Consider the Wolfer Sunspot Data (Anderson 1971, p. 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. An ARMA(2,1) model is fitted to these data using the Method of Moments. With BackwardOrigin
= 3, the Forecast
method is used to obtain forecasts starting from 1866, 1867, 1868, and 1869, respectively. Note that the values in the first row of the matrix returned by this method are the one-step ahead forecasts for 1867, 1868, ..., 1870. The values in the second row are the two-step ahead forecasts for 1868, 1869, ..., 1871, etc.
Method GetForecast
is used to compute the one-step ahead forecasts setting BackwardOrigin
= 10. This obtains the one-step ahead forecasts for the last 10 observations in the series, i.e. years 1860-1869, plus the next 5 years. The upper 90% confidence limits are computed for these forecasts using the GetDeviations
method.
using System;
using Imsl.Stat;
using PrintMatrix = Imsl.Math.PrintMatrix;
using PrintMatrixFormat = Imsl.Math.PrintMatrixFormat;
public class ARMAEx3
{
public static void Main(String[] args)
{
double[] z = new double[]{ 100.8, 81.6, 66.5, 34.8, 30.6,
7, 19.8, 92.5, 154.4, 125.9,
84.8, 68.1, 38.5, 22.8, 10.2,
24.1, 82.9, 132, 130.9, 118.1,
89.9, 66.6, 60, 46.9, 41,
21.3, 16, 6.4, 4.1, 6.8,
14.5, 34, 45, 43.1, 47.5,
42.2, 28.1, 10.1, 8.1, 2.5,
0, 1.4, 5, 12.2, 13.9,
35.4, 45.8, 41.1, 30.4, 23.9,
15.7, 6.6, 4, 1.8, 8.5,
16.6, 36.3, 49.7, 62.5, 67,
71, 47.8, 27.5, 8.5, 13.2,
56.9, 121.5, 138.3, 103.2, 85.8,
63.2, 36.8, 24.2, 10.7, 15,
40.1, 61.5, 98.5, 124.3, 95.9,
66.5, 64.5, 54.2, 39, 20.6,
6.7, 4.3, 22.8, 54.8, 93.8,
95.7, 77.2, 59.1, 44, 47,
30.5, 16.3, 7.3, 37.3, 73.9};
double[,] printEstimates = new double[1,4];
PrintMatrixFormat pmf = new PrintMatrixFormat();
PrintMatrix pm = new PrintMatrix();
pm.SetColumnSpacing(3);
ARMA arma = new ARMA(2, 1, z);
arma.RelativeError = 0.0;
arma.MaxIterations = 0;
arma.Compute();
printEstimates[0,0] = arma.Constant;
double[] ar = arma.GetAR();
printEstimates[0,1] = ar[0];
printEstimates[0,2] = ar[1];
double[] ma = arma.GetMA();
printEstimates[0,3] = ma[0];
String[] estimateLabels = {"Constant", "AR(1)", "AR(2)", "MA(1)"};
pmf.SetColumnLabels(estimateLabels);
pmf.NumberFormat = "0.0000";
pm.SetTitle("ARMA ESTIMATES");
pm.Print(pmf, printEstimates);
String[] labels = new String[]{"From 1866",
"From 1867",
"From 1868",
"From 1869"};
pmf.SetColumnLabels(labels);
pmf.FirstRowNumber = 1;
pmf.NumberFormat = "00.0";
arma.BackwardOrigin = 3;
new PrintMatrix("FORECASTS").Print(pmf, arma.Forecast(5));
double[,] printTable = new double[15,4];
/* FORECASTING - An example of forecasting using the ARMA estimates
* In this case, forecasts are returned for the last 10 values in the
* series followed by the forecasts for the next 5 values.
*/
String[] forecastLabels={"Observed", "Forecast",
"Residual", "UCL(90%)"};
pmf.SetColumnLabels(forecastLabels);
int backOrigin = 10;
int n_forecast = 5;
arma.BackwardOrigin = backOrigin;
arma.Confidence = 0.9;
double[] forecasts = arma.GetForecast(n_forecast);
double[] deviations = arma.GetDeviations();
for(int i=0; i<backOrigin; i++)
{
printTable[i,0] = z[z.Length-backOrigin+i];
printTable[i,1] = forecasts[i];
printTable[i,2] = z[z.Length-backOrigin+i]-forecasts[i];
printTable[i,3] = forecasts[i] + deviations[0];
}
for(int i=backOrigin; i<n_forecast+backOrigin; i++)
{
printTable[i,0] = Double.NaN;
printTable[i,1] = forecasts[i];
printTable[i,2] = Double.NaN;
printTable[i,3] = forecasts[i] + deviations[i-backOrigin];
}
pmf.FirstRowNumber = 1869-backOrigin+1;
pmf.NumberFormat = "000.0";
pm.SetTitle("ARMA ONE-STEP AHEAD FORECASTS");
pm.Print(pmf, printTable);
}
}
ARMA ESTIMATES
Constant AR(1) AR(2) MA(1)
0 15.5440 1.2443 -0.5751 -0.1241
FORECASTS
From 1866 From 1867 From 1868 From 1869
1 18.3 16.6 55.2 83.7
2 28.9 32.0 62.8 77.2
3 41.0 45.8 61.9 63.5
4 49.9 54.1 56.5 50.1
5 54.1 56.6 50.2 41.4
ARMA ONE-STEP AHEAD FORECASTS
Observed Forecast Residual UCL(90%)
1860 095.7 100.7 -005.0 128.6
1861 077.2 081.3 -004.1 109.2
1862 059.1 057.1 002.0 084.9
1863 044.0 044.4 -000.4 072.3
1864 047.0 036.4 010.6 064.2
1865 030.5 047.4 -016.9 075.3
1866 016.3 028.6 -012.3 056.4
1867 007.3 019.8 -012.5 047.7
1868 037.3 016.8 020.5 044.7
1869 073.9 055.2 018.7 083.1
1870 NaN 083.7 NaN 111.6
1871 NaN 077.2 NaN 124.5
1872 NaN 063.5 NaN 120.2
1873 NaN 050.1 NaN 109.4
1874 NaN 041.4 NaN 100.8
Link to C# source.