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align-x="6" align-y="6" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="2" /><link href="./dataTOC.xmcd" title="Links to Table of Contents" popup="false" /><rendering item-idref="3" /></region><region region-id="1448" left="174" top="44.25" width="68.25" height="12" align-x="174" align-y="54" 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="Titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="4" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="2056" left="36" top="90.75" width="210" height="13.5" align-x="36" align-y="102" 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">Comparing Splines with a Polynomial Fit</p></text><indexes><phrase>comparing splines with a polynomial fit</phrase><phrase>examples, comparing splines with a polynomial fit</phrase></indexes></region><region region-id="1450" left="36" top="104.25" width="76.5" height="12" align-x="36" 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="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">by Robert Adair</p></text></region><region region-id="2274" left="36" top="134.25" width="324" height="60" align-x="36" align-y="144" 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 <b><u><link href="./statspline.xmcd" title="Links to section" popup="false">statistical B-spline functions</link></u></b> introduced in the data analysis extension pack will calculate a string of knots using the Durbin-Watson statistic to accept or reject spline fits. In this way, statistical B-splines supply a minimal number of knots to reflect all of the data features. </p></text></region><region region-id="2275" left="66" top="213" width="127.5" height="12.75" align-x="76.5" align-y="222" 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>a</ml:id><ml:apply><ml:id>READPRN</ml:id><ml:str>example1.txt</ml:str></ml:apply></ml:define></math><rendering item-idref="4" /></region><region region-id="2276" left="66" top="231.75" width="33" height="18" align-x="75.75" align-y="246" 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>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="5" /></region><region region-id="2277" left="144" top="231.75" width="34.5" height="18" align-x="155.25" align-y="246" 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>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="6" /></region><region region-id="2278" left="222" top="231.75" width="36.75" height="18" align-x="235.5" align-y="246" 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>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="7" /></region><region region-id="2279" left="36" top="266.25" width="312" height="36" align-x="36" align-y="276" 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 <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">w</c></inlineAttr></b></f> is a vector of weights giving the estimated standard deviations of the random error in y. We'll fit with spline polynomials of order 3.</p></text></region><region region-id="2385" left="66" top="315" width="71.25" height="12.75" align-x="125.25" align-y="324" 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>SplineDegree</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="8" /></region><region region-id="2389" left="66" top="339" width="138.75" height="12.75" align-x="77.25" align-y="348" 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>b</ml:id><ml:apply><ml:id>Spline2</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>SplineDegree</ml:id><ml:id>w</ml:id></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="2303" left="36" top="368.25" width="300.75" height="57.75" align-x="43.5" align-y="378" 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 number of knots returned for the optimal spline fit is <region region-id="2304" left="36" top="381" width="36.75" height="18" align-x="51" align-y="390" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math 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="10" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">, which is a good compression of the data from the original </c></inlineAttr></f><region region-id="2305" left="72" top="399" width="427.5" height="12.75" align-x="114" align-y="408" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>length</ml:id><ml:id>x</ml:id></ml:apply></ml:eval></math><rendering item-idref="11" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">points. 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Do a global fit to the data using the same number of free parameters as the spline has knots, to see why the spline is a better option.</p></text></region><region region-id="2323" left="66" top="921" width="79.5" height="18" align-x="128.25" align-y="930" 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>NumFreeParm</ml:id><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="2329" left="222" top="921" width="132" height="12.75" align-x="274.5" align-y="930" 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>PolyDegree</ml:id><ml:apply><ml:minus /><ml:id>NumFreeParm</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="2325" left="66" top="945" width="124.5" height="12.75" align-x="81.75" align-y="954" 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>vs</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>PolyDegree</ml:id></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="2326" left="66" top="969" width="159.75" height="16.5" align-x="147" align-y="978" 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 subscript="global.regress">f</ml:id><ml:boundVars><ml:id>x1</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>vs</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>x1</ml:id></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="2327" left="66" top="999" width="118.5" height="22.5" align-x="106.5" align-y="1014" 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 subscript="regress">y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id subscript="global.regress">f</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2330" left="36" top="1040.25" width="318.75" height="24" align-x="36" align-y="1050" 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 <b><u><link href="./regress.xmcd" title="Links to section" popup="false">regress</link></u></b> function fails in this fit, which is not surprising given the high polynomial degree.