Fcomp_Model: Simulation for functional composition data.
In jiji6454/compReg: Regression with Compositional Covariates

Description Usage Arguments Details Value Author(s) References Examples

simulate functional compositional data.

Fcomp_Model(n, p, m = 0, intercept = TRUE,
            interval = c(0, 1), n_T = 100, obs_spar = 0.6, discrete = FALSE,
            SNR = 1, sigma = 2, Nzero_group = 4,
            rho_X, Corr_X = c("CorrCS", "CorrAR"),
            rho_T, Corr_T = c("CorrAR", "CorrCS"),
            range_beta = c(0.5, 1), beta_c = 1, beta_C ,
            theta.add = c(1, 2, 5, 6), gamma = 0.5,
            basis_beta = c("bs", "OBasis", "fourier"), df_beta = 5, degree_beta = 3,
            insert = c("FALSE", "X", "basis"), method = c("trapezoidal", "step"))

`n`	sample size.
`p`	number of the components in the functional compositional data.
`m`	size of unpenalized variables. The first `ceiling(m/2)` ones are generated with independent `bin(1,0.5)` entries; while the last `(m - ceiling(m/2))` ones are generated with independent `norm(0, 1)` entries. Default is 0.
`intercept`	whether to include an intercept. Default is `TRUE`.
`interval`	a vector of length 2 indicating the time domain. Default is `c(0, 1)`.
`n_T`	an integer specifying length of the equally spaced time sequence on domian `interval`.
`obs_spar`	a percentage used to get sparse ovbservation. Each time point is with probability `obs_spar` to be observed. It allows different subject to be observed on different time points. `obs_spar * n_T > 5` is required.
`discrete`	logical (default is `FALSE`) specifying whether the functional compositional data X is generated at different time points. If `distrete = TRUE`, generate X on dense sequence created by `max(ns_dense = 200 * diff(interval), 5 * n_T)` and then for each subject, randomly sample `n_T` points.
`SNR`	signal to noise ratio.
`sigma`	variance used to generate the covariance matrix `CovMIX = sigma^2 * kronecker(T.Sigma, X.Sigma)`. The "non-normalized" data w_i for each subject is genearted from multivariate normal distribution with covariance `CovMIX`. `T.Sigma` and `X.Sigma` are correlation matrices for time points and components, respectively.
`Nzero_group`	an even integer specifying that the first `Nzero_group` compositional predictors are with non-zero effects. Default is 4.
`rho_X, rho_T`	parameters used to generate correlation matrices.
`Corr_X, Corr_T`	character string specifying correlation structure bewteen components and between time points, respectively. `"CorrCS"`(Default for `Corr_X`) compound symmetry. `"CorrAR"`(Default for `Corr_T`) autoregressive.
`range_beta`	a sorted vector of length 2, specifying the range of coefficient matrix `B` of demension pk*. Specifically, each column of `B` is filled with `Nzero_group/2` values from the unifom distribution over `range_beta` and their negative counterparts. Default is `c(0.5, 1)`.
`beta_c`	value of coefficients for beta0 and beta_c (coefficients for intercept and time-invariant predictors). Default is 1.
`beta_C`	vectorized coefficient matrix. If missing, the program will generate `beta_C` according to `range_beta` and `Nzero_group`.
`theta.add`	logical or integer(s). If integer(s), a vector with value(s) in `[1,p]`, indicating which component(s) of compostions is of high level mean curve. If `TRUE`, the components `c(1:ceiling(Nzero_group/2)` and `Nzero_group + (1:ceiling(Nzero_group/2)))` are set to with high level mean. if `FALSE`, all mean curves are set to 0's.
`gamma`	for the high-level mean groups, log(p * gamma) is added on the "non-normalized" data w_i before the data are converted to be compositional.
`basis_beta, df_beta, degree_beta`	`basis_fun`, `k` and `degree` in `FuncompCGL` respectively.
`insert`	a character string sepcifying method to perform functional interpolation. `"FALSE"`(Default) no interpolation. `"X"` linear interpolation of functional compositional data along the time grid. `"basis"` the functional compositional data is interplolated as a step function along the time grid. If `insert` = `"X"` or `"basis"`, interplolation is conducted on `sseq`, where `sseq` is the sorted sequence of all the observed time points.
`method`	a character string sepcifying method used to approximate integral. `"trapezoidal"`(Default) Sum up areas under the trapezoids. `"step"` Sum up area under the rectangles.

The setup of this simulation follows Sun, Z., Xu, W., Cong, X., Li G. and Chen K. (2020) Log-contrast regression with functional compositional predictors: linking preterm infant's gut microbiome trajectories to neurobehavioral outcome, https://arxiv.org/abs/1808.02403 Annals of Applied Statistics.
Specifically, we first generate correlation matrix X.sigma for components of a composition based on rho_X and Corr_X, and correlation matrix T.sigma for time points based on rho_T and Corr_T. Then, the "non-normalized" data w_i=[w_i(t_1)^T,...,w_i(t_{n_T})^T] for each subject are generated from multivariate normal distrubtion with covariance CovMIX = sigma^2 * kronecker(T.Sigma, X.Sigma), and the mean vector is determined by theta.add and gamma. Each w_i(t_v) is a p-vector for each time point v =1,...,T_n. Finally, the compositional data are obtained as

x_{ij}(t_v) = exp(w_{ij}(t_v))/sum_{k=1}^{p} exp(w_{ik}(t_v)),

for each subject i=1,...,n, component of a composition j=1,...,p and time point v=1,...,n_T.

a list including

`data`	a list of observed data, `y` a vector of response variable, `Comp` a data frame of observed functional compositional data, a column of `Subject_ID`, and a column of `TIME`, `Zc` a matrix of unpenalized variables with dimension nm*, `intercept` whether an `intercept` is included.
`beta`	a length `p*df_beta + m + 1` vector of coefficients
`basis.info`	matrix of the basis function to generate the coefficient curves
`data.raw`	a list consisting of `Z_t.full` the functional compositional data. `Z_ITG` integrated functional compositional data. `Y.tru` true response vector without noise. `X` functional "non-normalized" data `W`.
`parameter`	a list of parameters used in the simulation.

Zhe Sun and Kun Chen

Sun, Z., Xu, W., Cong, X., Li G. and Chen K. (2020) Log-contrast regression with functional compositional predictors: linking preterm infant's gut microbiome trajectories to neurobehavioral outcome, https://arxiv.org/abs/1808.02403 Annals of Applied Statistics

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Data <- Fcomp_Model(n = 50, p = 30, m = 0, intercept = TRUE, Nzero_group = 4,
                    n_T = 20, SNR = 3, rho_X = 0, rho_T = 0.6,
                    df_beta = 5, obs_spar = 1, theta.add = FALSE)