## Recipe function for specifying dynamic functions using formulas

### Description

Recipe function for specifying dynamic functions using formulas

### Usage

isContinuousTime = FALSE, jacobian, ...)

### Arguments

 formula a list of formulas specifying the drift or state-transition equations for the latent variables in continuous or discrete time, respectively. startval a named vector of starting values of the parameters in the formulas for estimation with parameter names as its name. If there are no free parameters in the dynamic functions, leave startval as the default numeric(0). isContinuousTime if True, the left hand side of the formulas represent the first-order derivatives of the specified variables; if False, the left hand side of the formulas represent the current state of the specified variable while the same variable on the righ hand side is its previous state. jacobian (optional) a list of formulas specifying the analytic jacobian matrices containing the analytic differentiation function of the dynamic functions with respect to the latent variables. If this is not provided, dynr will invoke an automatic differentiation procedure to compute the jacobian functions. ... further named arguments. Some of these arguments may include: theta.formula specifies a list consisting of formula(s) of the form list (par ~ 1 * b_0 + covariate_1 * b_1 + ... + covariate_p * b_p + 1 * rand_par), where par is a parameter is a unit- (e.g., person-) specific that appears in a dynamic formula and is assumed to follow a linear mixed effects structure. Here, b_p are fixed effects parameters; covariate_1, ..., covariate_p are known covariates as predeclared in dynr.data, and rand_par is a random effect component representing unit i's random deviation in par value from that predicted by b_0 + covariate_1*b_1 + ... + covariate_p*b_p. random.names specifies names of random effect components in the theta.formula random.params.inicov specifies names of elements in the covariance matrix of the random effect components random.values.inicov specifies starting values of elements in the covariance matrix of the random effect components

### Details

This function defines the dynamic functions of the model either in discrete time or in continuous time. The function can be either linear or nonlinear, with free or fixed parameters, numerical constants, covariates, and other mathematical functions that define the dynamics of the latent variables. Every latent variable in the model needs to be defined by a differential (for continuous time model), or difference (for discrete time model) equation. The names of the latent variables should match the specification in prep.measurement(). For nonlinear models, the estimation algorithm generally needs a Jacobian matrix that contains elements of first differentiations of the dynamic functions with respect to the latent variables in the model. For most nonlinear models, such differentiations can be handled automatically by dynr. However, in some cases, such as when the absolute function (abs) is used, the automatic differentiation would fail and the user may need to provide his/her own Jacobian functions. When theta.formula and other accompanying elements in "..." are provided, the program automatically inserts the random effect components specified in random.names as additional latent (state) variables in the model, and estimate (cook) this expanded model. Do check that the expanded model satisfies conditions such as observability for the estimation to work.

### Value

Object of class 'dynrDynamicsFormula'

### Examples

# In this example, we present how to define the dynamics of a bivariate dual change score model
# (McArdle, 2009). This is a linear model and the user does not need to worry about
# providing any jacobian function (the default).

# We start by creating a list of formula that describes the model. In this model, we have four
# latent variables, which are "readLevel", "readSlope", "mathLevel", and "math Slope".  The right-
# hand side of each formula gives a function that defines the dynamics.

formula <- list(
mathLevel~ (1+beta.math)*mathLevel + mathSlope + gamma.math*readLevel,
mathSlope~ mathSlope
))

# Then we use prep.formulaDynamics() to define the formula, starting value of the parameters in
# the model, and state the model is in discrete time by setting isContinuousTime=FALSE.

startval=c(beta.read = -.5, beta.math = -.5,
gamma.read = .3, gamma.math = .03
), isContinuousTime=FALSE)

# For a full demo example of regime switching nonlinear discrete time model, you
# may refer to a tutorial on
# \url{https://quantdev.ssri.psu.edu/tutorials/dynr-rsnonlineardiscreteexample}

#Not run:
#For a full demo example that uses user-supplied analytic jacobian functions see:
#demo(RSNonlinearDiscrete, package="dynr")
formula <- list(
list(
x1 ~ a1*x1,
x2 ~ a2*x2),
list(
x1 ~ a1*x1 + c12*(exp(abs(x2)))/(1+exp(abs(x2)))*x2,
x2 ~ a2*x2 + c21*(exp(abs(x1)))/(1+exp(abs(x1)))*x1)
)
jacob <- list(
list(x1~x1~a1,
x2~x2~a2),
list(x1~x1~a1,
x1~x2~c12*(exp(abs(x2))/(exp(abs(x2))+1)+x2*sign(x2)*exp(abs(x2))/(1+exp(abs(x2))^2)),
x2~x2~a2,
x2~x1~c21*(exp(abs(x1))/(exp(abs(x1))+1)+x1*sign(x1)*exp(abs(x1))/(1+exp(abs(x1))^2))))
dynm <- prep.formulaDynamics(formula=formula, startval=c( a1=.3, a2=.4, c12=-.5, c21=-.5),
isContinuousTime=FALSE, jacobian=jacob)

#For a full demo example that uses automatic jacobian functions (the default) see:
#demo(RSNonlinearODE , package="dynr")
formula=list(prey ~ a*prey - b*prey*predator, predator ~ -c*predator + d*prey*predator)
startval=c(a = 2.1, c = 0.8, b = 1.9, d = 1.1),
isContinuousTime=TRUE)

#For a full demo example that includes unit-specific random effects in theta.formula see:
#demo(OscWithRand, package="dynr")
formula <- list(x ~ dx,
dx ~ eta_i * x + zeta*dx)
theta.formula  = list (eta_i ~ 1 * eta0  + u1 * eta1 + u2 * eta2 + 1 * b_eta)