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align-x="6" align-y="6" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="2" /><link href="./dataTOC.xmcd" title="Links to Table of Contents" popup="false" /><rendering item-idref="3" /></region><region region-id="1448" left="174" top="44.25" width="68.25" height="12" align-x="174" align-y="54" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="4" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="1495" left="36" top="84.75" width="206.25" height="13.5" align-x="36" align-y="96" 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">Statistical Analysis of Water Meter Data</p></text><indexes><phrase>statistical analysis of water meter data</phrase><phrase>examples, statistical analysis of water meter data</phrase><phrase>water meter data, statistical analysis of</phrase></indexes></region><region region-id="1597" left="36" top="98.25" width="126" height="24" align-x="36" align-y="108" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">by D. M. Griffin</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Louisiana Tech University</p></text></region><region region-id="2279" left="36" top="152.25" width="308.25" height="72" align-x="63.75" align-y="162" 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 data used here are daily water usage values expressed as gal/min.<sp count="2" />These are being collected as part of an ongoing research project between Louisiana Tech University and the Louisiana Department of Transportation and Development to quantify water usage and wastewater generation at Interstate Rest Areas in Louisiana.<sp count="2" /></p></text></region><region region-id="2280" left="36" top="242.25" width="306.75" height="108" align-x="36" align-y="252" 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 <b><u><link href="./smooth.xmcd" title="Links to section" popup="false">medsmooth</link></u></b> function was used to examine water flowrates over a period of record from 7/1/97 to the present.<sp count="2" />Using a smoothed data set (bandwidth = 7 days) it was found that daily flows after day 1200 were somewhat lower and less variable than those before day 1200.<sp count="2" />This results in substantial changes in less frequent flow rates before and after day 1200.<sp count="2" />A bandwidth of 7 days was used because significant autocorrelation occurs within a one-week period, less so after one week.<sp count="3" /></p></text></region><region region-id="2281" left="36" top="356.25" width="309.75" height="36" align-x="48" align-y="366" 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">Read in daily water usage rates at Grand Prairie Interstate rest area, I-49 in gal/min.<sp count="2" />Missing values given value of -1.<sp count="2" />There are 10 to 20 missing values in the data set.</p></text></region><region region-id="2809" left="66" top="399" width="75.75" height="21.75" align-x="100.5" align-y="408" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="4" disable-calc="false"><inputs hide="false" /><outputs hide="false"><ml:id>DATA</ml:id></outputs></component><rendering item-idref="5" /></region><region region-id="2814" left="66" top="429" width="104.25" height="12.75" align-x="74.25" align-y="438" 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:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="6" /></region><region region-id="2815" left="222" top="429" width="83.25" height="12.75" align-x="288" align-y="438" 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>DATA</ml:id></ml:apply></ml:eval></math><rendering item-idref="7" /></region><region region-id="2812" left="66" top="447" width="88.5" height="18" align-x="126.75" align-y="456" 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>elapsed_time</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:id>day</ml:id></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="2821" left="6" top="468" width="357" height="180.75" align-x="6" align-y="468" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="9" /><rendering item-idref="10" /></region><region region-id="2823" left="0" top="666" width="6000" height="6" align-x="0" align-y="666" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2014" left="36" top="680.25" width="273.75" height="36" align-x="45.75" align-y="690" 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 raw data exhibits no obvious changes in pattern.<sp count="2" />There were about 20 days where flow data were not available and these were given the value -1</p></text></region><region region-id="2063" left="36" top="728.25" width="314.25" height="108" align-x="43.5" align-y="738" 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 first need to deal with the -1 values.<sp count="2" />There is no generally accepted way of doing this.<sp count="2" />If they are removed, the data set is discontinuous, which is undesirable.<sp count="2" />In this case I will first remove the -1 values to create another set of "real" values.<sp count="2" />Then I will compute the mean of that data set. Finally, I will substitute the median value of that data set for all -1 values in the original data set.<sp count="2" />In this way I have a continuous data set (not a perfect solution, but this is the real world!) with no missing values.