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width="195.75" height="13.5" align-x="36" align-y="90" show-border="false" show-highlight="false" is-protected="true" 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">B-Splines and Least-Squares Splines</p></text><indexes><phrase>B-splines and least-squares splines</phrase><phrase>nonparametric fitting, B-splines and least-squares splines</phrase><phrase>splines, B-splines and least-squares</phrase></indexes></region><region region-id="702" left="66" top="103.5" width="81.75" height="12.75" align-x="66" align-y="114" show-border="false" show-highlight="false" is-protected="true" 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">bspline(vx,vy,u,n)</p></text><indexes><phrase>bspline</phrase><phrase>functions, bspline</phrase></indexes></region><region region-id="881" left="186" top="103.5" width="87.75" height="12.75" align-x="186" align-y="114" show-border="false" show-highlight="false" is-protected="true" 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">interp(vs,Mx,My,x)</p></text><indexes><phrase>interp</phrase><phrase>functions, interp</phrase></indexes></region><region region-id="876" left="66" top="121.5" width="145.5" height="12.75" align-x="66" align-y="132" show-border="false" show-highlight="false" is-protected="true" 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">Spline2(vx,vy,n,[vw],[u],[level])</p></text><indexes><phrase>Spline2</phrase><phrase>functions, Spline2</phrase></indexes></region><region region-id="1269" left="66" top="139.5" width="55.5" height="12.75" align-x="66" align-y="150" show-border="false" show-highlight="false" is-protected="true" 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">Binterp(x,b)</p></text><indexes><phrase>Binterp</phrase><phrase>functions, Binterp</phrase></indexes></region><region region-id="1291" left="186" top="139.5" width="36.75" height="12.75" align-x="196.5" align-y="150" show-border="false" show-highlight="false" is-protected="true" 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">DWS(b)</p></text><indexes><phrase>DWS</phrase><phrase>functions, DWS</phrase></indexes></region><region region-id="533" left="36" top="170.25" width="321.75" height="108" align-x="36" align-y="180" show-border="false" show-highlight="false" is-protected="true" 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">These functions construct basis splines. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">bspline</c></inlineAttr></b></f> function is a conventional B-spline interpolator, requiring a string of knots, u, as input. <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Spline2</c></inlineAttr></b></f> calculates a statistical b-spline with a minimal number of knots, or, optionally, takes a vector of defined knots, a vector of defined weights, or a rejection level for the statistical test. 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family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">bspline</c></inlineAttr></b></f>.<sp count="2" />There must be length(x) - n knots.</p></text></region><region region-id="738" left="66" top="531" width="23.25" height="12.75" align-x="77.25" align-y="540" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>n</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="8" /></region><region region-id="739" left="144" top="531" width="75.75" height="12.75" align-x="153" align-y="540" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>j</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id>N</ml:id><ml:parens><ml:apply><ml:minus 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tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>v</ml:id><ml:apply><ml:id>bspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>knots</ml:id><ml:id>n</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="742" left="222" top="597" width="91.5" height="12.75" align-x="244.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>G</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:sequence><ml:id>v</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>t</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="746" left="0" top="618" width="6000" height="6" align-x="0" align-y="618" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="743" left="30" top="630" width="243.75" height="147.75" align-x="30" align-y="630" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="13" /><rendering item-idref="14" /></region><region region-id="744" left="36" top="788.25" width="318" height="36" align-x="36" align-y="798" show-border="false" show-highlight="false" is-protected="true" 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 is a somewhat restrictive implementation of the B-spline.<sp count="2" />A more flexible version, with an optimal knot generator, is implemented as <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Spline2</c></inlineAttr></b></f>.