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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="185.25" 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">Multivariate Polynomial Regression </p></text><indexes><phrase>multivariate polynomial regression</phrase><phrase>regression, multivariate polynomial regression</phrase></indexes></region><region region-id="1320" left="66" top="115.5" width="99.75" 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(Mind,vDep,n)</p></text><indexes><phrase>regress</phrase><phrase>functions, regress</phrase></indexes></region><region region-id="1820" left="192" top="115.5" width="90.75" height="12.75" align-x="192" 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(Mind,vDep,<f family="Symbol" charset="2" size="11"><b><inlineAttr font-style="normal" underline="false" line-through="false"><c val="#800000">a</c></inlineAttr></b></f>)</p></text><indexes><phrase>loess</phrase><phrase>functions, loess</phrase></indexes></region><region region-id="1416" left="66" top="133.5" width="80.25" 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="2094" left="36" top="164.25" width="299.25" height="72" align-x="36" align-y="174" 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 multivariate regression minimizes the distance of a vector of dependent data points <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">vDep</c></inlineAttr></b></f> to a polynomial surface of degree <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">n</c></inlineAttr></b></f> (least-squares). <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Mind</c></inlineAttr></b></f> is a matrix with the same number of rows as there are data points, and one column for each independent variable. In the case of loess, no more than 4 independent variables can be fit.</p></text></region><region region-id="2098" left="36" top="248.25" width="316.5" height="72" align-x="36" align-y="258" 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">Keep in mind that you must always have enough data points for a regression to determine all the coefficients (at least as many independent measurements as coefficients). If you are fitting a polynomial surface of order n, it will have n+1 terms for each independent variable, and all the cross terms. For example, a surface of order 3 in two variables has the form </p></text></region><region region-id="2097" left="12" top="333" width="459.75" height="24" align-x="427.5" 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:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer 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/><ml:id>a</ml:id><ml:real>6</ml:real></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:parens><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>7</ml:real></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>8</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:real>9</ml:real></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></math><rendering item-idref="4" /></region><region region-id="2102" left="36" top="368.25" width="329.25" height="24" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">so it requires at least 10 data points (10 rows of data) in the first matrix argument for regress to fit. </p></text></region><region region-id="2056" left="36" top="404.25" width="154.5" height="12" align-x="36" align-y="414" 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 example, consider the data</p></text></region><region region-id="2057" left="66" top="433.5" width="54" height="111.75" align-x="89.25" align-y="492" 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>Age</ml:id><ml:matrix rows="7" cols="1"><ml:real>30</ml:real><ml:real>27</ml:real><ml:real>44</ml:real><ml:real>18</ml:real><ml:real>35</ml:real><ml:real>20</ml:real><ml:real>26</ml:real></ml:matrix></ml:define></math><rendering item-idref="5" /></region><region region-id="2058" left="144" top="433.5" width="63.75" height="111.75" align-x="177" align-y="492" 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>Height</ml:id><ml:matrix rows="7" cols="1"><ml:real>66</ml:real><ml:real>71</ml:real><ml:real>69</ml:real><ml:real>70</ml:real><ml:real>61</ml:real><ml:real>66</ml:real><ml:real>67</ml:real></ml:matrix></ml:define></math><rendering item-idref="6" /></region><region region-id="2060" left="228" top="433.5" width="71.25" height="111.75" align-x="264" align-y="492" 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>Weight</ml:id><ml:matrix rows="7" cols="1"><ml:real>165</ml:real><ml:real>204</ml:real><ml:real>160</ml:real><ml:real>158</ml:real><ml:real>163</ml:real><ml:real>175</ml:real><ml:real>152</ml:real></ml:matrix></ml:define></math><rendering item-idref="7" /></region><region region-id="2063" left="36" top="554.25" width="306.75" height="36" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">A regression can tell us if age can be expressed as a function of height and weight. Use regress to model the relationship as a polynomial surface.