this function by example.</p></text></region><region region-id="1998" left="36" top="476.25" width="315" height="36" align-x="36" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">These data are taken from an example discussed in Lanczos (1956). 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is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The individual parameters above are elements of a vector, <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f>.<f family="Verdana" charset="0"> Subscripts are made with the [ key. </f>You can, if you wish, specify the input function with individual variable names, rather than elements of a vector:</p></text></region><region region-id="2583" left="66" top="705" width="222.75" height="18.75" align-x="150" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="true"><ml:define><ml:function><ml:id 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xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">D</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">F</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="2024" left="36" top="734.25" width="79.5" height="12" align-x="36" align-y="744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Guess values:</p></text></region><region region-id="2718" left="66" top="753.75" width="68.25" height="95.25" align-x="101.25" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">βguess</ml:id><ml:matrix rows="6" cols="1"><ml:real>0.5</ml:real><ml:real>0.7</ml:real><ml:real>3.6</ml:real><ml:real>4.2</ml:real><ml:real>4</ml:real><ml:real>6.3</ml:real></ml:matrix></ml:define></math><rendering item-idref="11" /></region><region region-id="2104" left="36" top="860.25" width="81.75" height="12" align-x="36" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Fit the function:</p></text></region><region region-id="2029" left="66" top="884.25" width="186" height="13.5" align-x="87" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Res</ml:id><ml:apply><ml:id xml:space="preserve">LeastSquaresFit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">f</ml:id><ml:id xml:space="preserve">βguess</ml:id><ml:real>0.95</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="2036" left="36" top="908.25" width="309.75" height="48" align-x="36" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The x and y data are the first two arguments, followed by the fit function, f(x,<f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">). Next is the guess vector </c></inlineAttr></f><f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f>guess for the parameter values, followed by the desired confidence limit on the parameters, 0.95, or 95%. </p></text></region><region region-id="2534" left="36" top="968.25" width="322.5" height="132" align-x="42.75" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The solver uses sequential quadratic programming (SQP) to solve the resulting least-squares problem. By introducing additional variables, the original problem is transformed into a general equality constrained nonlinear programming problem, which is, in general, faster and more stable than some other methods. </p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" /><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The first column of output contains the values for the fitted parameters. The second column contains the left and the third column the right boundary for the confidence interval on the parameters.</p></text></region><region region-id="2047" left="66" top="1101.75" width="45.75" height="18" align-x="78" align-y="1116" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">β</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">Res</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="13" /></region><region region-id="2049" left="36" top="1130.25" width="192" height="12" align-x="36" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Correct values from the NIST website:</p></text></region><region region-id="2050" left="36" top="1143.75" width="442.5" height="24" align-x="63.75" align-y="1164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">NIST</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="6"><ml:apply><ml:mult style="default" /><ml:real>9.5100000027</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-2</ml:real></ml:apply></ml:apply><ml:real>1.0000000001</ml:real><ml:apply><ml:mult style="default" /><ml:real>8.6070000013</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-1</ml:real></ml:apply></ml:apply><ml:real>3.0000000002</ml:real><ml:real>1.5575999998</ml:real><ml:real>5.0000000001</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="14" /></region><region region-id="2584" left="0" top="1170" width="6000" height="6" align-x="0" align-y="1170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2051" left="36" top="1178.25" width="78" height="12" align-x="36" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Compare values</p></text></region><region region-id="2509" left="66" top="1197.75" width="78.75" height="95.25" align-x="93" align-y="1248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">NIST</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="6" cols="1"><ml:real>0.095100000027</ml:real><ml:real>1.0000000001</ml:real><ml:real>0.86070000013</ml:real><ml:real>3.0000000002</ml:real><ml:real>1.5575999998</ml:real><ml:real>5.0000000001</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="15" /></region><region region-id="2510" left="174" top="1197.75" width="167.25" height="95.25" align-x="194.25" align-y="1248" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#80ffff" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">Res</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="6" cols="3"><ml:real>0.036045289847462421</ml:real><ml:real>0.19092334213886544</ml:real><ml:real>1.5220328915524672</ml:real><ml:real>3.3101768908255851</ml:real><ml:real>0.956463224238258</ml:real><ml:real>5.7394177395984984</ml:real><ml:real>-0.095132039308963523</ml:real><ml:real>-2.1203155387944843</ml:real><ml:real>0.135641640054468</ml:real><ml:real>1.8943511927373098</ml:real><ml:real>-0.556824463098671</ml:real><ml:real>3.7561352330829414</ml:real><ml:real>0.16722261900388835</ml:real><ml:real>2.5021622230722151</ml:real><ml:real>2.9084241430504667</ml:real><ml:real>4.72600258891386</ml:real><ml:real>2.4697509115751872</ml:real><ml:real>7.722700246114055</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="16" /></region><region