Table 6.

Comparison of Exact and Approximate Methods.

The function g(T,ϕ) defined in Tiao et al. (1990) was computed using exact form of the information matrix and the approximation eqn. in Table 1 of our paper for selected parameter values given in Table 1 of Tiao et al. (1990).

Need to Initialize Code.m.

Reference

Tiao,G.C.,Reinsel,G.C.,Xu,D.,Pedrick,J.H.,Zhu,X.,Miller,A.J.,DeLuisi,J.J.,Mateer,C.L.and Wuebbles,D.J.(1990),``Effects of autocorrelation and temporal sampling schemes on estimation of trend and spatial correlation, Journal of Geophysical Research 95 D12,20,507--20,517.

Tiao et a. (1990), Table 1.  p.20,509

Let (Subscript[ℛ, t]^((T))) denote the ramp intervention defined in Tiao (1990) so we have (Subscript[ℛ, t]^((T))) = 
        (Subscript[R, t]^((T + 1)))/12.   Then assuming T = 0 the following expression was derived for \(TraditionalForm\`σ\_ω\&^\),

\(TraditionalForm\`\(\(σ\_\(ω\&^\)\%\((Tiao)\) = σ\ \(g(T, ϕ)\)\)\(,\)\)\)

where σ = (Subscript[σ, a]^2) /(1 - (ϕ^2)) and  \(TraditionalForm\`g(
          T, ϕ) = \((\((1 + ϕ)\) \((1 - ϕ)\))\)\^\(1/2\)\ 12/\((\(T(T\^2 - 1)\)/12)\)\^\(1/2\)\)

𝔤[
        T_, ϕ_] := ((((1 + ϕ)/(1 - ϕ)))^(1/2)) * 
        12/(((T ((T^2) - 1)/12))^(1/2))

((((1 + ϕ)/(1 - ϕ)))^(1/2)) * 
        12/(((T ((T^2) - 1)/12))^(1/2)) // Simplify

% /. {T -> 6*12, ϕ -> 0.6}

((((1 + ϕ) (1 - ϕ)))^(1/2)) 12/(((T((T^2) - 1)/12))^(1/2)) // Simplify

t = 
      Table[𝔤[12*n, ϕ], {n, 6, 15}, {ϕ, 0.0, 0.8, 
          0.2}]

𝔤[T_, ϕ_] := 
      24 (((3 (1 + ϕ)/((1 - ϕ) T ((T^2) - 1))))^(1/2))

\(TeXForm[
        24\ \((3\ \((1 + ϕ)\)/\((\((1 - ϕ)\) T \((T\^2 - 1)\))\))\)\^\(1/2\)] >> "e:/p/2003/tex/technometrics/gTiao.tex"\)

tTiao = 
        Table[𝔤[12*n, ϕ], {n, 6, 15}, {ϕ, 0.0, 0.8, 
            0.2}];

nyrs = Table[i, {i, 6, 15}];  
    phis = Table[ϕ, {ϕ, 0.0, 0.8, 0.2}];

MySy = 
        StyleForm[#, FontFamily -> Helvetica, 
            FontColor -> RGBColor[0, 0, 1]] &;

TableForm[tTiao, 
      TableHeadings -> {MySy /@ nyrs, MySy /@ phis}]

Remark:  Comparing with Tiao et al. (1990, Table 1) we see that columns 1 and 2 are completely accurate to 3 decimals (the accuracy shown in the published table).  The accuracy decreases as as ϕ increases.

Numerical and Symbolic Comparison With Our Result

Let \(TraditionalForm\`\(σ\_ω\&^\)(T, ϕ)\) denote our estimate.  Then taking into account (Subscript[ℛ, t]^((T))) = 
        (Subscript[R, t]^((T + 1)))/12 and

\(TraditionalForm\`\(\(σ\_\(ω\&^\)\%\((Tiao)\) = σ\ \(g(T, ϕ)\)\)\(,\)\)\)

where σ = (Subscript[σ, a]^2) /(1 - (ϕ^2)).  We obtain, \(TraditionalForm\`𝔤(
          T, ϕ) = \@\(\((1 - ϕ\^2)\)/12\)\ \ \(\(σ\_ω\&^\)(T + 1, ϕ)\)\)as the corresponding value in our notation.  In order to compare with our result we must assume that the mean is known.