</p></text></region><region region-id="2068" left="0" top="606" width="6000" height="6" align-x="0" align-y="606" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2103" left="66" top="669" width="140.25" height="12.75" align-x="95.25" 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:id>Indep</ml:id><ml:apply><ml:id>augment</ml:id><ml:id>Height</ml:id><ml:id>Weight</ml:id></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="2104" left="222" top="619.5" width="84.75" height="111.75" align-x="250.5" 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:eval><ml:id>Indep</ml:id></ml:eval></math><rendering item-idref="9" /></region><region region-id="2073" left="36" top="746.25" width="283.5" height="12" align-x="36" align-y="756" 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 number of independent variables of the polynomial is</p></text></region><region region-id="2076" left="66" top="771" width="85.5" height="12.75" align-x="96.75" align-y="780" 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>Nvars</ml:id><ml:apply><ml:id>cols</ml:id><ml:id>Indep</ml:id></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="2083" left="222" top="771" width="47.25" height="12.75" align-x="252" align-y="780" 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>Nvars</ml:id></ml:eval></math><rendering item-idref="11" /></region><region region-id="2079" left="36" top="800.25" width="162.75" height="12" align-x="36" align-y="810" 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">and the number of data points is</p></text></region><region region-id="2080" left="66" top="825" width="88.5" height="12.75" align-x="96.75" align-y="834" 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>Ndata</ml:id><ml:apply><ml:id>rows</ml:id><ml:id>Indep</ml:id></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="2082" left="222" top="825" width="47.25" height="12.75" align-x="252" align-y="834" 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>Ndata</ml:id></ml:eval></math><rendering item-idref="13" /></region><region region-id="2105" left="36" top="854.25" width="102.75" height="12" align-x="36" align-y="864" 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">Now call </c><b><c val="#000">regress</c></b><c val="#000"><sp /></c></p></text></region><region region-id="2107" left="66" top="879" width="110.25" height="12.75" align-x="78" align-y="888" 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:id>regress</ml:id><ml:id>Indep</ml:id><ml:id>Age</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="14" /></region><region region-id="2109" left="36" top="902.25" width="282.75" height="60" align-x="36" align-y="912" 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">This returns a vector of constants</c><b><c val="#000"><sp /></c></b><c val="#000">that does include the coefficients of the fitting multinomial, as well as other information about what function generated it. The vector is designed to be passed to </c><b><c val="#000">interp</c></b><c val="#000"> to give values from the fitting polynomial. </c></p></text></region><region region-id="2110" left="36" top="974.25" width="282" height="24" align-x="36" align-y="984" 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 argument list to </c><b><c val="#000">interp</c></b><c val="#000"> is the same as the </c><b><u><link href="./regress.xmcd" title="Links to section" popup="false">one-dimensional case</link></u></b><c val="#000">,</c></p></text></region><region region-id="2118" left="66" top="1014.75" width="49.5" height="29.25" align-x="80.25" align-y="1032" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIFunction"><ml:id>pt</ml:id><ml:matrix rows="2" cols="1"><ml:real>65</ml:real><ml:real>160</ml:real></ml:matrix></ml:define></math><rendering item-idref="15" /></region><region region-id="2119" left="66" top="1059" width="141.75" height="12.75" align-x="170.25" align-y="1068" 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>C</ml:id><ml:id>Indep</ml:id><ml:id>Age</ml:id><ml:id>pt</ml:id></ml:apply></ml:eval></math><rendering item-idref="16" /></region><region region-id="2120" left="36" top="1082.25" width="283.5" height="36" align-x="36" align-y="1092" 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">except that the last argument cannot be a single value because our polynomial is of two variables. We must evaluate at a point, written as a </c><c val="#000">vector.</c></p></text></region><region region-id="2121" left="36" top="1130.25" width="318.75" height="12" align-x="36" align-y="1140" 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">This is interpreted as a prediction of the age of someone who is</p></text></region><region region-id="2122" left="66" top="1155" width="39.75" height="18" align-x="84" align-y="1164" 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>pt</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="17" /></region><region region-id="2127" left="144" top="1154.25" width="50.25" height="12" align-x="144" align-y="1164" 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">inches tall</p></text></region><region region-id="2124" left="36" top="1178.25" width="55.5" height="12" align-x="36" align-y="1188" 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">and weighs</p></text></region><region region-id="2125" left="66" top="1203" width="44.25" height="18" align-x="84" align-y="1212" 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>pt</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="18" /></region><region region-id="2128" left="144" top="1202.25" width="35.25" height="12" align-x="144" align-y="1212" 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">pounds</p></text></region><region region-id="2227" left="36" top="1226.25" width="299.25" height="36" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Note that the order of the coordinates in our point (height, weight) had to match the order the factors were listed as columns of X.