region-id="2598" left="66" top="1311.75" width="78" height="18" align-x="91.5" align-y="1326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fit</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">Res</ml:id><ml:real>0</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="17" /></region><region region-id="2973" left="174" top="1317" width="95.25" height="12.75" align-x="221.25" align-y="1326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">NISTfit</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">NIST</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="2974" left="36" top="1340.25" width="302.25" height="60" align-x="36" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The 95% confidence limits on the parameters cover a fairly wide range, showing that the fit is difficult and the individual parameters can vary widely. The worksheet on the <b><u><link href="./quality.xmcd" title="Links to section" popup="false">confidence function</link></u></b> shows how to further analyze these limits.</p></text></region><region region-id="2517" left="66" top="1407" width="90.75" height="12.75" align-x="75.75" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">x</ml:id><ml:range><ml:sequence><ml:real>0</ml:real><ml:real>.05</ml:real></ml:sequence><ml:apply><ml:mult style="default" /><ml:real>1.1</ml:real><ml:apply><ml:id xml:space="preserve">max</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:apply></ml:range></ml:define></math><rendering item-idref="19" /></region><region region-id="2518" left="222" top="1407" width="114.75" height="12.75" align-x="290.25" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99999924677255736</ml:real></result></ml:eval></math><rendering item-idref="20" /></region><region region-id="2600" left="24" top="1434" width="307.5" height="203.25" align-x="24" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="21" /><rendering item-idref="22" /></region><region region-id="2604" left="66" top="1641" width="162" height="36" align-x="149.25" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.6032367166050749E-05</ml:real></result></ml:eval></math><rendering item-idref="23" /></region><region region-id="2800" left="36" top="1688.25" width="311.25" height="48" align-x="36" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You'll notice that this fit is approaching convergence, but could benefit from an adjustment in the convergence tolerance. That can be done through one of the optional arguments.</p></text></region><region region-id="2595" left="0" top="1746" width="6000" height="6" align-x="0" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2557" left="36" top="1753.5" width="213" height="12.75" align-x="36" align-y="1764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Constraints, standard deviation, and tolerance</p></text><indexes><phrase>constraints, standard deviation and tolerance</phrase><phrase>tolerance, constraints and standard deviation</phrase><phrase>standard deviation, constraints and tolerance</phrase></indexes></region><region region-id="2544" left="36" top="1778.25" width="326.25" height="72" align-x="36" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are several optional arguments to <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">LeastSquaresFit</c></inlineAttr></b></f>. Upper and lower bounds on fit parameters can be specified to constrain the fit values. Here they are set well outside the expected parameter values, because we don't know of a particular bound in this case. If we wanted to restrict one or more parameters, we could do so here.</p></text></region><region region-id="2555" left="66" top="1863.75" width="72" height="95.25" align-x="96" align-y="1914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">LBUB</ml:id><ml:matrix rows="6" cols="2"><ml:real>0</ml:real><ml:real>0</ml:real><ml:real>0</ml:real><ml:real>0</ml:real><ml:real>0</ml:real><ml:real>0</ml:real><ml:real>.1</ml:real><ml:real>10</ml:real><ml:real>10</ml:real><ml:real>10</ml:real><ml:real>10</ml:real><ml:real>10</ml:real></ml:matrix></ml:define></math><rendering item-idref="24" /></region><region region-id="2672" left="36" top="1970.25" width="297.75" height="48" align-x="36" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Additionally, you may set a vector of standard deviations for each y value. Standard deviations may be used to mask outlier points. For example, let's perturb one of the data values, to simulate a mismeasurement.</p></text></region><region region-id="2642" left="66" top="2031" width="75.75" height="18" align-x="123" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">StdY</ml:id><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:apply></ml:apply><ml:real>0.0</ml:real></ml:define></math><rendering item-idref="25" /></region><region region-id="2882" left="174" top="2031" width="56.25" height="18" align-x="207.75" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vy2old</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="2669" left="66" top="2055" width="51" height="18" align-x="87" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="2883" left="174" top="2049" width="51.75" height="24" align-x="204.75" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">StdY</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>8</ml:real></ml:apply></ml:define></math><rendering item-idref="28" /></region><region region-id="2649" left="36" top="2081.25" width="329.25" height="75" align-x="36" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The large value of 10<sup>8</sup> in the standard deviation effectively removes the value from the calculation, without removing it from the data. As with all least-squares solvers, we are trying to minimize, among other conditions, the sum of the residuals squared. In the case of standard deviations, the objective function is</p></text></region><region region-id="2803" left="174" top="2162.25" width="202.5" height="51" align-x="174" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">if the standard deviation is 0 for a point, then the original, undeviated function is used at that point, that is, StdY<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>i</sub></c></inlineAttr></f> is set to 1.