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Time series forecasting: double exponential smoothing
Time series forecasting: double exponential smoothing
In this example, the time series $y$ (a list of numbers) is stored in the array `data`. The dashed curve is a visualization of these (observed) values. Now, we try to forecast the nature of these values.
The blue curve are the predicted values and it is computed by the following rules:
Initial values:
$$ S_0 = y_0$$
$$ b_0 = y_1-y_0$$
The values are iterative computed by
$$S_t = \alpha\cdot y_t + (1-\alpha)\cdot(S_{t-1} + b_{t-1})$$
$$b_t = \gamma\cdot(S_t - S_{t-1}) + (1-\gamma)\cdot b_{t-1}$$
The values of $\alpha$ and $\gamma$ can be controlled with the two sliders.
// Define the id of your board in BOARDID
var data, datax, i, board;
//
// zurich.txt, originally from http://statistik.mathematik.uni-wuerzburg.de/timeseries/index.php
//
// global array data
data =
"406.60 428.50 429.30 426.30 434.70 415.90 419.00 408.80 410.10 408.30 420.40 415.20 409.70 408.90 411.00 410.60 409.60 409.50 409.80 413.00 417.90 415.80 415.50 421.30 423.50 426.80 426.60 427.20 433.30 435.00 442.50 447.00 450.60 448.90 446.20 443.60 446.30 448.20 452.40 451.30 451.80 459.90 464.70 467.30 463.50 466.60 461.10 464.90 467.30 458.40 458.80 463.20 462.40 461.10 465.50 461.50 458.20 460.80 459.30 445.70 425.10 437.60 438.00 436.60 437.60 437.60 438.60 443.10 446.40 445.90 450.80 451.60 457.30 456.70 455.60 454.75 453.90 451.20 450.70 446.80 443.40 448.40 451.80 449.80 449.10 447.60 448.40 450.00 443.00 440.60 437.40 435.40 432.00 430.80 429.60 437.10 440.00 438.30 435.20 436.60 435.25 433.90 436.50 436.30 437.40 441.00 445.40 450.10 449.20 450.50 455.60 452.00 451.80 456.80 455.30 457.40 457.40 461.10 459.60 462.40 463.40 464.60 469.00 472.20 471.80 470.10 465.20 470.40 468.50 468.70 469.70 472.50 474.70 472.40 475.00 476.10 473.20 471.50 472.20 471.10 472.80 470.40 470.50 472.10 471.10 468.50 465.50 465.70 465.40 466.90 468.85 470.80 474.00 478.10 480.50 481.00 479.10 476.40 469.80 471.60 470.60 467.20 473.10 471.70 474.80 477.20 474.60 475.10 475.90 475.80 472.00 470.80 469.10 464.30 463.70 467.20 467.30 467.10 465.60 462.70 449.45 436.20 466.00 467.40 467.00 471.50 469.80 474.20 476.10 477.10 480.30 478.70 478.80 479.30 479.30 478.30 477.20 480.20 484.10 488.70 492.70 492.60 491.90 491.90 495.10 494.50 494.50 496.90 496.20 498.40 498.00 496.00 497.90 495.40 497.30 495.20 499.20 500.60 497.90 499.60 497.00 498.10 496.70 491.40 487.60 486.70 487.40 489.30 485.30 501.80 485.40 491.30 495.50 501.80 504.50 502.50 505.80 510.30 511.90 509.90 508.70 510.70 512.90 512.90 513.80 516.10 512.10 511.10 505.30 505.10 505.20 508.40 510.70 511.30 514.90 517.30 519.70 521.80 524.40 526.80";
data = data.split(' ');
datax = [];
for (i = 0; i < data.length; i++) {
data[i] = parseFloat(data[i]);
datax[i] = i;
}
board = JXG.JSXGraph.initBoard(BOARDID, {
boundingbox: [-5, 550, data.length + 2, 380],
axis: false
});
// Create custom axes
board.create('axis', [[0, 0], [0, 1]]); // Vertical axis
board.create('axis', [[0, 400], [1, 400]]); // Horizontal axis
// The original data
board.create('curve', [datax, data], { strokeColor: 'gray', dash: 2 }); // plot the observed data
var alpha = board.create('slider', [[10, 520], [100, 520], [0, 0.1, 1.0]], { name: 'α' });
var gamma = board.create('slider', [[10, 510], [100, 510], [0, 0.1, 1.0]], { name: 'γ' });
// The double exponential smoothing
var estimate = board.create('curve', [[0], [0]]); // The filtered curve
estimate.updateDataArray = function() {
var t,
a = alpha.Value(), // Read the slider value of alpha
g = gamma.Value(), // Read the slider value of gamma
S = data[0], // Set the inital values for S and b
b = data[1] - data[0],
S_new;
this.dataX[0] = 0;
this.dataY[0] = S;
for (t = 1; t < data.length; t++) {
S_new = a * data[t] + (1 - a) * (S + b);
b = g * (S_new - S) + (1 - g) * b;
this.dataX[t] = t;
this.dataY[t] = S_new;
S = S_new;
}
}
board.update(); // This is necessary to trigger the first computation of the filtered curve.
This example is licensed under a
Creative Commons Attribution ShareAlike 4.0 International License.
Please note you have to mention
The Center of Mobile Learning with Digital Technology
in the credits.