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align-x="96" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./condition.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />PRECONDITIONING DATA</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="147.75" height="13.5" align-x="36" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">NaN Detection and Removal</p></text><indexes><phrase>NaN detection and removal</phrase><phrase>detection of, NaN</phrase><phrase>removal of, NaN</phrase><phrase>Not a Number notation</phrase><phrase>NaN</phrase><phrase>markNaN</phrase><phrase>filterNaN</phrase><phrase>IsNaN</phrase><phrase>MatchNaN</phrase><phrase>functions, NaN detection and removal</phrase><phrase>working with NaNs</phrase><phrase>filtering NaNs</phrase><phrase>swapping in NaNs</phrase></indexes></region><region region-id="923" left="66" top="97.5" width="21.75" height="12.75" align-x="75" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">NaN </p></text></region><region region-id="1141" left="66" top="115.5" width="93" height="12.75" align-x="66" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">markNaN(v, vindex)</p></text></region><region region-id="1273" left="174" top="115.5" width="39.75" height="12.75" align-x="174" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">IsNan(x)</p></text></region><region region-id="927" left="66" top="133.5" width="54" height="12.75" align-x="82.5" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">filterNan(v)</p></text></region><region region-id="1274" left="174" top="133.5" width="60" height="12.75" align-x="195.75" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">MatchNaN(v)</p></text></region><region region-id="928" left="36" top="164.25" width="280.5" height="96" align-x="36" align-y="174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">NaN</c></inlineAttr></b></f> stands for Not a Number, and is intended to represent points in data that were never recorded, because of operator or instrumentation errors, or were recorded with unphysical values. NaNs can either be read in using the <b><u><link href="./inout.xmcd" title="Links to section" popup="false">Data Wizard component</link></u></b>, in the case where the data file is missing column elements in one or more rows, or inserted into any Mathcad matrix element just as you would any other value.</p></text></region><region region-id="1120" left="36" top="271.5" width="81" height="12.75" align-x="36" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Swapping in NaNs</p></text></region><region region-id="1121" left="36" top="290.25" width="297" height="60" align-x="36" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It's possible to swap in NaNs for any Mathcad value. For example, you may wish to mark <b><u><link href="./outlier.xmcd" title="Links to section" popup="false">outliers</link></u></b> in data with NaNs, to retain the row (show that a measurement was made), but not use the measurement in data processing (filter out NaNs before processing). </p></text></region><region region-id="1122" left="66" top="357" width="115.5" height="46.5" align-x="95.25" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><component hide-arguments="false" clsid-buddy="577d93c8-ea65-4d05-b631-401168b146e4" item-idref="4" disable-calc="false"><inputs /><outputs><ml:id xml:space="preserve">spots</ml:id></outputs></component><rendering item-idref="5" /></region><region region-id="967" left="0" top="414" width="6000" height="6" align-x="0" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="974" left="60" top="420" width="285.75" height="135" align-x="60" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="6" /><rendering item-idref="7" /></region><region region-id="973" left="36" top="560.25" width="315" height="24" align-x="36" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Suppose I've decided that the points at 1957 and 1958 look suspiciously high by using an <b><u><link href="./outlier.xmcd" title="Links to section" popup="false">outlier test</link></u></b>,</p></text></region><region region-id="1090" left="66" top="598.5" width="134.25" height="23.25" align-x="93" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">index</ml:id><ml:apply><ml:matcol /><ml:apply><ml:id xml:space="preserve">ThreeSigma</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1097" left="222" top="600.75" width="66.75" height="29.25" align-x="248.25" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">index</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="1"><ml:real>257</ml:real><ml:real>258</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="9" /></region><region region-id="1100" left="66" top="633.75" width="95.25" height="24" align-x="135" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:parens><ml:apply><ml:matcol /><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:apply><ml:indexer /><ml:id xml:space="preserve">index</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>190</ml:real></result></ml:eval></math><rendering item-idref="10" /></region><region region-id="1160" left="222" top="633.75" width="95.25" height="24" align-x="291" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:indexer /><ml:parens><ml:apply><ml:matcol /><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:apply><ml:indexer /><ml:id xml:space="preserve">index</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>185</ml:real></result></ml:eval></math><rendering