Exercise SOME HINTS for a few exercises from Chap. 10

Transcript

1 SOME HINTS for a few exercises from Chap. 10 Exercise 10.8 data = {4, 10, 5, 11, 2, 6, 2, 8, 8, 4, 5, 5, 5, 3, 4, 4, 8, 4, 7, 1, 4, 6, 7, 5, 6, 7, 8, 3, 6, 4, 6, 5, 5, 7, 9, 5, 8, 6, 6, 5, 4, 2, 8, 4, 5, 8, 6, 6, 1, 3, 5, 5, 7, 9, 4, 6, 9, 7, 6, 6, 6, 9, 5, 3, 6, 8, 4, 6, 4, 6}; n = 100; p 0 = 0.05; k = n p 0 ; LCL = 3; CL = n p 0 ; UCL = 17; z[0] = 0; Do[z[i] = z[i - 1] + data[[i]] - k, {i, 1, Length[data]}] cusum = Table[z[i], {i, 1, Length[data]}]; list1 = Table[LCL, {i, 1, Length[data]}]; list2 = Table[CL, {i, 1, Length[data]}]; list3 = Table[UCL, {i, 1, Length[data]}]; TableForm[Transpose[{data, list1, cusum, list3}], TableHeadings {Automatic, {"y N ", "LCL", "z N ", "UCL"}}] G1 = ListPlot[cusum, Joined True, PlotStyle {Black, Thickness[0.002]}, PlotMarkers Automatic, PlotRange {- 2, 45}, DisplayFunction Identity]; G2 = ListPlot[list1, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, Joined True, DisplayFunction Identity]; G3 = ListPlot[list2, PlotStyle RGBColor[0, 1, 0], Joined True, DisplayFunction Identity]; G4 = ListPlot[list3, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, Joined True, DisplayFunction Identity]; Show[G1, G2, G3, G4, DisplayFunction $DisplayFunction] numberobsbeyondcontrollimits = Length[cusum] Mean[ Table[If[cusum[[i]] < LCL cusum[[i]] > UCL, 1, 0], {i, 1, Length[cusum]}]]; If[numberobsbeyondcontrollimits > 0, Print["There is (are) ", numberobsbeyondcontrollimits, " observation(s) beyond the control limits."], Print["There are no observations beyond the control limits."]] y N LCL z N UCL

2 2 Ex-Cap Version7-0.nb

3 Ex-Cap Version7-0.nb There is (are) 37 observation(s) beyond the control limits. Exercise n = 10; p = 0.5; χ[z_] = PDF[BinomialDistribution[n, p], z]; k = 6; LCL = 2; UCL = 10; (* The (infinite) probability transition matrix of the original Markov chain has entries P=[p ij ] i,j=...,lcl,...,ucl,... = [P Bin(n,p) (k+j- i)] i,j=...,lcl,...,ucl,... * ) (* Q block of the probability transition matrix of the absorbing Markov chain * ) Q = Table[χ[k + j - i], {i, LCL, UCL}, {j, LCL, UCL}]; MatrixForm[Round[N[Q, 5] ] * ]

4 4 Ex-Cap Version7-0.nb Exercise n = 80; λ = ; dist = PoissonDistribution[λ]; χ[z_] = PDF[dist, z]; β[z_] = CDF[dist, z]; k = 2; LCL = 0; UCL = 2; Q = Table[If[j == 0, β[k - i], χ[k + j - i]], {i, LCL, UCL}, {j, LCL, UCL}]; (* Check Equation (10.8). * ) MatrixForm[Round[N[Q, 5] ] * ] Example CUSUM chart Parameters n = 100; p 0 = 0.02; x = 6; k = 3; p 1 = FindRoot n Log 1- p 0 1- z Log (1- p 0) z (1- z) p 0 p 1 = {z} /. Dispatch[p 1 ]; p 1 = p 1 [[1]] == k, {z, 1.1 p 0, 0, 1} ; theta = {0, 0.001, , 0.005, , 0.01, p 1 - p 0, 0.03}; alpha = {0.05, 0.25, 0.5, 0.75, 0.9, 0.95}; χ[z_, δ_] = PDF[BinomialDistribution[n, δ], z]; β[z_, δ_] = CDF[BinomialDistribution[n, δ], z]; Q[δ_] = Table[If[j == 0, β[k - i, δ], χ[k + j - i, δ]], {i, 0, x}, {j, 0, x}]; Factorial moments E[RL (s) ]=E[RL (RL - 1)... (RL - s + 1)] ID = IdentityMatrix[x + 1]; FM[i_, δ_, s_] := Apply[Plus, Factorial[s] Table[If[j == i, 1, 0], {j, 0, x}]. MatrixPower[N[Q[δ], 10], s - 1].MatrixPower[Inverse[ID - N[Q[δ], 10]], s]];

