circdiff: Estimation of circular diffusion models

Description

circdiff returns the estimates of parameters of a discretely observed von-Mises process or Circular Brownian motion

Usage

circdiff(theta, dt, corr_process, iter=2000, lambda=0, lambda_1=0, lambda_2=0)

Value

A list containing the estimates of the model if corr_process=vmp then it returns

dt time step $\Delta t$
lambda_vm the drift parameter of the von Mises process
sigma_vm the volatility parameter of the von Mises process
mu_vm the mean direction of the von Mises process
lambda_1, lambda_2 value of the regularization parameters used for estimation

else if corr_process=cbm then it returns

dt time step $\Delta t$
sigma_cbm the volatility parameter of the circular Brownian motion
lambda value of the regularization parameter used for estimation

Arguments

theta: data of the discretely observed diffusion
dt: time step $\Delta t$
corr_process: vmp for von-Mises process; cbm for circular brownian motion
iter: number of iterations (Default is 2000)
lambda: regularization parameter for cbm (Default is 0)
lambda_1: regularization parameter for vmp (Default is 0)
lambda_2: regularization parameter for vmp (Default is 0)

Details

Let $\theta_0,\theta_{\Delta t},\theta_{2\Delta t}, \ldots, \theta_{n\Delta t}$ be a discretely observed circular diffusion at time step $\Delta t$. The circular diffusion could either be von-Mises process, $$d\theta_t=-\lambda\sin(\theta_t-\mu)dt+\sigma dW_t$$ or the circular brownian motion, $$d\theta_t=\sigma dW_t$$ under periodic boundary condition. The function returns the MLE of $\lambda,\sigma,\mu$ for von-Mises process or $\sigma$ for circular brownian motion.

We provide an option to perform penalised MLE using an iterative optimization procedure as following, we maximise for von Mises process, where $n$ is the number of observations, $$\text{Log-lik}-n\lambda_1\sum(1-cos(\theta_{i+1}-\theta_{i}))-\lambda_2\frac{2\lambda}{\sigma^2}$$ For Circular Brownian motion we maximise, $$\text{Log-lik}-n\lambda\sum(1-cos(\theta_{i+1}-\theta_{i}))$$

See section 3 of Majumdar and Laha (2024) doi:10.48550/arXiv.2412.06343.

Examples

Run this code

# \donttest{
library(stochcorr)

data(wind)
if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(wind, aes(x = Date, y = Dir)) +
  geom_line() +
  labs(title = "Sotavento Wind Farm",
       x = "Date",
       y = "Wind Direction") +
  scale_x_datetime(date_labels = "%d-%b", date_breaks = "2 day") +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )
 }

a<-circdiff(wind$Dir,10/1440,"vmp")
a

b<-circdiff(wind$Dir,10/1440,"cbm")
b# }