Revisions to Numerical Integration as Model for Nonlinear Fit

Fixed spelling and explained a bit of my work.

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edited Jul 10, 2013 at 16:54

51
3

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[pSum[NIntegrate[totalνLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalvLayertotalνLayer is a function I defined earlier. I just want the fit values for a and b and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

And the definition of totalvLayertotalνLayer:

totalvLayer[x_totalνLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - μ[mrad])^2/(
                          2 σ[mrad]^2)] + lin[mrad] x + (amp[mrad] γ[mrad]^2)/(
                          4 (γ[mrad]^2/4 + (x - ω[mrad])^2)))/(
                          crossSection[x] 10^-38*1*6300*6.022*10^23)

model[a_, b_, x_] := Sum[NIntegrate[
                           totalvLayer[ptotalνLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23,
                           {p, 0.2, 10}], 
                        {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[α, β, x], {α, β}, x, Weights -> 1/errors^2]

I apologize for all of the weird numbers and functions; this is neutrino data from my research.

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalvLayer is a function I defined earlier. I just want the fit values for a and b and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

And the definition of totalvLayer:

totalvLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - μ[mrad])^2/(
                          2 σ[mrad]^2)] + lin[mrad] x + (amp[mrad] γ[mrad]^2)/(
                          4 (γ[mrad]^2/4 + (x - ω[mrad])^2)))/(
                          crossSection[x] 10^-38*1*6300*6.022*10^23)

model[a_, b_, x_] := Sum[NIntegrate[
                           totalvLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23,
                           {p, 0.2, 10}], 
                        {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[α, β, x], {α, β}, x, Weights -> 1/errors^2]

model[a_, b_, x_] := Sum[NIntegrate[totalνLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalνLayer is a function I defined earlier. I just want the fit values for a and b and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

And the definition of totalνLayer:

totalνLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - μ[mrad])^2/(
                          2 σ[mrad]^2)] + lin[mrad] x + (amp[mrad] γ[mrad]^2)/(
                          4 (γ[mrad]^2/4 + (x - ω[mrad])^2)))/(
                          crossSection[x] 10^-38*1*6300*6.022*10^23)

model[a_, b_, x_] := Sum[NIntegrate[
                           totalνLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23,
                           {p, 0.2, 10}], 
                        {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[α, β, x], {α, β}, x, Weights -> 1/errors^2]

I apologize for all of the weird numbers and functions; this is neutrino data from my research.

deleted 31 characters in body

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edited Jul 10, 2013 at 8:39

feynmansafineman

51
3

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], 
                         {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], 
                         {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

added 205 characters in body

Source Link

edited Jul 10, 2013 at 4:43

Kuba

137.7k
13
289
751

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], 
                         {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalvLayertotalvLayer is a function I defined earlier. I just want the fit values for aa and bb and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

amp[x_] := 47.5111 - 1.79354 Abs[x]
\[Mu][mrad_]μ[mrad_] := 5.449517720787487 Exp[-0.0693211022591832 Abs[mrad]]
\[Sigma][mrad_]σ[mrad_] := 1.7679270770338407 Exp[-0.12510841305113626 Abs[mrad]]
\[Gamma][x_]γ[x_] := 2.5587996836804026 Exp[-0.20766051151975093 Abs[x]] + 0.5696086481954964
\[Omega][mrad_]ω[mrad_] := 7.63444841081276 Exp[-0.08637416159312404 Abs[mrad]]
lin[mrad_] := 0.9648001654360766 - 0.10434481940699446 Abs[mrad] + 0.003062473453032527 Abs[mrad]^2

crossSection[x_] := -600.089 + 1201.5577451256304 x - 1751.9101332336447` x^2 + 1899.988095515613` x^3 - 1556.709382526493` x^4 + 974.3641634962205` x^5 - 468.5129814014657` x^6 + 172.90071370757002` x^7 - 48.48328424613457` x^8 + 10.067733931487915` x^9 - 1.4558045159248463` x^10 + 0.1215353564558521` x^11 -0.00009554527467999401` x^12 - 0.001211849211458331` x^13 + 0.00009244381215772915` x^14 + 6.85533417357146`*^-6 x^15 - 1.3215490607411126`*^-6 x^16 - 1.213701585221995`*^-8 x^17 + 1.4963340473233928`*^-8 x^18 - 6.405376189932708`*^-10 x^19 - 1.3047056106410797`*^-10 x^20 + 2.0016864863803638`*^-11 x^21 - 1.2643359066392302`*^-12 x^22 + 4.036927989211094`*^-14 x^23 - 5.368234695690585`*^-16 x^24 + 214.8882519560799` x^-1 - 54.26505937368572` x^-2 + 9.480463574208194` x^-3 - 1.1135829819290948` x^-4 + 0.08354380635765371` x^-5 - 0.0036045838406917876` x^-6 + 0.0000678896567028218` x^-7

