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list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./regression.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />REGRESSION</link></c></inlineAttr></u></f></p></text></region><region region-id="1321" left="36" top="90.75" width="123" 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">Polynomial Regression </p></text><indexes><phrase>polynomial regression</phrase><phrase>regression, polynomial</phrase></indexes></region><region region-id="1320" left="66" top="115.5" width="80.25" height="12.75" align-x="66" align-y="126" 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">regress(vx, vy, n)</p></text><indexes><phrase>regress</phrase><phrase>functions, regress</phrase></indexes></region><region region-id="1459" left="168" top="115.5" width="84" height="12.75" align-x="168" align-y="126" 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">loess(vx, vy, span)</p></text><indexes><phrase>loess</phrase><phrase>functions, loess</phrase></indexes></region><region region-id="1416" left="66" top="133.5" width="89.25" height="12.75" align-x="77.25" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">interp(vs, vx, vy, x)</p></text><indexes><phrase>interp</phrase><phrase>functions, interp</phrase></indexes></region><region region-id="1470" left="36" top="158.25" width="309" height="120" align-x="36" align-y="168" 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">Polynomial regression minimizes the distance of all data points to a polynomial curve of degree <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">n</c></inlineAttr></b></f> (least-squares). The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress </c></inlineAttr></b></f>function creates a single polynomial that best approximates the entire data set. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess </c></inlineAttr></b></f><f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">function </c></inlineAttr></f>performs localized regression, fitting second-order (quadratic) polynomials piecewise along the curve. The <inlineAttr font-weight="normal" line-through="false"><c val="#000"><f family="Verdana" charset="0">span</f></c></inlineAttr> argument controls the size of the piece. As <f family="Verdana" charset="0"><c val="#000"><inlineAttr font-weight="normal" line-through="false">span</inlineAttr></c></f> gets larger, <f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><b>loess</b></c></inlineAttr></f> becomes equivalent to <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> with <f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">n = </c></inlineAttr></f><f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">2</c></inlineAttr></f>. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">interp</c></inlineAttr></b></f> function is used to calculate the value of the regression at a point x given the output, <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">vs</c></inlineAttr></b></f>, from <b><inlineAttr line-through="false"><c val="#000"><f family="Verdana" charset="0">regress</f></c></inlineAttr></b> or <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f>.</p></text></region><region region-id="1480" left="36" top="296.25" width="287.25" height="36" align-x="36" align-y="306" 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">Both <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress </c></inlineAttr></b></f>and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> can be used for <b><u><link href="./regress_multi.xmcd" title="Links to section" popup="false">multivariate regression</link></u></b> (more than one independent variable), as discussed in the next section. </p></text></region><region region-id="1483" left="36" top="349.5" width="99" height="12.75" align-x="36" align-y="360" 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">Polynomial of best-fit</p></text><indexes><phrase>polynomial of best-fit</phrase><phrase>best-fit, polynomial of</phrase></indexes></region><region region-id="1484" left="36" top="368.25" width="129" height="12" align-x="36" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Consider a set of data:</p></text></region><region region-id="1605" left="66" top="387" width="36.75" height="12.75" align-x="74.25" align-y="396" 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>30</ml:real></ml:range></ml:define></math><rendering item-idref="4" /></region><region region-id="1606" left="66" top="411" width="70.5" height="18" align-x="83.25" align-y="420" 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>vx</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id>rnd</ml:id><ml:real>10</ml:real></ml:apply><ml:real>5</ml:real></ml:apply></ml:define></math><rendering item-idref="5" /></region><region region-id="1978" left="162" top="411" width="108.75" height="18" align-x="180.75" align-y="420" 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>vy</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:id>sin</ml:id><ml:apply><ml:indexer /><ml:id>vx</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:id>rnd</ml:id><ml:real>0.5</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="6" /></region><region region-id="1490" left="36" top="434.25" width="304.5" height="51" align-x="36" align-y="444" 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">We can construct the polynomial of best-fit in the least squares sense much like we did for the <b><u><link href="./linreg.xmcd" title="Links to section" popup="false">linear case</link></u></b>. