Fit exponential distribution with noise I'm trying to fit an exponential with noise (which in this case is a constant $c$) like this one
$$ y(x) = \alpha e^{- \alpha x} + c \text{ ,}$$
having $(x_i, y_i)$ values (So $\alpha$ and $c$ are unknown and are the ones that I want to determine). Without the noise I would simply linearize the values and then apply the least squares method, but with the noise I have no idea how to do that. Are there any formulas to do it? Thank you.
 A: In the absence of a response to my questions relating to the variation about the signal, I'll explain a little about nonlinear least squares.
You can fit a model of the following form:
$y_i = c + \alpha \exp(-\alpha x_i)+\varepsilon_i$, where $E(\varepsilon_i)=0$.
If the $\varepsilon$ values are independent and of constant variance (or close to it), this should be quite a good approach (and would be my idea of a good starting point). If they're also normal it will also be maximum likelihood, and makes for simpler confidence intervals and tests (should you want those).
There's no closed form formula for the parameter estimates. They must be obtained iteratively, generally by taking a linear approximation at a current estimate to get the next estimate. Software to do this is in most stats packages.
Here's an example.
I made a tiny set of (x,y) data (here printed to 4 significant figures):
     x     y
 1.186 2.695
 2.805 2.677
 3.095 2.657
 1.399 2.661
 2.150 2.713
 7.989 2.547
 1.847 2.673
 3.867 2.588
 7.133 2.580
 6.136 2.581
 1.230 2.711
 7.272 2.581

I fitted your model in R (free statistical software), as follows:
expfnfit = nls( y ~ c+a*exp(-a*x) , start=list(c=2,a=.5))  # fits the model

summary(expfnfit) # shows information about the fit

Formula: y ~ c + a * exp(-a * x)

Parameters:
  Estimate Std. Error t value Pr(>|t|)    
c 2.529316   0.008608 293.848  < 2e-16 ***
a 0.229818   0.027285   8.423 7.48e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02448 on 10 degrees of freedom

Number of iterations to convergence: 6 
Achieved convergence tolerance: 1.39e-06


A: First of all, are you sure that the noise is additive? For instance, if your noise was multiplicative, then linearization would have worked.
For an additive noise you can't do much. Use nonlinear regression.
