Marked hawkes

[JoseKling]

Implementing marked and multivariate Hawkes processes

Because of the increased complexity, I don't think there is a way to keep everything in a single type, like with Poisson processes, but still trying to follow as best as I can. I am keeping consistent with the implementation of History, where unmarked univariate histories have marks nothing and multivariate histories have integer marks.

For now, I am considering multivariate mutually exciting processes and univariate processes where marks affect the jump size. All with exponential kernel.

Ideas for improvement or generalization welcome.

[codecov]

Codecov Report

❌ Patch coverage is 94.69027% with 18 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/hawkes/multivariate/intensity.jl	91.30%	6 Missing ⚠️
src/hawkes/multivariate/multivariate_hawkes.jl	80.95%	4 Missing ⚠️
src/hawkes/univariate/intensity.jl	93.44%	4 Missing ⚠️
src/hawkes/univariate/univariate_hawkes.jl	82.60%	4 Missing ⚠️

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[JoseKling]

Three variants of the Hawkes process.

Unmarked univariate Hawkes process

• parameters $\mu, \alpha, \omega \in R_{\geq 0}$
• event history - event times $t_1, \dots, t_n$

$$\lambda(t) = \mu + \sum_{t_j < t} \alpha ~ e^{-\omega (t - t_j)}$$

Marked univariate Hawkes process

• parameters $\mu, \alpha, \omega \in R_{\geq 0}$
• event history - marked times $(t_1, m_1), \dots, (t_n, m_n)$, $m_i \geq 0$

$$\lambda(t) = \mu + \sum_{t_j < t} \alpha ~ m_j ~ e^{-\omega (t - t_j)}$$

Unmarked multivariate Hawkes process

• parameters $\mu, \omega \in R^d_{\geq 0}$, $\alpha \in R^{d \times d}_{\geq 0}$
• event history - marked times $(t_1, m_1), \dots, (t_n, m_n)$, $m_i \in {1, \dots, d}$ (the marginal process)

$$\lambda_l(t) = \mu_l + \sum_{t_j < t} \alpha_{m_j, l} ~ e^{-\omega_l (t - t_j)}$$

Each marginal process has fixed base intensity and decay rate, and the jump intensity determines the influence of a marginal process on another.

Possible improvements (probably for a future PR)

• A fit method for the multivariate Hawkes process
• General kernel functions
• Optimize intensity, ground_intensity and integrated_ground_intensity when the t argument is a vector (if this becomes a problem)