I am trying to fit three time-evolution curves with two rate constants, k1 and k2. The system of equations I am trying to solve is:
A_t = A_0 * exp(-k1*t)
B_t = [A_0 * k1/(k2-k1)]* exp(-k1*t) - [A_0*(k1/(k2-k1)-B_0] * exp(-k2*t)
C_t = [A_0 * -k2/(k2-k1) ]* exp(-k1*t) + [A_0*(k1/(k2-k1)-B_0] * exp(-k2*t) + A_0 + B_0
where I want fit the best values of k1 and k2 to my data values of A,B and C, where A_t etc is the current population of A at time t, A_0=0.4 and B_0=0.6.
To solve this I am using the scipy.optimize.curve_fit function where I set up the equations as matrices u and w. In the following, I have manually entered the A_0=0.4 and B_0=0.6 into the function (which relates to part 2 of my question at the bottom):
def func(t,k1,k2):
u = np.array([[0.4,0,0],
[0.4*k1/(k2-k1),-0.4*(k1/(k2-k1))+0.6,0],
[0.4*(-k2/(k2-k1)),0.4*k1/(k2-k1)-0.6,1]])
w = np.array([np.exp(-t*k1),
np.exp(-t*k2),
np.ones_like(t)])
return np.dot(u,w).flatten()
To solve for some test data, I create a test set where I set k1=0.03 and k2=0.003:
t=np.arange(1000)*0.5
test=func(t,0.03,0.004).reshape((3,1000))
test+=np.random.normal(size=test.shape)*0.01
which produces the following plot:
When I then try to fit values of k1 and k2, I get the following error:
popt,popc=optimize.curve_fit(func,t,test.flatten(),method='lm')
/usr/local/lib/python3.6/site-packages/ipykernel_launcher.py:4: RuntimeWarning: divide by zero encountered in double_scalars after removing the cwd from sys.path. /usr/local/lib/python3.6/site-packages/ipykernel_launcher.py:5: RuntimeWarning: divide by zero encountered in double_scalars """ /usr/local/lib/python3.6/site-packages/scipy/optimize/minpack.py:785: OptimizeWarning: Covariance of the parameters could not be estimated category=OptimizeWarning)
I understand that there is a divide by zero error somewhere here, but I am not sure where it is or how to solve it. So, my questions are:
A_0 and B_0 into optimize.curve_fit, rather than manually entering the concentrations as I have done above? My understanding is that only the independent variable t and the unknowns k1 and k2 can be passed to the function? Thank you for any help that can be provided
Per the comments, here is an example of using scipy's differential_evolution genetic algorithm module for estimating initial parameters. This module uses the Latin Hypercube algorithm to ensure a thorough search of parameter space, and that algorithm requires bounds within which to search. In this example, those bounds are taken from the data minimum and maximum values. The fitting completes with a call to curve_fit() without passing bounds from the genetic algorithm search, in case the best parameters are outside those bounds.
import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings
xData = numpy.array([19.1647, 18.0189, 16.9550, 15.7683, 14.7044, 13.6269, 12.6040, 11.4309, 10.2987, 9.23465, 8.18440, 7.89789, 7.62498, 7.36571, 7.01106, 6.71094, 6.46548, 6.27436, 6.16543, 6.05569, 5.91904, 5.78247, 5.53661, 4.85425, 4.29468, 3.74888, 3.16206, 2.58882, 1.93371, 1.52426, 1.14211, 0.719035, 0.377708, 0.0226971, -0.223181, -0.537231, -0.878491, -1.27484, -1.45266, -1.57583, -1.61717])
yData = numpy.array([0.644557, 0.641059, 0.637555, 0.634059, 0.634135, 0.631825, 0.631899, 0.627209, 0.622516, 0.617818, 0.616103, 0.613736, 0.610175, 0.606613, 0.605445, 0.603676, 0.604887, 0.600127, 0.604909, 0.588207, 0.581056, 0.576292, 0.566761, 0.555472, 0.545367, 0.538842, 0.529336, 0.518635, 0.506747, 0.499018, 0.491885, 0.484754, 0.475230, 0.464514, 0.454387, 0.444861, 0.437128, 0.415076, 0.401363, 0.390034, 0.378698])
def func(t, n_0, L, offset): #exponential curve fitting function
return n_0*numpy.exp(-L*t) + offset
# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
val = func(xData, *parameterTuple)
return numpy.sum((yData - val) ** 2.0)
def generate_Initial_Parameters():
# min and max used for bounds
maxX = max(xData)
minX = min(xData)
maxY = max(yData)
minY = min(yData)
parameterBounds = []
parameterBounds.append([minX, maxX]) # seach bounds for n_0
parameterBounds.append([minX, maxX]) # seach bounds for L
parameterBounds.append([0.0, maxY]) # seach bounds for Offset
# "seed" the numpy random number generator for repeatable results
result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
return result.x
# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()
# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()
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()
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)