</p></text></region><region region-id="2715" left="66" top="2162.25" width="86.25" height="48.75" align-x="108" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">fit</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">β</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id xml:space="preserve">StdY</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda></ml:apply></math><rendering item-idref="29" /></region><region region-id="2569" left="36" top="2222.25" width="336.75" height="39" align-x="36" align-y="2232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you would like the solver to apply a more stringent convergence tolerance to the calculation, you can use a final optional argument. The default is 10<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>-7</sup></c></inlineAttr></f>.</p></text></region><region region-id="2608" left="66" top="2271" width="55.5" height="18.75" align-x="88.5" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Acc</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-12</ml:real></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="2655" left="36" top="2300.25" width="324" height="24" align-x="36" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Compare the calculation with and without the standard deviation that masks the bad point:</p></text></region><region region-id="2659" left="66" top="2336.25" width="235.5" height="13.5" align-x="87" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math error="non-positive covariance diagonal" optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Res</ml:id><ml:apply><ml:id xml:space="preserve">LeastSquaresFit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">f</ml:id><ml:id xml:space="preserve">βguess</ml:id><ml:real>0.90</ml:real><ml:id xml:space="preserve">LBUB</ml:id><ml:id xml:space="preserve">Acc</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="2713" left="66" top="2360.25" width="264.75" height="13.5" align-x="91.5" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Res1</ml:id><ml:apply><ml:id xml:space="preserve">LeastSquaresFit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">f</ml:id><ml:id xml:space="preserve">βguess</ml:id><ml:real>0.90</ml:real><ml:id xml:space="preserve">StdY</ml:id><ml:id xml:space="preserve">LBUB</ml:id><ml:id xml:space="preserve">Acc</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="2656" left="0" top="2382" width="6000" height="6" align-x="0" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2979" left="180" top="2390.25" width="165" height="36" align-x="180" align-y="2400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The unmasked calculation fails because the confidence limits are too large.</p></text></region><region region-id="2977" left="66" top="2385.75" width="96.75" height="95.25" align-x="102" align-y="2436" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:matcol /><ml:id xml:space="preserve">Res1</ml:id><ml:real>0</ml:real></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="6" cols="1"><ml:real>0.098529312110196332</ml:real><ml:real>1.0173205280677515</ml:real><ml:real>0.86952620483034726</ml:real><ml:real>3.0214941857590247</ml:real><ml:real>1.5453458320387405</ml:real><ml:real>5.0071901316262668</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="33" /></region><region region-id="2981" left="66" top="2499.75" width="87" height="18" align-x="96" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fit1</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">Res1</ml:id><ml:real>0</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="2872" left="60" top="2538" width="273.75" height="198" align-x="60" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="35" /><rendering item-idref="36" /></region><region region-id="2873" left="66" top="2745" width="56.25" height="18" align-x="87" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:real>2</ml:real></ml:apply><ml:id xml:space="preserve">vy2old</ml:id></ml:define></math><rendering item-idref="37" /></region><region region-id="2697" left="36" top="2774.25" width="301.5" height="36" align-x="36" align-y="2784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that order of arguments matters here. You are welcome to use any of the optional parameters alone, that is, you can calculate</p></text></region><region region-id="2754" left="66" top="2823" width="216" height="18.75" align-x="87" align-y="2838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Res</ml:id><ml:apply><ml:id xml:space="preserve">LeastSquaresFit</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">f</ml:id><ml:id xml:space="preserve">βguess</ml:id><ml:real>0.95</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-20</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="2755" left="36" top="2852.25" width="200.25" height="84" align-x="36" align-y="2862" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">if you just want to adjust the accuracy of the calculation. However, if you are using two or all three of the optional parameters, you must first enter the standard deviation vector, then the lower and upper bounds matrix, then the accuracy.</p></text></region><region region-id="2756" left="270" top="2841.75" width="83.25" height="95.25" align-x="301.5" align-y="2892" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:matcol /><ml:id xml:space="preserve">Res</ml:id><ml:real>0</ml:real></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="6" cols="1"><ml:real>0.095099999850107073</ml:real><ml:real>0.99999999920991989</ml:real><ml:real>0.86069999971155819</ml:real><ml:real>2.9999999991507971</ml:real><ml:real>1.5576000004383452</ml:real><ml:real>4.9999999997647446</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="39" /></region><region region-id="2761" left="0" top="2958" width="6000" height="6" align-x="0" align-y="2958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2764" left="36" top="2971.5" width="145.5" height="12.75" align-x="36" align-y="2982" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Strategies for nonlinear solving</p></text></region><region region-id="2772" left="36" top="3002.25" width="328.5" height="60" align-x="42.75" align-y="3012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">One of the difficulties of finding the unknown coefficients in a nonlinear least squares problem is figuring out how to specify the problem to give the nonlinear solver the best chance of finding a global solution with good numerical accuracy.