View source for Analyze data with the Statistics software R

<h1>Normal Location and Scale</h1>
This litte application sends the y-coordinates of the points which are normal distributed (pseudo-)random numbers to the server.<br /> 
There, location and scale of the sample are estimated using the [http://www.r-project.org Statistics software R].<br />
The return values are plotted and displayed.<br /><br />
<b>The computed estimates are:</b><br /> 
* mean, standard deviation: red (non-robust!)<br />
* median and MAD: black (most-robust!)<br /> 
* radius-minimax estimator: green (optimally robust; cf. Rieder et al. (2008))<br /><br />
By changing the y-position of the four movable points you should recognize the instability (non-robustness) of mean and standard deviation in contrast to the robust estimates; e.g., move one of the four movable points to the top of the plot.<br /><br />
===Online results:===
<html><script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script></html>
<pre id='output'>Statistics:<br></pre>
<jsxgraph width="700" height="400">
        brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-0.15, 60, 11.15, -20],axis:true});
        brd.suspendUpdate();
        var graph1,graph2,graph3,graph4,graph5,graph6,graph7,graph8,graph9;

        var x = [];
        var y = [];
        var p = [];
        var r1,r2,rad,phi,z1,z2;
        for (var i=0;i<6;i++) {
            x[i] = i;
            // Box-Muller-Method for standard normal random numbers
            r1 = Math.random();
            r2 = Math.random();
            rad = Math.sqrt(-2*Math.log(1-r1));
            phi = 2*Math.PI*r2;
            z1 = rad*Math.cos(phi);
            z2 = rad*Math.sin(phi);
            if(i < 5){
                y[i] = 10+z1*3;
                y[6+i] = 10+z2*3;
                x[6+i] = 6+i;
                p[i] = brd.createElement('point', [x[i],y[i]],{name:' ',fixed:true,size:2,face:'[]'}); 
                p[6+i] = brd.createElement('point', [x[6+i],y[6+i]],{name:' ',fixed:true,size:2,face:'[]'});
            }else{
                y[i] = 10+z1*3;
                p[i] = brd.createElement('point', [x[i],y[i]],{name:' ',fixed:true,size:2,face:'[]'}); 
            }
        }
        var hlow = [];
        var hup = [];
        var l = [];
        hlow[0] = brd.createElement('point', [x[0],0],{name:' ',visible:false}); 
        hlow[1] = brd.createElement('point', [x[10],0],{name:' ',visible:false}); 
        hlow[2] = brd.createElement('point', [x[8],0],{name:' ',visible:false}); 
        hlow[3] = brd.createElement('point', [x[9],0],{name:' ',visible:false}); 
        hup[0] = brd.createElement('point', [x[0],20],{name:' ',visible:false}); 
        hup[1] = brd.createElement('point', [x[10],20],{name:' ',visible:false}); 
        hup[2] = brd.createElement('point', [x[8],20],{name:' ',visible:false}); 
        hup[3] = brd.createElement('point', [x[9],20],{name:' ',visible:false}); 
        l[0] = brd.createElement('line', [hlow[0],hup[0]],{visible:false}); 
        l[1] = brd.createElement('line', [hlow[1],hup[1]],{visible:false}); 
        l[2] = brd.createElement('line', [hlow[2],hup[2]],{visible:false}); 
        l[3] = brd.createElement('line', [hlow[3],hup[3]],{visible:false}); 

        brd.removeObject(p[0]);
        brd.removeObject(p[10]);
        brd.removeObject(p[8]);
        brd.removeObject(p[9]);
        
        p[0] = brd.createElement('glider', [x[0],y[0],l[0]],{name:' ',size:4,face:'o'}); 
        p[10] = brd.createElement('glider', [x[10],y[10],l[1]],{name:' ',size:4,face:'o'}); 
        p[8] = brd.createElement('glider', [x[8],y[8],l[2]],{name:' ',size:4,face:'o'}); 
        p[9] = brd.createElement('glider', [x[9],y[9],l[3]],{name:' ',size:4,face:'o'}); 
        brd.unsuspendUpdate();

        doIt();
        
        function doIt() {
            brd.update();
            // Generate a ";"-separated list
            var t = '';
            for (var i=0;i<p.length;i++) {
                t += p[i].Y() + ';';
            }
            new Ajax.Request('/~alfred/jsxgraph/examples/rserv.php', {
                method:'post',
                parameters:'input='+escape(t),
                onComplete: function(transport) {
                    if (200 == transport.status) {
                        var t = transport.responseText;
                        var res = t.match(/"(.*)"/gi);
                        res = RegExp.$1;
                        var a = res.split(";");
                        var m = a[0]*1.0;
                        var sd = a[1]*1.0;
                        var med = a[2]*1.0;
                        var mad = a[3]*1.0;
                        var est1 = a[4]*1.0;
                        var est2 = a[5]*1.0;

