Example 2: Determination of an optimum model via Grid search

This is the same as Example 1, except now class AutoARIMA uses Method 2 with a possible seasonal adjustment. As a result, the unadjusted model with p = 3, q = 2, s = 1, d = 0 is chosen as optimum.

using System;
using Imsl.Stat;

public class AutoARIMAEx2
{
   public static void Main(String[] args)
   {
      int nOutliers;
      double aic, RSE, constant;
      int[] optimumModel;
      int[,] outlierStatistics;
      double[] outlierForecast, ar, ma;
      double[] psiWeights, probabilityLimits;
      int[] arOrders = { 0, 1, 2, 3 };
      int[] maOrders = { 0, 1, 2, 3 };
      int[] periods =  { 1, 2 };
      int[] orders =  { 0, 1, 2 };

      double[] x = 
      {
            12.8, 12.2, 11.9, 10.9, 10.6, 11.3, 11.1, 10.4, 10.0, 9.7, 9.7, 
            9.7, 11.1, 10.5, 10.3, 9.8, 9.8, 10.4, 10.4, 10.0, 9.7, 9.3, 9.6,
            9.7, 10.8, 10.7, 10.3, 9.7, 9.5, 10.0, 10.0, 9.3, 9.0, 8.8, 8.9, 
            9.2, 10.4, 10.0, 9.6, 9.0, 8.5, 9.2, 9.0, 8.6, 8.3, 7.9, 8.0, 
            8.2, 9.3, 8.9, 8.9, 7.7, 7.6, 8.4, 8.5, 7.8, 7.6, 7.3, 7.2, 7.3,
            8.5, 8.2, 7.9, 7.4, 7.1, 7.9, 7.7, 7.2, 7.0, 6.7, 6.8, 6.9, 7.8, 
            7.6, 7.4, 6.6, 6.8, 7.2, 7.2, 7.0, 6.6, 6.3, 6.8, 6.7, 8.1, 7.9, 
            7.6, 7.1, 7.2, 8.2, 8.1, 8.1, 8.2, 8.7, 9.0, 9.3, 10.5, 10.1,
            9.9, 9.4, 9.2, 9.8, 9.9, 9.5, 9.0, 9.0, 9.4, 9.6, 11.0, 10.8, 
            10.4, 9.8, 9.7, 10.6, 10.5, 10.0, 9.8, 9.5, 9.7, 9.6, 10.9, 10.3,
            10.4, 9.3, 9.3, 9.8, 9.8, 9.3, 8.9, 9.1, 9.1, 9.1, 10.2, 9.9, 9.4
      };

      double[] exactForecast = { 8.7, 8.6, 9.3, 9.1, 8.8, 8.5 };

      int[] times = 
      {
         1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 
         20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
         37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
         54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
         71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 
         88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 
         104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 
         117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 
         130, 131, 132, 133, 134, 135
      };

      AutoARIMA autoArima = new AutoARIMA(times, x);
      autoArima.CriticalValue = 3.8;
      autoArima.MaximumARLag = 5;
      autoArima.SetPeriods(periods);
      autoArima.SetDifferenceOrders(orders);
      autoArima.Compute(arOrders, maOrders);
      autoArima.Forecast(6);

      nOutliers = autoArima.NumberOfOutliers;
      aic = autoArima.AIC;
      optimumModel = autoArima.GetOptimumModelOrder();
      outlierStatistics = autoArima.GetOutlierStatistics();
      RSE = autoArima.ResidualStandardError;
      outlierForecast = autoArima.GetForecast();
      psiWeights = autoArima.GetPsiWeights();
      probabilityLimits = autoArima.GetDeviations();
      constant = autoArima.Constant;
      ar = autoArima.GetAR();
      ma = autoArima.GetMA();

      Console.Out.WriteLine("\nMethod 2: Grid search with " +
                            "differencing");
      Console.Out.WriteLine();
      Console.Out.WriteLine(
         "Optimum Model: p={0,1:d}, q={1,1:d}, s={2,1:d}, d={3,1:d}",
         optimumModel[0], optimumModel[1], optimumModel[2], optimumModel[3]);
      Console.Out.WriteLine("\nNumber of outliers:{0,2:d}", nOutliers);
      Console.Out.WriteLine();
      Console.Out.WriteLine("Outlier statistics:");
      Console.Out.WriteLine(" Time    Type");
      for (int i = 0; i < nOutliers; i++)
         Console.Out.WriteLine("{0,5:d}{1,8:d}", outlierStatistics[i, 0], 
            outlierStatistics[i, 1]);
      Console.Out.WriteLine("\nAIC:{0,12:f6}", aic);
      Console.Out.WriteLine("RSE:{0,12:f6}", RSE);
      Console.Out.WriteLine();
      Console.Out.WriteLine("     Parameters");
      Console.Out.WriteLine(" constant:{0,11:f6}", constant);
      for (int i = 0; i < ar.Length; i++)
         Console.Out.WriteLine(" ar[{0,1:d}]: {1,13:f6}", i, ar[i]);
      for (int i = 0; i < ma.Length; i++)
         Console.Out.WriteLine(" ma[{0,1:d}]: {1,13:f6}", i, ma[i]);
      Console.Out.WriteLine("\n\n      * * * Forecast Table * * *");
      Console.Out.WriteLine("  Exact   forecast     limits        psi");
      for (int i = 0; i < outlierForecast.Length; i++)
         Console.Out.WriteLine("{0,7:f4}{1,11:f4}{2,11:f4}{3,11:f4}",
            exactForecast[i], outlierForecast[i],
            probabilityLimits[i], psiWeights[i]);
   }
}

Output

Method 2: Grid search with differencing

Optimum Model: p=3, q=2, s=1, d=0

Number of outliers: 1

Outlier statistics:
 Time    Type
  109       0

AIC:  408.108176
RSE:    0.412456

     Parameters
 constant:   0.554459
 ar[0]:      1.940615
 ar[1]:     -1.898025
 ar[2]:      0.897791
 ma[0]:      1.115803
 ma[1]:     -0.911902


      * * * Forecast Table * * *
  Exact   forecast     limits        psi
 8.7000     9.1085     0.8084     0.8248
 8.6000     9.1715     1.0479     0.6145
 9.3000     9.5039     1.1597     0.5248
 9.1000     9.7677     1.2349     0.5926
 8.8000     9.7051     1.3245     0.7056
 8.5000     9.3817     1.4421     0.7157