</p></text></region><region region-id="2033" left="36" top="854.25" width="295.5" height="12" align-x="44.25" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Get all the indices of the data which are of value -1:</p></text></region><region region-id="2034" left="66" top="873" width="108" height="12.75" align-x="93" align-y="882" 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>index</ml:id><ml:apply><ml:id>match</ml:id><ml:real>-1</ml:real><ml:id>DATA</ml:id></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="2036" left="36" top="896.25" width="145.5" height="12" align-x="36" align-y="906" 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">Trim them from the data set:</p></text></region><region region-id="2041" left="66" top="915" width="132" height="12.75" align-x="114.75" align-y="924" 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>DATAtrim</ml:id><ml:apply><ml:id>trim</ml:id><ml:id>DATA</ml:id><ml:id>index</ml:id></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="2038" left="66" top="939" width="90.75" height="12.75" align-x="126" align-y="948" 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>rows</ml:id><ml:id>DATA</ml:id></ml:apply></ml:eval></math><rendering item-idref="13" /></region><region region-id="2040" left="222" top="939" width="105.75" height="12.75" align-x="297" align-y="948" 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>rows</ml:id><ml:id>DATAtrim</ml:id></ml:apply></ml:eval></math><rendering item-idref="14" /></region><region region-id="2044" left="36" top="968.25" width="311.25" height="24" align-x="45.75" 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">Now compute median of DATAtrim and substitute it for all -1 values.<sp count="3" /></p></text></region><region region-id="2055" left="66" top="1005" width="132.75" height="12.75" align-x="112.5" align-y="1014" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>medianval</ml:id><ml:apply><ml:id>median</ml:id><ml:id>DATAtrim</ml:id></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="2067" left="66" top="1035" width="89.25" height="12.75" align-x="99.75" 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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">As expected, the median changes less in this data transformation, since it's a more robust measure of <b><u><link href="./location.xmcd" title="Links to section" popup="false">centrality</link></u></b>.</p></text></region><region region-id="2087" left="66" top="1155" width="107.25" height="12.75" align-x="127.5" 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:id>mean</ml:id><ml:id>DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="20" /></region><region region-id="2088" left="222" top="1155" width="123" height="12.75" align-x="299.25" 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:id>mean</ml:id><ml:id>DATAorig</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="21" /></region><region region-id="2089" left="66" top="1179" width="114.75" height="12.75" align-x="135" align-y="1188" 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>median</ml:id><ml:id>DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="22" /></region><region region-id="2090" left="222" top="1179" width="130.5" height="12.75" align-x="306.75" align-y="1188" 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>median</ml:id><ml:id>DATAorig</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="23" /></region><region region-id="2820" left="0" top="1200" width="6000" height="6" align-x="0" align-y="1200" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2713" left="30" top="1207.5" width="72.75" height="12.75" align-x="30" align-y="1218" 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">Autocorrelation</p></text><indexes><phrase>autocorrelation</phrase></indexes></region><region region-id="2712" left="30" top="1226.25" width="339" height="60" align-x="30" 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">Autocorrelation is a procedure to determine if a data set is correlated with itself over time. Data which are correlated are not independent and standard statistical hypothesis tests are not valid, strictly speaking. Data which are not autocorrelated are said to be independent.</p></text></region><region region-id="2716" left="30" top="1304.25" width="321" height="36" align-x="30" align-y="1314" 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">First, compute and plot the residuals, the differences between each daily flow value and the overall mean flow for the data set.</p></text></region><region region-id="2744" left="66" top="1346.25" width="75.75" height="13.5" align-x="78" align-y="1356" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>μ</ml:id><ml:apply><ml:id>mean</ml:id><ml:id>DATA</ml:id></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2745" left="66" top="1364.25" width="84.75" height="13.5" align-x="98.25" align-y="1374" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>RESID</ml:id><ml:apply><ml:minus /><ml:id>DATA</ml:id><ml:id>μ</ml:id></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="2746" left="222" top="1365" width="38.25" height="12.75" align-x="239.25" align-y="1374" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>m1</ml:id><ml:real>320</ml:real></ml:define></math><rendering item-idref="26" /></region><region