5 Ex-Cap Version7-0.nb 5 4 Noncentral moments (in terms of the factorial moments) E[RL s ], s=1,...,4 NM[i_, δ_, 1] := FM[i, δ, 1]; NM[i_, δ_, 2] := FM[i, δ, 2] + FM[i, δ, 1]; NM[i_, δ_, 3] := FM[i, δ, 3] + 3 FM[i, δ, 2] + FM[i, δ, 1]; NM[i_, δ_, 4] := FM[i, δ, 4] + 6 FM[i, δ, 3] + 7 FM[i, δ, 2] + FM[i, δ, 1]; 3 Central moments (in terms of the factorial moments) E({RL - E(RL)} s ), s=1,...,4 CM[i_, δ_, 2] := NM[i, δ, 2] - NM[i, δ, 1] 2 ; CM[i_, δ_, 3] := NM[i, δ, 3] - 3 NM[i, δ, 2] NM[i, δ, 1] + 2 NM[i, δ, 1] 3 ; CM[i_, δ_, 4] := NM[i, δ, 4] - 4 NM[i, δ, 3] NM[i, δ, 1] + 6 NM[i, δ, 2] NM[i, δ, 1] 2-3 NM[i, δ, 1] 4 ; ARL (Average Run Length) ARL[i_, δ_] := NM[i, δ, 1]; ARL[0, p ] Plot[ARL[0, p 0 + δ], {δ, -.01,.1}, PlotRange {0, }] SDRL (Standard Deviation of the Run Length) SDRL[i_, δ_] := CM[i, δ, 2] ; CVRL (Coefficient of Variation of the Run Length) CVRL[i_, δ_] := SDRL[i, δ] ARL[i, δ] ; CSRL (Coefficient of Skewness of the Run Length) Assimetria CSRL[i_, δ_] := CM[i, δ, 3] SDRL[i, δ] 3 ;

6 6 Ex-Cap Version7-0.nb CKRL (Coefficient of Kurtosis of the Run Length) Achatamento CKRL[i_, δ_] := CM[i, δ, 4] - 3; 4 SDRL[i, δ] All RL related measures 0% HS TableForm[Transpose[Table[{theta[[i]], ARL[0, p 0 + theta[[i]]], SDRL[0, p 0 + theta[[i]]], CVRL[0, p 0 + theta[[i]]], CSRL[0, p 0 + theta[[i]]], CKRL[0, p 0 + theta[[i]]]}, {i, 1, Dimensions[theta][[1]]}]], TableAlignments - > {Right}] αα 100% Percentage points of the Run Length 0% HS S[i_, m_, δ_] := Apply[Plus, Table[If[j == i, 1, 0], {j, 0, x}]. MatrixPower[ N[Q[δ], 10], Floor[m]]]; MatrixForm[Transpose[Table[ If[j == 1, m = 0]; While[ If[S[0, m, p 0 + theta[[i]]] <= 1 - alpha[[j]], 1, 0] == 0, m++]; m, {i, 1, Dimensions[theta][[1]]}, {j, 1, Dimensions[alpha][[1]]}]]] Example np-chart Parameters n = 100; p 0 = 0.02; k = 3; p 1 = FindRoot n Log 1- p 0 1- z Log (1- p 0) z (1- z) p 0 p 1 = {z} /. Dispatch[p 1 ]; p 1 = p 1 [[1]] == k, {z, 1.1 p 0, 0, 1} ; theta = {0, 0.001, , 0.005, , 0.01, 0.02, p 1 - p 0, 0.03};

7 Ex-Cap Version7-0.nb 7 Parameter of RL β[dim_, δ_] = 1 - CDF[BinomialDistribution[dim, δ], UCL]; ARL (Average Run Length) ARL[dim_, δ_] := 1 N[β[dim, δ], 10] ; SDRL (Standard Deviation of the Run Length) SDRL[dim_, δ_] := 1 - N[β[dim, δ], 10] N[β[dim, δ], 10] ; CVRL (Coefficient of Variation of the Run Length) CVRL[dim_, δ_] := 1 - N[β[dim, δ], 10] ; CSRL (Coefficient of Skewness of the Run Length) Assimetria CSRL[dim_, δ_] := 2 - N[β[dim, δ], 10] 1 - N[β[dim, δ], 10] ; CKRL (Coefficient of Kurtosis of the Run Length) Achatamento 1 CKRL[dim_, δ_] := N[β[dim, δ], 10]; 1 - N[β[dim, δ], 10] All RL related measures TableForm[Transpose[Table[{theta[[i]], ARL[n, p 0 + theta[[i]]], SDRL[n, p 0 + theta[[i]]], CVRL[n, p 0 + theta[[i]]], CSRL[n, p 0 + theta[[i]]], CKRL[n, p 0 + theta[[i]]]}, {i, 1, Dimensions[theta][[1]]}]], TableAlignments - > {Right}] Survival Function S[m_, dim_, δ_] := Power[1 - N[β[dim, δ], 10], If[m < 0, 0, Floor[m]]];

8 8 Ex-Cap Version7-0.nb αα 100% Percentage points of the Run Length alpha = {0.05, 0.25, 0.5, 0.75, 0.9, 0.95}; MatrixForm[Transpose[Table[ If[j == 1, m = 0]; While[ If[S[m, n, p 0 + theta[[i]]] <= 1 - alpha[[j]], 1, 0] == 0, m++]; m, {i, 1, Dimensions[theta][[1]]}, {j, 1, Dimensions[alpha][[1]]}]]]

