# Time series forecasting: double exponential smoothing

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

The data is from 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:

$\displaystyle{ S_0 = y_0 }$
$\displaystyle{ b_0 = y_1-y_0 }$

Then, the values are iteratively computed by

$\displaystyle{ S_t = \alpha\cdot y_t + (1-\alpha)\cdot(S_{t-1} + b_{t-1}) }$
$\displaystyle{ b_t = \gamma\cdot(S_t - S_{t-1}) + (1-\gamma)\cdot b_{t-1} }$

Here, the time series $\displaystyle{ y }$ 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.create('axis',[[0,0],[0,1]]);
brd.create('axis',[[0,400],[1,400]]);

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

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

estimate = brd.create('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.