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finding a trend line where x axis semilog and y axis linear and trend line follow the eqn y=a*exp(b*x)

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Arvind
Arvind 2023-7-4
评论： Star Strider 2023-7-7
can you help to find a trend line and the coefficient values for semilox plot
where x- axis == semilog == time axis;
and y-axis == linear == depth axis;
and trend line follow the exponential eqn y = a*exp(b*x)

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回答（2 个）

Image Analyst
Image Analyst 2023-7-4
See attached demo for fitting data to exponentials. Adapt as needed.
Star Strider
Star Strider 2023-7-4
Ran in:
AA straight horizonta line would be the best model, since ‘Depth’ does not change at all over time —
T1 = readtable('data_mat_code.xlsx')
T1 = 914×2 table
time Depth ______ ______ 208.57 72.183 208.85 72.183 209.1 72.183 209.24 72.183 209.28 72.183 209.22 72.183 209.03 72.183 208.7 72.183 208.23 72.183 207.62 72.183 206.8 72.183 206.07 72.183 205.82 72.183 206 72.183 206.33 72.183 206.43 72.183
time = T1.time;
Depth = T1.Depth;
VN = T1.Properties.VariableNames;
DepthStats = [min(Depth); max(Depth); std(Depth)]
DepthStats = 3×1
72.1831 72.1831 0.0000
fcn = @(b,x) b(1).*exp(b(2).*x);
mdl = fitnlm(time,Depth,fcn,rand(2,1))
mdl =
Nonlinear regression model: y ~ b1*exp(b2*x) Estimated Coefficients: Estimate SE tStat pValue __________ __________ __________ ___________ b1 72.183 8.2362e-10 8.7641e+10 0 b2 2.0514e-12 5.7373e-14 35.756 1.0175e-175 Number of observations: 914, Error degrees of freedom: 912 Root Mean Squared Error: 6.87e-10 R-Squared: -4.75e+07, Adjusted R-Squared -4.76e+07 F-statistic vs. constant model: 0, p-value = 1
[y,yci] = predict(mdl, time);
figure
hp1 = semilogx(time, Depth, '.', 'DisplayName','Data');
hold on
hp2 = plot(time, y, '-r', 'DisplayName','Regression');
hp3 = plot(time, yci, '--r', 'DisplayName','95% Confidence Intervals');
hold off
grid
xlabel(VN{1})
ylabel(VN{2})
legend([hp1,hp2,hp3(1)], 'Location','best')
Perhaps this will work better on a different data set.
.
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Arvind
Arvind 2023-7-7
@Image Analyst Yes,You are right, I want to fit the line only on those data which have linearity or bets trend and the trend line what I want upto depth<510 , but also want to that trend line plot on all data
@Star Strider Thanks for your work
Star Strider
Star Strider 2023-7-7
Ran in:
My pleasure!
For the depths < 510, try these —
T1 = readtable('data_mat_code.xlsx', 'VariableNamingRule','preserve')
T1 = 1549×3 table
depth depth_1 transit time ______ _______ ____________ 0.0558 0.18307 213.4 0.2082 0.68307 213.74 0.3606 1.1831 213.71 0.513 1.6831 213.21 0.6654 2.1831 213.05 0.8178 2.6831 213.38 0.9702 3.1831 213.72 1.1226 3.6831 214.91 1.275 4.1831 216.66 1.4274 4.6831 216.73 1.5798 5.1831 215.62 1.7322 5.6831 214.53 1.8846 6.1831 213.2 2.037 6.6831 211.9 2.1894 7.1831 211.19 2.3418 7.6831 211.25
time = T1.('transit time');
Depth = T1.depth;
Depth_1 = T1.depth_1;
Lv1 = Depth < 510;
Lv2 = Depth_1 < 510;
VN = T1.Properties.VariableNames;
DepthStats = [min(Depth); max(Depth); max(Depth)-min(Depth); std(Depth)]
DepthStats = 4×1
0.0558 235.9710 235.9152 68.1688
Depth_1Stats = [min(Depth_1); max(Depth_1); max(Depth_1)-min(Depth_1); std(Depth_1)]
Depth_1Stats = 4×1
0.1831 774.1831 774.0000 223.6511
fcn = @(b,x) b(1).*exp(b(2).*x);
% ftns = @(b) norm(Depth - fcn(b,time));
% [B1,fv1,exitflag1,output1] = ga(ftns,2);
% B1
% fv1
% output1.generations
%
% mdl1 = fitnlm(time,Depth,fcn,B1)
% [y,yci] = predict(mdl1, time);
%
% figure
% hp1 = semilogx(time, Depth, '.', 'DisplayName','Data');
% hold on
% hp2 = plot(time, y, '-r', 'DisplayName','Regression');
% hp3 = plot(time, yci, '--r', 'DisplayName','95% Confidence Intervals');
% hold off
% grid
% title('All Data')
% xlabel(VN{3})
% ylabel(VN{1})
% legend([hp1,hp2,hp3(1)], 'Location','best')
%
% ftns = @(b) norm(Depth_1 - fcn(b,time));
