Power Comparison of IMA, ARMA(1,1) and Fractional Noise

Fractional Noise

?TacvfARMA

f = 0.2

cs = 
      TacvfARMA[{}, {}, 20,  FractionalDifference -> 0.2]

rs = cs/First[cs]

sdFD = First[cs] // Sqrt

f = 0.4

cs2 = 
      TacvfARMA[{}, {}, 20,  FractionalDifference -> 0.4]

rs2 = cs2/First[cs2]

sdFD2 = First[cs2] // Sqrt;

Initialization

g = TacvfARMA[{ϕ}, {θ}, 2]

g = Drop[g, 1]/First[g] // FullSimplify

ARMA for f = 0.2

sol = 
      Solve[{g[[1]] == 
            rs[[2]], 
          g[[2]] == 
            rs[[3]]}, {ϕ, θ}]

{phi, theta} = {{ϕ}, {θ}} /. 
        sol[[3]]

{phi, theta} = {{0.666667}, {0.451416}}

g

g0 = TacvfARMA[phi, theta, 7]

%/First[%]

sdARMA = First[g0] // Sqrt;

ARMA for f = 0.4

rs2

sol2 = 
      Solve[{g[[1]] == 
            rs2[[2]], 
          g[[2]] == 
            rs2[[3]]}, {ϕ, θ}]

{phi2, theta2} = {{ϕ}, {θ}} /. 
        sol2[[4]]

{phi2, theta2} = {{0.875}, {0.404831}}

g02 = TacvfARMA[phi2, theta2, 7]

%/First[%]

sdARMA2 = First[g02] // Sqrt;

Power Computations

?PowerIA

PowerIA[50, 25, 1.0/g0ARMA, phi, theta]

PowerIA[50, 25, 1.0/c0FD, {}, {}, 
      FractionalDifference -> 0.2]

PowerIA[50, 25, 1.0, {}, {}, 
      FractionalDifference -> -0.4]

PowerIA[50, 25, 1.0, {}, {}, FractionalDifference -> 0]

PowerIA[50, 25, 1.0, {}, {}]

Plot[{PowerIA[50, 25, del/g0ARMA, phi, theta], 
          PowerIA[50, 25, del/c0FD, {}, {}, 
            FractionalDifference -> 0.2]}, {del, -2, 2}, 
        PlotStyle -> {Dashing[{}], Dashing[{0.15, 0.05}]}];

f = 0.2

tf = 
      Table[{del, PowerIA[50, 25, del/sdARMA, phi, theta], 
          PowerIA[50, 25, del/sdFD, {}, {}, 
            FractionalDifference -> 0.2]}, {del, 0, 3, 0.5}]

TableForm[tf, TableHeadings -> {None, cn}]

f = 0.4

tf2 = 
      Table[{del, PowerIA[50, 25, del/sdARMA2, phi2, theta2], 
          PowerIA[50, 25, del/sdFD2, {}, {}, 
            FractionalDifference -> 0.4]}, {del, 0, 3, 0.5}]

TableForm[tf2, TableHeadings -> {None, cn}]

Combined Table

<< d:/math/2003/writetable.m

?MergeMatricies

tf

{x1, y1} = Drop[Transpose[tf], 1]

{x2, y2} = Drop[Transpose[tf2], 1]

xy = 
      MergeMatricies[Transpose[{y1, y2}], Transpose[{x1, x2}], 
        NumDecimals -> 3]

comCN = {"\delta", "f=0.2", "f=0.4"};

rowHeadings = Take[Transpose[tf], 1] // Flatten;

WriteTable["e:/p/2003/tex/technometrics/fractional.tex", xy, 
      NumDecimals -> False, ColumnNames -> comCN, 
      PrependColumn -> rowHeadings]