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> function returns the set of polynomial coefficients c<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>i</sub></c></inlineAttr></f> that make the sum-of-squares equation</p></text></region><region region-id="1551" left="66" top="493.5" width="119.25" height="49.5" align-x="124.5" align-y="522" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>Y</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>j</ml:id></ml:apply></ml:parens><ml:id>i</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:id>n</ml:id></ml:bounds></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:id>N</ml:id></ml:bounds></ml:apply></math><rendering item-idref="7" /></region><region region-id="1553" left="36" top="554.25" width="312.75" height="48" align-x="36" align-y="564" 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 minimum, by differentiating with respect to each coefficient and setting the results equal to zero. The normal equations obtained in this way are then solved by singular value decomposition.</p></text></region><region region-id="1554" left="0" top="612" width="6000" height="6" align-x="0" align-y="612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1637" left="36" top="626.25" width="273.75" height="12" align-x="36" align-y="636" 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 several polynomial fits, for order 3, 5, and 10.</p></text></region><region region-id="1994" left="66" top="651" width="92.25" height="12.75" align-x="81" align-y="660" 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>c3</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>3</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1993" left="222" top="651" width="109.5" height="12.75" align-x="248.25" align-y="660" 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>p3</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c3</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1990" left="66" top="669" width="92.25" height="12.75" align-x="81" align-y="678" 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>c5</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>5</ml:real></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="1983" left="222" top="669" width="109.5" height="12.75" align-x="248.25" align-y="678" 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>p5</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c5</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="1996" left="66" top="687" width="101.25" height="12.75" align-x="85.5" align-y="696" 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>c10</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>10</ml:real></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="1995" left="222" top="687" width="118.5" height="12.75" align-x="252.75" align-y="696" 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>p10</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c10</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="13" /></region><region region-id="1992" left="66" top="705" width="59.25" height="12.75" align-x="75" align-y="714" 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>t</ml:id><ml:range><ml:sequence><ml:real>-6</ml:real><ml:real>-5.9</ml:real></ml:sequence><ml:real>6</ml:real></ml:range></ml:define></math><rendering item-idref="14" /></region><region region-id="2010" left="30" top="726" width="294.75" height="191.25" align-x="30" align-y="726" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="15" /><rendering item-idref="16" /></region><region region-id="1722" left="36" top="943.5" width="171.75" height="12.75" align-x="36" align-y="954" 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">Choosing the order of the polynomial</p></text><indexes><phrase>choosing the order of the polynomial</phrase><phrase>polynomial, choosing the order of the</phrase></indexes></region><region region-id="1639" left="36" top="968.25" width="302.25" height="156" align-x="36" align-y="978" 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 plot brings up an important point. The order of a polynomial specifies the number of bumps in the function. While you want to include enough bumps to adequately model your data, the extra wiggles will probably make your function behave more unpredictably, that is, less like the general trend of your data, at the ends of the data range. As with all regression, using the polynomial fit function outside the range of the data carries risk. In addition, calculating polynomials of higher order requires more computations, since you are solving a larger system of equations. You must balance the need for accuracy against the speed of the computation and the overall behavior of the function. </p></text></region><region region-id="1702" left="36" top="1136.25" width="309" height="24" align-x="36" align-y="1146" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">It may be worthwhile to investigate how much your estimates improve with higher order:</p></text></region><region region-id="2008" left="0" top="1170" width="6000" height="6" align-x="0" align-y="1170" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1698" left="66" top="1185" width="339" height="93" align-x="100.5" align-y="1194" 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>polyerr</ml:id><ml:program><ml:for><ml:id>n</ml:id><ml:range><ml:real>1</ml:real><ml:real>10</ml:real></ml:range><ml:program><ml:localDefine><ml:id>c</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>n</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:apply><ml:indexer /><ml:id>SSE</ml:id><ml:apply><ml:minus /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>vx</ml:id></ml:apply><ml:id>n</ml:id></ml:apply></ml:apply><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>vy</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:id>interp</ml:id><ml:id>c</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:apply><ml:indexer /><ml:id>vx</ml:id><ml:id>j</ml:id></ml:apply></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:id>last</ml:id><ml:id>vx</ml:id></ml:apply></ml:bounds></ml:apply></ml:apply></ml:localDefine></ml:program></ml:for><ml:id>SSE</ml:id></ml:program></ml:define></math><rendering item-idref="17" /></region><region region-id="1705" left="66" top="1287" width="32.25" height="12.75" align-x="74.25" align-y="1296" show-border="false" show-highlight="false" is-protected="true" 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:real>9</ml:real></ml:range></ml:define></math><rendering