How do you fit non-negative exponential decay that is biased with non-uniform noise over time?

We often make assumptions when we model exponential decays.

The most common assumptions that I know of are, 1) the signal is contaminated with zero-mean, gaussian distributed noise and 2) the noise distribution is uniform across time with the decay.

What do we do when we know these assumptions are both false. In my particular case, the signal is biased by a positive, non-zero mean, noise distribution that is dependent on the signal intensity, and thus the noise distribution changes with time as the signal decays.

Answer 1

[John D'Errico]

Ok, so postulate some sort of behavior for that distribution as a function of time. Use a couple of parameters to model the behavior, choosing some logical candidate distribution. It need not be perfect, just as good as you can do.

Then use maximum likelihood estimation to find the parameters for the noise model, as well as the exponential decay model. This is just an optimization, although you will probably need constraints on the parameters. And most of the time, you will need to optimize the log of the likelihood function to make things numerically tractable.

So, for example, you might postulate some sort of gamma distribution for the noise model, where the gamma parameters are a function of time. Or, pick some other distribution, anything from lognormal, to beta, even uniform. Whatever makes sense in context.

Sorry, I won't/can't write it for you, since you need to do the work up front to choose these models. But MLE is simple in concept. It ends up as a product of probabilities (thus sum of logs when you log it), which you need to optimize over.

[Eric Diaz]

I need to think about your answer for a little while before I really understand it, but I think it was helpful.

Say for example,

Signal(t)=A*exp(-Bt)
Sigma(t)=Postulated behavior
NoiseBiasedSignal(Signal(t),Sigma(t))

[John D'Errico]

Yes. So assume the signal is of the form of a decaying exponential, PLUS an additive noise term. Or, you might assume the noise variance (or whatever scale parameter you use) is proportional to the signal, in which case you would have a naturally decreasing noise.

There is one issue that you will need to deal with - the one-sided noise. That will cause issues once you start optimizing, forcing you to use a constrained optimizer, perhaps fmincon. For example, some sets of parameters for the signal(t) function could result in a prediction for some point that MUST have a negative noise value. Since that would clearly be impossible, conflicting with your assumptions, it will cause the MLE objective function to blow up. Again, a constrained optimizer will be necessary.

[Eric Diaz]

What if I can't postulate a behavior for the distribution over time but I can calculate the Sigma (and the NoiseBiasedSignal) specifically for each time point based on knowledge of the noise distribution at a given time point?

Is there a way to plug that in?

[John D'Errico]

Sorry, but no. If all that you know is the standard deviation, there are potentially many possible different distributions that have the same standard deviations. Each of those potential distributions will have very different characteristics in the modeling.

So many people assume that since they know the standard deviation, then they know everything important about the distribution under study. This probably comes from the prevalence of the Normal (Gaussian) distribution, since then knowing the mean and variance of a Normal distribution tells you all that you need to know. In this case, your distribution is VERY non-normal.

At best, you might ASSUME the noise distribution follows some specific distribution. Pick a gamma distribution, or a lognormal, for example. Once the variance of that distribution is known, then you can make inferences about the actual distribution parameters of that distribution. Depending on the distribution you choose, this may not be sufficient information to completely compute all the parameters of the distribution.

[Eric Diaz]

Thank you for your input.

I'm not quite sure that I understand your response.

Let me clarify my previous question.

My data is a set of images which decay over time. For each image at each time point, I assume a noise distribution, which is related to a non-central chi distribution, however the degrees of freedom are slightly more restricted. At any rate, I can compute the effective sigma for a particular image with pretty good accuracy.

I can calculate the sigma for each time point, which would bias the signal at that particular time point. The problem is that I don't know of a clear temporal behavior of the noise distribution between time points. What I do know, is that the noise distribution depends on the overall signal strength. Seems kind of circular, that the signal depends on the noise, and the noise depends on the signal, but that is how I understand it.

In my particular case, the pure (non-noise biased) signal decays via a mono- or bi-exponential decay. I am able to compute a noise-biased signal, assuming uniform noise across time, by incorporating the signal equation into the equation for a specific noise distribution.

The more complex problem is incorporating the signal equation into the noise distribution which is non-uniform over time. I am able to compute the Sigmas for each time point using different equations based on the SNR of a particular image.

Answer 2

[Eric Diaz]

If anyone can clarify John's last response or help in replying to my last comment, I would greatly appreciate it.