<sp count="2" /></p></text></region><region region-id="784" left="36" top="841.5" width="90.75" height="12.75" align-x="36" align-y="852" show-border="false" show-highlight="false" is-protected="true" 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">Statistical B-splines</p></text><indexes><phrase>statistical B-splines</phrase><phrase>B-splines, statistical</phrase></indexes></region><region region-id="789" left="36" top="860.25" width="308.25" height="72" align-x="36" align-y="870" show-border="false" show-highlight="false" is-protected="true" 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 statistical B-spline functions, based on Thijsse and Hollanders' Spline2 software, will accept weights for data values, reflecting the level of relative error, and will calculate a string of knots using the Durbin-Watson statistic to accept or reject spline fits.<sp count="2" />In this way, statistical B-splines supply a minimal number of knots to reflect all of the data features.<sp count="3" /></p></text></region><region region-id="820" left="66" top="957" width="127.5" height="12.75" align-x="76.5" align-y="966" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="unable to open file" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>a</ml:id><ml:apply><ml:id>READPRN</ml:id><ml:str>example1.txt</ml:str></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="821" left="66" top="975.75" width="33" height="18" align-x="75.75" align-y="990" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>x</ml:id><ml:apply><ml:matcol /><ml:id>a</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="16" /></region><region region-id="822" left="144" top="975.75" width="34.5" height="18" align-x="155.25" align-y="990" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>y</ml:id><ml:apply><ml:matcol /><ml:id>a</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="17" /></region><region region-id="823" left="222" top="975.75" width="36.75" height="18" align-x="235.5" align-y="990" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>w</ml:id><ml:apply><ml:matcol /><ml:id>a</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="874" left="36" top="1004.25" width="312" height="48" align-x="36" align-y="1014" show-border="false" show-highlight="false" is-protected="true" 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 w is a vector of weights giving the estimated standard deviations of the random error in y. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Spline2</c></inlineAttr></b></f> function takes the x and y data, the vector of weights, and the degree of the desired spline polynomials, in this case, 3.</p></text></region><region region-id="870" left="66" top="1065" width="90" height="12.75" align-x="77.25" align-y="1074" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:id>w</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="19" /></region><region region-id="819" left="36" top="1088.25" width="347.25" height="36" align-x="36" align-y="1098" show-border="false" show-highlight="false" is-protected="true" 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 output vector, b, contains a variety of information.<sp count="2" />The first element is the Bspline order, and the second element gives the number of intervals (knots - 1).<sp count="2" />This is followed by the knot values:</p></text></region><region region-id="826" left="66" top="1137" width="32.25" height="18" align-x="81" align-y="1146" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="20" /></region><region region-id="827" left="144" top="1137" width="32.25" height="18" align-x="159" align-y="1146" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="21" /></region><region region-id="828" left="66" top="1161" width="37.5" height="18" align-x="74.25" align-y="1170" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="22" /></region><region region-id="829" left="144" top="1161" width="56.25" height="18" align-x="174.75" align-y="1170" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>knots</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:apply><ml:plus /><ml:id>i</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="831" left="36" top="1190.25" width="346.5" height="24" align-x="36" align-y="1200" show-border="false" show-highlight="false" is-protected="true" 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 first and last knots, when automatically generated, match the endpoints of the original x-data:</p></text></region><region region-id="832" left="66" top="1227" width="48" height="12.75" align-x="96.75" align-y="1236" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>min</ml:id><ml:id>x</ml:id></ml:apply></ml:eval></math><rendering item-idref="24" /></region><region region-id="838" left="162" top="1227" width="48.75" height="12.75" align-x="193.5" align-y="1236" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>max</ml:id><ml:id>x</ml:id></ml:apply></ml:eval></math><rendering item-idref="25" /></region><region region-id="841" left="66" top="1245" width="66.75" height="12.75" align-x="115.5" align-y="1254" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>min</ml:id><ml:id>knots</ml:id></ml:apply></ml:eval></math><rendering item-idref="26" /></region><region region-id="842" left="162" top="1245" width="67.5" height="12.75" align-x="212.25" align-y="1254" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>max</ml:id><ml:id>knots</ml:id></ml:apply></ml:eval></math><rendering item-idref="27" /></region><region region-id="867" left="0" top="1272" width="6000" height="6" align-x="0" align-y="1272" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="844" left="36" top="1286.25" width="297.75" height="60" align-x="36" align-y="1296" show-border="false" show-highlight="false" is-protected="true" 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 rest of the vector contains the coefficients for the B-spline basis functions. The spline can be plotted at interpolated points using the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Binterp</c></inlineAttr></b></f> function, which returns the value at the requested point, and the first, second, and third derivatives at that point.