<sp count="2" />Consider the following function:</p></text></region><region region-id="2795" left="66" top="3075" width="153" height="18.75" align-x="102.75" align-y="3090" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fnew</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>10000</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:neg /><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>.0001</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:real>-100</ml:real><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="2796" left="36" top="3110.25" width="335.25" height="48" align-x="36" align-y="3120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where x ranges from 0 to 1.<sp count="2" />This function has coefficients that range in value from .0001 to 10000 which can cause nonlinear solvers to fail to converge, or give poor solutions.<sp count="2" />The obvious functional form to fit is:</p></text></region><region region-id="2784" left="66" top="3171" width="180" height="18.75" align-x="156.75" align-y="3186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fit1new</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">C</ml:id><ml:id xml:space="preserve">D</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">A</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">B</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">C</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">D</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="2782" left="36" top="3206.25" width="324" height="72" align-x="36" align-y="3216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The solver may not be able to converge to a solution because the dynamic range of A,B,C and D is so large. If possible, it is best to scale the coefficients to be of similar size.<sp count="2" />While the true values may not be known, close initial guesses and appropriate upper and lower bounds can help convergence.<sp count="2" />A better functional fit would be of the form:</p></text></region><region region-id="2785" left="66" top="3291" width="243.75" height="18.75" align-x="156.75" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">fit2new</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">C</ml:id><ml:id xml:space="preserve">D</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>5000</ml:real><ml:id xml:space="preserve">A</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>-1</ml:real><ml:id xml:space="preserve">B</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>.001</ml:real><ml:id xml:space="preserve">C</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>-200</ml:real><ml:id xml:space="preserve">D</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="42" /></region><region region-id="2781" left="36" top="3320.25" width="333.75" height="48" align-x="36" align-y="3330" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The unknown coefficients now only have about a factor of 10 dynamic range which will greatly increase the likelihood that the solver will converge to a solution and that the solution will be the global one.</p></text></region><region region-id="2790" left="36" top="3380.25" width="318.75" height="132" align-x="36" align-y="3390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It also helps to scale the data into a reasonable range, which is really equivalent to the type of parameter scaling mentioned above.<sp count="2" />One common technique for this is to scale the y data by the x values.<sp count="2" />There is an example of this sort of transformation in the application document on <b><u><link href="./thermocouple.xmcd" title="Links to Application" popup="false">thermocouple coefficients</link></u></b>.You could also scale it to a range from 0 to 1 or from -10 to 10, or some other appropriate set of values for your problem.<sp count="2" />Remember to scale the parameters along with your data set.<sp count="2" />There are examples of this sort of scaling in the application document on <b><u><link href="./refrindex.xmcd" title="Links to Application" popup="false">index of refraction</link></u></b>. You may wish to use the <b><u><link href="./scale.xmcd" title="Links to section" popup="false">scale</link></u></b> function in this type of calculation.</p></text></region><region region-id="2789" left="36" top="3524.25" width="323.25" height="60" align-x="36" align-y="3534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Finally, if your convergence isn't good at first try, or you suspect you are at a local minimum rather than a global one, you should try adjusting the optional accuracy parameter, and providing lower and upper bounds on parameters, if you know them. </p></text></region><region region-id="2792" left="0" top="3588" width="6000" height="6" align-x="0" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2793" left="36" top="3596.25" width="325.5" height="90" align-x="36" align-y="3606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Setting accuracy is, to a certain extent, a function of the scale of your data.<sp count="2" />If the data is of very small scale, perhaps 10<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>-8</sup></c></inlineAttr></f> or smaller, and you do not wish to scale it before fitting, you may require a very small accuracy before your problem converges.<sp count="2" />For a problem with data over a small dynamic range and of order 1 or 10, a setting somewhere in the range of 10<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>-5</sup></c></inlineAttr></f> to 10<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>-12</sup></c></inlineAttr></f> is probably sufficient and more meaningful computationally.</p></text></region><region region-id="2762" left="36" top="3692.25" width="311.25" height="72" align-x="36" align-y="3702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Any nonlinear optimization method can be applied to linear problems. The method just may not be the most efficient for that type of problem. You may wish to use the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">LeastSquaresFit</c></inlineAttr></b></f> function to solve linear models, so that you can include constraint functions, standard deviations, bounds, and confidence limits on these fits as well.</p></text></region><region region-id="2178" left="36" top="3775.5" width="115.5" height="12.75" align-x="36" align-y="3786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">General nonlinear fitting</p></text><indexes><phrase>general nonlinear fitting</phrase><phrase>fitting, general nonlinear</phrase><phrase>nonlinear fitting, general</phrase></indexes></region><region region-id="2179" left="36" top="3794.25" width="310.5" height="96" align-x="36" align-y="3804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You may, of course, use Mathcad's general nonlinear fitting function, <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">genfit</c></inlineAttr></b></f>. