Time series forecasting: double exponential smoothing

The data is the file zurich.txt from http://statistik.mathematik.uni-wuerzburg.de/timeseries/index.php.

The dashed curve are the observed values. The blue curve are the predicted values and it is computed by the following rules:

Initial values:

[math]\displaystyle{ S_0 = y_0 }[/math]

[math]\displaystyle{ b_0 = y_1-y_0 }[/math]

Then, the values are iteratively computed by

[math]\displaystyle{ S_t = \alpha\cdot y_t + (1-\alpha)\cdot(S_{t-1} + b_{t-1}) }[/math]

[math]\displaystyle{ b_t = \gamma\cdot(S_t - S_{t-1}) + (1-\gamma)\cdot b_{t-1} }[/math]

Here, the time series [math]\displaystyle{ y }[/math] is stored in the array data.

The JavaScript code

    var data, datax, i, brd;
    
    //
    // zurich.txt from http://statistik.mathematik.uni-wuerzburg.de/timeseries/index.php
    //
    // global array data
    data = "406.60 428.50 ... 521.80 524.40 526.80";
    data = data.split(' ');
    datax = [];
    for (i = 0; i < data.length; i++)  { 
        data[i] = parseFloat(data[i]); 
        datax[i] = i;
        }
    
    
    brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-2, 550, data.length+2, 380], grid: false});
    brd.createElement('axis',[[0,0],[0,1]]);
    brd.createElement('axis',[[0,400],[1,400]]);

    brd.createElement('curve',[datax,data],{strokeColor:'gray',dash:2});                    // plot the observed data

    alpha = brd.createElement('slider', [[10,520],[100,520],[0,0.1,1.0]],{name:'&alpha;'});   
    gamma = brd.createElement('slider', [[10,510],[100,510],[0,0.1,1.0]],{name:'&gamma;'});

    estimate = brd.createElement('curve',[[0],[0]]);                                        // The filtered curve

    estimate.updateDataArray = function() { 
            var t,
                alphalocal = alpha.Value(),  // Read the slider value of alpha
                gammalocal = gamma.Value(),  // Read the slider value of gamma 
                S = data[0],                 // Set the inital values for S and b
                b = data[1]-data[0],
                Snew;
                
            this.dataX[0] = 0;
            this.dataY[0] = S;
            for (t=1; t<data.length; t++) {
                Snew = alphalocal*data[t] + (1-alphalocal)*(S + b);
                b    = gammalocal*(Snew - S) + (1-gammalocal)*b;
                this.dataX[t] = t;
                this.dataY[t] = Snew;
                S = Snew;
            }
        }
     brd.update(); // first computation of the filtered curve.