As previously described, NMath 5.1 and NMath Stats 3.4 include classes for plotting NMath types using the Microsoft Chart Controls for .NET. (Free adapter code is also available for using NMath with Syncfusion Essential Chart.) Let’s use Anscombe’s data to explore how NMath’s new visualization capabilities can be used to reveal the differences in the data sets.

First, we’ll load the data into a DoubleMatrix.

DoubleMatrix A = new DoubleMatrix( @"11x8 [
  10.0 8.04  10.0 9.14 10.0 7.46  8.0  6.58
  8.0  6.95  8.0  8.14 8.0  6.77  8.0  5.76
  13.0 7.58  13.0 8.74 13.0 12.74 8.0  7.71
  9.0  8.81  9.0  8.77 9.0  7.11  8.0  8.84
  11.0 8.33  11.0 9.26 11.0 7.81  8.0  8.47
  14.0 9.96  14.0 8.10 14.0 8.84  8.0  7.04
  6.0  7.24  6.0  6.13 6.0  6.08  8.0  5.25
  4.0  4.26  4.0  3.10 4.0  5.39  19.0 12.50
  12.0 10.84 12.0 9.13 12.0 8.15  8.0  5.56
  7.0  4.82  7.0  7.26 7.0  6.42  8.0  7.91
  5.0  5.68  5.0  4.74 5.0  5.73  8.0  6.89 ]" );

Now let’s perform some simple descriptive statistics.

 int groups = 4;
 Slice rows = Slice.All;
 Slice xCols = new Slice( 0, groups, 2 );
 Slice yCols = new Slice( 1, groups, 2 );
 double unbiased = (double)A.Rows / ( A.Rows - 1 );

 Console.WriteLine( "Mean of x: {0}",
   NMathFunctions.Mean( A[ rows, xCols ] ) );
 Console.WriteLine( "Variance of x: {0}",
   NMathFunctions.Variance( A[rows, xCols] ) * unbiased );
 Console.WriteLine( "Mean of y: {0}",
   NMathFunctions.Round( NMathFunctions.Mean( A[rows, yCols] ), 2 ) );
 Console.WriteLine( "Variance of y: {0}",
   NMathFunctions.Round(
     NMathFunctions.Variance( A[rows, yCols] ) * unbiased, 3 ) );

 Console.Write( "Correlation of x-y: " );
 for (int i = 0; i < A.Cols; i += 2 )
 {
   Console.Write( NMathFunctions.Round(
    StatsFunctions.Correlation( A.Col(i), A.Col(i + 1) ), 3 ) + " " );
 }
 Console.WriteLine();

You can see from the output that the statistics are nearly identical for all four data sets:

Mean of x: [ 9 9 9 9 ]
Variance of x: [ 11 11 11 11 ]
Mean of y: [ 7.5 7.5 7.5 7.5 ]
Variance of y: [ 4.127 4.128 4.123 4.123 ]
Correlation of x-y: 0.816 0.816 0.816 0.817

Now let's fit a linear model to each data set.

 LinearRegression[] lrs = new LinearRegression[groups];

 for( int i = 0; i < groups; i ++ )
 {
   Console.WriteLine( "Group {0}", i + 1 );

   bool addIntercept = true;
   lrs[i] = new LinearRegression( new DoubleMatrix( A.Col( 2 * i ) ),
     A.Col( 2 * i + 1 ), addIntercept );
   Console.WriteLine( "equation of regression line: Y = {0} + {1}X",
     Math.Round( lrs[i].Parameters[0], 2 ),
     Math.Round( lrs[i].Parameters[1], 3 ) );

   LinearRegressionParameter param =
     new LinearRegressionParameter( lrs[i], 1 );
   Console.WriteLine( "standard error of estimate of slope: {0}",
     Math.Round( param.StandardError, 3 ) );
   Console.WriteLine( "t-statistic: {0}",
     Math.Round( param.TStatistic( 0 ), 2 ) );

   LinearRegressionAnova anova = new LinearRegressionAnova( lrs[i] );
   Console.WriteLine( "regression sum of squares: {0}",
     Math.Round( anova.RegressionSumOfSquares, 2 ) );
   Console.WriteLine( "residual Sum of squares: {0}",
     Math.Round( anova.ResidualSumOfSquares, 2 ) );
   Console.WriteLine( "r2: {0}", Math.Round( anova.RSquared, 2 ) );

   Console.WriteLine();
 }

Again, the output is nearly identical for each data set:

