ARSeasonalFit Example

Example: ARSeasonalFit

Consider the Airline Data (Box, Jenkins and Reinsel 1994, p. 547) consisting of the monthly total number of international airline passengers from January 1949 through December 1960. Class ARSeasonalFit is used to compute the optimum seasonality representation of the adjusted series

$W_t(s,d) = \Delta_{s_1}^{d_1}\Delta_{s_2}^{d_2}Z_t = {(1-B^{s_1})}^{d_1}{(1-B^{s_2})}^{d_2}Z_t\rm{,}$ where

and $d=(1,1)\rm{.}$

As differenced series with minimum AIC,

$W_t = \Delta_1^1\Delta_{12}^2Z_t = (Z_t - Z_{t-12}) - (Z_{t-1} - Z_{t-13})\rm{,}$ is identified.

using System;
using Imsl.Stat;
using Imsl.Math;
using PrintMatrix = Imsl.Math.PrintMatrix;
using Imsl;

public class ARSeasonalFitEx1
{
    
    public static void  Main(String[] args)
    {
        double[] z = 
            new double[]{112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 
                            104, 118, 115, 126, 141, 135, 125, 149, 170, 
                            170, 158, 133, 114, 140, 145, 150, 178, 163, 
                            172, 178, 199, 199, 184, 162, 146, 166, 171, 
                            180, 193, 181, 183, 218, 230, 242, 209, 191, 
                            172, 194, 196, 196, 236, 235, 229, 243, 264, 
                            272, 237, 211, 180, 201, 204, 188, 235, 227, 
                            234, 264, 302, 293, 259, 229, 203, 229, 242, 
                            233, 267, 269, 270, 315, 364, 347, 312, 274, 
                            237, 278, 284, 277, 317, 313, 318, 374, 413, 
                            405, 355, 306, 271, 306, 315, 301, 356, 348, 
                            355, 422, 465, 467, 404, 347, 305, 336, 340, 
                            318, 362, 348, 363, 435, 491, 505, 404, 359, 
                            310, 337, 360, 342, 406, 396, 420, 472, 548, 
                            559, 463, 407, 362, 405, 417, 391, 419, 461, 
                            472, 535, 622, 606, 508, 461, 390, 432};
        
        int[,] sInit = {{1, 1}, {1, 12}};
        
        ARSeasonalFit seasFit = new ARSeasonalFit(10, sInit, z);
        seasFit.Compute();
        Console.Out.WriteLine("NLost = " + seasFit.NLost);
        Console.Out.WriteLine("aic =" + seasFit.AIC);
        new PrintMatrix("Best Periods (Optimum S)").Print(seasFit.GetOptimumS());
        new PrintMatrix("Best Orders (Optimum D)").Print(seasFit.GetOptimumD());
        Console.Out.WriteLine("Best AR order selected = " + seasFit.AROrder);
        new PrintMatrix("AR Coefficients").Print(seasFit.GetAR());
        
        Console.Out.WriteLine("");
        double[] w = seasFit.GetTransformedTimeSeries();
        double[][] pack = new double[30][];
        for (int i = 0; i < 30; i++)
        {
            pack[i] = new double[2];
            pack[i][0] = z[i];
            pack[i][1] = w[i];
        }
        PrintMatrix pm = new PrintMatrix();
        String str = "First 30 elements of the original time series and " +
         "differenced series";
        pm.SetTitle(str);
        PrintMatrixFormat fmt = new PrintMatrixFormat();
        fmt.SetColumnLabels(new String[]{"Original", "Differenced"});
        pm.Print(fmt, pack);
    }
}

Output

NLost = 13
aic =606.397245653491
Best Periods (Optimum S)
   0   
0   1  
1  12  

Best Orders (Optimum D)
   0  
0  1  
1  1  

Best AR order selected = 1
    AR Coefficients
           0           
0  -0.309834454941166  


First 30 elements of the original time series and differenced series
    Original  Differenced  
 0    112          NaN     
 1    118          NaN     
 2    132          NaN     
 3    129          NaN     
 4    121          NaN     
 5    135          NaN     
 6    148          NaN     
 7    148          NaN     
 8    136          NaN     
 9    119          NaN     
10    104          NaN     
11    118          NaN     
12    115          NaN     
13    126          5       
14    141          1       
15    135         -3       
16    125         -2       
17    149         10       
18    170          8       
19    170          0       
20    158          0       
21    133         -8       
22    114         -4       
23    140         12       
24    145          8       
25    150         -6       
26    178         13       
27    163         -9       
28    172         19       
29    178        -18

Link to C# source.