# Time Rescaling Theorem and Residual Analysis

Let $\mathcal{P}$ an homogeneous unit rate Poisson process. It's conditional intensity function (star indicating conditioning on the history) can be written as $$\lambda^*(t) = \lambda = 1$$ meaning that it does not depend on either the history, and is constant with time. It's compensator is by definition $$\Lambda^*(t) = \int_0^t \lambda^*(s)ds = t \lambda = t$$

Let now $[0, T)$ some interval, and $t_1, \dots, t_n$ some point pattern observed on that interval. By the time rescaling theorem, $$\{ \Lambda^*(t_1), \dots, \Lambda^*(t_n) \} = \{ t_1, \dots, t_n \}$$ is a unit rate homogeneous Poisson process.

So far so good, the time rescaling let the pattern unchanged, as I would expect.

The likelihood of the point pattern on $[0, T)$ is given by

$$\mathcal{L} = \left(\prod_{i = 1}^{n} \lambda^*(t_i)\right) \exp \left( - \int_0^T\lambda^*(s)ds \right) = exp(-T)$$

which is independent of $t_1, \dots, t_n$, saying that any point pattern on $[0, T)$ is equally likely.

But if so then it is independent of the inter-arrival times as well,

$$\{\tau_1, \dots, \tau_{n-1}\} = \{t_2 - t_1 , \dots, t_n - t_{n -1}\}$$

which are also supposed to be exponentially distributed $\tau_i \sim exp(1)$

Question: I have trouble understanding how testing for the $\tau_i$ being exponentially distributed can help determining goodness of fit for a unit rate Poisson process, if they can indeed be all equally likely.

In their work, Ogata (1988)[1] and Brown (2002)[2] perform residual analysis by testing for the inter-arrival times being exponentially distributed (Kolmogorov-Smirnoff, ...) after applying the time rescaling theorem to some other point process.

[1] Ogata, Yosihiko. "Statistical models for earthquake occurrences and residual analysis for point processes." Journal of the American Statistical association 83.401 (1988): 9-27.

[2] Brown, Emery N., et al. "The time-rescaling theorem and its application to neural spike train data analysis." Neural computation 14.2 (2002): 325-346.

I'm confused by the likelihood terminology when you consider the parameter $\lambda$ fixed and talk about different point patterns (data). What is the reference measure here? I work with spatial point processes and we specify densities and likelihoods with respect to the unit rate Poisson process. Is that what you are doing here?
In that case the likelihood/density just says something about how likely an given point pattern is compared to how likely it is for a unit rate Poisson process. The fact that if you set $\lambda=1$ all point patterns are equally likely just says that for a process with $\lambda=1$ each pattern has the same likelihood as for the unit rate Poisson process, which makes sense since it is a unit rate Poisson process.
If instead you e.g. set $\lambda=2$ you should see that different patterns have different likelihoods compared to the unit rate Poisson process.