item-idref="18" /></region><region region-id="1718" left="30" top="1314" width="249.75" height="196.5" align-x="30" align-y="1314" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="19" /><rendering item-idref="20" /></region><region region-id="1720" left="270" top="1297.5" width="105.75" height="166.5" align-x="303.75" align-y="1380" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>polyerr</ml:id></ml:eval><resultFormat><table item-idref="21" /></resultFormat></math><rendering item-idref="22" /></region><region region-id="1707" left="36" top="1502.25" width="306" height="24" align-x="36" align-y="1512" 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">So, in the case of this data, a polynomial order of 5 or 6 is probably more than sufficient. </p></text></region><region region-id="1753" left="36" top="1538.25" width="327.75" height="24" align-x="36" align-y="1548" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">Finally, recall that you can't fit a polynomial with an order greater than one less than the total number of data points:</p></text></region><region region-id="2009" left="0" top="1566" width="6000" height="6" align-x="0" align-y="1566" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1750" left="66" top="1569.75" width="107.25" height="18" align-x="81" align-y="1584" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>vx</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="5"><ml:real>0.5</ml:real><ml:real>1</ml:real><ml:real>1.5</ml:real><ml:real>2</ml:real><ml:real>2.5</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="1772" left="222" top="1575" width="58.5" height="12.75" align-x="230.25" align-y="1584" show-border="false" show-highlight="false" is-protected="true" 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>vx</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="24" /></region><region region-id="1752" left="66" top="1593.75" width="120.75" height="18" align-x="82.5" align-y="1608" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>vy</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="5"><ml:real>-1.7</ml:real><ml:real>3.9</ml:real><ml:real>2.7</ml:real><ml:real>3.1</ml:real><ml:real>0</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="1773" left="222" top="1599" width="48.75" height="12.75" align-x="231" align-y="1608" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>t</ml:id><ml:range><ml:sequence><ml:real>0</ml:real><ml:real>0.1</ml:real></ml:sequence><ml:real>3</ml:real></ml:range></ml:define></math><rendering item-idref="26" /></region><region region-id="1756" left="66" top="1623" width="92.25" height="12.75" align-x="81" align-y="1632" 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>c1</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="1771" left="222" top="1623" width="109.5" height="12.75" align-x="248.25" align-y="1632" 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>P1</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c1</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="28" /></region><region region-id="1758" left="66" top="1641" width="92.25" height="12.75" align-x="81" align-y="1650" 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>c2</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="1770" left="222" top="1641" width="109.5" height="12.75" align-x="248.25" align-y="1650" 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>P2</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c2</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="1760" left="66" top="1659" width="92.25" height="12.75" align-x="81" align-y="1668" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>c3</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>3</ml:real></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="1769" left="222" top="1659" width="109.5" height="12.75" align-x="248.25" align-y="1668" 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>P3</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c3</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="1762" left="66" top="1677" width="92.25" height="12.75" align-x="81" align-y="1686" 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>c4</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>4</ml:real></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="1768" left="222" top="1677" width="109.5" height="12.75" align-x="248.25" align-y="1686" 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>P4</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c4</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="1655" left="24" top="1698" width="284.25" height="167.25" align-x="24" align-y="1698" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="35" /><rendering item-idref="36" /></region><region region-id="1774" left="36" top="1874.25" width="318.75" height="48" align-x="36" align-y="1884" 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 see the parabolic, cubic, and quartic fits to the data. In the case of 5 data points, the quartic is sufficient to pass through every point, and is, in fact, the same as the interpolating polynomial.</p></text></region><region region-id="1675" left="66" top="1941" width="92.25" height="12.75" align-x="81" align-y="1950" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>c4</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>4</ml:real></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="1676" left="222" top="1941" width="109.5" height="12.75" align-x="248.25" align-y="1950" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id>P4</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>c4</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1997" left="66" top="1965" width="108.75" height="18" align-x="95.25" align-y="1974" 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>pol</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:indexer /><ml:apply><ml:id>polyint</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>t</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="1665" left="36" top="1992" width="286.5" height="167.25" align-x="36" align-y="1992" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="40" /><rendering item-idref="41" /></region><region region-id="1775" left="0" top="2166" width="6000" height="6" align-x="0" align-y="2166" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1685" left="36" top="2180.25" width="300.75" height="108" align-x="36" align-y="2190" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">Note that the polynomial of best fit calculated by <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> is, in general, different than the interpolating polynomial given by <b><u><link href="./polint.xmcd" title="Links to section" popup="false">polyint</link></u></b>. While <b>polyint</b> returns a line that will pass through every point, the interpolating polynomial will be of degree one less than the number of points, and so may oscillate between points unrealistically or undesirably. The<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"> regress</c></inlineAttr></b></f> function will allow you to constrain the order of the polynomial, and will return the coefficients of the fit. It is more like <inlineAttr line-through="false"><c val="#000"><f family="Verdana" charset="0"><b>polyiter</b></f></c></inlineAttr> in this respect.</p></text></region><region region-id="1686" left="36" top="2300.25" width="180.75" height="12" align-x="36" align-y="2310" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Any requests for higher degree fits, </p></text></region><region region-id="1688" left="66" top="2319" width="23.25" height="12.75" align-x="77.25" align-y="2328" 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>5</ml:real></ml:define></math><rendering item-idref="42" /></region><region region-id="1689" left="66" top="2337" width="88.5" height="12.75" align-x="76.5" align-y="2346" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="There must be at least as many data points as parameters" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>c</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>n</ml:id></ml:apply></ml:define></math><rendering item-idref="43" /></region><region region-id="1687" left="36" top="2360.25" width="274.5" height="48" align-x="36" align-y="2370" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><c val="#000">result in an error message that there must be as many data points than polynomial coefficients. Now let's look at what the </c><b><c val="#000">regress</c></b><c val="#000"> function returns and why the </c><b><c val="#000">interp</c></b><c val="#000"> function enters the picture.</c></p></text></region><region region-id="1777" left="36" top="2420.25" width="309" height="60" align-x="36" align-y="2430" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><c val="#000">The </c><b><c val="#000">regress</c></b><c val="#000"> function is used like the various spline functions (</c><b><c val="#000">pspline</c></b><c val="#000">, </c><b><c val="#000">lspline</c></b><c val="#000">, </c><b><c val="#000">cspline</c></b><c val="#000">) discussed in the section on </c><b><u><link href="./splines_1.xmcd" title="Links to section" popup="false">spline interpolation</link></u></b><c val="#000">. </c><c val="#000">T</c><c val="#000">he output of </c><b><c val="#000">regress</c></b><c val="#000"> will be a vector. </c>The first three components contain codes for <b>interp</b>. They are:</p></text></region><region region-id="1803" left="144" top="2499" width="36" height="18" align-x="162.75" align-y="2508" 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>c4</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="44" /></region><region region-id="1811" left="192" top="2498.25" width="179.25" height="24" align-x="192" align-y="2508" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><c val="#000">A code telling </c><b><c val="#000">interp</c></b><c val="#000"> that this is the output of </c><b><c val="#000">regress</c></b><f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">.</c></inlineAttr></f><b><c val="#000"><sp /></c></b></p></text></region><region region-id="1813" left="144" top="2529" width="36" height="18" align-x="162.75" align-y="2538" 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>c4</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="45" /></region><region region-id="1814" left="192" top="2528.25" width="180.75" height="24" align-x="192" align-y="2538" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The index within the vector where the polynomial coefficients begin. </p></text></region><region region-id="1971" left="60" top="2501.25" width="69" height="128.25" align-x="77.25" align-y="2568" 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>Co</ml:id><ml:matrix rows="8" cols="1"><ml:real>3</ml:real><ml:real>3</ml:real><ml:real>4</ml:real><ml:real>-36</ml:real><ml:real>118.65</ml:real><ml:real>-125.95</ml:real><ml:real>56.2</ml:real><ml:real>-9</ml:real></ml:matrix></ml:define></math><rendering item-idref="46" /></region><region region-id="1817" left="144" top="2559" width="36" height="18" align-x="162.75" align-y="2568" 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>c4</ml:id><ml:real>2</ml:real></ml:apply></ml:eval></math><rendering item-idref="47" /></region><region region-id="1818" left="192" top="2558.25" width="75.75" height="12" align-x="192" align-y="2568" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The value of n.</p></text></region><region region-id="1819" left="144" top="2588.25" width="202.5" height="48" align-x="144" align-y="2598" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The remaining n+1 terms are the coefficients of the fitting polynomial from the lowest-degree term to the highest-degree term.