notebooks/time_to_event.R

In ku-awdc/HexScape: Aggregation of Spatial Data into Discrete Patches using Simple Features

## Parameters for the time to event:
rho <- 1.5
mean <- 50

# Calculate parameters for Weibull:
intercept <- (log(mean) - log(gamma(1 + 1/rho))) * -rho
scale <- exp(intercept)^(-1/rho)

# Just needed for plotting:
MaxT <- 200

# This is the density of distances travelled:
curve(dweibull(x,rho,scale),from=0,to=MaxT,ylab='',xlab='distance')

# And this is the cumulative proportion by distances travelled:
curve(pweibull(x,rho,scale),from=0,to=MaxT,ylab='',xlab='distance')

# So the cumulative hazard of settling by x km is:
x <- 10
(p_x <- pweibull(x,rho,scale))

# And the integrated (total) hazard of settling between x km and y km is:
y <- 15
p_y <- pweibull(y,rho,scale)
(p_y-p_x)

# So the total hazard of settling between x km and y km conditional on having made it as far as x km is:
(p_y-p_x) / (1-p_x)

# And the instantaneous hazard of settling at x km conditional on having made it as far as x km is:
dweibull(x,rho,scale) / (1-pweibull(x,rho,scale))

# With small enough x/rho then this is the same as:
rho * x^(rho-1) * exp(intercept)
# BUT as x gets larger, they diverge...