Spread event times recorded to a resolution across their interval

Daily (or hourly, or second) timestamps put every event of an interval at one time, but a Hawkes process never has two events at one instant, and fitting the rounded times as exact biases the estimates (Filimonov & Sornette 2015). This function spreads each recorded time uniformly over the interval it stands for, so the times can be passed to core_hawkes_fit().

Usage

core_hawkes_jitter(times, resolution = 1, seed = 42, horizon = Inf)

Arguments

times: Numeric event times, each the start of its resolution interval (for example whole days since an origin).
resolution: The width of the interval a recorded time stands for, one positive number in the unit of times: 1 (the default) for daily dates in days, 1/24 for hourly times in days.
seed: Seed of the uniforms, as in core_uniforms(); default 42.
horizon: End of the observation window, when it cuts an interval short: an event recorded at t is spread over [t, min(t + resolution, horizon)], so a time dated on the horizon itself stays there (Inf, the default: no window).

Value

A numeric vector of the jittered times, the same length as times, sorted increasingly.

Details

Each time becomes t + w * U, with w = min(resolution, horizon - t) and U uniform on [0, 1), and the result is sorted; this is the remedy Filimonov and Sornette use. The times are sorted (radix order) before the uniforms are assigned, so the result does not depend on the input order. The uniforms are the splitmix64 stream of core_uniforms(), so the result is reproducible and, on daily dates with seed = 42, the same as the jitter rmorie and morie (Python) apply before their TPS Hawkes fits. R's random number generator is not used.

Errors: times with NA or non-finite values; resolution not one positive finite number; horizon not one number; a time past horizon.

References

Filimonov V, Sornette D (2015). Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data. Quant. Finance 15(8), 1293–1314. doi:10.1080/14697688.2015.1032544

Examples

days <- c(0, 0, 0, 1, 3, 3)
core_hawkes_jitter(days)
#> [1] 0.1599104 0.2786011 0.7415649 1.3441907 3.0380302 3.8682281
anyDuplicated(core_hawkes_jitter(days))
#> [1] 0

# a horizon keeps the last day inside the window
core_hawkes_jitter(c(0, 1, 2), horizon = 2)
#> [1] 0.7415649 1.1599104 2.0000000

# hourly resolution, in days, and another seed
core_hawkes_jitter(c(0, 0, 0), resolution = 1 / 24, seed = 7)
#> [1] 0.0006995123 0.0162429062 0.0375316950

# times past the horizon are refused
try(core_hawkes_jitter(c(0, 5), horizon = 3))
#> Error : `times` must not exceed `horizon`