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I have a time series, e.g. the daily number of visitors on my blog. I have a set of events of some class, like the days when I made a new posting. I want to measure the effect of a new posting on the visitors. However, the effect of a new posting is not limited to the same day the posting was made. It could be days later.

So to generalize it: There is a (either discrete time or continous time) series of data points. There is a set of events given by their time instants. All events are believed to have a similar effect on the series, beginning from the event time and lasting for some time t_L.

Is there a statictical method to approach this problem? Am I missing some assumptions?

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1 Answer 1

This probably belongs on CrossValidated. It seems like a reasonable question but not quite as easy as you might like to answer (without shortcuts).

A statistician's model of this would probably start with a Poisson-distributed response (reasonable for count data). It is most straightforward to assume that posting has an effect on the log-density of the Poisson response for each days. So let's say we count the number of postings on the same day, one day before, two days before, $i$ days before, as $k_i$. Then we could say something like

$$ \begin{split} y_i & \sim \textrm{Poisson}(\eta) \\ \eta & = \exp\left(\beta_0 + \beta_1 \sum_{i=0}^{t_L} k_i \beta_2^i\right) \end{split} $$

That is, the effect of postings on the log-density declines geometrically with time since posting (set $\beta_2=1$ if you want the effects to remain constant over time until $t_L$ at which time they disappear, but this seems less realistic ...)

You could then estimate the parameters by maximum likelihood estimation (write computer code to compute the probability $L$ of the observed outcomes for a given set of parameters $\{\beta_j\}$, then find $\textrm{argmax}_{\{\beta_j\}} L$.

Some of the biggest things left out here:

  • possibility of overdispersion, i.e. there is more variability in the daily counts than expected from the Poisson distribution, even after accounting for the posting pattern. Using a negative binomial rather than a Poisson response is fairly straightforward.
  • temporal autocorrelation. This is a big one; we are implicitly assuming that the $y_i$ are conditionally independent. To allow for autocorrelation we would either have to add a (probably Gaussian) latent variable, or figure out what the marginal variance-covariance matrix would look like in the presence of autocorrelation ...
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