# Create random GMAR, StMAR, or G-StMAR model compatible parameter vector `random_ind` creates a random GMAR, StMAR, or G-StMAR model compatible mean-parametrized parameter vector. `smart_ind` creates a random GMAR, StMAR, or G-StMAR model compatible parameter vector close to argument `params`. Sometimes returns exactly the given parameter vector. ```r random_ind( p, M, model = c("GMAR", "StMAR", "G-StMAR"), restricted = FALSE, constraints = NULL, mu_scale, sigma_scale, forcestat = FALSE ) smart_ind( p, M, params, model = c("GMAR", "StMAR", "G-StMAR"), restricted = FALSE, constraints = NULL, mu_scale, sigma_scale, accuracy, which_random = numeric(0), forcestat = FALSE ) ``` ## Arguments - `p`: a positive integer specifying the autoregressive order of the model. - `M`: - **For GMAR and StMAR models:**: a positive integer specifying the number of mixture components. - **For G-StMAR models:**: a size $(2\times 1)$ integer vector specifying the number of **GMAR type** components `M1` in the first element and **StMAR type** components `M2` in the second element. The total number of mixture components is `M=M1+M2`. - `model`: is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first `M1` components are **GMAR type** and the rest `M2` components are **StMAR type**. - `restricted`: a logical argument stating whether the AR coefficients $\phi_{m,1},...,\phi_{m,p}$ are restricted to be the same for all regimes. - `constraints`: specifies linear constraints imposed to each regime's autoregressive parameters separately. - **For non-restricted models:**: a list of size $(p \times q_{m})$ constraint matrices C_{m} of full column rank satisfying phi_{m} $=$C_{m}psi_{m} for all $m=1,...,M$, where phi_{m} $=(\phi_{m,1},...,\phi_{m,p})$ and psi_{m} $=(\psi_{m,1},...,\psi_{m,q_{m}})$. - **For restricted models:**: a size $(p\times q)$ constraint matrix C of full column rank satisfying phi $=$Cpsi , where phi $=(\phi_{1},...,\phi_{p})$ and psi $=\psi_{1},...,\psi_{q}$. The symbol $\phi$ denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always `p` for all regimes. Ignore or set to `NULL` if applying linear constraints is not desired. - `mu_scale`: a real valued vector of length two specifying the mean (the first element) and standard deviation (the second element) of the normal distribution from which the $\phi_{m,0}$ or $\mu_{m}$ (depending on the desired parametrization) parameters (for random regimes) should be generated. - `sigma_scale`: a positive real number specifying the standard deviation of the (zero mean, positive only by taking absolute value) normal distribution from which the component variance parameters (for random regimes) should be generated. - `forcestat`: use the algorithm by Monahan (1984) to force stationarity on the AR parameters (slower) for random regimes? Not supported for constrained models. - `params`: a real valued parameter vector specifying the model. - **For non-restricted models:**: Size $(M(p+3)+M-M1-1\times 1)$ vector theta $=$(upsilon_{1} $,...,$upsilon_{M} , $\alpha_{1},...,\alpha_{M-1},$nu ) where * upsilon_{m} $=(\phi_{m,0},$phi_{m} $,$$\sigma_{m}^2)$ * phi_{m} $=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M$ * nu $=(\nu_{M1+1},...,\nu_{M})$ * $M1$ is the number of GMAR type regimes. In the GMAR model, $M1=M$ and the parameter nu dropped. In the StMAR model, $M1=0$. If the model imposes linear constraints on the autoregressive parameters: Replace the vectors phi_{m} with the vectors psi_{m} that satisfy phi_{m} $=$C_{m}psi_{m} (see the argument `constraints`). - **For restricted models:**: Size $(3M+M-M1+p-1\times 1)$ vector theta $=(\phi_{1,0},...,\phi_{M,0},$phi $,$ $\sigma_{1}^2,...,\sigma_{M}^2,$$\alpha_{1},...,\alpha_{M-1},$nu ), where phi =$(\phi_{1},...,\phi_{p})$ contains the AR coefficients, which are common for all regimes. If the model imposes linear constraints on the autoregressive parameters: Replace the vector phi with the vector psi that satisfies phi $=$Cpsi (see the argument `constraints`). Symbol $\phi$ denotes an AR coefficient, $\sigma^2$ a variance, $\alpha$ a mixing weight, and $\nu$ a degrees of freedom parameter. If `parametrization=="mean"`, just replace each intercept term $\phi_{m,0}$ with the regimewise mean $\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})$. In the G-StMAR model, the first `M1` components are **GMAR type** and the rest `M2` components are **StMAR type**. Note that in the case M=1 , the mixing weight parameters $\alpha$ are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters $\nu$ have to be larger than $2$. - `accuracy`: a real number larger than zero specifying how close to `params` the generated parameter vector should be. Standard deviation of the normal distribution from which new parameter values are drawn from will be corresponding parameter value divided by `accuracy`. - `which_random`: a numeric vector of maximum length `M` specifying which regimes should be random instead of "smart" when using `smart_ind`. Does not affect mixing weight parameters. Default in none. ## Returns Returns estimated parameter vector with the form described in `initpop`. ## Details These functions can be used, for example, to create initial populations for the genetic algorithm. Mean-parametrization (instead of intercept terms $\phi_{m,0}$) is assumed. ## Examples ```r set.seed(1) # GMAR model parameter vector params22 <- random_ind(p=2, M=2, mu_scale=c(0, 1), sigma_scale=1) smart22 <- smart_ind(p=2, M=2, params22, accuracy=10) cbind(params22, smart22) # Restricted GMAR parameter vector params12r <- random_ind(p=1, M=2, restricted=TRUE, mu_scale=c(-2, 2), sigma_scale=2) smart12r <- smart_ind(p=1, M=2, params12r, restricted=TRUE, accuracy=20) cbind(params12r, smart12r) # StMAR parameter vector: first regime is random in the "smart individual" params13t <- random_ind(p=1, M=3, model="StMAR", mu_scale=c(3, 1), sigma_scale=3) smart13t <- smart_ind(p=1, M=3, params13t, model="StMAR", accuracy=15, mu_scale=c(3, 3), sigma_scale=3, which_random=1) cbind(params13t, smart13t) # Restricted StMAR parameter vector params22tr <- random_ind(p=2, M=2, model="StMAR", restricted=TRUE, mu_scale=c(3, 2), sigma_scale=0.5) smart22tr <- smart_ind(p=2, M=2, params22tr, model="StMAR", restricted=TRUE, accuracy=30) cbind(params22tr, smart22tr) # G-StMAR parameter vector params12gs <- random_ind(p=1, M=c(1, 1), model="G-StMAR", mu_scale=c(0, 1), sigma_scale=1) smart12gs <- smart_ind(p=1, M=c(1, 1), params12gs, model="G-StMAR", accuracy=20) cbind(params12gs, smart12gs) # Such StMAR(3,2) that the AR coefficients are restricted to be # the same for both regimes and that the second AR coefficients are # constrained to zero. Second regime is random in the "smart individual". params32trc <- random_ind(p=3, M=2, model="StMAR", restricted=TRUE, constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2), mu_scale=c(-2, 0.5), sigma_scale=4) smart32trc <- smart_ind(p=3, M=2, params32trc, model="StMAR", restricted=TRUE, constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2), mu_scale=c(0, 0.1), sigma_scale=0.1, which_random=2, accuracy=20) cbind(params32trc, smart32trc) ``` ## References * Monahan J.F. 1984. A Note on Enforcing Stationarity in Autoregressive-Moving Average Models. **Biometrica** 71 , 403-404.

random_ind function