Time-Rescaling & Point Process Tests

• The paper highlights how time-rescaling transforms event times into i.i.d. exponential intervals, enabling classical tests like Kolmogorov–Smirnov for model validation.
• Thinning and complementing methods offer alternative diagnostics by selectively retaining or synthesizing events, thus probing model fit in both event-dense and sparse intervals.
• Extensions using surrogate point processes and innovation martingale transforms overcome biases in discrete models, optimizing local and global goodness-of-fit assessments.

Time-rescaling and point process-based goodness-of-fit tests constitute a foundational diagnostic toolkit in the statistical analysis of temporal, spatial, and spatiotemporal event data, especially for models—such as Generalized Linear Models (@@@@1@@@@) and conditional intensity-based point processes—used extensively in the neurosciences and in fields such as seismology. The central aim is to rigorously evaluate whether the fitted model adequately captures the data-generating mechanism, relying on constructions that reduce model residual analysis to tractable canonical forms where classical statistical tests (e.g., Kolmogorov–Smirnov) apply. Three main types dominate: time-rescaling (random time change), thinning, and complementing/superposition, supported by recent distribution-free approaches built upon innovation martingale transforms. These methodologies, subject to precise regularity conditions, enable comprehensive omnibus and local assessments of model adequacy across a range of data regimes and modeling scenarios (Gerhard et al., 2010, Clements et al., 2012, Baars et al., 31 Mar 2025).

1. Foundations: Conditional Intensity and Model Likelihood

The fundamental object for temporal point processes is the conditional intensity function $\lambda(t\mid H_t)$, which encodes the instantaneous event rate given history $H_t$: $$\lambda(t\mid H_t) = \lim_{\Delta \to 0} \frac{P\{\text{event in }(t, t+\Delta] \mid H_t\}}{\Delta}$$ with the constraint $\lambda(t\mid H_t) \ge 0$. This function uniquely determines both the likelihood for observed spike/event sequences $\{u_i\}_{i=1}^n$,

$$\log L = \sum_{i=1}^n \log \lambda(u_i \mid H_{u_i}) - \int_0^T \lambda(t\mid H_t)\,dt,$$

and the entirety of the classical residual toolkit for goodness-of-fit analysis. For multivariate and spatiotemporal processes, the intensity generalizes to $\lambda(s,t)$ or $\lambda_\theta^{(k)}(t)$ per component (Gerhard et al., 2010, Baars et al., 31 Mar 2025).

2. Time-Rescaling Theorem and Application

Let $\{u_i\}$ denote the observed event times. The time-rescaling (random time change) theorem states that, for a correctly specified $\lambda(t\mid H_t)$, the transformed cumulative intensity times

$t_i = \int_0^{u_i} \lambda(s\mid H_s)\,ds$

have i.i.d. exponential inter-arrivals: $\tau_i = t_i - t_{i-1}$, $t_0=0$, $\tau_i \sim \mathrm{Exp}(1)$, and the variables $z_i = 1 - e^{-\tau_i}$ are i.i.d. $\mathrm{Uniform}(0,1)$. The empirical CDF of $\{z_i\}$ is then compared to the uniform via the Kolmogorov–Smirnov test as an omnibus goodness-of-fit diagnostic. The theorem critically requires nonnegative, integrable intensity, no event coalescence, and precise time observations (Gerhard et al., 2010).

For spatial or spatiotemporal point processes $\lambda(s,t)$, the time-rescaling transform is augmented using the marginal temporal intensity $\Lambda(t) = \int_S \lambda(s,t)\,ds$ and conditional spatial transforms (e.g., Rosenblatt transform) to jointly achieve uniformity in higher dimensions (Clements et al., 2012).

Time-rescaling is sensitive to the integrated intensity but may be insensitive to local deviations if the area under $\lambda(t)$ is preserved between events. For discretized models (e.g., binned GLM spike counts), direct application induces bias; instead, surrogate continuous-time event sequences are constructed before rescaling (see Section 4) (Gerhard et al., 2010).

3. Thinning, Complementing, and Superposition Diagnostics

Beyond rescaling, thinning and complementing construct (under $H_0$) processes where residual points are distributed as homogeneous Poisson under the fitted model:

• Thinning retains each observed event $u_i$ independently with probability $p_i = B / \lambda(u_i)$ for $B \leq \lambda(t)$ (lower bound), yielding a homogeneous Poisson process of rate $B$. Multi-threshold approaches aggregate evidence by evaluating at several $B^*_k$ and combining the resulting p-values via Simes’ method. Thinning probes $\lambda(t)$ only at observed events and is thus insensitive to intervals devoid of events (Gerhard et al., 2010, Clements et al., 2012).
• Complementing (or superposition) simulates, over intervals with $\lambda(t) \leq C^*$, auxiliary Poisson events with intensity $\lambda^c(t) = C^* - \lambda(t)$, and merges these synthetic events with observed ones. The total is homogeneous Poisson rate $C^*$. This method is sensitive in non-event intervals, complementing the weaknesses of thinning (Gerhard et al., 2010).
• Super-thinning (hybrid): For highly volatile intensities where thinning loses power ($B \to 0$) or complementing is dominated by simulation ($C \gg \lambda$), super-thinning targets a desired residual rate $k$ by retaining or synthesizing events as needed, optimizing sensitivity balance (Clements et al., 2012).