9 Ex-Cap Version7-0.nb 9 Exercise 10.17(a) data = {34.5, 34.2, 31.6, 31.5, 35.0, 34.1, 32.6, 33.8, 34.8, 33.6, 31.9, 38.6, 35.4, 34.0, 37.1, 34.9, 33.5, 31.7, 34.0, 35.1, 33.7, 32.8, 33.5, 34.2}; Length[data] n = 5; μ 0 = 32.5; σ 0 = 49 ; k = μ 0 ; h = ; (* The in- control ARL of this standard CUSUM chart for μ is approximately 500. Its control statistic is a deviation between the sample mean and the target value measured in in- control standard deviations of X. * ) LCL = - h; CL = 0; UCL = h; z[0] = 0; data[[i]] - k Do z[i] = z[i - 1] +, {i, 1, Length[data]} cusum = Table[z[i], {i, 1, Length[data]}]; list1 = Table[LCL, {i, 1, Length[data]}]; list2 = Table[CL, {i, 1, Length[data]}]; list3 = Table[UCL, {i, 1, Length[data]}]; σ 0 n TableForm[Transpose[{data, list1, cusum, list3}], TableHeadings {Automatic, {"y N ", "LCL", "z N ", "UCL"}}] G1 = ListPlot[cusum, Joined True, PlotStyle {Black, Thickness[0.002]}, PlotMarkers Automatic, PlotRange {- 25, 25}, DisplayFunction Identity]; G2 = ListPlot[list1, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; G3 = ListPlot[list2, Joined True, PlotStyle RGBColor[0, 1, 0], DisplayFunction Identity]; G4 = ListPlot[list3, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; Show[G1, G2, G3, G4, DisplayFunction $DisplayFunction] numberobsbeyondcontrollimits = Length[cusum] Mean[ Table[If[cusum[[i]] < LCL cusum[[i]] > UCL, 1, 0], {i, 1, Length[cusum]}]]; If[numberobsbeyondcontrollimits > 0, Print["There is (are) ", numberobsbeyondcontrollimits, " observation(s) beyond the control limits."], Print["There are no observations beyond the control limits."]] 24

10 10 Ex-Cap Version7-0.nb y N LCL z N UCL There are no observations beyond the control limits. Exercise 10.17(b) Sample size and distribution function of the standard normal and the chi-square with (n-1) degrees of freedom n = 5; Φ[z_] = CDF[NormalDistribution[0, 1], z]; χ[z_] = CDF[ChiSquareDistribution[n - 1], z]; Definition of the sub-stochastic matrix Q μμ (δδ, θθ) of the chart CUSUM-μμ

11 Ex-Cap Version7-0.nb 11 h μ = ; k μ = 0.0; x μ = 20; A μ [δ_, θ_] = Table Φ 1 θ 2 h μ j i δ, {i, - x μ, x μ }, {j, - x μ - 1, x μ } ; 2 x μ + 1 T μ = Table[If[i == j, - 1, If[i == j + 1, 1, 0]], {i, - x μ, x μ + 1}, {j, - x μ, x μ }]; Q μ [δ_, θ_] = A μ [δ, θ].t μ ; Definition of ARLi C- μμ (δδ, θθ) ID μ = IdentityMatrix[2 x μ + 1]; M μ [δ_, θ_] := Inverse[ID μ - Q μ [δ, θ]]; um μ = Table[1, {i, - x μ, x μ }]; b μ [i_] = Table[If[j == i, 1, 0], {j, - x μ, x μ }]; ARL μ [i_, δ_, θ_] := b μ [i]. M μ [δ, θ]. um μ ; In-control ARL0 C- μμ (δδ, θθ) and plot of ARL0 C- μμ (δδ, 1) ARL μ [0, 0, 1] Plot[ARL μ [0, δ, 1], {δ, - 2, 2}, PlotRange Full, AxesOrigin {0, 0}]

12 12 Ex-Cap Version7-0.nb Exercise λ = {0.05, 0.1, 1.}; G1 = ListPlot Table λ[[1]] (1 - λ[[1]]) j, {j, 1, 10}, PlotStyle {RGBColor[1, 0, 0], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; G2 = ListPlot Table λ[[2]] (1 - λ[[2]]) j, {j, 1, 10}, PlotStyle {RGBColor[0, 1, 0], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; G3 = ListPlot Table λ[[3]] (1 - λ[[3]]) j, {j, 1, 10}, PlotStyle {RGBColor[0, 0, 1], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; Show[G1, G2, G3, DisplayFunction $DisplayFunction] λ = {0.05, 0.1, 1.}; G1 = ListPlot Table λ[[1]] (1 - λ[[1]]) j, {j, 1, 10}, PlotStyle {RGBColor[1, 0, 0], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; G2 = ListPlot Table λ[[2]] (1 - λ[[2]]) j, {j, 1, 10}, PlotStyle {RGBColor[0, 1, 0], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; G3 = ListPlot Table λ[[3]] (1 - λ[[3]]) j, {j, 1, 10}, PlotStyle {RGBColor[0, 0, 1], PointSize[0.025]}, PlotRange {0,.11}, DisplayFunction Identity ; Show[G1, G2, G3, DisplayFunction $DisplayFunction]