<sp count="2" /></p></text></region><region region-id="1823" left="36" top="2648.25" width="318" height="36" align-x="36" align-y="2658" 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 means you can extract the polynomial coefficients for the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> fit directly, if you need them, and construct the fitting polynomial as</p></text></region><region region-id="1843" left="222" top="2697" width="33" height="12.75" align-x="236.25" align-y="2706" 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>x0</ml:id><ml:real>2.2</ml:real></ml:define></math><rendering item-idref="48" /></region><region region-id="1974" left="66" top="2700" width="94.5" height="45" align-x="86.25" align-y="2724" 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>f</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>Co</ml:id><ml:apply><ml:plus /><ml:id>i</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:indexer /><ml:id>Co</ml:id><ml:real>2</ml:real></ml:apply></ml:bounds></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="1837" left="222" top="2715" width="57" height="12.75" align-x="246" align-y="2724" 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>f</ml:id><ml:id>x0</ml:id></ml:apply></ml:eval></math><rendering item-idref="50" /></region><region region-id="1838" left="222" top="2733" width="119.25" height="12.75" align-x="308.25" align-y="2742" 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>interp</ml:id><ml:id>c4</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>x0</ml:id></ml:apply></ml:eval></math><rendering item-idref="51" /></region><region region-id="1975" left="66" top="2754" width="247.5" height="45" align-x="140.25" align-y="2778" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:symEval><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>Co</ml:id><ml:apply><ml:plus /><ml:id>i</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:indexer /><ml:id>Co</ml:id><ml:real>2</ml:real></ml:apply></ml:bounds></ml:apply><ml:symResult><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:real>-36</ml:real><ml:apply><ml:minus /><ml:apply><ml:mult style="narrow-dot" /><ml:apply><ml:div /><ml:real>2373</ml:real><ml:real>20</ml:real></ml:apply><ml:id>x</ml:id></ml:apply><ml:apply><ml:mult style="narrow-dot" /><ml:apply><ml:div /><ml:real>2519</ml:real><ml:real>20</ml:real></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:mult style="narrow-dot" /><ml:apply><ml:div /><ml:real>281</ml:real><ml:real>5</ml:real></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="narrow-dot" /><ml:real>9</ml:real><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>4</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:symResult></ml:symEval></math><rendering item-idref="52" /></region><region region-id="1822" left="0" top="2814" width="6000" height="6" align-x="0" align-y="2814" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1820" left="36" top="2827.5" width="120.75" height="12.75" align-x="36" align-y="2838" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag="loess"><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">Piecewise quadratic fitting</p></text><indexes><phrase>piecewise quadratic fitting</phrase><phrase>fitting, piecewise quadratic</phrase></indexes></region><region region-id="1867" left="36" top="2846.25" width="311.25" height="72" align-x="36" align-y="2856" 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">loess</c></inlineAttr></b></f> function fits a series of quadratic functions to data using local regression. A percentage of the total number of points in a weighted window around each data point will be used for successive quadratic fits. The percentage is defined by the third argument, span, or <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">a</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">,</c></inlineAttr></f> which is governed by the general guideline that </p></text></region><region region-id="1926" left="66" top="2931" width="75.75" height="12.75" align-x="130.5" align-y="2940" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:greaterThan /><ml:apply><ml:mult style="default" /><ml:id>span</ml:id><ml:id>Numpts</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></math><rendering item-idref="53" /></region><region region-id="1927" left="36" top="2954.25" width="316.5" height="48" align-x="36" align-y="2964" 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">since you want some number of points greater than 1 to be averaged into each least-squares weighted fit. Using only a few points around each value of x would be roughly equivalent to doing a quadratic spline. So, for the data </p></text></region><region region-id="1917" left="66" top="3021" width="123.75" height="36" align-x="78" align-y="3030" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="54" disable-calc="false"><inputs hide="false" /><outputs hide="false"><ml:id>S</ml:id></outputs></component><rendering item-idref="55" /></region><region region-id="1918" left="66" top="3069.75" width="37.5" height="18" align-x="79.5" align-y="3084" 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>S</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="1919" left="174" top="3069.75" width="36" height="18" align-x="186" align-y="3084" 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>S</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="1900" left="66" top="3098.25" width="27" height="13.5" align-x="78.75" align-y="3108" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>α</ml:id><ml:real>.5</ml:real></ml:define></math><rendering item-idref="58" /></region><region region-id="1936" left="144" top="3098.25" width="84" height="13.5" align-x="199.5" align-y="3108" 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:mult style="default" /><ml:id>α</ml:id><ml:apply><ml:id>length</ml:id><ml:id>X</ml:id></ml:apply></ml:apply></ml:eval></math><rendering item-idref="59" /></region><region region-id="2004" left="240" top="3099" width="67.5" height="12.75" align-x="285.75" align-y="3108" 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="60" /></region><region region-id="1895" left="66" top="3123" width="99.75" height="12.75" align-x="94.5" align-y="3132" 