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">genfit</c></inlineAttr></b></f> function lets you choose a Levenberg-Marquardt or an Optimized Levenberg-Marquard method for solving - right click on the function name to pick. You can give it the analytical forms of the derivatives with respect to each parameter to improve accuracy. Live symbolics will help in constructing the vector of partial derivatives.</p></text></region><region region-id="2223" left="66" top="3903.75" width="221.25" height="34.5" align-x="243" align-y="3924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:symEval style="default" hide-keywords="false" hide-lhs="false"><ml:apply><ml:derivative style="derivative" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">β1</ml:id></ml:boundVars><ml:parens><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β1</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β2</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β3</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β4</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β5</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β6</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:parens></ml:lambda></ml:apply><ml:symResult><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:neg /><ml:id xml:space="preserve">β2</ml:id></ml:apply></ml:parens><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:symResult></ml:symEval></math><rendering item-idref="43" /></region><region region-id="2224" left="66" top="3943.5" width="252.75" height="30" align-x="243" align-y="3960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:symEval style="default" hide-keywords="false" hide-lhs="false"><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">β2</ml:id></ml:boundVars><ml:parens><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β1</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β2</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β3</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β4</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">β5</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:id xml:space="preserve">β6</ml:id></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:parens></ml:lambda></ml:apply><ml:symResult><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:neg /><ml:id xml:space="preserve">β1</ml:id></ml:apply></ml:parens><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:neg /><ml:id xml:space="preserve">β2</ml:id></ml:apply></ml:parens><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:symResult></ml:symEval></math><rendering item-idref="44" /></region><region region-id="2205" left="246" top="3986.25" width="73.5" height="12" align-x="246" align-y="3996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fitting function</p></text></region><region region-id="2206" left="246" top="4010.25" width="159" height="15" align-x="246" align-y="4020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">deriv w.r.t. parameter <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sub>0</sub></c></inlineAttr></f></p></text></region><region region-id="2207" left="36" top="3984" width="201.75" height="139.5" align-x="68.25" align-y="4056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id xml:space="preserve">F</ml:id><ml:boundVars><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">β</ml:id></ml:boundVars></ml:function><ml:matrix rows="7" cols="1"><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>4</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">e</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>5</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>5</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>4</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">exp</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:apply><ml:indexer /><ml:id xml:space="preserve">β</ml:id><ml:real>5</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply></ml:apply></ml:matrix></ml:define></math><rendering item-idref="45" /></region><region region-id="2213" left="282" top="4046.25" width="3.75" height="36" align-x="282" align-y="4056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">.</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">.</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">.</p></text></region><region region-id="2212" left="246" top="4106.25" width="159" height="15" align-x="246" align-y="4116" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">deriv w.r.t. parameter <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f><f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sub>5</sub></c></inlineAttr></f></p></text></region><region region-id="2226" left="36" top="4136.25" width="319.5" height="60" align-x="36" align-y="4146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This vector contains the fit function in the first element, and derivatives with respect to each parameter in the subsequent elements. Note that the parameters must be specified as elements in a vector of parameters, using index subscripts (press [ on your keyboard).</p></text></region><region region-id="2918" left="0" top="4206" width="6000" height="6" align-x="0" align-y="4206" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2229" left="66" top="4214.25" width="117.75" height="13.5" align-x="78" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">β</ml:id><ml:apply><ml:genfit method="optimized levenberg-marquardt" /><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">βguess</ml:id><ml:id xml:space="preserve">F</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="2324" left="222" top="4209" width="160.5" height="18.75" align-x="238.5" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:transpose /><ml:id xml:space="preserve">β</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="1" cols="6"><ml:real>0.09510000003814785</ml:real><ml:real>1.0000000001827472</ml:real><ml:real>0.86070000016152748</ml:real><ml:real>3.00000000030172</ml:real><ml:real>1.5575999998003896</ml:real><ml:real>5.0000000001336433</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="47" /></region><region