Summary of sensitivity:

Method	Probes $\lambda(t)$ at	Insensitive to errors at
Time-rescaling	Integrated between	Local shape mismatch
Thinning	Observed events	Gaps (no events)
Complementing	Event gaps	At event times

4. Adapting to Discrete Models: Surrogate Point Processes

To bridge gap between continuous-time point processes and discrete-time GLMs (Poisson or Bernoulli with binwidth $\Delta$), surrogate spike times are sampled within time bins:

• For Poisson-count–based models ($$c_i \sim \mathrm{Pois}(\mu_i), \mu_i = \Delta \lambda_i$$), uniformly sample $c_i$ times within bin $i$.
• For Bernoulli models ($b_i \sim \mathrm{Bernoulli}(p_i)$), infer $\mu_i = -\ln(1-p_i)$, draw $c_i$ from the zero-truncated Poisson, and sample $c_i$ times as above.

Once surrogate event times and piecewise-constant $\lambda(t)$ are obtained, any point-process–based test can be validly applied, removing discretization bias (Gerhard et al., 2010).

5. Extensions: Distribution-Free Martingale Transformation and Residual Analysis

Asymptotically distribution-free tests have been introduced to overcome limitations of rescaling-based methods with parameter uncertainty and non-stationarity. Compensated (innovation) martingales,

$$M(t) := N_{\theta_0}(t) - \int_0^t \lambda_{\theta_0}(s)\,ds$$

admit empirical plug-in processes aligned with an innovation martingale transformation (Khmaladze transform), resulting in a process converging weakly to a standard Wiener process $W(u)$. Classical functionals (KS, CvM, Anderson–Darling) applied to this process yield exact null distributions, addressing the issue that time-rescaling tests may be undersized or biased under parameter estimation error. Monte Carlo studies show that this approach achieves nominal type I error and greater power under alternatives than classical rescaling, especially for complex (e.g., Hawkes, self-correcting) point processes (Baars et al., 31 Mar 2025).

6. Comparative Power, Practical Recommendations, and Limitations

Empirical studies systematically compare these tests in simulations:

• For inhomogeneous Poisson and neural spike response models, thinning and complementing generally possess higher sensitivity to intensity misfit than naive time-rescaling.
• For renewal-process violations (e.g., Gamma ISI), time-rescaling often retains higher power owing to direct ISI distributional sensitivity.
• Discrete-time rescaling is systematically biased at large $\Delta$ or high firing rates; surrogate spike times are necessary.
• In "spiky" models (volatile $\lambda(t)$), pure thinning and superposition may lose sensitivity, with super-thinning maintaining residual density and thus power (Gerhard et al., 2010, Clements et al., 2012).

Guidelines:

• Use time-rescaling as a global fit check when precise times and intensities are available, supplementing with thinning or complementing as required for local misfit detection.
• For discrete data or high-rate regimes, form surrogate spike sequences.
• For volatile conditional intensities, employ super-thinning for optimal balance.
• Weighted K-functions and deviance residuals, although not pointwise tests, serve as valuable global diagnostics for clustering/inhibition and for model comparison in binned space–time frameworks (Clements et al., 2012).

7. Spatiotemporal and Model-Comparison Extensions

Time-rescaling can be extended spatially via the Rosenblatt transform, evaluating whether rescaled data points are uniform over the spatiotemporal unit hypercube, contingent on the fitted $\lambda(s,t)$ (Clements et al., 2012). Weighted K-functions enable global pattern analysis of clustering or inhibition in the distribution of residuals. Deviance residuals allow for local model-comparison on a pre-defined spatial/temporal lattice, highlighting where one intensity model outperforms another in likelihood. Composite testing frameworks leveraging these multiple diagnostics, with power and localization varied according to substantive model features and application context, form the contemporary standard for comprehensive assessment of goodness-of-fit in point process applications (Clements et al., 2012).

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References:

• "Rescaling, thinning or complementing? On goodness-of-fit procedures for point process models and Generalized Linear Models" (Gerhard et al., 2010)
• "Residual analysis methods for space--time point processes with applications to earthquake forecast models in California" (Clements et al., 2012)
• "Asymptotically distribution-free goodness-of-fit testing for point processes" (Baars et al., 31 Mar 2025)