Group 1
equation of regression line: Y = 3 + 0.5X
standard error of estimate of slope: 0.118
t-statistic: 4.24
regression sum of squares: 27.51
residual Sum of squares: 13.76
r2: 0.67

Group 2
equation of regression line: Y = 3 + 0.5X
standard error of estimate of slope: 0.118
t-statistic: 4.24
regression sum of squares: 27.5
residual Sum of squares: 13.78
r2: 0.67

Group 3
equation of regression line: Y = 3 + 0.5X
standard error of estimate of slope: 0.118
t-statistic: 4.24
regression sum of squares: 27.47
residual Sum of squares: 13.76
r2: 0.67

Group 4
equation of regression line: Y = 3 + 0.5X
standard error of estimate of slope: 0.118
t-statistic: 4.24
regression sum of squares: 27.49
residual Sum of squares: 13.74
r2: 0.67

Finally, let's use the new NMath charting functionality to plot each linear regression fit. Note that we make use of the Compose() method to combine multiple charts into a single composite Chart control.

 List charts = new List();
 for( int i = 0; i < lrs.Length; i++ )
 {
   charts.Add( NMathStatsChart.ToChart( lrs[i], 0 ) );
 }
 Chart all = NMathStatsChart.Compose( charts, 2, 2,
   NMathChart.AreaLayoutOrder.RowMajor );
 for( int i = 0; i < groups; i++ )
 {
   all.ChartAreas[i].AxisX.Title = "x" + ( i + 1 );
   all.ChartAreas[i].AxisX.Minimum = 2;
   all.ChartAreas[i].AxisX.Maximum = 22;
   all.ChartAreas[i].AxisX.Interval = 4;

   all.ChartAreas[i].AxisY.Title = "y" + ( i + 1 );
   all.ChartAreas[i].AxisY.Minimum = 2;
   all.ChartAreas[i].AxisY.Maximum = 14;
   all.ChartAreas[i].AxisY.Interval = 4;

   all.Series[2 * i].Color = Color.DarkOrange;
   all.Series[2 * i + 1].Color = Color.SteelBlue;
 }
 NMathStatsChart.Show( all );

The charts reveal  dramatic differences between the data sets, despite the identical fitted models. Group 1 shows a linear relationship, while in Group 2 the releationship is clearly non-linear.  Groups 3 and 4 demonstrate how a single outlier can have a large effect on simple statistics.

Ken

References

Anscombe, F. J. (1973). "Graphs in Statistical Analysis". American Statistician 27 (1): 17–21. JSTOR 2682899.
Tufte, Edward R. (2001). The Visual Display of Quantitative Information (2nd ed.). Cheshire, CT: Graphics Press

The post The Importance of Graphing Your Data appeared first on CenterSpace.

In an earlier post, we described how NMath types can be plotted using the Microsoft Chart Controls for .NET, but programmatically constructing Chart objects with data series bound to NMath objects. The latest release of the NMath libraries makes this process much simpler by providing adapter classes which easily construct basic math charts for you from NMath types:

• NMath 5.1 includes assembly NMathChartMicrosoft.dll containing class NMathChart, which provides convenience methods for plotting NMath types using the Microsoft Chart Controls
• NMath Stats 3.4 includes assembly NMathStatsChartMicrosoft.dll containing NMathStatsChart which provides convenience methods for plotting NMath Stats types using the Microsoft Chart Controls

The generated charts can then be customized to meet your needs.

Using the Math Chart Adapters

The Microsoft Chart Controls for .NET are available as a separate download for .NET 3.5. Beginning in .NET 4.0, the Chart controls are part of the .NET Framework. To use the Chart controls, add a reference to System.Windows.Forms.DataVisualization.
To use the adapters, add a reference to NMathChartMicrosoft.dll and/or NMathStatsChartMicrosoft.dll, and using statement:

using CenterSpace.NMath.Charting.Microsoft;

A previous blog article provides a simple app.config solution for deploying your chart dependent applications to target both .NET 3.5 or .NET 4.0. Without this app.config modification, your application will only target .NET 3.5.

The Math Chart Adapter API

Both NMathChart and NMathStatsChart classes share a common API. Overloads of the ToChart() function are provided for our common math and stats types. ToChart() returns an instance of System.Windows.Forms.DataVisualization.Charting.Chart, which can be customized as desired. For example

Polynomial poly = new Polynomial( new DoubleVector( 4, 2, 5, -2, 3 ) );
Chart chart = NMathChart.ToChart( poly, -1, 1 );
chart.Titles.Add("Hello World");

For prototyping and debugging console applications, the Show() function shows a given chart in a default form.