% [B2,fv2,exitflag2,output2] = ga(ftns,2);
% B2
% fv2
% output2.generations
%
% mdl2 = fitnlm(time,Depth_1,fcn,B2)
% [y,yci] = predict(mdl2, time);
%
% figure
% hp1 = semilogx(time, Depth_1, '.', 'DisplayName','Data');
% hold on
% hp2 = plot(time, y, '-r', 'DisplayName','Regression');
% hp3 = plot(time, yci, '--r', 'DisplayName','95% Confidence Intervals');
% hold off
% grid
% title('All Data')
% xlabel(VN{3})
% ylabel(strrep(VN{2},'_','\_'))
% legend([hp1,hp2,hp3(1)], 'Location','best')
ftns = @(b) norm(Depth(Lv1) - fcn(b,time(Lv1)));
[B3,fv3,exitflag3,output3] = ga(ftns,2);
Optimization terminated: average change in the fitness value less than options.FunctionTolerance.
B3
B3 = 1×2
10.2959 0.0117
fv3
fv3 = 2.8890e+03
output3.generations
ans = 114
mdl3 = fitnlm(time(Lv1),Depth(Lv1),fcn,B3)
Warning: Some columns of the Jacobian are effectively zero at the solution, indicating that the model is insensitive to some of its parameters. That may be because those parameters are not present in the model, or otherwise do not affect the predicted values. It may also be due to numerical underflow in the model function, which can sometimes be avoided by choosing better initial parameter values, or by rescaling or recentering. Parameter estimates may be unreliable.
mdl3 =
Nonlinear regression model: y ~ b1*exp(b2*x) Estimated Coefficients: Estimate SE tStat pValue __________ __________ __________ ______ b1 1.8762e+07 1.5549e-14 1.2066e+21 0 b2 -0.059993 5.7701e-05 -1039.7 0 Number of observations: 1549, Error degrees of freedom: 1548 Root Mean Squared Error: 55.8 R-Squared: 0.33, Adjusted R-Squared 0.33 F-statistic vs. zero model: 7.7e+03, p-value = 0
[y,yci] = predict(mdl3, time(Lv1));
figure
hp1 = semilogx(time(Lv1), Depth(Lv1), '.', 'DisplayName','Data');
hold on
hp2 = plot(time(Lv1), y, '-r', 'DisplayName','Regression');
hp3 = plot(time(Lv1), yci, '--r', 'DisplayName','95% Confidence Intervals');
hold off
grid
xlabel(VN{3})
ylabel(VN{1})
title('Depth < 510')
legend([hp1,hp2,hp3(1)], 'Location','best')
ftns = @(b) norm(Depth_1(Lv2) - fcn(b,time(Lv2)));
[B4,fv4,exitflag4,output4] = ga(ftns,2);
Optimization terminated: average change in the fitness value less than options.FunctionTolerance.
B4
B4 = 1×2
10.3339 0.0156
fv4
fv4 = 5.4074e+03
output4.generations
ans = 93
mdl4 = fitnlm(time(Lv2),Depth_1(Lv2),fcn,B4)
Warning: Rank deficient, rank = 1, tol = 4.103161e-07.
Warning: Some columns of the Jacobian are effectively zero at the solution, indicating that the model is insensitive to some of its parameters. That may be because those parameters are not present in the model, or otherwise do not affect the predicted values. It may also be due to numerical underflow in the model function, which can sometimes be avoided by choosing better initial parameter values, or by rescaling or recentering. Parameter estimates may be unreliable.
mdl4 =
Nonlinear regression model: y ~ b1*exp(b2*x) Estimated Coefficients: Estimate SE tStat pValue __________ __________ __________ ______ b1 1.1436e+09 1.378e-16 8.2993e+24 0 b2 -0.076496 3.0998e-05 -2467.8 0 Number of observations: 1020, Error degrees of freedom: 1019 Root Mean Squared Error: 56 R-Squared: 0.855, Adjusted R-Squared 0.855 F-statistic vs. zero model: 2.71e+04, p-value = 0
[y,yci] = predict(mdl4, time(Lv2));
figure
hp1 = semilogx(time(Lv2), Depth_1(Lv2), '.', 'DisplayName','Data');
hold on
hp2 = plot(time(Lv2), y, '-r', 'DisplayName','Regression');
hp3 = plot(time(Lv2), yci, '--r', 'DisplayName','95% Confidence Intervals');
hold off
grid
title('Depth\_1 < 510')
xlabel(VN{3})
ylabel(strrep(VN{2},'_','\_'))
legend([hp1,hp2,hp3(1)], 'Location','best')
That is the best I can do with these data.
(The ‘All Data’ regressions are in the earlier series, and since it is very difficult to get all of these working in the same run, I am only calculating and plotting the ‘Depth < 510’ and ‘Depth_1 < 510’ regressions here.
.

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