Definition of Function

InformationMatrixAR1Ramp[n_, 
        T_,  ϕ_] := 
With[{κ = ϕ - 
              1, 
b12 = (1 - ϕ) (n + 1 - 
                  T) (n + 2 - T - ϕ (n - T))/
                2, 
b22 = 
            (1/6) (1 + n - T) (6 + 7 n + 2 (n^2) - 7 T - 4 n T + 
                  2 (T^2) - 8 n ϕ - 4 (n^2) ϕ + 8 T ϕ + 
                  8 n T ϕ - 4 (T^2) ϕ + n (ϕ^2) + 
                  2 (n^2) (ϕ^2) - T (ϕ^2) - 4 n T (ϕ^2) + 
                  2 (T^2) (ϕ^2))}, {{n (κ^2), b12}, {b12, 
            b22}}
]

mygPierce[
        T_, ϕ_] := (Sqrt[Inverse[
                  InformationMatrixAR1Ramp[T, 1, ϕ]] // Flatten // 
              Last])*Sqrt[1 - (ϕ^2)]*12

Symbolic Expression

gsym = mygPierce[T, ϕ] // FullSimplify

Denominator[Part[gsym, 1]^2] // FullSimplify

Denominator[Part[gsym, 1]^2] // FullSimplify >> 
      e:/p/2003/tex/technometrics/myg.tex

gsym

Is Tiao's expression identical to ours

gT = 𝔤[T, ϕ] // FullSimplify

gT^2 - gsym^2 // FullSimplify

So we see that it is identical.

Comparison Table

t = 
      Table[mygPierce[12*n, ϕ], {n, 6, 15}, {ϕ, 0.0, 0.8, 
          0.2}]

TableForm[t, 
      TableHeadings -> {MySy /@ nyrs, MySy /@ phis}]

OK.  So this is just exactly what Tiao et al have produced.

tTiao/t

Comparison Using SEOmega

Consistency

We show that the estimate is consistent, even in the case of unknown mean.

Table[InformationMatrixAR1Ramp[m + T, 
              1,  ϕ] /. {ϕ -> 0.8, m -> 0} // Flatten // 
        Last, {T, 12, 12*10, 12}]

Limit[
      InformationMatrixAR1Ramp[m + T, 1,  ϕ] /. {ϕ -> 0.8, 
          m -> 0}, T -> ∞]

Table[InformationMatrixAR1Ramp[m + T, 
              1,  ϕ] /. {ϕ -> 0.8, m -> 0} // Flatten // 
        Last, {T, 120, 120*10, 120}]

Computation Using General Purpose Function

<< d:/math/2004/poweria/iapower.m

?Se*

?IAPower`*

?SdωARMA

Options[SdωARMA]

SdωARMA[50, 100, {0.5}, {}]

Options[SdωARMA]

nyrs = Table[i, {i, 6, 15}];  
    phis = Table[ϕ, {ϕ, 0.0, 0.8, 0.2}];

\(myg[
        T_?NumericQ, ϕ_?
          NumericQ] := 
Module[{}, 
SdωARMA[Round[T], T + 30*12, {ϕ}, {}, 
            InterventionType -> "Ramp"]*
          Sqrt[\((1 - ϕ\^2)\)]*12
]\)

myg[12, 0.5]

SdωARMA[12, 1, {0.5}, {}]

texact = 
      Table[myg[12*n, ϕ], {n, 6, 15}, {ϕ, 0. , 0.8, 
          0.2}]

texact = 
      Table[myg[12*n, ϕ,  PierceApproximation -> False], {n, 6, 
          15}, {ϕ, 0. , 0.8, 0.2}]

MySy = 
        StyleForm[#, FontFamily -> Helvetica, 
            FontColor -> RGBColor[0, 0, 1]] &;

TableForm[texact, 
      TableHeadings -> {MySy /@ nyrs, MySy /@ phis}]

percent = (texact - tTiao)/texact*100

TableForm[Round[percent] // Chop, 
      TableHeadings -> {MySy /@ nyrs, MySy /@ phis}]

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

per = 
        Take[Transpose[Take[Transpose[percent], -2]], 5];

phipars = {0.6,  0.8}; npars = Range[6, 10];

TableForm[Round[per] // Chop, 
      TableHeadings -> {MySy /@ npars, MySy /@ phipars}]

cn = Join[{Number of Years}, ToString /@ phipars];  
    rn =  ToString /@ npars;

WriteTable["e:/p/2003/tex/Technometrics/tableTiao.tex", Round[per], 
      NumDecimals -> False,  ColumnNames -> cn, 
      PrependColumn -> rn]