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The function returns the value of the ETAS triggering function at a specified time t for the points in the history th, mh

Usage

gt(theta, t, th, mh, M0)

Arguments

theta

ETAS parameters as list with names K, alpha, c, p

t

Time at which the function is calculated, scalar or vector

th

Time of events in the history in [days, months,...], scalar or vector

mh

Magnitude of events in the history, scalar or vector

M0

Minimum magnitude threshold, scalar

Value

value of the ETAS triggering function evaluated at t with history th, mh.

Details

The ETAS triggering function to be evaluated is $$g(t - t_h | m_h) = K e^{\alpha(m_h - M_0)} \left( \frac{t - t_h}{c} + 1\right)^{-p}$$ Where \(K, \alpha, c > 0\), and \(p \geq 1\) are the ETAS parameters, \(t\) (t) is the time at which the function is evaluated, considering the past observation \(t_h, m_h\) (th, mh). The function returns 0 if \(t_h > t\). If \(t\) is a scalar and \(t_h, m_h\) are vectors than the function returns a vector, same if \(t\) is a vector and \(t_h, m_h\) are scalars, or if \(t, t_h, m_h\) are vectors of the same length.

Do not use if \(t\) and \(t_h, m_h\) are vectors of different lengths.