I have data points distributed in this manner:
Now, I want to fit a smooth Gaussian over it. Can anyone tell me what would be the best way to fit a smooth Gaussian for these data points
There can't be. It is obviously not a gaussian at all.
You have to find another model. For example, a constant + a gaussian.
In that case, you could, for example, imitate Gaussian fit for Python . Except that in this example, the model is a Gaussian, when in your case it is not.
So what you want to fit with curve_fit looks more like
def GaussPlusConst(x, c, a, x0, sigma):
return c + a * np.exp(-(x - x0)**2 / (2 * sigma**2))
Then, since you have an extra parameter (the constant), you need to call curve_fit with that extra parameter
popt,pcov = curve_fit(Gauss, x, y, p0=[min(y), max(y)-min(y), mean, sigma])
I used a rough adaptation of the initial guess (anyway, it is just an initial guess). Constant is the minimum, since, gaussian is so sharp that it is practically 0 on large part of the chart.
Then amplitude of the gaussian is not just max(y) but max-min (it is the amplitude of what is added to this constant).
Then estimates for mean and sigma should also be adjusted to concern only the gaussian part (what is over the constant)
mean = sum(x*(y-min(y)))/n
sigma = sum((y-min(y))*(x-mean)**2)/n
But that is just an example (my estimate of the constant is very rough. Plus, you may want to change the model. Even that constant+gaussian is not that realistic)
Related
I am trying to understand whether I am carrying out curve fitting correctly and appropriately using the "curve_fit()" module from scipy within python. I have a script that iterates through each column of my dataframe, plots the data, and then fits a curve to it, and then judges the fit of the equation across the entire data set by calculating the average root mean square error across all of the rows.
First, here is my dataframe:
I would have tried to actually provide the data here but I am not sure how. As an example description of what this data means, I am looking at how pollution levels decay as distance away from a factory site increases, with a distance column and each subsequent column representing a different factory (factory "0", factory "1", factory "2", factory "3", etc.). And so we see pollution levels recorded at each distance moving away from each factory in 30-meter increments, with 0-meters representing the location at the factory site itself. And so I am going by the hypothesis that as move away from a factory site, the pollution levels will start to decay. The research question I am trying to discover here though is, from my specific data set, overall, across all of my studied factories, how does pollution decay with increasing distance away? Is this linear decay? Exponential decay? Logarithmic decay? polynomial decay? Inverse-logistic decay? That is what I am trying to find out.
And so I am trying to fit different curve equations to my data to see which type of curve best represents the decay I am looking at. Here is my code:
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
#Un-comment each of these individual curve functions to try them out individually
# def func(x, a, b):
# return a * x + b
# def func(x, a, b):
# return -a *np.log(x) + b
#def func(x, a, b, c):
# return a * np.exp(-b * x) + c
# def func(x, a, b, c):
# return a / (1 + b*c**(-x))
# def func(x, a, b, c):
# return a*x**2+b*x+c
RMSE_list = []
xdata = data['Distance'].to_numpy()
c = range(8)
for i in c:
# plt.figure(figsize=(3, 3))
greenspace = str(i)
ydata = data[greenspace].to_numpy()
plt.plot(xdata, ydata, 'b-', label='data')
popt, pcov = curve_fit(func, xdata, ydata)
plt.plot(xdata, func(xdata, *popt), 'g--')
#Calculate RMSE
params, _ = curve_fit(func, xdata, ydata)
a, b = params[0], params[1]
yfit1 = a * xdata + b
rmse = np.sqrt(np.mean((yfit1 - ydata) ** 2))
RMSE_list.append(rmse)
RMSE_Average = np.mean(RMSE_list)
print("RMSE Average:")
print(RMSE_Average)
And so here I am trying out curves of linear decay, logarithmic decay, exponential decay, inverse-logistic decay, and second-degree polynomial decay. And so I try out each of these curve fitting equations individually as a unique function and fit this to the data from each factory. I then find the RMSE of each of these fits for each factory, and then average them to get an average RMSE to describe all factories for each curve type. That's all my code does. What I am trying to seek some help here with is to ask whether I am actually doing this right or if it even sounds like my approach here is on the right track. This whole field of curve fitting is very new to me, and I am really just going as far as I know how to at my beginner level. The code works I suppose, but I get some strange outcomes.
