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layout="Title" style="Title"><Font executable="false">Analyzing Data Files in Maple</Font></Text-field></Input></Group><Group><Input><Text-field firstindent="0.0" layout="Author" leftmargin="0.0" linebreak="space" rightmargin="0.0" style="Author"><Font bold="false" encoding="ISO8859-1" executable="false" foreground="[0,0,0]" italic="false" subscript="false" superscript="false" underline="false">\251 Maplesoft, a division of Waterloo Maple Inc., 2005</Font></Text-field></Input></Group><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field></Title><Group><Input><Text-field layout="Normal" style="Text">One of Maple's greatest strengths is analyzing data. The data can come from many different sources such as the <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:sockets" style="Hyperlink" subscript="false" superscript="false">internet</Hyperlink>, a <Hyperlink bold="false" executable="false" italic="false" linktarget="http://www.maplesoft.com/products/toolboxes/database/index.aspx" style="Hyperlink" subscript="false" superscript="false">database</Hyperlink>, user input or a <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:FileTools" style="Hyperlink" subscript="false" superscript="false">data file</Hyperlink>. In this worksheet we will demonstrate loading data from a file, visualizing the data and then doing some simple analysis on it.</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Initialization</Text-field></Title><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">restart: with(plots): with(Optimization):</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Reading the data file</Text-field></Title><Text-field layout="Normal" style="Text">First we use Maple's <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:readdata" style="Hyperlink" subscript="false" superscript="false">readdata</Hyperlink> command to load in a file containing two columns of data which we will use as data points. This command takes two parameters, the name of the file, and the number of columns in the file.</Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">data := readdata("P11.dat",2);</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">Now we would like to find out how many points have been read in, we use <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:nops" style="Hyperlink" subscript="false" superscript="false">nops</Hyperlink> which returns the number of operands in the expression that are given to it.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">numPoints := nops ( data );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Visualizing the data</Text-field></Title><Text-field layout="Normal" style="Text">Next we use <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:plots,pointplot" style="Hyperlink" subscript="false" superscript="false">pointplot</Hyperlink> to display the data. Pointplot simply takes a list of data points. Here we also define the color and symbol to use.</Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">P1 := pointplot(data, color = blue, symbol = point):
P1;</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Find the maximum point</Text-field></Title><Group><Input><Text-field layout="Normal" style="Text">We would like to find the point in the data where the y value is the greatest.</Text-field><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" style="Text">First, we get a sequence of all of the y values. Use the <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:seq" style="Hyperlink" subscript="false" superscript="false">seq</Hyperlink> command to create a sequence of the second operand of each data point. </Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">yVals := seq( op(2, data[i] ), i= 1.. numPoints ):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">Then we use the <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:max" style="Hyperlink" subscript="false" superscript="false">max</Hyperlink> command to find the largest of the values.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">yMax := max ( yVals );</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" style="Text">Now we need to find the x value that corresponds to the max. We loop through all of the y values and find the index which corresponds to the maximum y value.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">for i from 1 to nops ( [ yVals ] ) do
	if yVals[i] = yMax then
		maxIndex := i;
		break;
	end if;
end do:
maxIndex;
</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">Now we grab the data point which matches that index.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">maxPoint := data[maxIndex];</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Fitting the data to a curve</Text-field></Title><Text-field layout="Normal" style="Text">Now, we would like to fit a 4th order polynomial to our data, let our polynomial be p(x) as defined below:</Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">p := a*x^4 + b*x^3 +c*x^2 +d*x+e;</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text"/></Input></Group><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" linebreak="space" linespacing="0.0" rightmargin="0.0" spaceabove="0.0" spacebelow="0.0"><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false">Now we need to find a, b, c, d, and e such that </Font><Font background="[0,0,0]" executable="false" family="Times New Roman" underline="false">  </Font><Equation input-equation="Float(5, -1)*(Sum((p(x[i])-y[i])^2, i = 1 .. 12))" style="2D Comment">NiMqJi1JJkZsb2F0R0kqcHJvdGVjdGVkR0YmNiQiIiYhIiIiIiItSSRTdW1HNiI2JCokLCYtSSJwR0YtNiMmSSJ4R0YtNiNJImlHRi1GKiZJInlHRi1GNkYpIiIjL0Y3O0YqIiM3Rio=</Equation><Font background="[0,0,0]" family="Times New Roman">  <Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false">is a minimum.</Font></Font></Text-field><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" linebreak="space" linespacing="0.0" rightmargin="0.0" spaceabove="0.0" spacebelow="0.0"><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false">To achieve this we will use the </Font><Hyperlink linktarget="Help:LSSolve" style="Hyperlink">LSSolve</Hyperlink><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false"> function from the </Font><Hyperlink linktarget="Help:Optimization" style="Hyperlink">optimization</Hyperlink><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false"> package, LSSolve will find the values of a, b, c, d, and e that will minimize the expression above.</Font></Text-field><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" linebreak="space" linespacing="0.0" rightmargin="0.0" spaceabove="0.0" spacebelow="0.0"/><Text-field alignment="left" firstindent="0.0" leftmargin="0.0" linebreak="space" linespacing="0.0" rightmargin="0.0" spaceabove="0.0" spacebelow="0.0"><Font background="[0,0,0]" bold="false" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="false" size="12" underline="false">Let's define the an expression for the residues, and we'll pass it as an argument to LSSolve.</Font></Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">residues := map(d -&gt; eval(p, x=d[1])-d[2], data):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">sol := LSSolve(residues);</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text">We now substitute the values of a, b,  c, d, and e in our polynomial p and we get:</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">poly := eval(p, sol[2]);</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" style="Text">Next we create a plot of the fitted polynomial and store it to display later. </Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">P2 := plot(poly, x=0.00406..0.008):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" style="Text"/><Text-field layout="Normal" style="Text">Now we <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:plots, display" style="Hyperlink" subscript="false" superscript="false">display</Hyperlink> both plots with the fitted polynomial in red and the original data points in blue.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">display(P1, P2 );</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Finding the area under the curve</Text-field></Title><Group><Input><Text-field layout="Normal" style="Text">Finally, we <Hyperlink bold="false" executable="false" italic="false" linktarget="Help:int" style="Hyperlink" subscript="false" superscript="false">integrate</Hyperlink> the polynomial to find the area under the curve.</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">int(poly, x=0.004..0.008);</Text-field></Input></Group></Section><Section collapsed="true"><Title><Text-field layout="Heading 1" style="Heading 1">Conclusion</Text-field></Title><Text-field layout="Normal" style="Text">Maple gives the user many tools to help you to analyze all kinds of data. As shown in this example, it is simple to read in, visualize, and analyze your data.</Text-field></Section><Text-field/><Group><Input><Text-field layout="Normal" style="Text"><Font italic="true">
Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.  </Font>
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