region-id="2747" left="294" top="1365" width="38.25" height="12.75" align-x="311.25" align-y="1374" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>m2</ml:id><ml:real>360</ml:real></ml:define></math><rendering item-idref="27" /></region><region region-id="2718" left="6" top="1386" width="375" height="227.25" align-x="6" align-y="1386" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="28" /><rendering item-idref="29" /></region><region region-id="2722" left="36" top="1616.25" width="322.5" height="48" align-x="36" align-y="1626" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Comparing the data points to the y = 0 line, we see that the flow was generally larger than the mean for the first 90 days </p><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">or so; then was less than the mean until about day 240. Recently the flow has been increasing again. </p></text></region><region region-id="2727" left="0" top="1674" width="6000" height="6" align-x="0" align-y="1674" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2723" left="36" top="1688.25" width="321.75" height="156" align-x="36" align-y="1698" 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 high values occurring between 320 and 360 were caused by watering of trees and grass during May and June 1998. In addition, problems occurred with the well around 6/15/98, (day 349) which necessitated pumping the well for an extended period while working on it. High flows around the end of September (day 450) were probably a result of large numbers of evacuees heading north from New Orleans and surrounding area as hurricane Georges approached New Orleans (actually it turned out that the riser pipe in the well had split). High residuals occurring near day 880 and day 890 are due to the conduct of dye tests in the rock plants filters. All of these events tend to wash out legitimate long-term trends in the data, which is what we are interested in.</p></text></region><region region-id="2734" left="36" top="1856.25" width="327.75" height="36" align-x="36" align-y="1866" 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">Plot the correlogram for the daily flow. A correlogram is the sequence of autocorrelation coefficients of flow over a series of specified time intervals, called lag periods or lags. </p></text></region><region region-id="2833" left="66" top="1899" width="39.75" height="12.75" align-x="77.25" align-y="1908" 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:range><ml:real>0</ml:real><ml:real>28</ml:real></ml:range></ml:define></math><rendering item-idref="30" /></region><region region-id="2834" left="144" top="1899" width="81.75" height="12.75" align-x="157.5" align-y="1908" 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>N</ml:id><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="2835" left="246" top="1898.25" width="77.25" height="13.5" align-x="258" align-y="1908" 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>σ</ml:id><ml:apply><ml:id>stdev</ml:id><ml:id>DATA</ml:id></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="2836" left="66" top="1915.5" width="225.75" height="43.5" align-x="104.25" align-y="1938" 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>lagcorr</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:id>N</ml:id><ml:apply><ml:pow /><ml:id>σ</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></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:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>DATA</ml:id><ml:id>i</ml:id></ml:apply><ml:id>μ</ml:id></ml:apply></ml:parens><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>DATA</ml:id><ml:apply><ml:minus /><ml:id>i</ml:id><ml:id>n</ml:id></ml:apply></ml:apply><ml:id>μ</ml:id></ml:apply></ml:parens></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:id>n</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:bounds></ml:apply></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="2837" left="30" top="1962" width="355.5" height="204" align-x="30" align-y="1962" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="34" /><rendering item-idref="35" /></region><region region-id="2756" left="36" top="2168.25" width="344.25" height="72" align-x="36" align-y="2178" 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 autocorrelation coefficient over a lag period of 7 days is approximately 0.4. This means that the flow on any day is positively correlated with the the flow 7 days previous. Similarly the value at 21 days is somewhat over 0.2. This indicates that the flow is correlated with the flow 21 days previous but not as well as the flow 7 or 14 days ago. </p></text></region><region region-id="2765" left="36" top="2252.25" width="335.25" height="48" align-x="36" align-y="2262" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now look at the overall curve. It has a regular pattern that repeats every 7- 8 days. This is not immediately obvious from the raw data. It is also interesting to note that the value of the coefficients drop with each succeeding cycle.</p></text></region><region region-id="2766" left="0" top="2310" width="6000" height="6" align-x="0" align-y="2310" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2768" left="36" top="2318.25" width="317.25" height="36" align-x="36" align-y="2328" 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">Here are the upper and lower 95% confidence limits on the autocorrelation coefficients. Values inside this region cannot be assumed different than zero in a statistical sense.