Example 1: AutoCorrelation

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. This example computes the estimated autocovariances, estimated autocorrelations, and estimated standard errors of the autocorrelations using both Barlett and Moran formulas.

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

public class AutoCorrelationEx1
{
	public static void  Main(String[] args)
	{
		double[] x = 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};
		
		AutoCorrelation ac = new AutoCorrelation(x, 20);
		
		new PrintMatrix
			("AutoCovariances are: ").Print(ac.GetAutoCovariances());
		Console.Out.WriteLine();
		new PrintMatrix
			("AutoCorrelations are:  ").Print(ac.GetAutoCorrelations());
		Console.Out.WriteLine("Mean = " + ac.Mean);
		Console.Out.WriteLine();
		new PrintMatrix
			("Standard Error using Bartlett are:  ").Print
			(ac.GetStandardErrors(AutoCorrelation.StdErr.Bartletts));
		Console.Out.WriteLine();
		new PrintMatrix
			("Standard Error using Moran are:  ").Print
			(ac.GetStandardErrors(AutoCorrelation.StdErr.Morans));
		Console.Out.WriteLine();
		new PrintMatrix
			("Partial AutoCovariances:  ").
			Print(ac.GetPartialAutoCorrelations());
		ac.Mean = 50;
		new PrintMatrix
			("AutoCovariances are:  ").Print
			(ac.GetAutoCovariances());
		Console.Out.WriteLine();
		new PrintMatrix
			("AutoCorrelations are:  ").
			Print(ac.GetAutoCorrelations());
		Console.Out.WriteLine();
		new PrintMatrix
			("Standard Error using Bartlett are:  ").Print
			(ac.GetStandardErrors(AutoCorrelation.StdErr.Bartletts));
	}
}

Output

AutoCovariances are: 
          0        
 0  1382.908024    
 1  1115.02915024  
 2   592.00446848  
 3    95.29741072  
 4  -235.95179904  
 5  -370.0108088   
 6  -294.25541456  
 7   -60.44237232  
 8   227.63259792  
 9   458.38076816  
10   567.8407384   
11   546.12202864  
12   398.93728688  
13   197.75742912  
14    26.89107936  
15   -77.2807224   
16  -143.73279616  
17  -202.04799792  
18  -245.37223168  
19  -230.81567344  
20  -142.8788232   


 AutoCorrelations are:  
             0           
 0   1                   
 1   0.806293065691258   
 2   0.428086653780237   
 3   0.0689108813212006  
 4  -0.170620023128885   
 5  -0.267559955093586   
 6  -0.212780177317129   
 7  -0.043706718936501   
 8   0.164604293249802   
 9   0.331461500117813   
10   0.410613524938228   
11   0.394908424249623   
12   0.288477093166393   
13   0.143001143740562   
14   0.0194453129877855  
15  -0.0558827637549379  
16  -0.10393518127421    
17  -0.146103713633525   
18  -0.17743206881559    
19  -0.166906019369514   
20  -0.103317661565611   

Mean = 46.976

Standard Error using Bartlett are:  
            0           
 0  0.0347838253702384  
 1  0.0962419914340011  
 2  0.156783378574532   
 3  0.205766777086907   
 4  0.230955675779118   
 5  0.228994712235613   
 6  0.208621905639667   
 7  0.178475936561125   
 8  0.145727084432033   
 9  0.134405581638002   
10  0.150675803916788   
11  0.174348147103935   
12  0.190619474429408   
13  0.195490061669564   
14  0.195892530944597   
15  0.196285328179458   
16  0.196020624500033   
17  0.198716030900604   
18  0.205358590947539   
19  0.2093868822353     


Standard Error using Moran are:  
            0           
 0  0.0985184366143778  
 1  0.0980196058819607  
 2  0.0975182235357506  
 3  0.0970142500145332  
 4  0.0965076447241154  
 5  0.0959983659991659  
 6  0.0954863710632231  
 7  0.0949716159867634  
 8  0.094454055643212   
 9  0.0939336436627724  
10  0.0934103323839415  
11  0.0928840728025648  
12  0.0923548145182799  
13  0.0918225056781811  
14  0.0912870929175277  
15  0.090748521297303   
16  0.0902067342384192  
17  0.0896616734523426  
18  0.0891132788679007  
19  0.0885614885540095  


Partial AutoCovariances:  
             0            
 0   0.806293065691258    
 1  -0.634544877310468    
 2   0.0782508772709519   
 3  -0.0585660846582815   
 4  -0.00094221571933657  
 5   0.171719898229681    
 6   0.108591873581717    
 7   0.11000138764865     
 8   0.0785374339029981   
 9   0.0791563332964613   
10   0.0687065876031485   
11  -0.0378019674610775   
12   0.0811184838397538   
13   0.0334124214991749   
14  -0.0348467839607946   
15  -0.130648157884444    
16  -0.154900984829049    
17  -0.119085063160732    
18  -0.0161889037437313   
19  -0.00385175459253345  

AutoCovariances are:  
        0      
 0  1392.0526  
 1  1126.5241  
 2   604.1624  
 3   106.7545  
 4  -225.882   
 5  -361.0259  
 6  -286.5701  
 7   -53.7603  
 8   235.9665  
 9   470.7857  
10   584.0143  
11   564.7639  
12   418.3631  
13   216.1044  
14    43.125   
15   -63.4683  
16  -131.5012  
17  -189.0627  
18  -229.6888  
19  -212.1559  
20  -121.5693  


 AutoCorrelations are:  
             0           
 0   1                   
 1   0.809253975029392   
 2   0.434008312616923   
 3   0.0766885532917362  
 4  -0.162265420142888   
 5  -0.259347886710603   
 6  -0.205861545749061   
 7  -0.0386194458456527  
 8   0.16950975846746    
 9   0.338195338308337   
10   0.419534649768263   
11   0.405705861976767   
12   0.300536847530043   
13   0.155241547625427   
14   0.0309794328174093  
15  -0.04559332025241    
16  -0.0944656832651295  
17  -0.135815773053403   
18  -0.165000086922003   
19  -0.152405088715757   
20  -0.0873309672350025  


Standard Error using Bartlett are:  
            0           
 0  0.0344591054641365  
 1  0.0972222809088609  
 2  0.15947410033087    
 3  0.209799660647689   
 4  0.235599778243579   
 5  0.233236443705991   
 6  0.211657508693781   
 7  0.180412936841618   
 8  0.14689653606348    
 9  0.133747601649498   
10  0.148150190923942   
11  0.172282351100035   
12  0.190275929042947   
13  0.196791614240352   
14  0.197983743593071   
15  0.198474748794747   
16  0.198318159677368   
17  0.201022833791806   
18  0.207071652966429   
19  0.210217650328868   


Link to C# source.