# Minus loglikelihood of an (bivariate) MRHawkes model Calculates the minus loglikelihood of an (bivariate) MRHawkes model with given immigration hazard functions $\mu$, offspring density functions $h$ and bracnhing ratios $\eta$ for the event times and types `data` on the interval $[0,cens]$. ```r mllMRH(data, cens, par, h1.fn = function(x, p) 1 / p * exp( - x / p), h2.fn = function(x, p) 1 / p * exp( - x / p), mu1.fn = function(x, p){ exp(dweibull(x, shape = p[1], scale = p[2], log = TRUE) - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE)) }, mu2.fn = function(x, p){ exp(dweibull(x, shape = p[1], scale = p[2], log = TRUE) - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE)) }, H1.fn = function(x, p) pexp(x, rate = 1 / p), H2.fn = function(x, p) pexp(x, rate = 1 / p), Mu1.fn = function(x, p){ - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE) }, Mu2.fn = function(x, p){ - pweibull(x, shape = p[1], scale = p[2], lower.tail = FALSE, log.p = TRUE) }) ``` ## Arguments - `data`: A two column matrix. The first column contains the event times sorted in ascending order. The second column contains the corresponding event type with the label one or two. - `cens`: A scalar. The censoring time. - `par`: A numeric vector. Contains the ten parameters of the model, in order of the immigration parameters $\mu(.)$ for the two renewal distributions, the two offspring parameters $h(.)$ and lastly the four branching ratios $\eta$. - `h1.fn`: A (vectorized) function. The offspring density function for type one events. - `h2.fn`: A (vectorized) function. The offspring density function for type two events. - `mu1.fn`: A (vectorized) function. The immigration hazard function for events of type one. - `mu2.fn`: A (vectorized) function. The immigration hazard function for events of type two. - `H1.fn`: A (vectorized) function. Its value at `t` gives the integral of the offspring density function from 0 to `t` for type one events. - `H2.fn`: A (vectorized) function. Its value at `t` gives the integral of the offspring density function from 0 to `t` for type two events. - `Mu1.fn`: A (vectorized) function. Its value at `t` gives the integral of the immigrant hazard function from 0 to `t` for type one events. - `Mu2.fn`: A (vectorized) function. Its value at `t` gives the integral of the immigrant hazard function from 0 to `t` for type two events. ## Returns Value of the negative log-liklihood. ## Author(s) Tom Stindl Feng Chen ## Examples ```r ## Magnitude 5.5 or greater earthquakes over the 25 year period from ## 01/01/1991 to 31/12/2015. data(fivaqks); near.fiji <- grep("Fiji", fivaqks$place) near.vanuatu <- grep("Vanuatu", fivaqks$place) t.beg <- strptime("1991-01-01 00:00:00", "%Y-%m-%d %H:%M:%S", tz = "UTC") t.end <- strptime("2015-12-31 23:59:59", "%Y-%m-%d %H:%M:%S", tz = "UTC") t0 <- 0 t1 <- as.numeric(t.end - t.beg) tms <- strptime(fivaqks$time, "%Y-%m-%dT%H:%M:%OSZ", tz = "UTC") ts <- as.numeric(tms[-1] - t.beg) ts <- c(as.numeric(tms[1] - t.beg)/24, ts) ts.fi <- ts[near.fiji]; ts.fi <- ts.fi[ts.fi >= 0 & ts.fi <= t1] ts.va <- ts[near.vanuatu]; ts.va <- ts.va[ts.va >=0 & ts.va <= t1] ts.c <- c(ts.fi, ts.va) z.c <- c(rep(1, times = length(ts.fi)), rep(2, times = length(ts.va))) o <- order(ts.c) data <- cbind(ts.c[o], z.c[o]) ## calculate the minus loglikelihood of an (bivariate) MRHawkes with some ## parameters the default hazard functions and density functions are Weibull ## and exponential respectivley mllMRH(data, cens = t1, par = c(0.488, 20.10, 0.347, 9.53, 461, 720, 0.472, 0.293, 0.399, -0.0774)) ## calculate the MLE for the parameter assuming known parametric forms ## of the immigrant hazard function and offspring density functions. system.time(est <- optim(c(0.488, 20.10, 0.347, 9.53, 461, 720, 0.472, 0.293, 0.399, -0.0774), mllMRH, data = data, cens = t1, control = list(maxit = 5000, trace = TRUE), hessian = TRUE) ) ## point estimate by MLE est$par ## standard error estimates: diag(solve(est$hessian))^0.5 ```

mllMRH function