When I try this all out I get the following results for each curve fitting attempt:
Overall the idea makes sense to me here, but there are specific issues I encountered that I do not understand. I specifically had issues with the logarithmic, exponential and inverse-logistic curve fittings. For the logistic curve attempt I received the following error for each factory:
C:\Users\MyName\AppData\Local\Temp/ipykernel_7788/37186127.py:8: RuntimeWarning: divide by zero encountered in log return -a *np.log(x) + b
Also, just looking at the curve, the fitted logarithmic curve is fitted way below the actual data, it doesn't even touch!
And for the exponential curve attempt I received the following error for each factory:
C:\Users\MyName\AppData\Roaming\Python\Python39\site-packages\scipy\optimize\minpack.py:833: OptimizeWarning: Covariance of the parameters could not be estimated
I am thinking that perhaps a lot of this might be resolved by setting bounds. When I set the parameter bounds for the exponential attempt to "bounds=(0, [3., 1., 0.5])", based on this curve_fit() documention from scipy: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html
I get the same error message for each factory, but I receive this exponential curve fitting result:
So the visual fitting looks better, but the average RMSE looks the same, which is confusing me. And for the inverse-logistic fitting, it visually looks like the fitting is good, but the average RMSE is so high, which tells me the fit is really bad.
And so my question here is: how can I improve my approach and code here to most accurately and effectively fit these curves to my data and compare them using "curve_fit()" in python? I am thinking maybe this is a matter of methodically setting the bounds for each curve fitting, but I am not sure.
I have a set of data points which, according to the model I want to implement, could be modelled with a certain curve (in this case, a product between an exponential and a complementary error function).
For fitting these data into such a curve, I tried:
import numpy as np
from scipy.optimize import curve_fit
from scipy import special
x_fit = np.linspace(0,1,1000)
def fitted_function(x_fit, c, d, S):
return c*np.exp(((S*d/2)**2)-x_fit*d)*special.erfc(S*d/2-x_fit/S)
FitParameters, FitCovariance = curve_fit(fitted_function, x_data, y_data, maxfev = 100000)
It does not give me any particular error, but the result of the fitting is evidently wrong. I strongly suspect that it has to do with the the part x_fit/S, where the fitting parameter S appears as a denominator.
For example, I encounter the same problem while fitting a simple exponential: if I define the fitting curve with
return a*np.exp(-x_fit/b)
with a, b fitting parameters; since the fitting parameter b appears as a denominator, I find the same problem (i.e. the resulting fitted curve is a horizontal line for some reason).
For the case of a simple exponential I can simple bypass this by doing
return a*np.exp(-b*x_fit)
so that b is not a denominator anymore and the fitted curve is really an exponential curve. For my current case, instead, I cannot do this since S appears ad a numerator and a denominator in different part of the expression.
Any ideas? Thank you in advance!
I am experimenting with Python to fit curves to a series of data-points, summary below:
From the below, it would seem that polynomials of order greater than 2 are the best fit, followed by linear and finally exponential which has the overall worst outcome.
While I appreciate this might not be exponential growth, I just wanted to know whether you would expect the exponential function perform so badly (basically the coefficient of x,b, has been set to 0 and an arbitrary point has been picked on the curve to intersect) or if I have somehow done something wrong in my code to fit.
The code I'm using to fit is as follows:
# Fitting
def exponenial_func(x,a,b,c):
return a*np.exp(-b*x)+c
def linear(x,m,c):
return m*x+c
def quadratic(x,a,b,c):
return a*x**2 + b*x+c
def cubic(x,a,b,c,d):
return a*x**3 + b*x**2 + c*x + d
x = np.array(x)
yZero = np.array(cancerSizeMean['levelZero'].values)[start:]
print len(x)
print len(yZero)
popt, pcov = curve_fit(exponenial_func,x, yZero, p0=(1,1,1))
expZeroFit = exponenial_func(x, *popt)
plt.plot(x, expZeroFit, label='Control, Exponential Fit')
popt, pcov = curve_fit(linear, x, yZero, p0=(1,1))
linearZeroFit = linear(x, *popt)
plt.plot(x, linearZeroFit, label = 'Control, Linear')
popt, pcov = curve_fit(quadratic, x, yZero, p0=(1,1,1))
quadraticZeroFit = quadratic(x, *popt)
plt.plot(x, quadraticZeroFit, label = 'Control, Quadratic')
popt, pcov = curve_fit(cubic, x, yZero, p0=(1,1,1,1))
cubicZeroFit = cubic(x, *popt)
plt.plot(x, cubicZeroFit, label = 'Control, Cubic')
*Edit: curve_fit is imported from the scipy.optimize package
from scipy.optimize import curve_fit
curve_fit tends to perform poorly if you give it a poor initial guess with functions like the exponential that could end up with very large numbers. You could try altering the maxfev input so that it runs more iterations. otherwise, I would suggest trying with with something like:
p0=(1000,-.005,0)
-.01, since it ~doubles from 300 to 500 and you have -b in your eqn, 100 0 since it is ~3000 at 300 (1.5 doublings from 0). See how that turns out
As for why the initial exponential doesn't work at all, your initial guess is b=1, and x is in range of (300,1000) or range. This means python is calculating exp(-300) which either throws an exception or is set to 0. At this point, whether b is increased or decreased, the exponential is going to still be set to 0 for any value in the general vicinity of the initial estimate.