</p></text></region><region region-id="2332" left="6" top="1080" width="307.5" height="138" align-x="6" align-y="1080" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="25" /><rendering item-idref="26" /></region><region region-id="2390" left="0" top="1212" width="6000" height="6" align-x="0" align-y="1212" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2334" left="36" top="1226.25" width="320.25" height="24" align-x="36" align-y="1236" 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">Try fitting to Chebyshev polynomials for a better numerical stability.</p></text></region><region region-id="2351" left="66" top="1269" width="75.75" height="16.5" align-x="90" align-y="1278" 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 subscript="min">x</ml:id><ml:apply><ml:id>min</ml:id><ml:id>knots</ml:id></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="2393" left="228" top="1269" width="77.25" height="16.5" align-x="252.75" align-y="1278" 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 subscript="max">x</ml:id><ml:apply><ml:id>max</ml:id><ml:id>knots</ml:id></ml:apply></ml:define></math><rendering item-idref="28" /></region><region 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background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id>last</ml:id><ml:id>x</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="30" /></region><region region-id="2367" left="144" top="1347" width="75" height="12.75" align-x="153" align-y="1356" 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:id>PolyDegree</ml:id></ml:range></ml:define></math><rendering item-idref="31" /></region><region region-id="2371" left="66" top="1383" width="74.25" height="18" align-x="90.75" align-y="1392" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>M</ml:id><ml:sequence><ml:id>i</ml:id><ml:id>j</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>fPoly</ml:id><ml:id>j</ml:id><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="2370" left="144" top="1371.75" width="89.25" height="24" align-x="154.5" align-y="1392" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id>c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id>M</ml:id></ml:apply><ml:id>M</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply><ml:apply><ml:transpose /><ml:id>M</ml:id></ml:apply></ml:apply><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="2363" left="36" top="1412.25" width="313.5" height="24" align-x="36" align-y="1422" 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 matrix inversion is nearly singular in a numerical sense as can be seen by evaluating the determinant.</p></text></region><region region-id="2373" left="66" top="1449" width="116.25" height="18.75" align-x="106.5" align-y="1464" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:determinant /><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id>M</ml:id></ml:apply><ml:id>M</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="34" /></region><region region-id="2375" left="36" top="1484.25" width="312" height="36" align-x="36" align-y="1494" 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 testament to the numerical complexity of fitting a polynomial with enough coefficients to represent all the inflections in the data, as well as the large number of points.</p></text></region><region region-id="2357" left="66" top="1530.75" width="125.25" height="32.25" align-x="111" align-y="1542" 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 subscript="global">f</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:id>fPoly</ml:id><ml:id>j</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="2376" left="66" top="1575" width="82.5" height="22.5" align-x="102" align-y="1590" 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 subscript="global">y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id subscript="global">f</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="2379" left="24" top="1614" width="354.75" height="237" align-x="24" align-y="1614" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="37" /><rendering item-idref="38" /></region><region region-id="2396" left="0" top="1860" width="6000" height="6" align-x="0" align-y="1860" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2382" left="36" top="1868.25" width="315" height="72" align-x="36" align-y="1878" 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">Even the more stable Tchebychev polynomials will oscillate on the tails of the data because of the large number of terms, and fail to accurately represent the peaks and finer features of the data. While the splines are perhaps less informative as a physical model, they are a much better predictor of the behavior of the data at any arbitrary point in the range.</p></text></region><region region-id="2395" left="6" top="1956" width="427.5" height="8.25" align-x="6" align-y="1956" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="39" /><rendering item-idref="40" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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