item-idref="11" /></region><region region-id="1101" left="36" top="668.25" width="312" height="24" align-x="36" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">or by <b><u><link href="./trace.xmcd" title="Links to section" popup="false">tracing</link></u></b> on that region of the graph for the years at the high points, </p></text></region><region region-id="1076" left="66" top="705" width="135.75" height="12.75" align-x="165.75" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">vlookup</ml:id><ml:sequence><ml:real>1957</ml:real><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="1" cols="1"><ml:real>190</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="12" /></region><region region-id="1077" left="222" top="705" width="135.75" height="12.75" align-x="321.75" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">vlookup</ml:id><ml:sequence><ml:real>1958</ml:real><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="1" cols="1"><ml:real>185</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="13" /></region><region region-id="1102" left="36" top="728.25" width="303.75" height="24" align-x="36" align-y="738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">I can choose to mark any outlier values with a NaN, given the indices</p></text></region><region region-id="1167" left="66" top="759" width="84" height="72.75" align-x="93" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">index</ml:id><ml:matrix rows="2" cols="1"><ml:matrix rows="2" cols="1"><ml:apply><ml:indexer /><ml:id xml:space="preserve">index</ml:id><ml:real>0</ml:real></ml:apply><ml:real>1</ml:real></ml:matrix><ml:matrix rows="2" cols="1"><ml:apply><ml:indexer /><ml:id xml:space="preserve">index</ml:id><ml:real>1</ml:real></ml:apply><ml:real>1</ml:real></ml:matrix></ml:matrix></ml:define></math><rendering item-idref="14" /></region><region region-id="1168" left="162" top="788.25" width="180.75" height="24" align-x="162" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">the index values we found, in column 1 of the data.</p></text></region><region region-id="1154" left="66" top="837" width="147" height="12.75" align-x="113.25" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">NaNspots</ml:id><ml:apply><ml:id xml:space="preserve">markNaN</ml:id><ml:sequence><ml:id xml:space="preserve">spots</ml:id><ml:id xml:space="preserve">index</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="1259" left="36" top="854.25" width="308.25" height="36" align-x="36" align-y="864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Plotting results marked with NaNs plot real numbers, and skips the NaNs – note the two red dots from the original data set are not highlighted.</p></text></region><region region-id="1184" left="30" top="900" width="303.75" height="174" align-x="30" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="16" /><rendering item-idref="17" /></region><region region-id="1153" left="0" top="1080" width="6000" height="6" align-x="0" align-y="1080" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1257" left="36" top="1093.5" width="84" height="12.75" align-x="47.25" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Filtering out NaNs</p></text></region><region region-id="1258" left="36" top="1118.25" width="317.25" height="24" align-x="43.5" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The function <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">filterNaN</c></inlineAttr></b></f> will remove rows from a data set that contain NaN in one of the entries.</p></text></region><region region-id="1208" left="66" top="1155.75" width="136.5" height="18" align-x="93" align-y="1170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">index</ml:id><ml:apply><ml:id 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xml:space="preserve">filtspots</ml:id><ml:apply><ml:id xml:space="preserve">filterNaN</ml:id><ml:id xml:space="preserve">NaNspots</ml:id></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="1227" left="66" top="1221" width="91.5" height="12.75" align-x="131.25" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">rows</ml:id><ml:id xml:space="preserve">filtspots</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>294</ml:real></result></ml:eval></math><rendering item-idref="21" /></region><region region-id="1228" left="222" top="1221" width="81" height="12.75" align-x="276.75" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">rows</ml:id><ml:id xml:space="preserve">spots</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>296</ml:real></result></ml:eval></math><rendering item-idref="22" /></region><region region-id="1237" left="36" top="1244.25" width="331.5" height="24" align-x="36" align-y="1254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that this gives the same result as trimming the data with the indexes from matchNaN:</p></text></region><region region-id="1238" left="66" top="1281" width="129" height="12.75" align-x="168.75" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">rows</ml:id><ml:apply><ml:id xml:space="preserve">trim</ml:id><ml:sequence><ml:id xml:space="preserve">spots</ml:id><ml:id xml:space="preserve">index</ml:id></ml:sequence></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>294</ml:real></result></ml:eval></math><rendering item-idref="23" /></region><region region-id="1239" left="36" top="1310.25" width="313.5" height="84" align-x="36" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">filterNaN </c></inlineAttr></b></f>function just bundles it into a single routine. Once NaNs have been filtered out, you can find statistics on your data, or fit them without the presence of unwanted entries. In the example here, we have assigned NaN values, only to remove the rows. This makes more sense with a data set that had missing entries, and was imported from the <b><u><link href="./inout.xmcd" title="Links to section" popup="false">Data Wizard component</link></u></b> with NaNs.