13 Ex-Cap Version7-0.nb 13 Exercise μ 0 = 10.0; w 0 = μ 0 ; λ = 0.2; s = 2;(* target standard deviation of the sample mean* ) (* σ 0 =2;* ) γ = 3; (* n=length[data];* ) data = {10.5, 6, 10, 11, 12.5, 9.5, 6, 10, 10.5, 14.5, 9.5, 12, 12.5, 10.5, 8, 9.5, 7, 10, 13, 9, 12, 6, 12, 15, 11, 7, 9.5, 10, 12, 18}; W = Function N, (1 - λ) N N- 1 w 0 + λ (1 - λ) j data[[n - j]] ; (* Check Equation (10.19). * ) j=0 ewma = Table[W[i], {i, 1, Length[data]}]; LCLexact = Table μ 0 - γ s Sqrt λ 1 - (1 - λ)2 N, {N, 1, Length[data]} ; (2 - λ) CL = Table[μ 0, {N, 1, Length[data]}]; UCLexact = Table μ 0 + γ s Sqrt λ 1 - (1 - λ)2 N, {N, 1, Length[data]} ; (2 - λ) TableForm Transpose[{data, LCLexact, ewma, UCLexact}], TableHeadings Automatic, "x N ", "LCL N ", "w N ", "UCL N " G1 = ListPlot[ewma, Joined True, PlotStyle {Black, Thickness[0.002]}, PlotMarkers Automatic, PlotRange {8, 12}, DisplayFunction Identity]; G2 = ListPlot[LCLexact, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; G3 = ListPlot[CL, Joined True, PlotStyle RGBColor[0, 1, 0], DisplayFunction Identity]; G4 = ListPlot[UCLexact, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; Show[G1, G2, G3, G4, DisplayFunction $DisplayFunction] numberobsbeyondcontrollimits = Length[ewma] Mean[Table[If[ewma[[i]] < LCLexact[[i]] ewma[[i]] > UCLexact[[i]], 1, 0], {i, 1, Length[ewma]}]]; If[numberobsbeyondcontrollimits > 0, Print["There is (are) ", numberobsbeyondcontrollimits, " observation(s) beyond the control limits."], Print["There are no observations beyond the control limits."]]

14 14 Ex-Cap Version7-0.nb x N LCL N w N UCL N There are no observations beyond the control limits.

15 Ex-Cap Version7-0.nb 15 λ LCLasympt = Table μ 0 - γ s Sqrt, {N, 1, Length[data]} ; (2 - λ) CL = Table[μ 0, {N, 1, Length[data]}]; λ UCLasympt = Table μ 0 + γ s Sqrt, {N, 1, Length[data]} ; (2 - λ) TableForm Transpose[{data, LCLasympt, ewma, UCLasympt}], TableHeadings Automatic, "x N ", "LCL a ", "w N ", "UCL a " G1 = ListPlot[ewma, Joined True, PlotStyle {Black, Thickness[0.002]}, PlotMarkers Automatic, PlotRange {8, 12}, DisplayFunction Identity]; G2 = ListPlot[LCLasympt, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; G3 = ListPlot[CL, Joined True, PlotStyle RGBColor[0, 1, 0], DisplayFunction Identity]; G4 = ListPlot[UCLasympt, Joined True, PlotStyle {RGBColor[1, 0, 0], Thickness[0.008]}, DisplayFunction Identity]; Show[G1, G2, G3, G4, DisplayFunction $DisplayFunction] numberobsbeyondcontrollimits = Length[ewma] Mean[Table[If[ewma[[i]] < LCLasympt[[i]] ewma[[i]] > UCLasympt[[i]], 1, 0], {i, 1, Length[ewma]}]]; If[numberobsbeyondcontrollimits > 0, Print["There is (are) ", numberobsbeyondcontrollimits, " observation(s) beyond the control limits."], Print["There are no observations beyond the control limits."]] x N LCL a w N UCL a

16 16 Ex-Cap Version7-0.nb There are no observations beyond the control limits. Exercise Upper one-sided Shewhart chart Φ[z_] = CDF[NormalDistribution[0, 1], z]; n = 5; pfa μ = 1 / 500.0; γ S- μ = Quantile[NormalDistribution[0, 1], 1 - pfa μ ]; 1 ARL S- μ [δ_, θ_] = ARL S- μ [0, 1] Φ γ S- μ- δ θ ;

17 Ex-Cap Version7-0.nb 17 Upper one-sided EWMA chart (0% Head Start) γ E- μ = ; λ μ = 0.134; x μ = 40; Φ[z_] = CDF[NormalDistribution[0, 1], z]; n = 5; A μ [δ_, θ_] = Table If j == - 1, 0, Φ 1 θ γ E- μ j (1 - λ μ ) i (x μ + 1) λ μ (2 - λ μ ) - δ, {i, 0, x μ }, {j, - 1, x μ } ; T μ = Table[If[i == j, - 1, If[i == j + 1, 1, 0]], {i, 0, x μ + 1}, {j, 0, x μ }]; Q μ [δ_, θ_] = A μ [δ, θ].t μ ; ID μ = IdentityMatrix[x μ + 1]; M μ [δ_, θ_] := Inverse[ID μ - Q μ [δ, θ]]; um μ = Table[1, {i, 0, x μ }]; ARL E- μ [i_, δ_, θ_] := Table[If[j == i, 1, 0], {j, 0, x μ }]. M μ [δ, θ]. um μ ; ARL E- μ [0, 0, 1] shift = {0, 0.05, 0.10, 0.20, 0.30, 0.40, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.5, 2.0, 3.0}; TableForm[Table[{shift[[i]], ARL E- μ [0, shift[[i]], 1], ARL S- μ [shift[[i]], 1]}, {i, 1, Length[shift]}], TableHeadings {Automatic, {"δ", "ARL E- μ ", "ARL S- μ "}}] (* ARL values for the EWMA and Shewhart charts * ) δ ARL E- μ ARL S- μ

18 18 Ex-Cap Version7-0.nb EWMA vs. Shewhart chart (percentual reduction in the ARL when the Shewhart chartis replaced by a EWMA chart) Plot 1 - ARL E- μ[0, δ, 1] ARL S- μ [δ, 1] 100, {δ, 0, 10}