Basically, python uses a numerical method with limited precision, and the exponential estimate went outside of the range of values it can handle
I'm not sure how you're fitting the curves -- are you using polynomial least squares? In that case, you'd expect the fit to improve with each additional degree of flexibility, and you choose the power based on diminishing marginal improvement / outside theory.
The improving fit should look something like this.
I actually wrote some code to do Polynomial Least Squares in python for a class a while back, which you can find here on Github. It's a bit hacky though and loosely commented since I was just using it to solve exercises. Hope it's helpful.
Suppose 'h' is a function of x,y,z and t and it gives us a graph line (t,h) (simulated). At the same time we also have observed graph (observed values of h against t). How can I reduce the difference between observed (t,h) and simulated (t,h) graph by optimizing values of x,y and z? I want to change the simulated graph so that it imitates closer and closer to the observed graph in MATLAB/Python. In literature I have read that people have done same thing by Lavenberg-marquardt algorithm but don't know how to do it?
You are actually trying to fit the parameters x,y,z of the parametrized function h(x,y,z;t).
MATLAB
You're right that in MATLAB you should either use lsqcurvefit of the Optimization toolbox, or fit of the Curve Fitting Toolbox (I prefer the latter).
Looking at the documentation of lsqcurvefit:
x = lsqcurvefit(fun,x0,xdata,ydata);
It says in the documentation that you have a model F(x,xdata) with coefficients x and sample points xdata, and a set of measured values ydata. The function returns the least-squares parameter set x, with which your function is closest to the measured values.
Fitting algorithms usually need starting points, some implementations can choose randomly, in case of lsqcurvefit this is what x0 is for. If you have
h = #(x,y,z,t) ... %// actual function here
t_meas = ... %// actual measured times here
h_meas = ... %// actual measured data here
then in the conventions of lsqcurvefit,
fun <--> #(params,t) h(params(1),params(2),params(3),t)
x0 <--> starting guess for [x,y,z]: [x0,y0,z0]
xdata <--> t_meas
ydata <--> h_meas
Your function h(x,y,z,t) should be vectorized in t, such that for vector input in t the return value is the same size as t. Then the call to lsqcurvefit will give you the optimal set of parameters:
x = lsqcurvefit(#(params,t) h(params(1),params(2),params(3),t),[x0,y0,z0],t_meas,h_meas);
h_fit = h(x(1),x(2),x(3),t_meas); %// best guess from curve fitting
Python
In python, you'd have to use the scipy.optimize module, and something like scipy.optimize.curve_fit in particular. With the above conventions you need something along the lines of this:
import scipy.optimize as opt
popt,pcov = opt.curve_fit(lambda t,x,y,z: h(x,y,z,t), t_meas, y_meas, p0=[x0,y0,z0])
Note that the p0 starting array is optional, but all parameters will be set to 1 if it's missing. The result you need is the popt array, containing the optimal values for [x,y,z]:
x,y,z = popt
h_fit = h(x,y,z,t_meas)
I have two lists .
import numpy
x = numpy.array([7250, ... list of 600 ints ... ,7849])
y = numpy.array([2.4*10**-16, ... list of 600 floats ... , 4.3*10**-16])
They make a U shaped curve.
Now I want to fit a gaussian to that curve.
from scipy.optimize import curve_fit
n = len(x)
mean = sum(y)/n
sigma = sum(y - mean)**2/n
def gaus(x,a,x0,sigma,c):
return a*numpy.exp(-(x-x0)**2/(2*sigma**2))+c
popt, pcov = curve_fit(gaus,x,y,p0=[-1,mean,sigma,-5])
pylab.plot(x,y,'r-')
pylab.plot(x,gaus(x,*popt),'k-')
pylab.show()
I just end up with the noisy original U-shaped curve and a straight horizontal line running through the curve.
I am not sure what the -1 and the -5 represent in the above code but I am sure that I need to adjust them or something else to get the gaussian curve. I have been playing around with possible values but to no avail.