                        if (!graph2) { 
                            graph2 = brd.createElement('curve', [[x[0],x[x.length-1]],[m,m]], {strokecolor:'red'}); 
                            graph3 = brd.createElement('curve', [[x[0],x[x.length-1]],[m+sd,m+sd]], {strokecolor:'red',dash:2}); 
                            graph4 = brd.createElement('curve', [[x[0],x[x.length-1]],[m-sd,m-sd]], {strokecolor:'red',dash:2}); 
                            graph5 = brd.createElement('curve', [[x[0],x[x.length-1]],[med,med]], {strokecolor:'black'}); 
                            graph1 = brd.createElement('curve', [[x[0],x[x.length-1]],[med-mad,med-mad]], {strokecolor:'black',dash:3}); 
                            graph6 = brd.createElement('curve', [[x[0],x[x.length-1]],[med+mad,med+mad]], {strokecolor:'black',dash:3}); 
                            graph8 = brd.createElement('curve', [[x[0],x[x.length-1]],[est1,est1]], {strokecolor:'green'}); 
                            graph7 = brd.createElement('curve', [[x[0],x[x.length-1]],[est1-est2,est1-est2]], {strokecolor:'green',dash:4}); 
                            graph9 = brd.createElement('curve', [[x[0],x[x.length-1]],[est1+est2,est1+est2]], {strokecolor:'green',dash:4}); 
                        } else {
                            graph2.dataY = [m,m];
                            graph3.dataY = [m+sd,m+sd];
                            graph4.dataY = [m-sd,m-sd];
                            graph5.dataY = [med,med];
                            graph1.dataY = [med-mad,med-mad];
                            graph6.dataY = [med+mad,med+mad];
                            graph8.dataY = [est1,est1];
                            graph7.dataY = [est1-est2,est1-est2];
                            graph9.dataY = [est1+est2,est1+est2];
                        }

                        document.getElementById('output').innerHTML = '<b><font size="+1">Normal location and scale:</font></b><br /><br />'+ 
                                '<b>Estimates for location (true value = 10):</b><br />' +
                                'Mean = ' + Math.round(m,2) + '<br />' +
                                'Median = ' + Math.round(med,2) + '<br />' +
                                'Radius-minimax estimator = ' + Math.round(est1,2) + '<br /><br />' +
                                '<b>Estimates for scale (true value = 3):</b><br />' +
                                'Standard deviation = ' + Math.round(sd,2) + '<br />' +
                                'MAD = ' + Math.round(mad,2) + '<br />' + 
                                'Radius-minimax estimator = ' + Math.round(est2,2) + '<br />';
                        brd.update();
                    };
                }});
        }
brd.addHook(doIt, 'mouseup');
</jsxgraph>

=== The underlying source code ===
[[The underlying JavaScript and PHP code]]

The R script can be [http://jsxgraph.uni-bayreuth.de/~alfred/jsxgraph/branches/0.70/examples/LokSkala.R downloaded here].

=== References ===
* ''The Costs of not Knowing the Radius'', Helmut Rieder, Matthias Kohl and Peter Ruckdeschel, Statistical Methods and Application 2008 Feb; 17(1): p.13-40; cf. also [http://www.stamats.de/RRlong.pdf] for an extended version.
* ''Robust Asymptotic Statistics'', Helmut Rieder, Springer, 1994.
* ''Numerical Contributions to the Asymptotic Theory of Robustness'', Matthias Kohl, PhD-Thesis, University of Bayreuth, 2005; cf. also [http://www.stamats.de/ThesisMKohl.pdf].

=== External links ===
* [http://www.r-project.org The R Project for statistical computing.]
* [https://r-forge.r-project.org/projects/robast/ RobASt project on R-Forge]
* [http://en.wikipedia.org/wiki/Robust_statistics Wikipedia on Robust Statistics]

[[Category:Examples]]
[[Category:Statistics]]