</p></text></region><region region-id="1136" left="66" top="1365" width="41.25" height="12.75" align-x="74.25" align-y="1374" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:real>100</ml:real></ml:range></ml:define></math><rendering item-idref="28" /></region><region region-id="1137" left="144" top="1358.25" width="154.5" height="27.75" align-x="174.75" align-y="1374" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>range</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:parens><ml:apply><ml:minus /><ml:apply><ml:id>max</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:id>min</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:parens></ml:apply><ml:real>101</ml:real></ml:apply><ml:apply><ml:id>min</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="1254" left="198" top="1400.25" width="154.5" height="51" align-x="198" align-y="1410" show-border="false" show-highlight="false" is-protected="true" 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">Each column of output contains the interpolated value and the first 3 derivatives at the point range<f family="Times New Roman" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>i</sub></c></inlineAttr></f>.</p></text></region><region region-id="1248" left="66" top="1413" width="107.25" height="12.75" align-x="96.75" align-y="1422" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline</ml:id><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:id>range</ml:id><ml:id>b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="1249" left="66" top="1437.75" width="68.25" height="18" align-x="96.75" align-y="1452" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline</ml:id><ml:apply><ml:transpose /><ml:id>Spline</ml:id></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="1252" left="66" top="1503" width="47.25" height="12.75" align-x="96" align-y="1512" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>Spline</ml:id></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="3" show-trailing-zeros="false" exponential-threshold="3" /><matrix display-style="auto" expand-nested-arrays="false" /><unit format-units="false" simplify-units="false" /><table item-idref="32" /></resultFormat></math><rendering item-idref="33" /></region><region region-id="1415" left="6" top="1560" width="374.25" height="236.25" align-x="6" align-y="1560" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="34" /><rendering item-idref="35" /></region><region region-id="1213" left="36" top="1820.25" width="282.75" height="60" align-x="36" align-y="1830" show-border="false" show-highlight="false" is-protected="true" 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 determination of the optimal number of knots and their spacing uses the Durbin-Watson statistic.<sp count="2" />You can see what the value of this statistic was for the final fit using the DWS function, or by pulling off the appropriate entry in the vector b:</p></text></region><region region-id="1270" left="0" top="1884" width="6000" height="6" align-x="0" align-y="1884" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1214" left="66" top="1893" width="57" height="12.75" align-x="105.75" align-y="1902" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>b</ml:id></ml:apply></ml:eval></math><rendering item-idref="36" /></region><region region-id="1215" left="144" top="1893" width="63" height="18" align-x="189.75" align-y="1902" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:apply><ml:minus /><ml:apply><ml:id>last</ml:id><ml:id>b</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="37" /></region><region region-id="1216" left="36" top="1916.25" width="202.5" height="12" align-x="36" align-y="1926" show-border="false" show-highlight="false" is-protected="true" 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">A good fit should have a value around 2.</p></text></region><region region-id="1234" left="36" top="1945.5" width="81.75" height="12.75" align-x="36" align-y="1956" show-border="false" show-highlight="false" is-protected="true" 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">Spline derivatives</p></text><indexes><phrase>spline derivatives</phrase><phrase>derivatives, spline</phrase></indexes></region><region region-id="1229" left="36" top="1970.25" width="332.25" height="24" align-x="36" align-y="1980" show-border="false" show-highlight="false" is-protected="true" 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">Here are the derivatives, scaled so you can see them on the same graph.