# cocoReg The function fits first and second order (Generalized) Poisson Autoregressive (G)PAR models presented in (Jung and Tremayne, 2010). Autoregressive dependence on past counts is modelled by a special random operator that not only preserve integer status, but also, via the property of closure under convolution, ensure that the marginal distribution of the observed counts is from the same family as the innovations. The models can be thought of as stationary Markov chains of finite order, where the distribution of the innovations can either be Poisson or Generalized Poisson, where the latter can account for overdispersed data. Maximum likelihood is used for estimation and the user can choose to include linear constraints or not. If linear constraints are not included, it cannot be guaranteed that the parameters will lie in the theoretically feasible parameter space, but the optimization process might be faster. The function uses method of moments estimators to obtain starting values for the numerical optimization, but the user can also specify their own starting values if desired. If Julia is installed, the user can choose whether the optimization is run in Julia which might faster yield results and increased numeric stability due to the use of automatic differentiation. See details for more information on the Julia implementation. ```r cocoReg( type, order, data, xreg = NULL, constrained.optim = TRUE, b.beta = -10, start = NULL, start.val.adjust = TRUE, method_optim = "Nelder-Mead", replace.start.val = 1e-05, iteration.start.val = 0.6, method.hessian = "Richardson", cores = 2, julia = FALSE, julia_installed = FALSE ) ``` ## Arguments - `type`: character string indicating the type of model to be fitted - `order`: integer vector indicating the order of the model - `data`: time series data to be used in the analysis - `xreg`: optional matrix of explanatory variables for use in a regression model - `constrained.optim`: logical indicating whether optimization should be constrained, currently only available in the R version - `b.beta`: numeric value indicating the lower bound for the parameters of the explanatory variables for the optimization, currently only available in the R version - `start`: optional numeric vector of starting values for the optimization - `start.val.adjust`: logical indicating whether starting values should be adjusted, currently only available in the R version - `method_optim`: character string indicating the optimization method to be used, currently only available in the R version. In the julia implementation this is by default the LBFGS algorithm - `replace.start.val`: numeric value indicating the value to replace any invalid starting values, currently only available in the R version - `iteration.start.val`: numeric value indicating the proportion of the interval to use as the new starting value, currently only available in the R version - `method.hessian`: character string indicating the method to be used to approximate the Hessian matrix, currently only available in the R version - `cores`: numeric indicating the number of cores to use, currently only available in the R version - `julia`: if TRUE, the model is estimated with Julia. This can improve the speed significantly since Julia makes use of derivatives using autodiff. In this case, only type, order, data, xreg, and start are used as other inputs. - `julia_installed`: if TRUE, the model R output will contain a Julia compatible output element. ## Returns an object of class coco. It contains the parameter estimates, standard errors, the log-likelihood, and information on the model specifications. If Julia is used for parameter estimation or the Julia installation parameter is set to TRUE, the results contain an additional Julia element that is called from the model Julia assessment tools if they are run with the Julia implementation. ## Details Let a time series of counts be $\{X_t\}$ and be $R(\cdot)$ a random operator that differs between model specifications. For more details on the random operator, see Jung and Tremayne (2011) and Joe (1996). The general first-order model is of the form $$ X_t = R(X_{t-1}) + I_t, $$ and the general second-order model of the form $$ X_t = R(X_{t-1}, X_{t-2}) + I_t, $$ where $I_t$ are i.i.d Poisson ($I_t \sim Po(\lambda_t)$) or Generalized Poisson ($I_t \sim GP(\lambda_t, \eta)$) innovations. Through closure under convolution the marginal distributions of $\{X_t\}$ are therefore Poisson or Generalized Poisson distributions, respectively. If no covariates are used $\lambda_t = \lambda$ and if covariates are used $$ \lambda_t = \exp{\left(\beta_0 + \sum_{j = 1}^k \beta_j \cdot z_{t,j} \right)}, $$ whereby $z_{t,j}$ is the $j$-th covariate at time $t$. Standard errors are computed by the square root of the diagonal elements of the inverse Hessian. This function is implemented in 2 versions. The default runs on RCPP. An alternative version uses a Julia implementation which can be chosen by setting the argument julia to TRUE. In order to use this feature, a running Julia installation is required on the system. The RCPP implementation uses the derivative-free Nelder-Mead optimizer to obtain parameter estimates. The Julia implementation makes use of Julia's automatic differentiation in order to obtain gradients such that it can use the LBFGS algorithm for optimization. This enhances the numeric stability of the optimization and yields an internal validation if both methods yield qualitatively same parameter estimates. Furthermore, the Julia implementation can increase the computational speed significantly, especially for large models. The model assessment tools `cocoBoot`, `cocoPit`, and `cocoScore` will use a Julia implementation as well, if the `cocoReg` was run with Julia. Additionally, one can make the RCPP output of `cocoReg` compatible with the Julia model assessments by setting julia_installed to true. In this case, the user can choose between the RCPP and the Julia implementation for model assessment. ## Examples ```r ## GP2 model without covariates length <- 1000 par <- c(0.5,0.2,0.05,0.3,0.3) data <- cocoSim(order = 2, type = "GP", par = par, length = length) fit <- cocoReg(order = 2, type = "GP", data = data) ##Poisson1 model with covariates length <- 1000 period <- 50 sin <- sin(2*pi/period*(1:length)) cos <- cos(2*pi/period*(1:length)) cov <- cbind(sin, cos) par <- c(0.2, 0.2, -0.2) data <- cocoSim(order = 1, type = "Poisson", par = par, xreg = cov, length = length) fit <- cocoReg(order = 1, type = "Poisson", data = data, xreg = cov) ``` ## References Jung, R. C. and Tremayne, A. R. (2011) Convolution-closed models for count timeseries with applications. **Journal of Time Series Analysis**, 32 , 3, 268--280. Joe, H. (1996) Time series models with univariate margins in the convolution-closed infinitely divisible class. **Journal of Applied Probability**, 664--677. ## Author(s) Manuel Huth

cocoReg function