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>cpoly</ml:id><ml:apply><ml:id>regress</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="61" /></region><region region-id="2005" left="240" top="3123" width="133.5" height="12.75" align-x="282.75" align-y="3132" 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>fitpoly</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>cpoly</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="62" /></region><region region-id="1897" left="66" top="3146.25" width="83.25" height="13.5" align-x="84" align-y="3156" 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>clo</ml:id><ml:apply><ml:id>loess</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>α</ml:id></ml:apply></ml:define></math><rendering item-idref="63" /></region><region region-id="2006" left="240" top="3147" width="126" height="12.75" align-x="285.75" align-y="3156" 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>fitloess</ml:id><ml:boundVars><ml:id>t</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>clo</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>t</ml:id></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="1877" left="66" top="3171" width="135.75" height="12.75" align-x="75" align-y="3180" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>t</ml:id><ml:range><ml:sequence><ml:apply><ml:id>min</ml:id><ml:id>X</ml:id></ml:apply><ml:apply><ml:plus /><ml:apply><ml:id>min</ml:id><ml:id>X</ml:id></ml:apply><ml:real>.1</ml:real></ml:apply></ml:sequence><ml:apply><ml:id>max</ml:id><ml:id>X</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="65" /></region><region region-id="1878" left="54" top="3198" width="300.75" height="165.75" align-x="54" align-y="3198" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="66" /><rendering item-idref="67" /></region><region region-id="1904" left="36" top="3380.25" width="306.75" height="48" align-x="36" align-y="3390" 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">In the case where <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">a</c></inlineAttr></f> is large, it means that all data are being considered together in a single quadratic fit with almost all weights equal. This will approach the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> solution for a quadratic polynomial. Try setting <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">a</c></inlineAttr></f> to 2<f family="Verdana" charset="0"> or 3</f>.</p></text></region><region region-id="2007" left="0" top="3438" width="6000" height="6" align-x="0" align-y="3438" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1911" left="36" top="3444.75" width="312" height="64.5" align-x="36" align-y="3456" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">There are no real exact guidelines in choosing a value for <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr></f>. As <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr></f> gets larger, the plot of the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> fit gets smoother, because fewer pieces are being connected in the fit. But the fit may fail to follow data features appropriately. Conversely, as <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">a</c></inlineAttr></f> gets small, you may find that <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> doesn't converge.</p></text></region><region region-id="1943" left="36" top="3518.25" width="313.5" height="36" align-x="36" align-y="3528" 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 loess algorithm is not designed for extrapolation. If you try to evaluate <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">interp</c></inlineAttr></b></f> at values beyond the original x-range, you'll get an error.</p></text></region><region region-id="1945" left="66" top="3573" width="69.75" height="12.75" align-x="122.25" align-y="3582" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="loess cannot extrapolate" optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>fitloess</ml:id><ml:real>-15</ml:real></ml:apply></ml:eval></math><rendering item-idref="68" /></region><region region-id="1951" left="36" top="3602.25" width="315" height="48" align-x="36" align-y="3612" 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 <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> when your data has a somewhat discontinuous characteristic, but when you do not wish to piecewise fit by hand, or use complex or physically unrealistic models and nonlinear fitting methods.</p></text></region><region region-id="1952" left="36" top="3662.25" width="309.75" height="24" align-x="36" align-y="3672" 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">Both <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">regress</c></inlineAttr></b></f> and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> are capable of multivariate fitting, as described in the <b><u><link href="./regress_multi.xmcd" title="Links to section" popup="false">next section</link></u></b>.</p></text></region><region region-id="1950" left="36" top="3698.25" width="298.5" height="96" align-x="36" align-y="3708" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">The Mathcad implementation of <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> is a variation (some approximations made for the sake of speed) on the algorithm described in <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">Smoothing by Local Regression: Principles and Methods</c></inlineAttr></i></f>. W. S. Cleveland and C. L. Loader (1996), <b><u><link href="http://cm.bell-labs.com/cm/ms/departments/sia/doc/smoothing.springer.pdf" title="Links to Web Site" popup="false">http://cm.bell-labs.com/cm/ms/departments/sia/doc/smoothing.springer.pdf.</link></u></b></p></text></region><region region-id="1794" left="6" top="3804" width="427.5" height="5.25" align-x="6" align-y="3804" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="69" /><rendering item-idref="70" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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