</p></text></region><region region-id="1230" left="36" top="2001.75" width="89.25" height="18" align-x="81.75" align-y="2016" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>FirstDeriv</ml:id><ml:apply><ml:matcol /><ml:id>Spline</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1231" left="144" top="2001.75" width="101.25" height="18" align-x="201.75" align-y="2016" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>SecondDeriv</ml:id><ml:apply><ml:matcol /><ml:id>Spline</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="1232" left="270" top="2001.75" width="93" height="18" align-x="319.5" align-y="2016" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>ThirdDeriv</ml:id><ml:apply><ml:matcol /><ml:id>Spline</ml:id><ml:real>3</ml:real></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="966" left="6" top="2034" width="355.5" height="224.25" align-x="6" align-y="2034" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="41" /><rendering item-idref="42" /></region><region region-id="972" left="36" top="2288.25" width="318.75" height="60" align-x="36" align-y="2298" show-border="false" show-highlight="false" is-protected="true" 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 wish to provide a significancy level, or reject level, for the Durbin Watson test, use the optional last argument to <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Spline2</c></inlineAttr></b></f>, a percentage between 0 and 1. In general, although not always, higher reject levels produce more knots and longer calculations.</p></text></region><region region-id="1271" left="0" top="2358" width="6000" height="6" align-x="0" align-y="2358" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1272" left="66" top="2373" width="39" height="12.75" align-x="90.75" align-y="2382" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>level</ml:id><ml:real>.1</ml:real></ml:define></math><rendering item-idref="43" /></region><region region-id="1273" left="66" top="2397" width="113.25" height="12.75" align-x="77.25" align-y="2406" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:id>w</ml:id><ml:id>level</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="44" /></region><region region-id="1416" left="210" top="2397" width="32.25" height="18" align-x="225" align-y="2406" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="45" /></region><region region-id="1417" left="258" top="2396.25" width="80.25" height="12" align-x="258" align-y="2406" show-border="false" show-highlight="false" is-protected="true" 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">number of knots</p></text></region><region region-id="1276" left="66" top="2421" width="111.75" height="12.75" align-x="101.25" align-y="2430" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline1</ml:id><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:id>range</ml:id><ml:id>b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="1418" left="210" top="2421" width="57" height="12.75" align-x="249.75" align-y="2430" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>b</ml:id></ml:apply></ml:eval></math><rendering item-idref="47" /></region><region region-id="1419" left="300" top="2415.75" width="77.25" height="18" align-x="335.25" align-y="2430" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline1</ml:id><ml:apply><ml:transpose /><ml:id>Spline1</ml:id></ml:apply></ml:define></math><rendering item-idref="48" /></region><region region-id="1421" left="24" top="2436" width="296.25" height="134.25" align-x="24" align-y="2436" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="49" /><rendering item-idref="50" /></region><region region-id="1048" left="36" top="2594.25" width="156.75" height="12" align-x="36" align-y="2604" show-border="false" show-highlight="false" is-protected="true" 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">To interpolate without weights,</p></text></region><region region-id="1049" left="66" top="2619" width="78" height="12.75" align-x="77.25" align-y="2628" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="51" /></region><region region-id="1050" left="66" top="2643" width="57" height="12.75" align-x="105.75" align-y="2652" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>b</ml:id></ml:apply></ml:eval></math><rendering item-idref="52" /></region><region region-id="1051" left="36" top="2666.25" width="276" height="12" align-x="36" align-y="2676" show-border="false" show-highlight="false" is-protected="true" 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">To interpolate without weights and with a reject level, </p></text></region><region region-id="1052" left="66" top="2691" width="93.75" height="12.75" align-x="77.25" align-y="2700" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:real>0.5</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="53" /></region><region region-id="1053" left="66" top="2715" width="57" height="12.75" align-x="105.75" align-y="2724" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>b</ml:id></ml:apply></ml:eval></math><rendering item-idref="54" /></region><region region-id="1056" left="36" top="2743.5" width="119.25" height="12.75" align-x="36" align-y="2754" show-border="false" show-highlight="false" is-protected="true" 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">Supplying your own knots</p></text><indexes><phrase>supplying