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Analysis with Maple\251 Maplesoft, a division of Waterloo Maple Inc., 2004

Maple 9.5 is a general-purpose environment for performing fast, accurate technical analysis in any field of engineering, science or mathematics.Maple can greatly facilitate all stages of statistical analysis, including: Manipulation of data setsStatistical plotsDescriptive statisticsLinear and nonlinear regressionHandling outliersStatistical tests

restart; interface( warnlevel = 0 ): with( FileTools[Text] ): with( plots ): with( stats ): with( describe ): with( transform ): with( random ): with( statplots ): with( statevalf ): with( RandomTools ): with( CurveFitting ): with( Optimization ):

Read in the data file create the data sets we want to work withfile := "Stock Price.txt": n := CountLines( file ): maxTime := month[n]: for i from 1 to n do month[i] := parse( ReadString( file ) ); Price[i] := parse( ReadString( file ) ); end do: prices := [seq( Price[i], i=1..n)]: data := [seq( [month[i], Price[i]], i=1..n)];

Plot the data setp1 := pointplot( data, color=blue ): p1;Compute a two-week moving averagemovingAvg := moving[ 5 ]( prices );movingAvgData := [seq( [ month[i], movingAvg[i] ], i=1..n-5 ) ]: Plot the moving average on top of the data plotp2 := pointplot( movingAvgData, connect=true, color=green, thickness=2 ): display( p1, p2 );Create histograms with relative and absolute frequency histogram( prices ); histogram( prices, area=count );Create a box plot and superimpose it onto the data plotb1 := boxplot( prices, shift=maxTime/2, width=maxTime, axes=boxed, color=red, thickness=2 ): display( b1, p1 );

Mean, median and rangemean( prices ); median( prices ); range( prices );Quartiles and percentilesquartile[ 1 ]( prices ); quartile[ 2 ]( prices ); quartile[ 3 ]( prices ); percentile[ 28 ]( prices ); Standard deviation and variancestandarddeviation( prices ); variance( prices );The kth moment of NiM7JkkiWEc2IjYjIiIiJkYlNiNJIm5HRiY= about t is defined as NiMqJi1JJFN1bUc2IjYkKSwmJkkiWEdGJjYjSSJpR0YmIiIiSSJ0R0YmISIiSSJrR0YmL0YtO0YuSSJuR0YmRi5GNEYw In Maple, call moment[ k, t ]( data )moment[ 1 ]( prices ); moment[ 2 ]( prices ); moment[ 2, mean(prices) ]( prices ); # same as the variance moment[ 3 ]( prices ); kurtosis( prices );

p1;Quadratic best fitquadraticFit := LeastSquares( data, t, curve = c*t^2+b*t+a );Plot the quadratic best fit on top of the data setp4 := plot( quadraticFit, t=1..maxTime, thickness=2 ): display( p1, p4 );quadraticError := add( (eval( quadraticFit, t=month[i] ) - prices[i])^2, i=1..n);Let's fit a non-linear model NiMqJkkiYUc2IiIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMsJiomSSJiR0YlRiZJInRHRiVGJkYmKiZJImNHRiVGJiokRjAiIiNGJiEiIkYm to the data, with parameters a, b and cmodel := t-> a*exp( b*t + c*t^2 );Use Maple's non-linear regression solverresiduals := [seq( model(month[i])-prices[i], i=1..n)]:sol := LSSolve( residuals, initialpoint={ a = prices[1], b = .2, c = -.2 });The fitted modelfit := eval( model(t), sol[2] );Plot the fit against the datap3 := plot( fit, t=0..maxTime+1, thickness=2): display( p3, p1 );

Maple supports 19 built-in probability distributions, including their Pdfs, Cdfs and Inverse Cdfs.plot( [ statevalf[ pdf, normald[0,1] ], statevalf[ pdf, studentst[5] ], statevalf[ pdf, chisquare[5] ], statevalf[ pdf, lognormal[0,1] ], statevalf[ pdf, gamma[3,2] ]], -8..6, color=[red, blue, cyan, green,black], thickness=2); Generate 200 normally distributed random data with NiM2JC9JI211RzYiIiNYL0kmc2lnbWFHRiYiIiQ=data := [ random[ normald[45,3] ]( 200 ) ]; Check that the data set is approximately normally distributed histogram( data );Draw a random subsample of 75dataSample := [ seq( Generate( choose( data ) ), i=1..75 ) ]; mean( dataSample );How close is the sample mean to the population mean, assuming we don't know the population mean?We can add to Maple's library by defining our own commands z_value := proc( alpha ) icdf[ normald ]( 1 - alpha/2 ); end proc: t_value := proc( alpha, n ) icdf[ studentst[n-1] ]( 1 - alpha/2 ); end proc: z-values and t-values for NiM2JS9JJmFscGhhRzYiLUkmRmxvYXRHSSpwcm90ZWN0ZWRHRik2JCIjNSEiIy1GKDYkIiImRiwtRig2JCIiIkYsz_value( .10 ); z_value( .05 ); z_value( .01 ); t_value( .10, 75 ); t_value( .05, 75 ); t_value( .01, 75 );Build on these shortcuts to create routines for computing confidence intervals z_confidenceInterval := proc( conf_level, stddev, N ) evalf( z_value( 1 - conf_level ) * stddev / sqrt( N ) ); end proc: t_confidenceInterval := proc( conf_level, stddev, N ) evalf( t_value( 1 - conf_level, N ) * stddev / sqrt( N ) ); end proc:Now we can compute confidence intervals just as if the routines were built in to Maple!ss := standarddeviation( dataSample ); mu_bar := mean( dataSample ); z_confidenceInterval( .90, ss, 75 ); z_confidenceInterval( .95, ss, 75 ); z_confidenceInterval( .99, ss, 75 ); t_confidenceInterval( .90, ss, 75 ); t_confidenceInterval( .95, ss, 75 ); t_confidenceInterval( .99, ss, 75 );How does the width of the confidence interval depend on the sample size N?plot([ z_confidenceInterval( .90, ss, size ), z_confidenceInterval( .95, ss, size ), z_confidenceInterval( .99, ss, size)], size=100..1000, title="Confidence interval size vs. Sample size", legend=["90% confidence level", "95% confidence level","99% confidence level"], color=[red,blue,green,black], thickness=2 );

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