Ezyfit: Single vs. Two-Step Fitting Problem

Ezfit (actually I am using the updated version from Walter Roberson) fails to accurately determine all 4 parameters in a single curve fit (see Method A). A workaround, setting 1 parameter initially and then fitting the remaining 3 in a second step (Method B), looks instead successful.
Is there a method or setting within Ezfit to obtain accurate 4-parameter results in a single fitting run (Method A)?
% (1) Input Data
x = [6, 16, 26, 36, 46, 56, 66, 66, 76, 86, 96, 126, 176, 186, 246, 286, 306, 336, 366, 396, 426, 486, 496, 506, 546, 546, 616, 656, 726, 756, 806, 816, 836, 946, 966, 976, 1056, 1066, 1116, 1136, 1176, 1296, 1326, 1386, 1426, 1456, 1476, 1586, 1596, 1736, 1806, 1836, 1846, 1886, 2016, 2046, 2106, 2196, 2296, 2346, 2356, 2366, 2556, 2666, 2806, 2856, 2916, 2976, 3116, 3266, 3366, 3416, 3526, 3726, 3876, 3976, 4386, 4636, 4686, 5246, 5346, 6966];
y = [22.932, 13.063, 10.877, 8.7408, 7.4074, 5.8344, 12.693, 4.8185, 3.5444, 2.6637, 2.2515, 4.5284, 1.7597, 1.5207, 1.1754, 0.74499, 0.75637, 0.28157, 0.35862, 0.37542, 0.27759, 0.25008, 0.11389, 0.23569, 0.16777, 0.084717, 0.11865, 0.03168, 0.085227, 0.042803, 0.038632, 0.016028, 0.049869, 0.034689, 0.025157, 0.0064107, 0.017007, 0.0067062, 0.012242, 0.0065558, 0.014677, 0.0048147, 0.0048443, 0.0045972, 0.0059391, 0.0024968, 0.0050464, 0.0024622, 0.0053329, 0.0038574, 0.0035999, 0.0022632, 0.00147, 0.00098146, 0.0012814, 0.0024717, 0.00073639, 0.00073862, 0.0013448, 0.00088996, 0.00096546, 0.00084, 0.00062343, 0.0014789, 0.00050959, 0.00027922, 0.00055839, 0.0019602, 0.00098146, 0.00020941, 0.00028421, 0.00075468, 0.00031831, 0.00055524, 0.00055844, 0.00039789, 0.00031207, 0.00072048, 0.00021507, 0.0004718, 0.00018294, 0.00018086];

% (2) Method A: Perform a simple curve fitting with Ezfit
ft1 = ezfit(x,log(y),’log(a*x^(-b) + c*exp(-d*x))’,’MaxFunEvals’, 1e6, ‘MaxIter’, 1e6); %, ‘usega’, usega);
array2table([ft1.m(1) ft1.m(2) ft1.m(3) ft1.m(4) (ft1.r)^2],’VariableNames’,{‘a’,’b’,’c’,’d’,’R2′})
% a b c d R2
% _____ ______ ______ _____ _______
%
% 56106 2.2236 -85026 49582 0.85991

% (3) Method B: Perform curve fitting by using Ezfit 2 times
% (3.1) With the first Ezfit, find the exponent "b" (among a range of values between 1.5 and 2.5)
% which maximise the coefficient of determination R^2 (I set it as R^2 = 0.99).
% The resulting exponent will be "b_range(idx_max_b) = 1.86".
i = 1;
b_range = 1.5 : 0.01 : 2.5;
for b = b_range
ft2 = ezfit(x,log(y),sprintf(‘log(a*x^(-%.2f) + c*exp(-d*x))’,b),’options’,optimset(‘MaxFunEvals’, 1e6, ‘MaxIter’, 1e6));
ftr(i) = (ft2.r)^2;
i = i + 1;
end
idx_max_b = find(ftr == max(ftr(ftr < 0.99)));

% (3.2) Following the initial determination of the optimal exponent "b" (i.e. "b_range(idx_max_b)"),
% the second Ezfit step finds the remaining parameters
ft3 = ezfit(x,log(y),sprintf(‘log(a*x^(-%.2f) + c*exp(-d*x))’,b_range(idx_max_b)),’options’,optimset(‘MaxFunEvals’, 1e6, ‘MaxIter’, 1e6));
array2table([ft3.m(1) b_range(idx_max_b) ft3.m(2) ft3.m(3) (ft3.r)^2],’VariableNames’,{‘a’,’b’,’c’,’d’,’R2′})
% a b c d R2
% ______ ____ _____ _________ _______
%
% 2084.4 1.86 4.806 0.0064196 0.98323