region-id="2307" left="66" top="4238.25" width="75" height="15.75" align-x="102" align-y="4248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">genf</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:indexer /><ml:apply><ml:id xml:space="preserve">F</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">β</ml:id></ml:sequence></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="48" /></region><region region-id="2523" left="222" top="4239" width="96" height="12.75" align-x="300.75" align-y="4248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:apply><ml:id xml:space="preserve">genf</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99999999999999944</ml:real></result></ml:eval></math><rendering item-idref="49" /></region><region region-id="2512" left="54" top="4260" width="283.5" height="173.25" align-x="54" align-y="4260" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="50" /><rendering item-idref="51" /></region><region region-id="2816" left="36" top="4447.5" width="135.75" height="12.75" align-x="36" align-y="4458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Weighting functions in genfit</p></text></region><region region-id="2843" left="36" top="4466.25" width="335.25" height="84" align-x="36" align-y="4476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It is possible, using local function definitions in programs, to modify the genfit routine to perform some of the extended functionality of LeastSquaresFit using the Levenberg-Marquardt solver instead of the SQP method. Customize the modifications to suit your problem. For example, to weight data with a standard deviation and calculate numerical derivatives (using the Optimized Levenberg-Marquardt choice), the new merit function is</p></text></region><region region-id="2844" left="66" top="4557" width="139.5" height="36" align-x="90.75" align-y="4572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:id xml:space="preserve">Error</ml:id><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">wt</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:parens><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id 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cols="1"><ml:real>0</ml:real><ml:real>2.004</ml:real><ml:real>123.3</ml:real><ml:real>13.72</ml:real><ml:real>21.92</ml:real><ml:real>32.57</ml:real><ml:real>35.48</ml:real><ml:real>50.11</ml:real><ml:real>57.67</ml:real><ml:real>82.47</ml:real><ml:real>70</ml:real><ml:real>105</ml:real><ml:real>111.4</ml:real><ml:real>124.7</ml:real><ml:real>137</ml:real></ml:matrix></ml:define></math><rendering item-idref="53" /></region><region region-id="2851" left="126" top="4603.5" width="54" height="243.75" align-x="138" align-y="4728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">σ</ml:id><ml:matrix rows="15" 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height="18" align-x="237.75" align-y="4758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">X</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">i</ml:id></ml:define></math><rendering item-idref="56" /></region><region region-id="2865" left="222" top="4776.75" width="56.25" height="29.25" align-x="252" align-y="4794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">guess</ml:id><ml:matrix rows="2" cols="1"><ml:real>2</ml:real><ml:real>2</ml:real></ml:matrix></ml:define></math><rendering item-idref="57" /></region><region region-id="2866" left="0" top="4866" width="6000" height="6" align-x="0" align-y="4866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2898" left="6" top="4892.25" width="375.75" height="279" align-x="112.5" align-y="4902" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">genfitwt</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">σ</ml:id><ml:id xml:space="preserve">guess</ml:id><ml:id xml:space="preserve">f</ml:id></ml:boundVars></ml:function><ml:program><ml:str xml:space="preserve">Compute the weighting factor from standard deviation</ml:str><ml:str xml:space="preserve">Does not handle divide by zero cases</ml:str><ml:localDefine><ml:id xml:space="preserve">wt</ml:id><ml:program><ml:localDefine><ml:id xml:space="preserve">s</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:id xml:space="preserve">σ</ml:id></ml:apply></ml:localDefine><ml:apply><ml:div /><ml:id xml:space="preserve">s</ml:id><ml:apply><ml:absval /><ml:id xml:space="preserve">s</ml:id></ml:apply></ml:apply></ml:program></ml:localDefine><ml:str xml:space="preserve">Multiply the y values by the weights</ml:str><ml:localDefine><ml:id xml:space="preserve">ywt</ml:id><ml:apply><ml:vectorize /><ml:parens><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">wt</ml:id></ml:apply></ml:parens></ml:apply></ml:localDefine><ml:str xml:space="preserve">Create a variable to replace x which is simply the index of x</ml:str><ml:for><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:range><ml:localDefine><ml:apply><ml:indexer /><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">i</ml:id></ml:localDefine></ml:for><ml:str xml:space="preserve">Create a function that will return the genfit derivative matrix</ml:str><ml:str xml:space="preserve">ith x,y data pair.  Derivatives are computed numerically</ml:str><ml:localDefine><ml:function><ml:id xml:space="preserve" subscript="wt">f</ml:id><ml:boundVars><ml:id xml:space="preserve">i</ml:id><ml:id xml:space="preserve">vc</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">wt</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">vc</ml:id></ml:sequence></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Call genfit using the new variables and local function</ml:str><ml:return><ml:apply><ml:genfit method="optimized levenberg-marquardt" /><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">ywt</ml:id><ml:id xml:space="preserve">guess</ml:id><ml:id xml:space="preserve" subscript="wt">f</ml:id></ml:sequence></ml:apply></ml:return></ml:program></ml:define></math><rendering item-idref="58" /></region><region region-id="2909" left="6" top="5180.25" width="325.5" height="24" align-x="6" align-y="5190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Just supply the fit function - the Optimized Levenberg-Marquardt routine in genfit will calculate the derivatives numerically.