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Exponential fit with the least squares Python

I have a very specific task, where I need to find the slope of my exponential function.
I have two arrays, one denoting the wavelength range between 400 and 750 nm, the other the absorption spectrum. x = wavelengths, y = absorption.
My fit function should look something like that:
y_mod = np.float(a_440) * np.exp(-S*(x - 440.))
where S is the slope and in the image equals 0.016, which should be in the range of S values I should get (+/- 0.003). a_440 is the reference absorption at 440 nm, x is the wavelength.
Modelled vs. original plot:
I would like to know how to define my function in order to get an exponential fit (not on log transformed quantities) of it without guessing beforehand what the S value is.
What I've tried so far was to define the function in such way:
def func(x, a, b):
return a * np.exp(-b * (x-440))
And it gives pretty nice matches
fitted vs original.
What I'm not sure is whether this approach is correct or should I do it differently?
How would one use also the least squares or the absolute differences in y approaches for minimization in order to remove the effect of overliers?
Is it possible to also add random noise to the data and recompute the fit?

Your situation is the same as the one described in the documentation for scipy's curve_fit.
The problem you're incurring is that your definition of the function accepts only one argument when it should receive three: x (the independent variable where the function is evaluated), plus a_440 and S.
Cleaning a bit, the function should be more like this.
def func(x, A, S):
return A*np.exp(-S*(x-440.))
It might be that you run into a warning about the covariance matrix. you solve that by providing a decent starting point to the curve_fit through the argument p0 and providing a list. For example in this case p0=[1,0.01] and in the fitting call it would look like the following
curve_fit(func, x, y, p0=[1,0.01])

Using scipy optimize for MLE estimate and curve fitting

I randomly generated 1000 data points using the weights I know are true for the normal distribution. Now I am trying to minimize the -log likelihood function to estimate the values of sig^2 and the weights. I sort of get the process conceptually, but when I try to code it I'm just lost.
This is my model:
p(y|x, w, sig^2) = N(y|w0+w1x+...+wnx^n, sig^2)
I've been googling for a while now and I've learned the scipy.stats.optimize.minimize function is good for this, but I can't get it to work right. Every solution I have tried has worked for the example I got the solution from, but I'm unable to extrapolate it to my problem.
x = np.linspace(0, 1000, num=1000)
data = []
for y in x:
data.append(np.polyval([.5, 1, 3], y))
#plot to confirm I do have a normal distribution...
data.sort()
pdf = stats.norm.pdf(data, np.mean(data), np.std(data))
plt.plot(test, pdf)
plt.show()
#This is where I am stuck.
logLik = -np.sum(stats.norm.logpdf(data, loc=??, scale=??))
I have found that the equation error(w) = .5*sum(poly(x_n, w) - y_n)^2 is relevant for minimizing the error of the weights, which therefore maximizes my likelihood for the weights, but I don't understand how to code this... I have found a similar relationship for sig^2, but have the same problem. Can somebody clarify how to do this to help my curve fitting? Maybe go as far to post psuedo code I can use?

Yes, implementing likelihood fitting with minimize is tricky, I spend a lot of time on it. Which is why I wrapped it. If I may shamelessly plug my own package symfit, your problem can be solved by doing something like this:
from symfit import Parameter, Variable, Likelihood, exp
import numpy as np
# Define the model for an exponential distribution
beta = Parameter()
x = Variable()
model = (1 / beta) * exp(-x / beta)
# Draw 100 samples from an exponential distribution with beta=5.5
data = np.random.exponential(5.5, 100)
# Do the fitting!
fit = Likelihood(model, data)
fit_result = fit.execute()
I have to admit I don't exactly understand your distribution, since I don't understand the role of your w, but perhaps with this code as an example, you'll know how to adapt it.
If not, let me know the full mathematical equation of your model so I can help you further.
For more info check the docs. (For a more technical description of what happens under the hood, read here and here.)

I think there's an issue with your setup. With maximum likelihood, you obtain the parameters that maximize the probability of observing your data (given a certain model). Your model seems to be:
where epsilon is N(0, sigma).
So you maximize it:
or equivalently take logs to get:
The f in this case is the log-normal probability density function which you can get with stats.norm.logpdf. You should then use scipy.minimize to maximize an expression that will be the summation of stats.norm.logpdf evaluated at each of the i points, from 1 to your sample size.
If I've understood you correctly, your code is missing having a y vector plus an x vector! Show us a sample of those vectors and I can update my answer to include a sample code for estimating MLE with that date.

Reducing difference between two graphs by optimizing more than one variable in MATLAB/Python?

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)

Gaussian fit in Python - parameters estimation

I want to fit an array of data (in the program called "data", of size "n") with a Gaussian function and I want to get the estimations for the parameters of the curve, namely the mean and the sigma. Is the following code, which I found on the Web, a fast way to do that? If so, how can I actually get the estimated values of the parameters?
import pylab as plb
from scipy.optimize import curve_fit
from scipy import asarray as ar,exp
x = ar(range(n))
y = data
n = len(x) #the number of data
mean = sum(x*y)/n #note this correction
sigma = sum(y*(x-mean)**2)/n #note this correction
def gaus(x,a,x0,sigma,c):
return a*exp(-(x-x0)**2/(sigma**2))+c
popt,pcov = curve_fit(gaus,x,y,p0=[1,mean,sigma,0.0])
print popt
print pcov
plt.plot(x,y,'b+:',label='data')
plt.plot(x,gaus(x,*popt),'ro:',label='fit')
plt.legend()
plt.title('Fig. 3 - Fit')
plt.xlabel('q')
plt.ylabel('data')
plt.show()

To answer your first question, "Is the following code, which I found on the Web, a fast way to do that?"
The code that you have is in fact the right way to proceed with fitting your data, when you believe is Gaussian and know the fitting function (except change the return function to
a*exp(-(x-x0)**2/(sigma**2)).
I believe for a Gaussian function you don't need the constant c parameter.
A common use of least-squares minimization is curve fitting, where one has a parametrized model function meant to explain some phenomena and wants to adjust the numerical values for the model to most closely match some data. With scipy, such problems are commonly solved with scipy.optimize.curve_fit.
To answer your second question, "If so, how can I actually get the estimated values of the parameters?"
You can go to the link provided for scipy.optimize.curve_fit and find that the best fit parameters reside in your popt variable. In your example, popt will contain the mean and sigma of your data. In addition to the best fit parameters, pcov will contain the covariance matrix, which will have the errors of your mean and sigma. To obtain 1sigma standard deviations, you can simply use np.sqrt(pcov) and obtain the same.

Fitting gaussian to a curve in Python II

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.