19 Ex-Cap Version7-0.nb 19 Exercise Upper one-sided EWMA chart (0% Head Start) n = 5; sigma 0 = 1; λ σ = 0.05; LCL E- σ = 0; UCL E- σ = ; γ E- σ = UCL E- σ - Log sigma 0 2 ; PolyGamma 1, n- 1 2 λ σ 2- λ σ χ[z_] = CDF[ChiSquareDistribution[n - 1], z]; x σ = 20; A σ [θ_] = Table If j == - 1, 0, χ n - 1 θ 2 Exp γ E- σ PolyGamma 1, n j (1 - λ σ ) i + 1 2, {i, 0, x σ }, {j, - 1, x σ } ; (x σ + 1) λ σ (2 - λ σ ) T σ = Table[If[i == j, - 1, If[i == j + 1, 1, 0]], {i, 0, x σ + 1}, {j, 0, x σ }]; Q σ [θ_] = A σ [θ].t σ ; ID σ = IdentityMatrix[x σ + 1]; M σ [θ_] := Inverse[ID σ - Q σ [θ]]; um σ = Table[1, {i, 0, x σ }]; b σ [i_] = Table[If[j == i, 1, 0], {j, 0, x σ }]; ARL E- σ [i_, θ_] := b σ [i]. M σ [θ]. um σ ; ARL E- σ [0, 1] Upper one-sided Shewhart chart UCL S- σ = ; γ S- σ = UCL S- σ n ; ARL S- σ [θ_] = ARL S- σ [1] 1 ; 1 - χ γ S- σ θ

20 20 Ex-Cap Version7-0.nb EWMA vs. Shewhart chart (percentual reduction in the ARL when the Shewhart chartis replaced by a EWMA chart) Plot 1 - ARL E- σ[0, θ] ARL S- σ [θ] 100, {θ, 1, 10}, AxesOrigin {1, 0}

21 Ex-Cap Version7-0.nb 21 Exercise n = 5; Φ[x_] = CDF[NormalDistribution[0, 1], x]; χ[x_] = CDF[ChiSquareDistribution[n - 1], x]; pfa μ = 1 / 1000.; γ μ = Quantile NormalDistribution[0, 1], 1 - pfa μ 2 ; 1 pfa σ = ; γ σ = Quantile[ChiSquareDistribution[n - 1], 1 - pfa σ ]; 1 ARL μ [δ_, θ_] = ARL σ [θ_] = 1 - Φ γ μ- δ θ 1 ; 1 - χ γ σ θ 2 - Φ - γ μ- δ θ ARL μ,σ [δ_, θ_] = Φ γ μ - δ ; - Φ - γ μ - δ θ θ ARL μ [0, 1] ARL σ [1] ARL μ,σ [0, 1] Plot3D[Log[ARL μ,σ [δ, θ]], {δ, 0, 3}, {θ, 1, 2}] χ γ σ θ 2 ;

22 22 Ex-Cap Version7-0.nb ARL μ,σ [δ_, θ_] = ARL μ [δ, θ] ARL σ [θ] ARL μ [δ, θ] + ARL σ [θ] - 1 ; (* Relating the ARL of the joint scheme and ssthe ones of the individual charts... * ) ARL μ [0, 1] ARL σ [1] ARL μ,σ [0, 1] Exercise n = 10; Φ[x_] = CDF[NormalDistribution[0, 1], x]; χ[x_] = CDF[ChiSquareDistribution[n - 1], x]; (* data μ ={1.04,1.06,1.09,1.05,1.07,1.06,1.05,1.10, 1.09,1.05,0.99,1.06,1.05,1.07,1.11,1.04,1.03,1.05,1.06,1.04}; data σ ={0.87,0.85,0.90,0.85,0.73,0.80,0.78,0.83,0.87, , 0.79,0.82,0.75,0.76,0.89,0.91,0.85,0.83,0.79,0.85}; * ) pfa μ = 0.002; γ μ = Quantile NormalDistribution[0, 1], 1 - pfa μ 2 ; pfa σ = 0.002; γ σ = Quantile[ChiSquareDistribution[n - 1], 1 - pfa σ ]; μ 0 = Mean[data μ ]; σ 0 = Mean[data σ ] ; LCL μ = μ 0 - γ μ σ 0 n UCL μ = μ 0 + γ μ σ 0 LCL σ = 0 UCL σ = σ 0 2 n - 1 γ σ n probsignal μ,σ [δ_, θ_] = 1 - probsignal μ,σ [0, 1] Φ γ μ - δ θ - Φ - γ μ - δ θ χ γ σ θ 2 ;

23 Ex-Cap Version7-0.nb 23 probvalidsignal = probsignal μ,σ [0.1, 1.1]; dist = GeometricDistribution[probvalidsignal] ; (* geometric distribution for the number of trials before the first success, where the probability of success in a trial is p * ) CDF[dist, 10-1] ARL μ [δ_, θ_] = 1 - Φ γ μ- δ θ 1 - Φ - γ μ- δ θ ARL σ [θ_] = χ γ σ θ 2 ; ARL μ,σ [δ_, θ_] = ARL μ [δ, θ] ARL σ [θ] ARL μ [δ, θ] + ARL σ [θ] - 1 ; ARL μ [0, 1] ARL σ [1] ARL μ,σ [0, 1] Plot3D[Log[ARL μ,σ [δ, θ]], {δ, 0, 3}, {θ, 1, 2}] ;