your own knots</phrase><phrase>knots, supplying your own</phrase></indexes></region><region region-id="1004" left="36" top="2768.25" width="224.25" height="12" align-x="36" align-y="2778" show-border="false" show-highlight="false" is-protected="true" 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">To interpolate with a defined string of knots:</p></text></region><region region-id="1006" left="66" top="2793" width="60.75" height="12.75" align-x="96.75" align-y="2802" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Knots</ml:id><ml:id>range</ml:id></ml:define></math><rendering item-idref="55" /></region><region region-id="1008" left="66" top="2817" width="148.5" height="12.75" align-x="106.5" align-y="2826" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>SplineW</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:id>w</ml:id><ml:id>Knots</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="1013" left="66" top="2841" width="151.5" height="12.75" align-x="111.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline_uk</ml:id><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:id>range</ml:id><ml:id>SplineW</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="1284" left="222" top="2841" width="86.25" height="12.75" align-x="291" align-y="2850" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>SplineW</ml:id></ml:apply></ml:eval></math><rendering item-idref="58" /></region><region region-id="1302" left="66" top="2859.75" width="98.25" height="18" align-x="111.75" align-y="2874" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline_uk</ml:id><ml:apply><ml:transpose /><ml:id>Spline_uk</ml:id></ml:apply></ml:define></math><rendering item-idref="59" /></region><region region-id="1292" left="0" top="2898" width="6000" height="-6" align-x="0" align-y="2892" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1015" left="24" top="2904" width="320.25" height="206.25" align-x="24" align-y="2904" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="60" /><rendering item-idref="61" /></region><region region-id="1286" left="36" top="3134.25" width="272.25" height="12" align-x="36" align-y="3144" show-border="false" show-highlight="false" is-protected="true" 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 also supply knots without a weighting vector:</p></text></region><region region-id="1077" left="66" top="3159" width="143.25" height="12.75" align-x="113.25" align-y="3168" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>SplineNW</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:id>Knots</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="62" /></region><region region-id="1264" left="66" top="3183" width="93" height="12.75" align-x="141.75" align-y="3192" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>SplineNW</ml:id></ml:apply></ml:eval></math><rendering item-idref="63" /></region><region region-id="1281" left="36" top="3212.25" width="310.5" height="36" align-x="36" align-y="3222" show-border="false" show-highlight="false" is-protected="true" 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 an example of why one would use splines for this data rather than a parametric fit, see the example on <b><u><link href="./splinesvspoly.xmcd" title="Links to Application" popup="false">Comparing Splines with a Polynomial Fit</link></u></b> in the Applications section.</p></text></region><region region-id="1315" left="36" top="3265.5" width="96" height="12.75" align-x="36" align-y="3276" show-border="false" show-highlight="false" is-protected="true" 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">Weights and outliers</p></text></region><region region-id="1313" left="36" top="3284.25" width="315.75" height="24" align-x="36" align-y="3294" show-border="false" show-highlight="false" is-protected="true" 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 use the vector of weights in conjunction with the <b><u><link href="./outlier.xmcd" title="Links to section" popup="false">outlier detection</link></u></b> functions.</p></text></region><region region-id="1367" left="66" top="3321" width="110.25" height="18" align-x="93" align-y="3330" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>index</ml:id><ml:apply><ml:indexer /><ml:apply><ml:id>Grubbs</ml:id><ml:sequence><ml:id>y</ml:id><ml:real>0.55</ml:real></ml:sequence></ml:apply><ml:sequence><ml:real>0</ml:real><ml:real>0</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="1372" left="222" top="3321" width="43.5" height="12.75" align-x="248.25" align-y="3330" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>index</ml:id></ml:eval></math><rendering item-idref="65" /></region><region region-id="1361" left="36" top="3344.25" width="300" height="36" align-x="36" align-y="3354" show-border="false" show-highlight="false" is-protected="true" 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 is a suspicious point at the very top of the second data peak.