# Fractional Gaussian Noise versus FARIMA Noise

A FARIMAProcess with no autoregressive and no moving-average components and a discrete time restriction of a FractionalGaussianNoiseProcess are frequently used to model colored noises.

 In[1]:= XFGN[h_] := FractionalGaussianNoiseProcess[h]; FARIMA[h_] := FARIMAProcess[{}, h - 1/2, {}, 1];
 In[2]:= Xrange = {0.1, 0.5, 0.9};

Simulate both processes for integer times.

 In[3]:= Xn = 1000; FGNsamples = RandomFunction[FGN[#], {0, n, 1}, n] & /@ range; FARIMAsamples = RandomFunction[FARIMA[#], {0, n}, n] & /@ range;

FractionalGaussianNoiseProcess simulation.

 In[4]:= XMatrixPlot[#["ValueList"], AspectRatio -> 1, ImageSize -> 150, Frame -> False, MaxPlotPoints -> 256] & /@ FGNsamples
 Out[4]=
 In[5]:= XListPlot[PeriodogramArray[Flatten[#["ValueList"]], n], ImageSize -> 150] & /@ FGNsamples
 Out[5]=

FARIMA noise simulation.

 In[6]:= XMatrixPlot[#["ValueList"], AspectRatio -> 1, ImageSize -> 150, Frame -> False, MaxPlotPoints -> 256] & /@ FARIMAsamples
 Out[6]=
 In[7]:= XListPlot[PeriodogramArray[Flatten[#["ValueList"]], n], ImageSize -> 150] & /@ FARIMAsamples
 Out[7]=

Compare periodogram plots.

 Out[8]=

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