And the definition of totalvLayertotalvLayer:

totalvLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - \[Mu][mrad]μ[mrad])^2/(
                          2 \[Sigma][mrad]^2σ[mrad]^2)] + 
 lin[mrad] x + (amp[mrad] \[Gamma][mrad]^2γ[mrad]^2)/(
                          4 (\[Gamma][mrad]^2γ[mrad]^2/4 + (x - \[Omega][mrad]ω[mrad])^2)))/(
                          crossSection[x] 10^-38*1*6300*6.022*10^23)

data = {{0, 1185.2847069383793`}, {4, 780.6026466248076`}, {8, 
 493.5502399832769`}, 
        {12, 312.1838499130911`}, {16, 
 193.16881196193143`}, {20, 113.41366552793787`},
        {24, 64.20654682074144`}};

errors = {132.12627600307064`, 90.55918065608893`, 60.884899393246535`,
         41.88064141772561`, 29.076284534300132`, 20.049728493077968`, 
         13.950371635495452`};

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[pSum[NIntegrate[
                           totalvLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23,
                           {p, 0.2, 10}], 
                        {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[\[Alpha]model[α, \[Beta]β, x], {\[Alpha]α, \[Beta]β}, x, Weights -> 1/errors^2]

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),{p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalvLayer is a function I defined earlier. I just want the fit values for a and b and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

amp[x_] := 47.5111 - 1.79354 Abs[x]
\[Mu][mrad_] := 5.449517720787487 Exp[-0.0693211022591832 Abs[mrad]]
\[Sigma][mrad_] := 1.7679270770338407 Exp[-0.12510841305113626 Abs[mrad]]
\[Gamma][x_] := 2.5587996836804026 Exp[-0.20766051151975093 Abs[x]] + 0.5696086481954964
\[Omega][mrad_] := 7.63444841081276 Exp[-0.08637416159312404 Abs[mrad]]
lin[mrad_] := 0.9648001654360766 - 0.10434481940699446 Abs[mrad] + 0.003062473453032527 Abs[mrad]^2

crossSection[x_] := -600.089 + 1201.5577451256304 x - 1751.9101332336447` x^2 + 1899.988095515613` x^3 - 1556.709382526493` x^4 + 974.3641634962205` x^5 - 468.5129814014657` x^6 + 172.90071370757002` x^7 - 48.48328424613457` x^8 + 10.067733931487915` x^9 - 1.4558045159248463` x^10 + 0.1215353564558521` x^11 -0.00009554527467999401` x^12 - 0.001211849211458331` x^13 + 0.00009244381215772915` x^14 + 6.85533417357146`*^-6 x^15 - 1.3215490607411126`*^-6 x^16 - 1.213701585221995`*^-8 x^17 + 1.4963340473233928`*^-8 x^18 - 6.405376189932708`*^-10 x^19 - 1.3047056106410797`*^-10 x^20 + 2.0016864863803638`*^-11 x^21 - 1.2643359066392302`*^-12 x^22 + 4.036927989211094`*^-14 x^23 - 5.368234695690585`*^-16 x^24 + 214.8882519560799` x^-1 - 54.26505937368572` x^-2 + 9.480463574208194` x^-3 - 1.1135829819290948` x^-4 + 0.08354380635765371` x^-5 - 0.0036045838406917876` x^-6 + 0.0000678896567028218` x^-7

And the definition of totalvLayer:

totalvLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - \[Mu][mrad])^2/(2 \[Sigma][mrad]^2)] + 
 lin[mrad] x + (amp[mrad] \[Gamma][mrad]^2)/(
4 (\[Gamma][mrad]^2/4 + (x - \[Omega][mrad])^2)))/(crossSection[x] 10^-38*1*6300*6.022*10^23)

data = {{0, 1185.2847069383793`}, {4, 780.6026466248076`}, {8, 
 493.5502399832769`}, {12, 312.1838499130911`}, {16, 
 193.16881196193143`}, {20, 113.41366552793787`}, {24, 64.20654682074144`}};

errors = {132.12627600307064`, 90.55918065608893`, 60.884899393246535`, 41.88064141772561`, 29.076284534300132`, 20.049728493077968`, 13.950371635495452`};