</p></text></region><region region-id="2226" 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="2135" left="36" top="1286.25" width="153.75" height="12" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Define the fitting polynomial as</p></text></region><region region-id="2136" left="66" top="1308.75" width="166.5" height="29.25" align-x="109.5" align-y="1326" 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>Q</ml:id><ml:boundVars><ml:id>ht</ml:id><ml:id>wt</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>C</ml:id><ml:id>Indep</ml:id><ml:id>Age</ml:id><ml:matrix rows="2" cols="1"><ml:id>ht</ml:id><ml:id>wt</ml:id></ml:matrix></ml:apply></ml:define></math><rendering item-idref="19" /></region><region region-id="2141" left="36" top="1352.25" width="97.5" height="12" align-x="36" align-y="1362" 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 sum of squares</p></text></region><region region-id="2139" left="66" top="1381.5" width="207.75" height="43.5" align-x="231.75" align-y="1404" 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:summation /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>Age</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:id>Q</ml:id><ml:apply><ml:indexer /><ml:id>Height</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>Weight</ml:id><ml:id>i</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>Age</ml:id></ml:apply></ml:bounds></ml:apply></ml:eval></math><rendering item-idref="20" /></region><region region-id="2142" left="36" top="1436.25" width="253.5" height="24" align-x="36" align-y="1446" 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">is the smallest it could be for all polynomials in two variables with degree 2.</p></text></region><region region-id="2143" left="66" top="1476" width="231" height="217.5" align-x="180.75" align-y="1476" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="21" disable-calc="false"><inputs hide="false"><ml:sequence><ml:parens><ml:sequence><ml:id>Height</ml:id><ml:id>Weight</ml:id><ml:id>Age</ml:id></ml:sequence></ml:parens><ml:id>Q</ml:id></ml:sequence></inputs><outputs hide="false" /></component><rendering item-idref="22" /></region><region region-id="2146" left="36" top="1706.25" width="308.25" height="24" align-x="36" align-y="1716" 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 that all the data points bury themselves in the surface, indicating a reasonable fit.</p></text></region><region region-id="2223" left="0" top="1746" width="6000" height="6" align-x="0" align-y="1746" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1970" left="36" top="1759.5" width="117.75" height="12.75" align-x="36" align-y="1770" 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">Loess multivariate fitting</p></text><indexes><phrase>loess multivariate fitting</phrase><phrase>fitting, loess multivariate</phrase></indexes></region><region region-id="2147" left="36" top="1784.25" width="304.5" height="36" align-x="36" align-y="1794" 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><u><link href="./regress.xmcd#loess" title="Links to section" popup="false">loess function</link></u></b><c val="#000"> also has an extension to higher-order data sets. In the case of loess, no more that 4 independent variables may be fit.</c></p></text></region><region region-id="2184" left="66" top="1833" width="41.25" height="12.75" align-x="74.25" align-y="1842" 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="23" /></region><region region-id="2185" left="144" top="1826.25" width="35.25" height="27.75" align-x="164.25" align-y="1842" 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>X1</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:div /><ml:id>i</ml:id><ml:real>4</ml:real></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2186" left="66" top="1863" width="61.5" height="18" align-x="86.25" align-y="1872" 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>X2</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>10</ml:real><ml:apply><ml:id>sin</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="2187" left="144" top="1856.25" width="70.5" height="27.75" align-x="164.25" align-y="1872" 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>X3</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>i</ml:id><ml:real>10</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="2156" left="66" top="1893" width="69.75" height="18" align-x="80.25" align-y="1902" 