</p></text></region><region region-id="2781" left="66" top="2364.75" width="157.5" height="29.25" align-x="77.25" align-y="2382" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>u</ml:id><ml:parens><ml:apply><ml:div /><ml:apply><ml:plus /><ml:real>-1</ml:real><ml:apply><ml:mult style="default" /><ml:real>1.96</ml:real><ml:apply><ml:sqrt /><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens></ml:define></math><rendering item-idref="36" /></region><region region-id="2782" left="252" top="2373" width="43.5" height="12.75" align-x="262.5" align-y="2382" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>u</ml:id></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="3" show-trailing-zeros="true" exponential-threshold="3" /><matrix display-style="matrix" expand-nested-arrays="false" /><unit format-units="false" simplify-units="false" /></resultFormat></math><rendering item-idref="37" /></region><region region-id="2783" left="66" top="2406.75" width="141.75" height="29.25" align-x="74.25" align-y="2424" 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>l</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:real>-1</ml:real><ml:apply><ml:mult style="default" /><ml:real>1.96</ml:real><ml:apply><ml:sqrt /><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="2784" left="252" top="2415" width="45.75" height="12.75" align-x="259.5" align-y="2424" 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>l</ml:id></ml:eval></math><rendering item-idref="39" /></region><region region-id="2092" left="36" top="2449.5" width="86.25" height="12.75" align-x="36" align-y="2460" 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">Median Smoothing</p></text><indexes><phrase>median smoothing</phrase><phrase>smoothing, median</phrase></indexes></region><region region-id="2060" left="36" top="2474.25" width="309.75" height="36" align-x="45.75" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now use <b><u><link href="./smooth.xmcd" title="Links to section" popup="false">medsmooth</link></u></b> to smooth the data.<sp count="2" />Based on the serial correlation analysis, it was decided to use a bandwidth of 7 days.</p></text></region><region region-id="2093" left="66" top="2529" width="115.5" height="12.75" align-x="83.25" 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:define><ml:id>Q7</ml:id><ml:apply><ml:id>medsmooth</ml:id><ml:id>DATA</ml:id><ml:real>7</ml:real></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="2095" left="36" top="2558.25" width="333.75" height="48" align-x="36" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <link href="location.xmcd" title="Links to section" popup="false">median</link> of the 7 points surrounding each data point is used to replace each data point. The procedure is followed for each point so there are as many point in the smoothed data set as the original.<sp count="2" /></p></text></region><region region-id="2838" left="6" top="2610" width="377.25" height="216.75" align-x="6" align-y="2610" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="41" /><rendering item-idref="42" /></region><region region-id="2839" left="0" top="2832" width="6000" height="6" align-x="0" align-y="2832" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2142" left="36" top="2840.25" width="328.5" height="144" align-x="44.25" align-y="2850" 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">Note the change in water use highlighted by the smoothed data. The curve above suggests that daily water use appeared to increase up until about day 1200, which would correspond to about October of 2000.<sp count="2" />After that time, water usage dropped and appeared to be less variable. This is not at all evident in the original data. The main reason for this change is the fact that before day 1200, testing was occurring at the site, requiring extra water to be run through the treatment system. Based on these results it could be argued that we have, in effect, 2 data sets, one before day 1200, and one from day 1200 to the present.<sp count="2" />It seems reasonable to plot probability curves for the data before and after day 1200. </p></text></region><region region-id="2144" left="36" top="3001.5" width="60.75" height="12.75" align-x="36" align-y="3012" 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">Weibull plots</p></text><indexes><phrase>Weibull plots</phrase><phrase>plots, Weibull</phrase></indexes></region><region region-id="2145" left="36" top="3026.25" width="267.75" height="12" align-x="36" align-y="3036" 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">First, we extract the data before and after day 1200.