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23, {p, 0.2, 10}], {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[\[Alpha], \[Beta], x], {\[Alpha], \[Beta]}, x, Weights -> 1/errors^2]

model[a_, b_, x_] := Sum[NIntegrate[totalvLayer[p, m] (a (1 - b^-p)),
                                   {p, 0.2, 10}], 
                         {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[a, b, x], {a,b}, x, Weights -> 1/errors^2]

Where totalvLayer is a function I defined earlier. I just want the fit values for a and b and their covariance matrix, but I'm clearly doing something wrong. Any help is much appreciated!

amp[x_] := 47.5111 - 1.79354 Abs[x]
μ[mrad_] := 5.449517720787487 Exp[-0.0693211022591832 Abs[mrad]]
σ[mrad_] := 1.7679270770338407 Exp[-0.12510841305113626 Abs[mrad]]
γ[x_] := 2.5587996836804026 Exp[-0.20766051151975093 Abs[x]] + 0.5696086481954964
ω[mrad_] := 7.63444841081276 Exp[-0.08637416159312404 Abs[mrad]]
lin[mrad_] := 0.9648001654360766 - 0.10434481940699446 Abs[mrad] + 0.003062473453032527 Abs[mrad]^2

crossSection[x_] := -600.089 + 1201.5577451256304 x - 1751.9101332336447` x^2 + 1899.988095515613` x^3 - 1556.709382526493` x^4 + 974.3641634962205` x^5 - 468.5129814014657` x^6 + 172.90071370757002` x^7 - 48.48328424613457` x^8 + 10.067733931487915` x^9 - 1.4558045159248463` x^10 + 0.1215353564558521` x^11 -0.00009554527467999401` x^12 - 0.001211849211458331` x^13 + 0.00009244381215772915` x^14 + 6.85533417357146`*^-6 x^15 - 1.3215490607411126`*^-6 x^16 - 1.213701585221995`*^-8 x^17 + 1.4963340473233928`*^-8 x^18 - 6.405376189932708`*^-10 x^19 - 1.3047056106410797`*^-10 x^20 + 2.0016864863803638`*^-11 x^21 - 1.2643359066392302`*^-12 x^22 + 4.036927989211094`*^-14 x^23 - 5.368234695690585`*^-16 x^24 + 214.8882519560799` x^-1 - 54.26505937368572` x^-2 + 9.480463574208194` x^-3 - 1.1135829819290948` x^-4 + 0.08354380635765371` x^-5 - 0.0036045838406917876` x^-6 + 0.0000678896567028218` x^-7

And the definition of totalvLayer:

totalvLayer[x_, mrad_] := .3*3.78*10^-5*810^2*5 (amp[mrad] Exp[-(x - μ[mrad])^2/(
                          2 σ[mrad]^2)] + lin[mrad] x + (amp[mrad] γ[mrad]^2)/(
                          4 (γ[mrad]^2/4 + (x - ω[mrad])^2)))/(
                          crossSection[x] 10^-38*1*6300*6.022*10^23)

data = {{0, 1185.2847069383793`}, {4, 780.6026466248076`}, {8, 493.5502399832769`}, 
        {12, 312.1838499130911`}, {16, 193.16881196193143`}, {20, 113.41366552793787`},
        {24, 64.20654682074144`}};

errors = {132.12627600307064`, 90.55918065608893`, 60.884899393246535`,
         41.88064141772561`, 29.076284534300132`, 20.049728493077968`, 
         13.950371635495452`};

model[a_, b_, x_] := Sum[NIntegrate[
                           totalvLayer[p, m] (a (1 - b^-p))*10^-38*1.4*150*6.022*10^23,
                           {p, 0.2, 10}], 
                        {m, x - .45, x + .45, .02}]

nlm = NonlinearModelFit[data, model[α, β, x], {α, β}, x, Weights -> 1/errors^2]

Added code to replicate problem

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edited Jul 10, 2013 at 4:35

feynmansafineman

51
3

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asked Jul 10, 2013 at 4:20

feynmansafineman

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