# Generate Data Design Matrix $X$ and Coefficient Vector $\beta$ Generate a random design matrix $X$ and coefficient vector $\beta$ useful for simulations of (high dimensional) linear models. In particular, the function `rXb()` can be used to exactly recreate the reference linear model datasets of the hdi paper. ```r rXb(n, p, s0, xtype = c("toeplitz", "exp.decay", "equi.corr"), btype = "U[-2,2]", permuted = FALSE, iteration = NA, do2S = TRUE, x.par = switch(xtype, "toeplitz" = 0.9, "equi.corr" = 0.8, "exp.decay" = c(0.4, 5)), verbose = TRUE) rX(n, p, xtype, permuted, do2S = TRUE, par = switch(xtype, "toeplitz" = 0.9, "equi.corr" = 0.8, "exp.decay" = c(0.4, 5))) ``` ## Arguments - `n`: integer; the sample size $n$ (paper had always `n = 100`). - `p`: integer; the number of coefficients in the linear model. (paper had always `p = 500`). - `s0`: integer number of **nonzero** coefficients desired in the model; hence at most `p`. - `xtype`: a `character` string specifying the type of design matrix one wants to generate. Must be one of `"toeplitz"`, `"equi.corr"` or `"exp.decay"`. - `btype`: a `character` string specifying the type of nonzero coefficients (beta ) one wants to generate. In the hdi paper, this has been one of "U[-2,2]", "U[0,2]", "U[0,4]", "bfix1", "bfix2" and "bfix10". In general, any string of the form `"U[a,b]"` or `"bfix"` is allowed, where `a`, `b`, and `` must be numbers (with $a <= b$). - `permuted`: logical specifying if the columns of the design matrix should be permuted. - `iteration`: integer or `NA` specifying if seeds should be set to generate reproducible realizations of the design type and coefficients type. `NA` corresponds to not setting seeds. Iteration numbers 1 to 50 correspond to the setups from the paper. If a seed is set, the original `.Random.seed` at the point of entering the function is saved and is restored upon exit of the data generation. If `NA`, the current `.Random.seed` is taken as usual in . - `do2S`: logical indicating if in the case of `xtype`s `"toeplitz"` or `"equi.corr"`, the $p * p$ covariance matrix should be inverted twice. Must be true, to regenerate the $X$ matrices from the hdi paper exactly to the last bit . - `x.par,par`: the parameters to be used for the design matrix. Must be a numeric vector of length one or two. The default uses the parameters also used in the hdi paper. - `verbose`: should the function give a message if seeds are being set? (logical). ## Details Generation of the design matrix X: For all `xtype`'s, the $X$ matrix will be multivariate normal, with mean zero and (co)variance matrix $\Sigma = $`C` determined from `xtype`, `x.par` and $p$ as follows: - **`xtype = "toeplitz"`:**: `C <- par ^ abs(toeplitz(0:(p-1)))` - **`xtype = "equi.corr"`:**: $\Sigma_{i,j} = \code{par}$ for $i != j$, and $= 1$ for $i = j$, i.e., on the diagonal. - **`xtype = "exp.decay"`:**: `C <- solve(par[1] ^ abs(toeplitz(0:(p-1)) / par[2]))` ## Returns - **For `rXb()`:**: A `list` with components - **x**: the generated $n * p$ design matrix $X$. - **beta**: the generated coefficient vector $\beta$ (beta ). - **For `rX()`:**: the generated $n * p$ design matrix $X$. ## References Dezeure, R., , P., Meier, L. and Meinshausen, N. (2015) High-dimensional inference: confidence intervals, p-values and R-software hdi. **Statistical Science** 30 , 533--558. ## Author(s) Ruben Dezeure dezeure@stat.math.ethz.ch ## Examples ```r ## Generate the first realization of the linear model with design matrix ## type Toeplitz and coefficients type uniform between -2 and 2 dset <- rXb(n = 80, p = 20, s0 = 3, xtype = "toeplitz", btype = "U[-2,2]") x <- dset$x beta <- dset$beta ## generate 100 response vectors of this linear model y <- as.vector( x %*% beta ) + replicate(100, rnorm(nrow(x))) ## Use 'beta_min' fulfilling beta's (non standard 'btype'): str(ds2 <- rXb(n = 50, p = 12, s0 = 3, xtype = "exp.decay", btype = "U[0.1, 5]")) ## Generate a design matrix of type "toeplitz" set.seed(3) # making it reproducible X3 <- rX(n = 800, p = 500, xtype = "toeplitz", permuted = FALSE) ## permute the columns set.seed(3) Xp <- rX(n = 800, p = 500, xtype = "toeplitz", permuted = TRUE) ```

rXb.rd function