</p></text></region><region region-id="2147" left="66" top="3051" width="52.5" height="12.75" align-x="81" align-y="3060" 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>k1</ml:id><ml:range><ml:real>0</ml:real><ml:real>1199</ml:real></ml:range></ml:define></math><rendering item-idref="43" /></region><region region-id="2149" left="144" top="3051" width="113.25" height="12.75" align-x="159" align-y="3060" 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>k2</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id>last</ml:id><ml:id>DATA</ml:id></ml:apply><ml:real>1200</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="44" /></region><region region-id="2162" left="66" top="3075" width="123" height="21" align-x="144.75" 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:apply><ml:indexer /><ml:id subscript="bef_1200">DATA</ml:id><ml:id>k1</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>DATA</ml:id><ml:id>k1</ml:id></ml:apply></ml:define></math><rendering item-idref="45" /></region><region region-id="2151" left="66" top="3105" width="144.75" height="21" align-x="142.5" align-y="3114" 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 subscript="aft_1200">DATA</ml:id><ml:id>k2</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>DATA</ml:id><ml:apply><ml:plus /><ml:id>k2</ml:id><ml:real>1200</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="2230" left="66" top="3141" width="143.25" height="16.5" align-x="163.5" align-y="3150" 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>mean</ml:id><ml:id subscript="bef_1200">DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="47" /></region><region region-id="2828" left="234" top="3141" width="141" height="16.5" align-x="329.25" align-y="3150" 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>mean</ml:id><ml:id subscript="aft_1200">DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="48" /></region><region region-id="2231" left="66" top="3171" width="150.75" height="16.5" align-x="171" 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:eval><ml:apply><ml:id>median</ml:id><ml:id subscript="bef_1200">DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="49" /></region><region region-id="2829" left="234" top="3171" width="148.5" height="16.5" align-x="336.75" 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:eval><ml:apply><ml:id>median</ml:id><ml:id subscript="aft_1200">DATA</ml:id></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="50" /></region><region region-id="2298" left="36" top="3200.25" width="306" height="48" align-x="45.75" align-y="3210" 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 sort the data and create the Weibull plotting variable for three probability curves. The first is for all data, the second for data collected before day 1200, and the third for data collected after day 1200.</p></text></region><region region-id="2622" left="66" top="3260.25" width="75.75" height="27.75" align-x="80.25" align-y="3276" 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>x1</ml:id><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:id>DATA</ml:id></ml:apply><ml:id>gpm</ml:id></ml:apply></ml:define></math><rendering item-idref="51" /></region><region region-id="2621" left="204" top="3260.25" width="121.5" height="27.75" align-x="219.75" align-y="3276" 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>y1</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:apply><ml:id>Rank</ml:id><ml:id>DATA</ml:id></ml:apply></ml:apply><ml:apply><ml:id>length</ml:id><ml:id>DATA</ml:id></ml:apply></ml:apply><ml:real>100</ml:real></ml:apply></ml:define></math><rendering item-idref="52" /></region><region region-id="2620" left="66" top="3316.5" width="111.75" height="31.5" align-x="80.25" align-y="3336" 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>x2</ml:id><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:id subscript="bef_1200">DATA</ml:id></ml:apply><ml:id>gpm</ml:id></ml:apply></ml:define></math><rendering item-idref="53" /></region><region region-id="2619" left="204" top="3316.5" width="157.5" height="35.25" align-x="219.75" align-y="3336" 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>y2</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:apply><ml:id>Rank</ml:id><ml:id subscript="bef_1200">DATA</ml:id></ml:apply></ml:apply><ml:apply><ml:id>length</ml:id><ml:id subscript="bef_1200">DATA</ml:id></ml:apply></ml:apply><ml:real>100</ml:real></ml:apply></ml:define></math><rendering item-idref="54" /></region><region region-id="2618" left="66" top="3376.5" width="109.5" height="31.5" align-x="80.25" align-y="3396" 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>x3</ml:id><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:id subscript="aft_1200">DATA</ml:id></ml:apply><ml:id>gpm</ml:id></ml:apply></ml:define></math><rendering item-idref="55" /></region><region region-id="2617" left="204" top="3376.5" width="155.25" height="35.25" align-x="219.75" align-y="3396" 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>y3</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:apply><ml:id>sort</ml:id><ml:apply><ml:id>Rank</ml:id><ml:id subscript="aft_1200">DATA</ml:id></ml:apply></ml:apply><ml:apply><ml:id>length</ml:id><ml:id subscript="aft_1200">DATA</ml:id></ml:apply></ml:apply><ml:real>100</ml:real></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="2695" left="0" top="3432" width="6000" height="-6" align-x="0" align-y="3426" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2696" left="60" top="3444" width="316.5" height="225.75" align-x="60" align-y="3444" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="57" /><rendering item-idref="58" /></region><region region-id="2241" left="36" top="3680.25" width="298.5" height="84" align-x="44.25" align-y="3690" 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 curves may be interpreted as follows:</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit" /><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1. The curve for all the data combined shows a flow rate with a 50% probability of occurrence is 3.42 gal/min.