Any ideas?
First of all, your variable sigma is actually variance, i.e. sigma squared --- http://en.wikipedia.org/wiki/Variance#Definition.
This confuses the curve_fit by giving it a suboptimal starting estimate.
Then, your fitting ansatz, gaus, includes an amplitude a and an offset, is this what you actually need? And the starting values are a=-1 (negated bell shape) and offset c=-5. Where do they come from?
Here's what I'd do:
fix your fitting model. Do you want just a gaussian, does it need to be normalized. If it does, then the amplitude a is fixed by sigma etc.
Have a look at the actual data. What's the tail (offset), what's the sign (amplitude sign).
If you're actually want just a gaussian without any bells and whistles, you might not actually need curve_fit: a gaussian is fully defined by two first moments, mean and sigma. Calculate them as you do, plot them over the data and see if you're not all set.
p0 in your call to curve_fit gives the initial guesses for the additional parameters of you function in addition to x. In the above code you are saying that I want the curve_fit function to use -1 as the initial guess for a, -5 as the initial guess for c, mean as the initial guess for x0, and sigma as the guess for sigma. The curve_fit function will then adjust these parameters to try and get a better fit. The problem is your initial guesses at your function parameters are really bad given the order of (x,y)s.
Think a little bit about the order of magnitude of your different parameters for the Gaussian. a should be around the size of your y values (10**-16) as at the peak of the Gaussian the exponential part will never be larger than 1. x0 will give the position within your x values at which the exponential part of your Gaussian will be 1, so x0 should be around 7500, probably somewhere in the centre of your data. Sigma indicates the width, or spread of your Gaussian, so perhaps something in the 100's just a guess. Finally c is just an offset to shift the whole Gaussian up and down.
What I would recommend doing, is before fitting the curve, pick some values for a, x0, sigma, and c that seem reasonable and just plot the data with the Gaussian, and play with a, x0, sigma, and c until you get something that looks at least some what the way you want the Gaussian to fit, then use those as the starting points for curve_fit p0 values. The values I gave should get you started, but may not do exactly what you want. For instance a probably needs to be negative if you want to flip the Gaussian to get a "U" shape.
Also printing out the values that curve_fit thinks are good for your a,x0,sigma, and c might help you see what it is doing and if that function is on the right track to minimizing the residual of the fit.
I have had similar problems doing curve fitting with gnuplot, if the initial values are too far from what you want to fit it goes in completely the wrong direction with the parameters to minimize the residuals, and you could probably do better by eye. Think of these functions as a way to fine tune your by eye estimates of these parameters.
hope that helps
I don't think you are estimating your initial guesses for mean and sigma correctly.
Take a look at the SciPy Cookbook here
I think it should look like this.
x = numpy.array([7250, ... list of 600 ints ... ,7849])
y = numpy.array([2.4*10**-16, ... list of 600 floats ... , 4.3*10**-16])
n = len(x)
mean = sum(x*y)/sum(y)
sigma = sqrt(abs(sum((x-mean)**2*y)/sum(y)))
def gaus(x,a,x0,sigma,c):
return a*numpy.exp(-(x-x0)**2/(2*sigma**2))+c
popy, pcov = curve_fit(gaus,x,y,p0=[-max(y),mean,sigma,min(x)+((max(x)-min(x)))/2])
pylab.plot(x,gaus(x,*popt))
If anyone has a link to a simple explanation why these are the correct moments I would appreciate it. I am going on faith that SciPy Cookbook got it right.
Here is the solution thanks to everyone .
x = numpy.array([7250, ... list of 600 ints ... ,7849])
y = numpy.array([2.4*10**-16, ... list of 600 floats ... , 4.3*10**-16])
n = len(x)
mean = sum(x)/n
sigma = math.sqrt(sum((x-mean)**2)/n)
def gaus(x,a,x0,sigma,c):
return a*numpy.exp(-(x-x0)**2/(2*sigma**2))+c
popy, pcov = curve_fit(gaus,x,y,p0=[-max(y),mean,sigma,min(x)+((max(x)-min(x)))/2])
pylab.plot(x,gaus(x,*popt))
Maybe it is because I use matlab and fminsearch or my fits have to work on much fewer datapoints (~ 5-10), I have much better results with the following starter values (as simple as they are):
a = max(y)-min(y);
imax= find(y==max(y),1);
mean = x(imax);
avg = sum(x.*y)./sum(y);
sigma = sqrt(abs(sum((x-avg).^2.*y) ./ sum(y)));
c = min(y);
The sigma works fine.