<sp count="2" />Lets try dropping this point out of the interpolation by setting its weight to <b><u><link href="./nan.xmcd" title="Links to section" popup="false">NaN</link></u></b>:</p></text></region><region region-id="1377" left="66" top="3393" width="58.5" height="18" align-x="98.25" align-y="3402" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>w</ml:id><ml:id>index</ml:id></ml:apply><ml:id>NaN</ml:id></ml:define></math><rendering item-idref="66" /></region><region region-id="1378" left="144" top="3393" width="105.75" height="12.75" align-x="171" align-y="3402" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b_no</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id><ml:real>3</ml:real><ml:id>w</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="67" /></region><region region-id="1394" left="270" top="3393" width="72.75" height="12.75" align-x="325.5" align-y="3402" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>DWS</ml:id><ml:id>b_no</ml:id></ml:apply></ml:eval></math><rendering item-idref="68" /></region><region region-id="1397" left="36" top="3416.25" width="302.25" height="24" align-x="36" align-y="3426" show-border="false" show-highlight="false" is-protected="true" 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 results of the interpolation at the suspicious data value:</p></text></region><region region-id="1398" left="66" top="3453" width="96" height="18" align-x="144.75" align-y="3462" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>index</ml:id></ml:apply><ml:id>b</ml:id></ml:sequence></ml:apply><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="69" /></region><region region-id="1399" left="222" top="3453" width="111.75" height="18" align-x="316.5" align-y="3462" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>index</ml:id></ml:apply><ml:id>b_no</ml:id></ml:sequence></ml:apply><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="70" /></region><region region-id="1400" left="36" top="3482.25" width="306" height="36" align-x="36" align-y="3492" show-border="false" show-highlight="false" is-protected="true" 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 value drops slightly, indicating that the spline doesn't have to go as high to fit the extra point.<sp count="2" />Look at a zoomed plot:</p></text></region><region region-id="1388" left="0" top="3528" width="6000" height="6" align-x="0" align-y="3528" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1401" left="66" top="3537" width="41.25" height="12.75" align-x="74.25" align-y="3546" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:real>200</ml:real></ml:range></ml:define></math><rendering item-idref="71" /></region><region region-id="1402" left="144" top="3537" width="66" height="18" align-x="174.75" align-y="3546" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>range</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:real>700</ml:real><ml:id>i</ml:id></ml:apply></ml:define></math><rendering item-idref="72" /></region><region region-id="1405" left="66" top="3561" width="138.75" height="12.75" align-x="112.5" align-y="3570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline_no</ml:id><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:id>range</ml:id><ml:id>b_no</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="73" /></region><region region-id="1411" left="222" top="3561" width="107.25" height="12.75" align-x="252.75" align-y="3570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline</ml:id><ml:apply><ml:id>Binterp</ml:id><ml:sequence><ml:id>range</ml:id><ml:id>b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="74" /></region><region region-id="1406" left="66" top="3579.75" width="99.75" height="18" align-x="112.5" align-y="3594" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline_no</ml:id><ml:apply><ml:transpose /><ml:id>Spline_no</ml:id></ml:apply></ml:define></math><rendering item-idref="75" /></region><region region-id="1412" left="222" top="3579.75" width="68.25" height="18" align-x="252.75" align-y="3594" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_variable" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Spline</ml:id><ml:apply><ml:transpose /><ml:id>Spline</ml:id></ml:apply></ml:define></math><rendering item-idref="76" /></region><region region-id="1414" left="24" top="3612" width="344.25" height="234" align-x="24" align-y="3612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="77" /><rendering item-idref="78" /></region><region region-id="1420" left="36" top="3878.25" width="327" height="60" align-x="36" align-y="3888" show-border="false" show-highlight="false" is-protected="true" 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">Spline2</c></inlineAttr></b></f> functions are based on Spline2 software, developed by B. J. Thijsse and M. A. Hollanders. This software is documented at <f family="Verdana" charset="0"><b><u><inlineAttr line-through="false"><c val="#000"><link href="http://dutsm183.stm.tudelft.nl/software/software.htm" title="Links to web site" popup="false">http://dutsm183.stm.tudelft.nl/software/software.htm</link></c></inlineAttr></u></b></f>, and is used by permission. </p></text></region><region region-id="1103" left="6" top="3954" width="427.5" height="5.25" align-x="6" align-y="3954" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="79" display-width="435" display-height="3.75" /><rendering item-idref="80" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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