<sp count="2" />That is, 50% of future flows will be less than this value. Corresponding values for data collected before and after day 1200 are 3.8 and 3.1 gal/min respectively.</p></text></region><region region-id="2616" left="66" top="3783" width="147" height="12.75" align-x="167.25" align-y="3792" 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>percentile</ml:id><ml:id>DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id>%</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="59" /></region><region region-id="2625" left="66" top="3807" width="183" height="16.5" align-x="203.25" align-y="3816" 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>percentile</ml:id><ml:id subscript="bef_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id>%</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="60" /></region><region region-id="2626" left="66" top="3831" width="180.75" height="16.5" align-x="201" align-y="3840" 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>percentile</ml:id><ml:id subscript="aft_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id>%</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="61" /></region><region region-id="2615" left="36" top="3860.25" width="268.5" height="48" align-x="36" align-y="3870" 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">2. Using the curve for data collected after day 1200, 90% of future flows will be less than 5.8 gal/min. The corresponding value for data collected before day 1200 is 8.9 gal/min<sp count="4" /></p></text></region><region region-id="2632" left="66" top="3921" width="183" height="16.5" align-x="203.25" align-y="3930" 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>percentile</ml:id><ml:id subscript="bef_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>90</ml:real><ml:id>%</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="62" /></region><region region-id="2633" left="66" top="3951" width="180.75" height="16.5" align-x="201" align-y="3960" 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>percentile</ml:id><ml:id subscript="aft_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>90</ml:real><ml:id>%</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id>gpm</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="63" /></region><region region-id="2697" left="0" top="3978" width="6000" height="6" align-x="0" align-y="3978" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2245" left="36" top="3992.25" width="318.75" height="36" align-x="45.75" align-y="4002" 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">Now, lets examine changes in flowrate as a function of how often the flow occurs.<sp count="2" />We will look at 20 percentile flows, median flows, and 90 percentile flows. </p></text></region><region region-id="2246" left="36" top="4045.5" width="81" height="12.75" align-x="40.5" align-y="4056" 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">20 percentile flow</p></text></region><region region-id="2645" left="36" top="4070.25" width="85.5" height="12" align-x="51" align-y="4080" 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">Before day 1200:</p></text></region><region region-id="2644" left="144" top="4071" width="178.5" height="16.5" align-x="183" align-y="4080" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="20">B1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="bef_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>20</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="2639" left="36" top="4112.25" width="78.75" height="12" align-x="36" align-y="4122" 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">After day 1200:</p></text></region><region region-id="2642" left="144" top="4113" width="177.75" height="16.5" align-x="184.5" align-y="4122" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="20">A1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="aft_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>20</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="65" /></region><region region-id="2248" left="66" top="4144.5" width="138.75" height="35.25" align-x="110.25" align-y="4164" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="20">change</ml:id><ml:apply><ml:neg /><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id subscript="20">B1200</ml:id><ml:id subscript="20">A1200</ml:id></ml:apply><ml:id subscript="20">A1200</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="66" /></region><region region-id="2646" left="222" top="4155" width="85.5" height="16.5" align-x="265.5" align-y="4164" 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 subscript="20">change</ml:id><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="67" /></region><region region-id="2669" left="36" top="4201.5" width="60" height="12.75" align-x="53.25" align-y="4212" 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">Median flows</p></text></region><region region-id="2662" left="36" top="4232.25" width="85.5" height="12" align-x="51" align-y="4242" 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">Before day 1200:</p></text></region><region