NMathChart.Show( chart );

Note that when the window is closed, the chart is disposed.

If you do not need to customize the chart, overloads of Show() are also provided for common NMath types.

NMathChart.Show( poly );

This is equivalent to calling:

NMathChart.Show( NMathChart.ToChart( poly ) );

The Save() function saves a chart to a file or stream.

NMathChart.Save( chart, "chart.png", ChartImageFormat.Png );

If you are developing a Windows Forms application using the Designer, add a Microsoft Chart control to your form, then update it with an NMath object using the appropriate Update() function after initialization.

public Form1()
{
  InitializeComponent();

  Polynomial poly =
    new Polynomial( new DoubleVector( 4, 2, 5, -2, 3 ) );
  NMathChart.Update( ref this.chart1, poly, -1, 1 );
}

This has the following effect on your existing Chart object:

• a new, default ChartArea is added if one does not exist, otherwise chart.ChartAreas[0] is used
• axis titles, and DefaultAxisTitleFont and DefaultMajorGridLineColor, only have an effect if a new ChartArea is added
• titles are added only if the given Chart does not already contain any titles
• chart.Series[0] is replaced, or added if necessary

Charting Examples

Here are a few examples of usage in C# (Of course, as with all NMath functionality, these examples will also work, given the necessary syntactic changes, from VB.NET or F#, or any other .NET language.) :

double xmin = -1;
double xmax = 1;
int numInterpolatedValues = 100;
Polynomial poly = new Polynomial( new DoubleVector( 4, 2, 5, -2, 3 ) );
NMathChart.Show( poly, xmin, xmax, numInterpolatedValues );

DoubleVector x = new DoubleVector( 100, 0, 0.1 );
DoubleVector cos = x.Apply( NMathFunctions.CosFunction );
DoubleVector sin = x.Apply( NMathFunctions.SinFunction );

DoubleVector[] data = new DoubleVector[] { cos, sin };
NMathChart.Unit unit = new NMathChart.Unit( 0, 0.1, "x" );
Chart chart = NMathChart.ToChart( data, unit );

chart.Series[0].Name = "cos(x)";
chart.Series[1].Name = "sin(x)";
NMathChart.Show( chart );

double lambda = 4.0;
PoissonDistribution poisson = new PoissonDistribution( lambda );

NMathStatsChart.Show( poisson,
  NMathStatsChart.DistributionFunction.PDF );

DoubleMatrix predictors =
  new DoubleMatrix( 100, 3, new RandGenUniform() ); ;
DoubleVector observations = 2 * predictors.Col(0) +
  -0.75 * predictors.Col(1) + 1.25 * predictors.Col(2) + 0.5;
bool addIntercept = true;

LinearRegression lr =
  new LinearRegression( predictors, observations, addIntercept );
NMathStatsChart.Show( lr, 2 );

More Information

For more information, and many more examples, see:

• Whitepaper: NMath Visualization Using the Microsoft Chart Controls
• Whitepaper: NMath Stats Visualization Using the Microsoft Chart Controls
• the API docs for the CenterSpace.NMath.Charting.Microsoft namespace

The post .NET Math with Microsoft Chart Controls, Revisited appeared first on CenterSpace.

<configuration>
   ...
   <runtime>
      ...
      <assemblyBinding xmlns="urn:schemas-microsoft-com:asm.v1">
         ...
         <dependentAssembly>
            <assemblyIdentity name="System.Windows.Forms.DataVisualization"
              publicKeyToken="31bf3856ad364e35"/>
            <bindingRedirect oldVersion="3.5.0.0-3.5.0.0" newVersion="4.0.0.0"/>
         </dependentAssembly>
         ...
      </assemblyBinding>
      ...
   </runtime>
   ...
</configuration>

In this way, your application with charts can be deployed to both .NET 3.5 and .NET 4.0. Please note that all other NMath assemblies (with non-visual functionality) can be used in .NET 3.5 and .NET 4.0 without this step.

See this tutorial for more information.

The post Using NMath Charts with .NET 4.0 appeared first on CenterSpace.

A Chart object contains one or more ChartAreas, each of which contain one or more data Series. Each Series has an associated chart type, and a DataPoint collection. DataPoints can be manually appended or inserted into the collection, or added automatically when a series is bound to a datasource using either the DataBindY() or DataBindXY() method. Since any IEnumerable can act as a datasource, it’s easy to use an NMath vector or vector view (of a matrix row or column, for example) as a datasource.