24 24 Ex-Cap Version7-0.nb Exercise (with n=9, ARL μμ (0,1)=ARL σσ (1)=500) n = 9; Φ[x_] = CDF[NormalDistribution[0, 1], x]; χ[x_] = CDF[ChiSquareDistribution[n - 1], x]; pfa μ = 1 / 500.0; γ μ = Quantile NormalDistribution[0, 1], 1 - pfa μ 2 ; pfa σ = 1 / 500.0; γ σ = Quantile[ChiSquareDistribution[n - 1], 1 - pfa σ ]; PMS III [θ_] = 1 - Φ γ μ θ - Φ - γ μ θ 1 χ γ σ θ 2 - Φ γ μ θ - Φ - γ μ θ ; theta = {1.01, 1.03, 1.05, 1.10, 1.20, 1.30, 1.40, 1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 3.00}; TableForm[Table[{theta[[i]], PMS III [theta[[i]]]}, {i, 1, Length[theta]}], TableHeadings {Automatic, {"θ", "PMS III "}}] Plot[{PMS III [θ]}, {θ, 1.001, 3}, AxesOrigin - > {1, 0}] 1 - χ[γ σ ] PMS IV [δ_] = ; 1 - χ[γ (Φ[γ μ - δ]- Φ[- γ μ - δ]) σ] delta = {0.05, 0.10, 0.20, 0.30, 0.40, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.5, 2.0, 3.0}; TableForm[Table[{delta[[i]], PMS IV [delta[[i]]]}, {i, 1, Length[delta]}], TableHeadings {Automatic, {"δ", "PMS IV "}}] Plot[{PMS IV [δ]}, {δ, 0.001, 3}, AxesOrigin - > {0, 0}] θ PMS III

25 Ex-Cap Version7-0.nb δ PMS IV

26 26 Ex-Cap Version7-0.nb Exercise n = 5; Φ[x_] = CDF[NormalDistribution[0, 1], x]; χ[x_] = CDF[ChiSquareDistribution[n - 1], x]; pfa μ = 1 / 500.0; γ μ = Quantile[NormalDistribution[0, 1], 1 - pfa μ ]; pfa σ = 1 / 500.0; γ σ = Quantile[ChiSquareDistribution[n - 1], 1 - pfa σ ]; PMS III [θ_] = 1 - Φ γ μ θ ; 1 - Φ γμ χ γσ θ θ 2 theta = {1.01, 1.03, 1.05, 1.10, 1.20, 1.30, 1.40, 1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 3.00}; TableForm[Table[{theta[[i]], PMS III [theta[[i]]]}, {i, 1, Length[theta]}], TableHeadings {Automatic, {"θ", "PMS III "}}] Plot[{PMS III [θ]}, {θ, 1.001, 3}, AxesOrigin - > {1, 0}] 1 - χ[γ σ ] PMS IV [δ_] = ; 1 - χ[γ Φ[γ μ - δ] σ] delta = {0.05, 0.10, 0.20, 0.30, 0.40, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.5, 2.0, 3.0}; TableForm[Table[{delta[[i]], PMS IV [delta[[i]]]}, {i, 1, Length[delta]}], TableHeadings {Automatic, {"δ", "PMS IV "}}] Plot[{PMS IV [δ]}, {δ, 0.001, 3}, AxesOrigin - > {0, 0}] θ PMS III

27 Ex-Cap Version7-0.nb δ PMS IV Sample size and distribution function of the standard normal and the chi-square with (n-1) degrees of freedom n = 5; Φ[z_] = CDF[NormalDistribution[0, 1], z]; χ[z_] = CDF[ChiSquareDistribution[n - 1], z]; Definition of the sub-stochastic matrix Q μμ (δδ, θθ) of the chart EWMA-μμ

28 28 Ex-Cap Version7-0.nb γ μ = ; λ μ = 0.134; x μ = 40; A μ [δ_, θ_] = Table If j == - 1, 0, Φ 1 θ γ μ j (1 - λ μ ) i {i, 0, x μ }, {j, - 1, x μ } ; (x μ + 1) λ μ (2 - λ μ ) - δ, T μ = Table[If[i == j, - 1, If[i == j + 1, 1, 0]], {i, 0, x μ + 1}, {j, 0, x μ }]; Q μ [δ_, θ_] = A μ [δ, θ].t μ ; Definition of ARLi E- μμ (δδ, θθ) ID μ = IdentityMatrix[x μ + 1]; M μ [δ_, θ_] := Inverse[ID μ - Q μ [δ, θ]]; um μ = Table[1, {i, 0, x μ }]; b μ [i_] = Table[If[j == i, 1, 0], {j, 0, x μ }]; ARL μ [i_, δ_, θ_] := b μ [i]. M μ [δ, θ]. um μ ; ARL μ [0, 0, 1] Definition of the sub-stochastic matrix Q σσ (θθ) of the chart EWMA-σσ γ σ = ; λ σ = 0.043; x σ = 40; A σ [θ_] = Table If j == - 1, 0, χ n - 1 θ 2 Exp γ σ PolyGamma 1, n j (1 - λ σ ) i + 1 2, {i, 0, x σ }, {j, - 1, x σ } ; (x σ + 1) λ σ (2 - λ σ ) T σ = Table[If[i == j, - 1, If[i == j + 1, 1, 0]], {i, 0, x σ + 1}, {j, 0, x σ }]; Q σ [θ_] = A σ [θ].t σ ; Definition of ARLi E- σσ (θθ) ID σ = IdentityMatrix[x σ + 1]; M σ [θ_] := Inverse[ID σ - Q σ [θ]]; um σ = Table[1, {i, 0, x σ }]; b σ [i_] = Table[If[j == i, 1, 0], {j, 0, x σ }]; ARL E- σ [i_, θ_] := b σ [i]. M σ [θ]. um σ ; ARL E- σ [0, 1] i Survival function of RL E- μμ (δδ, θθ)