</p></text></region><region region-id="2911" left="36" top="5209.5" width="112.5" height="25.5" align-x="92.25" align-y="5226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id 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disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">LeastSquaresFit</ml:id><ml:sequence><ml:id xml:space="preserve">X</ml:id><ml:id xml:space="preserve">Y</ml:id><ml:id xml:space="preserve">fexp</ml:id><ml:id xml:space="preserve">guess</ml:id><ml:real>0.95</ml:real><ml:id xml:space="preserve">σ</ml:id></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="3"><ml:real>2.9776140251105678</ml:real><ml:real>1.4459825978534595</ml:real><ml:real>-4.2432111868481313</ml:real><ml:real>0.44778016066729376</ml:real><ml:real>10.198439237069266</ml:real><ml:real>2.4441850350396255</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="62" /></region><region region-id="2995" left="36" top="5312.25" width="105" height="13.5" align-x="82.5" align-y="5322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">wtgenf</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">fexp</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">βexp</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="63" /></region><region region-id="2988" left="36" top="5334" width="290.25" height="173.25" align-x="36" align-y="5334" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="64" /><rendering item-idref="65" /></region><region region-id="2912" left="0" top="5514" width="6000" height="6" align-x="0" align-y="5514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2462" left="36" top="5521.5" width="136.5" height="12.75" align-x="36" align-y="5532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Minimizing with a solve block</p></text><indexes><phrase>minimizing with a solve block</phrase><phrase>solve block, minimizing with</phrase></indexes></region><region region-id="2463" left="36" top="5546.25" width="303.75" height="24" align-x="36" align-y="5556" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You may wish to write out the equation for a sum of squares explicitly and solve using a solve block for notational clarity.</p></text></region><region region-id="2464" left="36" top="5582.25" width="149.25" height="12" align-x="53.25" align-y="5592" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Minimize the sums of squares:</p></text></region><region region-id="2707" left="228" top="5583" width="58.5" height="12.75" align-x="236.25" align-y="5592" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="66" /></region><region region-id="2466" left="66" top="5606.25" width="126" height="36.75" align-x="101.25" align-y="5622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">SSE</ml:id><ml:boundVars><ml:id xml:space="preserve">β</ml:id></ml:boundVars></ml:function><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">β</ml:id></ml:sequence></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda></ml:apply></ml:define></math><rendering item-idref="67" /></region><region region-id="2804" left="228" top="5607" width="58.5" height="18.75" align-x="258" align-y="5622" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">CTOL</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-9</ml:real></ml:apply></ml:define></math><rendering item-idref="68" /></region><region region-id="2467" left="36" top="5642.25" width="185.25" height="12" align-x="36" align-y="5652" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Use the same guess values as before</p></text></region><region region-id="2469" left="36" top="5667" width="24" height="12.75" align-x="47.25" align-y="5676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:id xml:space="preserve">Given</ml:id></math><rendering item-idref="69" /></region><region region-id="2470" left="66" top="5684.25" width="69" height="13.5" align-x="123.75" align-y="5694" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">SSE</ml:id><ml:id xml:space="preserve">βguess</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="70" /></region><region region-id="2936" left="66" top="5708.25" width="87" 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cols="6"><ml:real>0.033942887025533816</ml:real><ml:real>0.48169408686602405</ml:real><ml:real>0.73758866002015389</ml:real><ml:real>2.6093538744561391</ml:real><ml:real>1.7418552103490581</ml:real><ml:real>4.8927122378973849</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="72" /></region><region region-id="2934" left="66" top="5763" width="102" height="18.75" align-x="89.25" align-y="5778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">ERR</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>2.0338290915357887E-08</ml:real></result></ml:eval></math><rendering item-idref="73" /></region><region region-id="2935" left="192" top="5763" width="117" height="18.75" align-x="230.25" align-y="5778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">SSE</ml:id><ml:id xml:space="preserve">β1</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>2.0338290915359783E-08</ml:real></result></ml:eval></math><rendering item-idref="74" /></region><region region-id="2474" left="36" top="5792.25" width="298.5" height="48" align-x="36" align-y="5802" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">ERR</c></inlineAttr></b></f> variable is defined within Mathcad whenever a solve block is called. It returns the sum of the squared <b><u><link href="./quality.xmcd" title="Links to section" popup="false">residuals</link></u></b>, and it is the quantity that is being minimized when calculating the vector of parameters.</p></text></region><region region-id="2426" left="36" top="5852.25" width="303.75" height="156" align-x="36" align-y="5862" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can manipulate the method used to minimize this solve block by right-clicking on the function <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Minerr</c></inlineAttr></b></f> above. You'll be able to choose between the Levenberg-Marquardt method, Quasi-Newton, and Conjugate Gradient solvers. For nonlinear problems, Minerr will automatically choose the first one that converges, but you may wish to experiment with the different options to see which converges to the smallest error for your problem. </p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit" /><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that, if you wish to use Levenberg-Marquardt optimally, you should supply it with the vector of residual equations set to zero, rather than the sum of squared residuals.