# Semi-discrete probability distribution

I am somewhat new to statistics, so bear with me.

I have a dataset that contains integer values. I created a model for this dataset such that

y(x) = f(x) + c


where f(x) is my model and c is the error in my model. My f function is continuous, and therefore f(x) is continuous. I am assuming that c is noise, and I am trying to compute the distribution of c.

The problem is that, while c is real-valued, it is not continuous, since it is defined as a discrete value minus a real-value (c = y(x) - f(x), where y(x) is an integer).

The histogram of c looks like an exponential distribution, except there are spaces in between (this site wouldn't let me upload my image for lack of reputation). My goal is to create a synthetic dataset that is essentially y(x) = f(x) + c_2, where c_2 are random values from the same distribution as c. What distribution should I use for this?

EDIT: Here's some more information. My data is a time-series that are essentially the number of times an action occurs in a given time bin (Poisson-like). So I have data y(t), but I have noticed that I can form pretty good estimates from another variable x, such that I can form good models using y(x) = f(x). The model is not perfect, however, hence the error term c. My goal is to find the distribution of c so that I can create an estimated new dataset with an underlying f(x) model with the same distribution of noise as the original dataset.

• You should really post a sample of the data you are trying to analyze, as well as give a little more information about where your data is coming from. Commented Feb 27, 2013 at 20:55

If your response is discrete, you rather than express your model as $y(x) = f(x) + c$, you may instead find it better to think in the form $E(y|x) = f(x)$. Then you might for example specify some distribution for $y$ and thereby estimate $f$. For example, you might fit a GLM - say a Poisson or quasi-Poisson model, if you know the form of $f$, or perhaps a GAM if you don't.
As for creating a simulated data set, once you fit your model, you could for example sample from the distribution you used when you fitted, with the mean set to $\hat{f}(x)$ (though this ignores the uncertainty in the estimate of $f$ - that can be taken into account with some additional effort). Or if it suits your circumstance, you could use a bootstrapping scheme (such as resampling the $(x,y)$ pairs.