29 Ex-Cap Version7-0.nb 29 F i RLE- μμ δδ,θθ (k) S μ [i_, k_, δ_, θ_] := b μ [i].matrixpower[ Q μ [δ, θ], If[k 0, Floor[k], 0]]. um μ ; j Survival function of RL E- σσ (θθ) F j RLE- σσ (θθ) (k) S σ [j_, k_, θ_] := b σ [j].matrixpower[ Q σ [θ], If[k 0, Floor[k], 0]]. um σ ; "Probability of Misleading Signals" EE + Type III PMS EE +(III; θθ) HS μμ = 0 %, HS σσ = 0 % theta = {1.01, 1.03, 1.05, 1.10, 1.20, 1.30, 1.40, 1.50, 1.60, 1.70, 1.80, 1.90, 2.00, 3.00}; head μ = {0}; head σ = {0};

30 30 Ex-Cap Version7-0.nb Do[Do[Do[{ δ = 0; θ = theta[[l]]; i = head μ [[m]]; j = head σ [[p]]; cu μ = Q μ [δ, θ]; cu σ = Q σ [θ]; err = ; erel = 2 err; oldsum = 0; pot1 μ = ID μ ; pot2 μ = cu μ ; pot σ = cu σ ; oldparc μ = b μ [i].(pot1 μ - pot2 μ ). um μ ; oldparc σ = b σ [j].pot σ. um σ ; oldparc = oldparc μ * oldparc σ ; nparc = 1; While[erel > err, {soma = oldsum + oldparc, erel = Abs[(soma - oldsum) / soma], oldsum = soma, oldpot2 μ = pot2 μ, oldpot σ = pot σ, pot1 μ = pot2 μ, pot2 μ = cu μ.oldpot2 μ, pot σ = cu σ.oldpot σ, oldparc μ = b μ [i].(pot1 μ - pot2 μ ). um μ, oldparc σ = b σ [j].pot σ. um σ, oldparc = oldparc μ * oldparc σ, nparc = nparc + 1, If[Mod[nparc - 1, 500] == 0, Print[ "iter=", nparc - 1, " erro relat=", erel, " valor aprox=", oldsum, " δ=", δ, " θ=", θ]]}], PMS III [i, j, θ] = oldsum; iter[i, j, θ] = nparc - 1; Print[ " δ=", δ, " θ=", θ, " i=", i, " j=", j, " aproximate value=", oldsum, " iterations=", nparc - 1, " relative error=", erel]}, {p, 1, Dimensions[head σ ][[1]]}], {l, 1, Dimensions[theta][[1]]}], {m, 1, Dimensions[head μ ][[1]]}]

31 Ex-Cap Version7-0.nb 31 iter=500 erro relat= valor aprox= δ=0 θ=1.01 iter=1000 erro relat= valor aprox= δ=0 θ=1.01 iter=1500 erro relat= valor aprox= δ=0 θ=1.01 δ=0 θ=1.01 i=0 j=0 aproximate value= iterations=1772 relative error= iter=500 erro relat= valor aprox= δ=0 θ=1.03 iter=1000 erro relat= valor aprox= δ=0 θ=1.03 δ=0 θ=1.03 i=0 j=0 aproximate value= iterations=1320 relative error= iter=500 erro relat= valor aprox= δ=0 θ=1.05 δ=0 θ=1.05 i=0 j=0 aproximate value= iterations=991 relative error= iter=500 erro relat= valor aprox= δ=0 θ=1.1 δ=0 θ=1.1 i=0 j=0 aproximate value= iterations=517 relative error= δ=0 θ=1.2 i=0 j=0 aproximate value= iterations=194 relative error= δ=0 θ=1.3 i=0 j=0 aproximate value= iterations=101 relative error= δ=0 θ=1.4 i=0 j=0 aproximate value= iterations=64 relative error= δ=0 θ=1.5 i=0 j=0 aproximate value= iterations=45 relative error= δ=0 θ=1.6 i=0 j=0 aproximate value= iterations=35 relative error= δ=0 θ=1.7 i=0 j=0 aproximate value= iterations=28 relative error= δ=0 θ=1.8 i=0 j=0 aproximate value= iterations=24 relative error= δ=0 θ=1.9 i=0 j=0 aproximate value= iterations=20 relative error= δ=0 θ=2. i=0 j=0 aproximate value= iterations=18 relative error= δ=0 θ=3. i=0 j=0 aproximate value= iterations=9 relative error=

32 32 Ex-Cap Version7-0.nb MatrixForm[Table[{ theta[[l]], (* head μ [[i]], head σ [[j]], iter[head μ [[i]],head σ [[j]],theta[[l]]], * ) PMS III [head μ [[i]], head σ [[j]], theta[[l]]]}, {j, 1, Dimensions[head σ ][[1]]}, {i, 1, Dimensions[head μ ][[1]]}, {l, 1, Dimensions[theta][[1]]}]] Type IV PMS EE +(IV; δδ) HS μμ = 0 %, HS σσ = 0 % delta = {0.05, 0.10, 0.20, 0.30, 0.40, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.5, 2.0, 3.0}; head μ = {0}; head σ = {0};