</p></text></region><region region-id="2942" left="66" top="6020.25" width="146.25" height="52.5" align-x="105" align-y="6030" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id 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height="12.75" align-x="47.25" align-y="6084" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:id xml:space="preserve">Given</ml:id></math><rendering item-idref="76" /></region><region region-id="2949" left="66" top="6098.25" width="72.75" height="13.5" align-x="127.5" align-y="6108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">resid</ml:id><ml:id xml:space="preserve">βguess</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="77" /></region><region region-id="2967" left="66" top="6122.25" width="97.5" height="17.25" align-x="92.25" align-y="6132" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="LM">β</ml:id><ml:apply><ml:Minerr auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false" /><ml:id xml:space="preserve">βguess</ml:id></ml:apply></ml:define></math><rendering item-idref="78" /></region><region region-id="2968" left="0" top="6150" width="6000" height="6" align-x="0" align-y="6150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2954" left="66" top="6158.25" width="131.25" height="33.75" align-x="132" align-y="6180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:pow /><ml:id 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cols="1"><ml:real>0.095100000027463952</ml:real><ml:real>1.0000000001278848</ml:real><ml:real>0.8607000001345475</ml:real><ml:real>3.0000000002348348</ml:real><ml:real>1.5575999998380492</ml:real><ml:real>5.0000000001116236</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="81" /></region><region region-id="2970" left="150" top="6213.75" width="78.75" height="95.25" align-x="177" align-y="6264" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">NIST</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="6" cols="1"><ml:real>0.095100000027</ml:real><ml:real>1.0000000001</ml:real><ml:real>0.86070000013</ml:real><ml:real>3.0000000002</ml:real><ml:real>1.5575999998</ml:real><ml:real>5.0000000001</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="82" /></region><region region-id="2495" left="36" top="6326.25" width="305.25" height="84" align-x="36" align-y="6336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving nonlinear problems is extremely difficult. One solving technique may work very well on certain problems but break down on others. The search for a minimum, which happens as steps are taken away from the guess values, may get sidetracked and the method may wander off, finding neither a min or max. In this case we say the method "does not converge." </p></text></region><region region-id="2432" left="36" top="6422.25" width="305.25" height="72" align-x="36" align-y="6432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Since the solve block/Minerr method of nonlinear regression allows you to put any equation you want into the block for minimization, you have the option of creating your own regression specification. For example, if you want to minimize a weighted sum of squares, so that you can include standard deviations, use an equation like:</p></text></region><region region-id="2434" left="66" top="6505.5" width="178.5" height="89.25" align-x="113.25" align-y="6552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">SSE</ml:id><ml:boundVars><ml:id xml:space="preserve">β</ml:id><ml:id xml:space="preserve">w</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">w</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">f</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:id xml:space="preserve">β</ml:id></ml:sequence></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:bounds></ml:apply><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:indexer /><ml:id xml:space="preserve">w</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">vx</ml:id></ml:apply></ml:bounds></ml:apply></ml:apply></ml:define></math><rendering item-idref="83" /></region><region region-id="2436" left="36" top="6608.25" width="306" height="36" align-x="42.75" align-y="6618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and supply a vector of weights. You can also include boolean constraint equations within the body of the solve block to limit the search space on parameters.</p></text></region><region region-id="2174" left="36" top="6662.25" width="311.25" height="48" align-x="36" align-y="6672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The data in this example were drawn from Lanczos, C. (1956), <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">Applied Analysis, </c></inlineAttr></i></f>Prentice Hall, pp. 272-280, as documented in the NIST Statistical Reference Dataset Archive, <link href="http://www.itl.nist.gov/div898/strd/" title="Links to Web Site" popup="false"><b><u>http://www.itl.nist.gov/div898/strd/</u></b><f family="Verdana" charset="0"><inlineAttr font-weight="normal" underline="false" line-through="false"><c val="#000">.</c></inlineAttr></f></link></p></text></region><region region-id="2869" left="0" top="6720" width="6000" height="6" align-x="0" align-y="6720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2708" left="36" top="6728.25" width="330.75" height="132" align-x="36" align-y="6738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For further information on the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">LeastSquaresFit</c></inlineAttr></b></f> sequential quadratic programming method, see </p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit" /><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="decimal" tabs="inherit">Schittkowski K. (1988), <i>Solving nonlinear least squares problems by a general purpose SQP-method</i>, in Trends in Mathematical Optimization K.-H. Hoffmann, et. al., eds., International Series of Numerical Mathematics, Vol 84, Birkhäuser, pp. 295-309.</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="decimal" tabs="inherit">Schittowski, K. (1985), "NLQPL: A FORTRAN-Subroutine Solving Constrained Nonlinear Programming Problems," <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">Annals of Operations Research</c></inlineAttr></i></f>, Vol. 5, pp. 485-500.</p></text></region><region region-id="2709" left="6" top="6876" width="427.5" height="5.25" align-x="6" align-y="6876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><png item-idref="84" display-width="426" display-height="3.75" /><rendering item-idref="85" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/
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