7.5 Residual Analysis

There isn’t an obvious or unique notion for what is the residual of a point process model and different people have come up with different ideas for what could be done in practice. One attractive idea involves transforming the original data via a time shift that is governed by the conditional intensity model.

If \(\lambda(t\mid H_t)\) is the conditional intensity of a general point process, then we can define the compensator as

\[ \Lambda(t)=\int_0^t\lambda(t\mid H_t)\,dt. \]

Then the residual process for a point process governed by \(\lambda(t\mid H_t)\) is defined by

\[ \tau_i = \Lambda(t_i). \]

Furthermore, the residual process is a stationary Poisson process with rate \(1\), regardless of the nature of the original point process.

Intuitively, we can see how the time transformation modifies the original point process. If we look at the inter-event time of the transformed process, we have

\[ \tau_i - \tau_{i-1} = \Lambda(t_i) - \Lambda(t_{i-1}) \]

If the conditional intensity is high right around \(t_{i-1}\), perhaps because there is a cluster of points right around there and \(t_i\) is close by, then the compensator at \(t_{i-1}\) and \(t_i\) will be large because the integral of \(\lambda(t)\) is large. This will cause the \(\tau_i\)s to be more spaced out in that neighborhood because the differences in the compensator will be large. In other words, areas of high event concentration will be stretched out.

Similarly, if \(t_{i-1}\) and \(t_i\) are very far apart, suggesting an area of low rate, then the compensator values will be small and hence the corresponding \(\tau_{i-1}\) and \(\tau_i\) will be close together.

For example, take the model \(\lambda(t)=\alpha+\beta t\). The compensator is then

\[ \Lambda(t)=\int_0^t\alpha+\beta t\, dt = \alpha t+\beta\frac{t^2}{2} \] Therefore, the inter-event time for the residual process is

\[\begin{eqnarray*} \tau_i-\tau_{i-1} & = & \Lambda(t_i)-\Lambda(t_{i-1})\\ & = & \alpha t_i+\beta \frac{t_i^2}{2}-\alpha t_{i-1}-\beta\frac{t_{i-1}^2}{2}\\ & = & \alpha(t_i-t_{i-1})+\frac{1}{2}\beta(t_i^2-t_{i-1}^2) \end{eqnarray*}\]

The basic idea with residual analysis for point process models is that we construct the residual process and then we treat it as another point process. All of our usual tests and diagnostics for point process models can be applied to the residual process and we can check to see how different it is from a stationary Poisson process with rate 1.

There is a wrinkle though, because all of the above statements hold if we know the true conditional intensity function \(\lambda(t\mid H_t)\). Of course, we generally do not know the true model and must therefore plug in an estimate \(\hat{\lambda}(t\mid H_t)\). While that seems logical, because \(\hat{\lambda}\) depends on the data, plugging it in to compute the compensator results in a residual process that is not exactly stationary Poisson with rate 1. In fact it is a highly irregular self-correcting process. This is analogous to the residuals in a linear regression model which we know must sum to 0 because they are based on the fitted model. But, like with regression, for reasonably large samples, it seems that we can ignore this fact and treat the residuals as if we know the true \(\lambda\).