# Continuous-Time Fractionally Integrated ARMA Process for Irregularly Spaced Long-Memory Time Series Data The R package `carfima` provides a toolbox to fit a continuous-time fractionally integrated ARMA process (CARFIMA) on univariate and irregularly spaced time series data via both frequentist and Bayesian machinery. A general-order CARFIMA($p, H, q$) model for $p>q$ is specified in Tsai and Chan (2005). It involves $p+q+2$ unknown model parameters, i.e., $p$ AR parameters, $q$ MA parameters, Hurst parameter $H$, and process uncertainty (standard deviation) $\sigma$; see also `carfima`. Also, the model can account for heteroscedastic measurement errors, if the information about measurement error standard deviations is known. The package produces their maximum likelihood estimates and asymptotic uncertainties using a global optimizer called the differential evolution algorithm. It also produces posterior samples of the model parameters via Metropolis-Hastings within a Gibbs sampler equipped with adaptive Markov chain Monte Carlo. These fitting procedures, however, may produce numerical errors if $p>2$. The toolbox also contains a function to simulate discrete time series data from CARFIMA($p, H, q$) process given the model parameters and observation times. package ## Details ||| |:--|:--| |Package:|carfima| |Type:|Package| |Version:|2.0.2| |Date:|2020-03-20| |License:|GPL-2| |Main functions:|`carfima` , `carfima.loglik` , `carfima.sim`| ## Author(s) Hyungsuk Tak, Henghsiu Tsai, and Kisung You Maintainer: Hyungsuk Tak ## References H. Tsai and K.S. Chan (2005) "Maximum Likelihood Estimation of Linear Continuous Time Long Memory Processes with Discrete Time Data," Journal of the Royal Statistical Society (Series B), 67 (5), 703-716. DOI: 10.1111/j.1467-9868.2005.00522.x H. Tsai and K.S. Chan (2000) "A Note on the Covariance Structure of a Continuous-time ARMA Process," Statistica Sinica, 10, 989-998. Link: http://www3.stat.sinica.edu.tw/statistica/j10n3/j10n317/j10n317.htm ## Examples ```r ##### Irregularly spaced observation time generation. length.time <- 30 time.temp <- rexp(length.time, rate = 2) time <- rep(NA, length.time + 1) time[1] <- 0 for (i in 2 : (length.time + 1)) { time[i] <- time[i - 1] + time.temp[i - 1] } time <- time[-1] ##### Data genration for CARFIMA(1, H, 0) based on the observation times. parameter <- c(-0.4, 0.8, 0.2) # AR parameter alpha = -0.4 # Hurst parameter = 0.8 # Process uncertainty (standard deviation) sigma = 0.2 me.sd <- rep(0.05, length.time) # Known measurement error standard deviations 0.05 for all observations # If not known, remove the argument "measure.error = me.sd" in the following codes, # so that the default values (zero) are automatically assigned. y <- carfima.sim(parameter = parameter, time = time, measure.error = me.sd, ar.p = 1, ma.q = 0) ##### Fitting the CARFIMA(1, H, 0) model on the simulated data for MLEs. res <- carfima(Y = y, time = time, measure.error = me.sd, method = "mle", ar.p = 1, ma.q = 0) # It takes a long time due to the differential evolution algorithm (global optimizer). # res$mle; res$se; res$AIC; res$fitted.values ##### Fitting the CARFIMA(1, H, 0) model on the simulated data for Bayesian inference. res <- carfima(Y = y, time = time, measure.error = me.sd, method = "bayes", ar.p = 1, ma.q = 0, bayes.param.ini = parameter, bayes.param.scale = c(rep(0.2, length(parameter))), bayes.n.warm = 100, bayes.n.sample = 1000) # It takes a long time because the likelihood evaluation is computationally heavy. # The last number of bayes.param.scale is to update sigma2 (not sigma) on a log scale. # hist(res$param[, 1]); res$accept; res$AIC; res$fitted.values ##### Computing the log likelihood of the CARFIMA(1, H, 0) model given the parameters. loglik <- carfima.loglik(Y = y, time = time, ar.p = 1, ma.q = 0, measure.error = me.sd, parameter = parameter, fitted = FALSE) ```

carfima-package