如何对实验数据进行非线性数据拟合

Question

I have some experiment data. Hereby, I need to fit the following function to determine one of the variable. A Levenberg–Marquardt least-squares algorithm was used in this procedure.

I have used curve fitting option in Igor Pro software. I defined new fit function and tried to define independent and dependent variable. Nevertheless, I don't know what is the reason that I got the this error:

"The fitting function returned INF for at least one X variable"

My function is :

sin(theta) = -1+2*sqrt(alpha/x)*exp(-beta*(x-alpha)^2)

beta = 1.135e-4;

sin(theta) = [-0.81704 -0.67649 -0.83137 -0.73468 -0.66744 -0.43602 0.45368 0.75802 0.96705 0.99717 ]

x = [72.01 59.99 51.13 45.53 36.15 31.66 30.16 29.01 25.62 23.47 ]

Is there any suggestion to find alpha variable here?

Is there any handy software or program for nonlinear curve fitting?

Answer 1

In gnuplot, it would look like this. The fit is not great, but that's not the "fault" of gnuplot, but apparently this data cannot be fitted with this function very well.

Code:

### nonlinear curve fitting
reset session

$Data <<EOD
72.01 -0.81704
59.99 -0.67649
51.13 -0.83137
45.53 -0.73468
36.15 -0.66744
31.66 -0.43602
30.16 0.45368
29.01 0.75802
25.62 0.96705
23.47 0.99717
EOD

f(x) = -1+2*sqrt(alpha/x)*exp(-beta*(x-alpha)**2)

# initial guessed values
alpha = 25
beta = 1
set fit nolog results
fit f(x) $Data u 1:2 via alpha,beta

plot $Data u 1:2 w lp pt 7, \
    f(x) lc rgb "red"

print sprintf("alpha=%g, beta=%g",alpha,beta)
### end of code

Result:

alpha=25.818, beta=0.0195229

Answer 2

If it might be of some use, my equation search on your data turned up a good fit to a standard 4-parameter logistic equation "y = d + (a - d) / (1.0 + pow(x / c, b))" with parameters a = 0.96207949, b = 44.14292256, c = 30.67324939, and d = -0.74830947 yielding RMSE = 0.0565 and R-squared = 0.9943, and I have included code for a Python graphical fitter using this equation.

import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit


theta = [-0.81704, -0.67649, -0.83137, -0.73468, -0.66744, -0.43602, 0.45368, 0.75802, 0.96705, 0.99717]
x = [72.01, 59.99, 51.13, 45.53, 36.15, 31.66, 30.16, 29.01, 25.62, 23.47]

# rename to match previous example code
xData = numpy.array(x)
yData = numpy.array(theta)

# StandardLogistic4Parameter equation from zunzun.com
def func(x, a, b, c, d):
    return  d + (a - d) / (1.0 + numpy.power(x / c, b))


# these are the same as the scipy defaults
initialParameters = numpy.array([1.0, 1.0, 1.0, 1.0])

# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)

modelPredictions = func(xData, *fittedParameters) 

absError = modelPredictions - yData

SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))

print('Parameters:', fittedParameters)
print('RMSE:', RMSE)
print('R-squared:', Rsquared)

print()


##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    # first the raw data as a scatter plot
    axes.plot(xData, yData,  'D')

    # create data for the fitted equation plot
    xModel = numpy.linspace(min(xData), max(xData))
    yModel = func(xModel, *fittedParameters)

    # now the model as a line plot
    axes.plot(xModel, yModel)

    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot

graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)

Answer 3

Matlab

I slightly changed the function, -1 changed to -gamma and optimize to find gamma

The code is as follow

ydata =  [-0.81704 -0.67649 -0.83137 -0.73468 -0.66744 -0.43602 0.45368...
    0.75802 0.96705 0.99717 ];
xdata = [72.01 59.99 51.13 45.53 36.15 31.66 30.16 29.01 25.62 23.47 ];

sin_theta = @(alpha, beta, gamma, xdata) -gamma+2.*sqrt(alpha./xdata).*exp(beta.*(xdata-alpha).^2);

%Fitting function as function of array(x) required by lsqcurvefit
f = @(x,xdata) sin_theta(x(1),x(2), x(3),xdata);
% [alpha, beta, gamma]
x0 = [25, 0, 1] ;

options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt', 'FunctionTolerance', 1e-30);


[x,resnorm,residual,exitflag,output] = lsqcurvefit(f,x0,xdata,ydata,[], [], options);

% Accuracy 
RMSE = sqrt(sum(residual.^2)/length(residual));

alpha = x(1); beta = x(2); gamma = x(3);

%Plotting data
data = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ro',data,f(x,data),'b-', 'linewidth', 3)
legend('Data','Fitted exponential')
title('Data and Fitted Curve')
set(gca,'FontSize',20)

Result

alpha = 26.0582, beta = -0.0329, gamma = 0.7881 instead of 1, RMSE = 0.1498

Graph