region-id="2663" left="144" top="4233" width="178.5" height="16.5" align-x="183" align-y="4242" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="50">B1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="bef_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="68" /></region><region region-id="2664" left="36" top="4274.25" width="78.75" height="12" align-x="36" align-y="4284" 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">After day 1200:</p></text></region><region region-id="2665" left="144" top="4275" width="177.75" height="16.5" align-x="184.5" align-y="4284" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="50">A1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="aft_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="69" /></region><region region-id="2666" left="66" top="4306.5" width="146.25" height="35.25" align-x="117.75" align-y="4326" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="med">change</ml:id><ml:apply><ml:neg /><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id subscript="50">B1200</ml:id><ml:id subscript="50">A1200</ml:id></ml:apply><ml:id subscript="50">A1200</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="70" /></region><region region-id="2667" left="222" top="4317" width="97.5" height="16.5" align-x="273" align-y="4326" 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 subscript="med">change</ml:id><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="71" /></region><region region-id="2661" left="36" top="4351.5" width="84" height="12.75" align-x="40.5" align-y="4362" 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">90 percentile flow </p></text></region><region region-id="2682" left="36" top="4382.25" width="85.5" height="12" align-x="51" align-y="4392" 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">Before day 1200:</p></text></region><region region-id="2683" left="144" top="4383" width="178.5" height="16.5" align-x="183" align-y="4392" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="90">B1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="bef_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>90</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="72" /></region><region region-id="2684" left="36" top="4424.25" width="78.75" height="12" align-x="36" align-y="4434" 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">After day 1200:</p></text></region><region region-id="2685" left="144" top="4425" width="177.75" height="16.5" align-x="184.5" align-y="4434" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="90">A1200</ml:id><ml:apply><ml:id>percentile</ml:id><ml:id subscript="aft_1200">DATA</ml:id><ml:apply><ml:mult style="auto-select" /><ml:real>90</ml:real><ml:id>%</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="73" /></region><region region-id="2686" left="66" top="4456.5" width="138.75" height="35.25" align-x="110.25" align-y="4476" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id subscript="90">change</ml:id><ml:apply><ml:neg /><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id subscript="90">B1200</ml:id><ml:id subscript="90">A1200</ml:id></ml:apply><ml:id subscript="90">A1200</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="74" /></region><region region-id="2687" left="222" top="4467" width="85.5" height="16.5" align-x="265.5" align-y="4476" 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 subscript="90">change</ml:id><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval></math><rendering item-idref="75" /></region><region region-id="2688" left="36" top="4502.25" width="56.25" height="12" align-x="36" align-y="4512" 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"><f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Summary:</c></inlineAttr></b></f></p></text></region><region region-id="2689" left="66" top="4527" width="76.5" height="16.5" align-x="109.5" align-y="4536" 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 subscript="20">change</ml:id><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="1" show-trailing-zeros="false" exponential-threshold="4" /><matrix display-style="auto" expand-nested-arrays="false" /><unit format-units="true" simplify-units="true" /></resultFormat></math><rendering item-idref="76" /></region><region region-id="2690" left="168" top="4527" width="88.5" height="16.5" align-x="219" align-y="4536" 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 subscript="med">change</ml:id><ml:unitOverride><ml:id>%</ml:id></ml:unitOverride></ml:eval><resultFormat><general radix="dec" complex-threshold="10" 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align-y="4560" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2277" left="36" top="4568.25" width="312" height="148.5" align-x="61.5" align-y="4578" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">These data suggest that changes in the more frequent flows before and after day 1200 are small, about <f family="Arial" charset="0" size="12"><inlineAttr line-through="false">-</inlineAttr></f> 5%.<sp count="2" />However, the change in median flows before and after day 1200 is <f family="Arial" charset="0" size="12"><inlineAttr line-through="false">-</inlineAttr></f>21% and the change in the 90 percentile flowrate is twice that, or <f 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