For example, suppose we want to create a scatter plot of the first two columns of a 20 x 5 DoubleMatrix (that is, column 0 vs. column 1).

DoubleMatrix data = new DoubleMatrix( 20, 5, new RandGenUniform() );
DoubleVector x = data.Col( 0 );
DoubleVector y = data.Col( 1 );

Begin by creating a new Chart object, and optionally adding a Title.

Chart chart = new Chart()
{
  Size = new Size( 500, 500 ),
};

Title title = new Title()
{
  Name = chart.Titles.NextUniqueName(),
  Text = "My Data",
  Font = new Font( "Trebuchet MS", 12F, FontStyle.Bold ),
};
chart.Titles.Add( title );

Next, add a ChartArea.

ChartArea area = new ChartArea()
{
  Name = chart.ChartAreas.NextUniqueName(),
};
area.AxisX.Title = "Col 0";
area.AxisX.TitleFont = new Font( "Trebuchet MS", 10F, FontStyle.Bold );
area.AxisX.MajorGrid.LineColor = Color.LightGray;
area.AxisX.RoundAxisValues();
area.AxisY.Title = "Col 1";
area.AxisY.TitleFont = new Font( "Trebuchet MS", 10F, FontStyle.Bold );
area.AxisY.MajorGrid.LineColor = Color.LightGray;
area.AxisY.RoundAxisValues();
chart.ChartAreas.Add( area );

Finally, add a new data Series, and bind the datasource to the NMath x,y vectors.

Series series = new Series()
{
  Name = "Points",
  ChartType = SeriesChartType.Point,
  MarkerStyle = MarkerStyle.Circle,
  MarkerSize = 8,
};
series.Points.DataBindXY( x, y );
chart.Series.Add( series );

To display the chart, we can use a utility function like this, which shows the given chart in a default form running in a new thread.

public static void Show( Chart chart )
{
  Form form = new Form();
  form.Size = new Size( chart.Size.Width + 20, chart.Size.Height + 40 );
  form.Controls.Add( chart );
  Thread t = new Thread( () => Application.Run( form ) );
   t.Start();
}

After calling

Show( chart );

the result looks like this:

Of course, you can easily plot more complicated data as well. For instance, suppose we fit a 4th degree polynomial to this (random) data:

int degree = 4;
PolynomialLeastSquares pls = new PolynomialLeastSquares( degree, x, y );

To add the fitted polynomial to the cart, just add a new data Series interpolating over the range of x values.

Series series2 = new Series()
{
  Name = "Polynomial",
  ChartType = SeriesChartType.Line,
  MarkerStyle = MarkerStyle.None,
};
int numInterpolatedValues = 100;
double xmin = NMathFunctions.MinValue( x );
double xmax = NMathFunctions.MaxValue( x );
double step = ( xmax - xmin ) / ( numInterpolatedValues - 1 );
DoubleVector xi = new DoubleVector( numInterpolatedValues, xmin, step );
series2.Points.DataBindXY( xi, pls.FittedPolynomial.Evaluate( xi ) );
chart.Series.Add( series2 );

Let’s also add a subtitle with the fitted function, and a Legend.

Title subtitle = new Title()
{
  Name = chart.Titles.NextUniqueName(),
  Text = "f(x) = " + pls.FittedPolynomial.ToString( "N2" ),
};
chart.Titles.Add( subtitle );

Legend legend = new Legend()
{
  Name = chart.Legends.NextUniqueName(),
  DockedToChartArea = chart.ChartAreas[0].Name,
};
chart.Legends.Add( legend );

Now the chart looks like this:

If you add a Chart control to your application using the visual designer (by dragging a Chart control from the Data category in the Toolbox), this generates code to create a default Chart, with a single ChartArea, Series, and Legend. You can then edit the properties in the Properties tab, and bind the generated Series to an NMath datasource. For example:

public Form1()
{
  InitializeComponent();

  DoubleMatrix data = new DoubleMatrix( 20, 5, new RandGenUniform() );
  this.chart1.Series[0].Points.DataBindXY( data.Col(0), data.Col(1) );
}

The Microsoft Chart Controls for .NET provide an easy (and free) solution for visualizing NMath numerical types.

Ken

The post Using NMath with Microsoft Chart Controls for .NET appeared first on CenterSpace.