33 Ex-Cap Version7-0.nb 33 Do[Do[Do[{ δ = delta[[k]]; θ = 1; i = head μ [[m]]; j = head σ [[p]]; cu μ = Q μ [δ, θ]; cu σ = Q σ [θ]; err = ; erel = 2 err; oldsum = 0; pot1 σ = ID σ ; pot2 σ = cu σ ; pot μ = cu μ ; oldparc σ = b σ [j].(pot1 σ - pot2 σ ). um σ ; oldparc μ = b μ [i].pot μ. um μ ; oldparc = oldparc μ * oldparc σ ; nparc = 1; While[erel > err, {soma = oldsum + oldparc, If[nparc > 1, erel = Abs[(soma - oldsum) / soma]], oldsum = soma, oldpot2 σ = pot2 σ, oldpot μ = pot μ, pot1 σ = pot2 σ, pot2 σ = cu σ.oldpot2 σ, pot μ = cu μ.oldpot μ, oldparc σ = b σ [j].(pot1 σ - pot2 σ ). um σ, oldparc μ = b μ [i].pot μ. um μ, oldparc = oldparc μ * oldparc σ, nparc = nparc + 1, If[Mod[nparc - 1, 500] == 0, Print[ "iter=", nparc - 1, " erro relat=", erel, " valor aprox=", oldsum, " δ=", δ, " θ=", θ]]}], PMS IV [i, j, δ] = oldsum; iter[i, j, δ] = nparc - 1; Print[ " δ=", δ, " θ=", θ, " i=", i, " j=", j, " aproximate value=", oldsum, " iterations=", nparc - 1, " relative error=", erel]}, {p, 1, Dimensions[head σ ][[1]]}], {k, 1, Dimensions[delta][[1]]}], {m, 1, Dimensions[head μ ][[1]]}]

34 34 Ex-Cap Version7-0.nb iter=500 erro relat= valor aprox= δ=0.05 θ=1 iter=1000 erro relat= valor aprox= δ=0.05 θ=1 iter=1500 erro relat= valor aprox= δ=0.05 θ=1 δ=0.05 θ=1 i=0 j=0 aproximate value= iterations=1707 relative error= iter=500 erro relat= valor aprox= δ=0.1 θ=1 iter=1000 erro relat= valor aprox= δ=0.1 θ=1 δ=0.1 θ=1 i=0 j=0 aproximate value= iterations=1388 relative error= iter=500 erro relat= valor aprox= δ=0.2 θ=1 δ=0.2 θ=1 i=0 j=0 aproximate value= iterations=885 relative error= iter=500 erro relat= valor aprox= δ=0.3 θ=1 δ=0.3 θ=1 i=0 j=0 aproximate value= iterations=561 relative error= δ=0.4 θ=1 i=0 j=0 aproximate value= iterations=364 relative error= δ=0.5 θ=1 i=0 j=0 aproximate value= iterations=246 relative error= δ=0.6 θ=1 i=0 j=0 aproximate value= iterations=174 relative error= δ=0.7 θ=1 i=0 j=0 aproximate value= iterations=128 relative error= δ=0.8 θ=1 i=0 j=0 aproximate value= iterations=97 relative error= δ=0.9 θ=1 i=0 j=0 aproximate value= iterations=76 relative error= δ=1. θ=1 i=0 j=0 aproximate value= iterations=62 relative error= δ=1.5 θ=1 i=0 j=0 aproximate value= iterations=28 relative error= δ=2. θ=1 i=0 j=0 aproximate value= iterations=17 relative error= δ=3. θ=1 i=0 j=0 aproximate value= iterations=9 relative error=

35 Ex-Cap Version7-0.nb 35 MatrixForm[Table[{ delta[[l]], (* head μ [[i]], head σ [[j]], iter[head μ [[i]],head σ [[j]],delta[[l]]], * ) PMS IV [head μ [[i]], head σ [[j]], delta[[l]]]}, {j, 1, Dimensions[head σ ][[1]]}, {i, 1, Dimensions[head μ ][[1]]}, {l, 1, Dimensions[delta][[1]]}]] Exercise n = 10; χ[x_] = CDF[ChiSquareDistribution[n - 1], x]; pfa σ = 1 / 100.0; γ σ = Quantile[ChiSquareDistribution[n - 1], 1 - pfa σ ]; 1 - χ γ σ pfa σ = 1 / 100.0; a = Quantile ChiSquareDistribution[n - 1], pfa σ 2 ; b = Quantile ChiSquareDistribution[n - 1], 1 - pfa σ 2 ; 1 - χ b a - χ

36 36 Ex-Cap Version7-0.nb probmisleadingsignal upper = 1 - χ γ σ.9 2 ; dist = GeometricDistribution[probmisleadingsignal upper ] ; (* geometric distribution for the number of trials before the first success, where the probability of success in a trial is p * ) CDF[dist, 100-1] probmisleadingsignal standard = 1 - χ b a - χ.92.9 ; 2 dist = GeometricDistribution[probmisleadingsignal standard ] ; (* geometric distribution for the number of trials before the first success, where the probability of success in a trial is p * ) CDF[dist, 100-1]