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>Y</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:apply><ml:id>cos</ml:id><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id>i</ml:id></ml:apply></ml:apply><ml:real>5</ml:real></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="2191" left="36" top="1928.25" width="183.75" height="12" align-x="36" align-y="1938" 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">With our fictitious data set, we have</p></text></region><region region-id="2194" left="66" top="1947" width="150" height="12.75" align-x="79.5" align-y="1956" 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:id>augment</ml:id><ml:id>X1</ml:id><ml:apply><ml:id>augment</ml:id><ml:id>X2</ml:id><ml:id>X3</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="28" /></region><region region-id="2195" left="66" top="1971" width="54" height="12.75" align-x="102.75" align-y="1980" 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>cols</ml:id><ml:id>X</ml:id></ml:apply></ml:eval></math><rendering item-idref="29" /></region><region region-id="2228" left="36" top="2000.25" width="304.5" height="24" align-x="36" align-y="2010" 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 loess function requires a smoothing parameter that has the same meaning as the one-factor case</c></p></text></region><region region-id="2202" left="66" top="2037" width="29.25" height="12.75" align-x="76.5" align-y="2046" 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>s</ml:id><ml:real>0.4</ml:real></ml:define></math><rendering item-idref="30" /></region><region region-id="2203" left="66" top="2061" width="79.5" height="12.75" align-x="82.5" align-y="2070" 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>LS</ml:id><ml:apply><ml:id>loess</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>s</ml:id></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="2205" left="36" top="2090.25" width="303.75" height="48" align-x="36" align-y="2100" 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">and again, the resulting vector is something you will never use directly. Compute values of the fit using </c><b><c val="#000">interp</c></b><c val="#000"> in the usual way, using a</c><c val="#000"> vector representing values of the independent factors for the last argument.</c></p></text></region><region region-id="2206" left="66" top="2152.5" width="120.75" height="45.75" align-x="158.25" align-y="2178" 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>LS</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:matrix rows="3" cols="1"><ml:real>2</ml:real><ml:real>3</ml:real><ml:real>1</ml:real></ml:matrix></ml:apply></ml:eval></math><rendering item-idref="32" /></region><region region-id="2224" left="0" top="2208" width="6000" height="6" align-x="0" align-y="2208" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2207" left="36" top="2222.25" width="267.75" height="24" align-x="36" align-y="2232" 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">More often, you will define the loess fit as a function, either with a vector argument</p></text></region><region region-id="2208" left="66" top="2259" width="123.75" height="12.75" align-x="106.5" align-y="2268" 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>Lvec</ml:id><ml:boundVars><ml:id>w</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>LS</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>w</ml:id></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="2209" left="66" top="2290.5" width="84" height="45.75" align-x="121.5" align-y="2316" 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>Lvec</ml:id><ml:matrix rows="3" cols="1"><ml:real>2</ml:real><ml:real>3</ml:real><ml:real>1</ml:real></ml:matrix></ml:apply></ml:eval></math><rendering item-idref="34" /></region><region region-id="2210" left="36" top="2354.25" width="264.75" height="24" align-x="36" align-y="2364" 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">or as a function of n (n = the number of independent factors) arguments:</p></text></region><region region-id="2222" left="66" top="2392.5" width="156" height="45.75" align-x="123.75" align-y="2418" 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>Lo</ml:id><ml:boundVars><ml:id>x1</ml:id><ml:id>x2</ml:id><ml:id>x3</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>interp</ml:id><ml:id>LS</ml:id><ml:id>X</ml:id><ml:id>Y</ml:id><ml:matrix rows="3" cols="1"><ml:id>x1</ml:id><ml:id>x2</ml:id><ml:id>x3</ml:id></ml:matrix></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="2214" left="66" top="2445" width="74.25" height="12.75" align-x="111.75" align-y="2454" 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>Lo</ml:id><ml:real>2</ml:real><ml:real>3</ml:real><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="36" /></region><region region-id="2217" left="36" top="2474.25" width="287.25" height="72" align-x="36" align-y="2484" 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 higher-order <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">loess</c></inlineAttr></b></f> fit cannot extrapolate in any direction, just as with the first-order case. 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