% (4) Plot
hold on
plot(x, y, ‘o’, ‘MarkerFaceColor’,[0.7 0.7 0.7],’DisplayName’, ‘Input Data’);
xx = 10 : 10000;
plot(xx, ft1.m(1) .* xx.^(-ft1.m(2)) + ft1.m(3) .* exp(-ft1.m(4) .* xx), ‘color’,’blue’, ‘Linewidth’,2, ‘DisplayName’, ‘Method A’);
plot(xx, ft3.m(1) .* xx.^(-b_range(idx_max_b)) + ft3.m(2) .* exp(-ft3.m(3) .* xx), ‘color’, ‘red’, ‘Linewidth’,2, ‘DisplayName’, ‘Method B’)
xlabel(‘x’);
ylabel(‘y’);
legend(‘show’);
set(gca, ‘XScale’, ‘log’, ‘YScale’, ‘log’)Ezfit (actually I am using the updated version from Walter Roberson) fails to accurately determine all 4 parameters in a single curve fit (see Method A). A workaround, setting 1 parameter initially and then fitting the remaining 3 in a second step (Method B), looks instead successful.
Is there a method or setting within Ezfit to obtain accurate 4-parameter results in a single fitting run (Method A)?
% (1) Input Data
x = [6, 16, 26, 36, 46, 56, 66, 66, 76, 86, 96, 126, 176, 186, 246, 286, 306, 336, 366, 396, 426, 486, 496, 506, 546, 546, 616, 656, 726, 756, 806, 816, 836, 946, 966, 976, 1056, 1066, 1116, 1136, 1176, 1296, 1326, 1386, 1426, 1456, 1476, 1586, 1596, 1736, 1806, 1836, 1846, 1886, 2016, 2046, 2106, 2196, 2296, 2346, 2356, 2366, 2556, 2666, 2806, 2856, 2916, 2976, 3116, 3266, 3366, 3416, 3526, 3726, 3876, 3976, 4386, 4636, 4686, 5246, 5346, 6966];
y = [22.932, 13.063, 10.877, 8.7408, 7.4074, 5.8344, 12.693, 4.8185, 3.5444, 2.6637, 2.2515, 4.5284, 1.7597, 1.5207, 1.1754, 0.74499, 0.75637, 0.28157, 0.35862, 0.37542, 0.27759, 0.25008, 0.11389, 0.23569, 0.16777, 0.084717, 0.11865, 0.03168, 0.085227, 0.042803, 0.038632, 0.016028, 0.049869, 0.034689, 0.025157, 0.0064107, 0.017007, 0.0067062, 0.012242, 0.0065558, 0.014677, 0.0048147, 0.0048443, 0.0045972, 0.0059391, 0.0024968, 0.0050464, 0.0024622, 0.0053329, 0.0038574, 0.0035999, 0.0022632, 0.00147, 0.00098146, 0.0012814, 0.0024717, 0.00073639, 0.00073862, 0.0013448, 0.00088996, 0.00096546, 0.00084, 0.00062343, 0.0014789, 0.00050959, 0.00027922, 0.00055839, 0.0019602, 0.00098146, 0.00020941, 0.00028421, 0.00075468, 0.00031831, 0.00055524, 0.00055844, 0.00039789, 0.00031207, 0.00072048, 0.00021507, 0.0004718, 0.00018294, 0.00018086];

% (4) Plot
hold on
plot(x, y, ‘o’, ‘MarkerFaceColor’,[0.7 0.7 0.7],’DisplayName’, ‘Input Data’);
xx = 10 : 10000;
plot(xx, ft1.m(1) .* xx.^(-ft1.m(2)) + ft1.m(3) .* exp(-ft1.m(4) .* xx), ‘color’,’blue’, ‘Linewidth’,2, ‘DisplayName’, ‘Method A’);
plot(xx, ft3.m(1) .* xx.^(-b_range(idx_max_b)) + ft3.m(2) .* exp(-ft3.m(3) .* xx), ‘color’, ‘red’, ‘Linewidth’,2, ‘DisplayName’, ‘Method B’)
xlabel(‘x’);
ylabel(‘y’);
legend(‘show’);
set(gca, ‘XScale’, ‘log’, ‘YScale’, ‘log’) Ezfit (actually I am using the updated version from Walter Roberson) fails to accurately determine all 4 parameters in a single curve fit (see Method A). A workaround, setting 1 parameter initially and then fitting the remaining 3 in a second step (Method B), looks instead successful.
Is there a method or setting within Ezfit to obtain accurate 4-parameter results in a single fitting run (Method A)?
% (1) Input Data
x = [6, 16, 26, 36, 46, 56, 66, 66, 76, 86, 96, 126, 176, 186, 246, 286, 306, 336, 366, 396, 426, 486, 496, 506, 546, 546, 616, 656, 726, 756, 806, 816, 836, 946, 966, 976, 1056, 1066, 1116, 1136, 1176, 1296, 1326, 1386, 1426, 1456, 1476, 1586, 1596, 1736, 1806, 1836, 1846, 1886, 2016, 2046, 2106, 2196, 2296, 2346, 2356, 2366, 2556, 2666, 2806, 2856, 2916, 2976, 3116, 3266, 3366, 3416, 3526, 3726, 3876, 3976, 4386, 4636, 4686, 5246, 5346, 6966];
y = [22.932, 13.063, 10.877, 8.7408, 7.4074, 5.8344, 12.693, 4.8185, 3.5444, 2.6637, 2.2515, 4.5284, 1.7597, 1.5207, 1.1754, 0.74499, 0.75637, 0.28157, 0.35862, 0.37542, 0.27759, 0.25008, 0.11389, 0.23569, 0.16777, 0.084717, 0.11865, 0.03168, 0.085227, 0.042803, 0.038632, 0.016028, 0.049869, 0.034689, 0.025157, 0.0064107, 0.017007, 0.0067062, 0.012242, 0.0065558, 0.014677, 0.0048147, 0.0048443, 0.0045972, 0.0059391, 0.0024968, 0.0050464, 0.0024622, 0.0053329, 0.0038574, 0.0035999, 0.0022632, 0.00147, 0.00098146, 0.0012814, 0.0024717, 0.00073639, 0.00073862, 0.0013448, 0.00088996, 0.00096546, 0.00084, 0.00062343, 0.0014789, 0.00050959, 0.00027922, 0.00055839, 0.0019602, 0.00098146, 0.00020941, 0.00028421, 0.00075468, 0.00031831, 0.00055524, 0.00055844, 0.00039789, 0.00031207, 0.00072048, 0.00021507, 0.0004718, 0.00018294, 0.00018086];

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