</p></text></region><region region-id="1240" left="66" top="1401.75" width="129" height="18" align-x="144" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">filtspots</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>48.768707482993193</ml:real></result></ml:eval></math><rendering item-idref="24" /></region><region region-id="1247" left="222" top="1401.75" width="118.5" height="18" align-x="289.5" align-y="1416" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">spots</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>49.706081081081081</ml:real></result></ml:eval></math><rendering item-idref="25" /></region><region region-id="1244" left="36" top="1430.25" width="314.25" height="36" align-x="36" align-y="1440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If there are no NaNs, the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">matchNaN</c></inlineAttr></b></f> function will give an error, and the<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"> filterNaN</c></inlineAttr></b></f> function will return the original data, since there are no NaNs to filter.</p></text></region><region region-id="1246" left="66" top="1479" width="94.5" height="12.75" align-x="143.25" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math error="no matching values found" optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">matchNaN</ml:id><ml:id xml:space="preserve">spots</ml:id></ml:apply></ml:eval></math><rendering item-idref="26" /></region><region region-id="1033" left="36" top="1501.5" width="85.5" height="12.75" align-x="36" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Working with NaN</p></text></region><region region-id="1255" left="36" top="1526.25" width="286.5" height="24" align-x="36" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can define new NaN aware functions that treat NaNs appropriately to your application, like <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Nmean</c></inlineAttr></b></f>:</p></text></region><region region-id="1261" left="66" top="1557.75" width="117.75" height="18" align-x="152.25" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">NaNspots</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>NaN</ml:real></result></ml:eval></math><rendering item-idref="27" /></region><region region-id="1266" left="66" top="1581.75" width="258" height="89.25" align-x="117" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">Nmean</ml:id><ml:boundVars><ml:id xml:space="preserve">M</ml:id></ml:boundVars></ml:function><ml:program><ml:localDefine><ml:id xml:space="preserve">newM</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">M</ml:id><ml:real>0</ml:real></ml:apply></ml:localDefine><ml:ifThen><ml:apply><ml:greaterThan /><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">M</ml:id></ml:apply><ml:real>1</ml:real></ml:apply><ml:for><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>1</ml:real><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">cols</ml:id><ml:id xml:space="preserve">M</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range><ml:localDefine><ml:id xml:space="preserve">newM</ml:id><ml:apply><ml:id xml:space="preserve">stack</ml:id><ml:sequence><ml:id xml:space="preserve">newM</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">M</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:sequence></ml:apply></ml:localDefine></ml:for></ml:ifThen><ml:localDefine><ml:id xml:space="preserve">newM</ml:id><ml:apply><ml:id xml:space="preserve">filterNaN</ml:id><ml:id xml:space="preserve">newM</ml:id></ml:apply></ml:localDefine><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">newM</ml:id></ml:apply></ml:program></ml:define></math><rendering item-idref="28" /></region><region region-id="1265" left="0" top="1680" width="6000" height="6" align-x="0" align-y="1680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1264" left="66" top="1689.75" width="144.75" height="18" align-x="159.75" align-y="1704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">Nmean</ml:id><ml:apply><ml:matcol /><ml:id xml:space="preserve">NaNspots</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>48.768707482993193</ml:real></result></ml:eval></math><rendering item-idref="29" /></region><region region-id="1251" left="36" top="1718.25" width="313.5" height="12" align-x="36" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you redefine the variable name NaN, it will no longer be NaN.</p></text></region><region region-id="1252" left="66" top="1743" width="36.75" height="12.75" align-x="90.75" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">NaN</ml:id><ml:real>1</ml:real></ml:define></math><rendering item-idref="30" /></region><region region-id="1253" left="144" top="1743" width="74.25" height="12.75" align-x="201" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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