# Locally D-Optimal Designs Finds locally D-optimal designs for linear and nonlinear models. It should be used when a vector of initial estimates is available for the unknown model parameters. Locally optimal designs may not be efficient when the initial estimates are far away from the true values of the parameters. ```r locally( formula, predvars, parvars, family = gaussian(), lx, ux, iter, k, inipars, fimfunc = NULL, ICA.control = list(), sens.control = list(), initial = NULL, npar = length(inipars), plot_3d = c("lattice", "rgl"), x = NULL, crtfunc = NULL, sensfunc = NULL ) ``` ## Arguments - `formula`: A linear or nonlinear model `formula`. A symbolic description of the model consists of predictors and the unknown model parameters. Will be coerced to a `formula` if necessary. - `predvars`: A vector of characters. Denotes the predictors in the `formula`. - `parvars`: A vector of characters. Denotes the unknown parameters in the `formula`. - `family`: A description of the response distribution and the link function to be used in the model. This can be a family function, a call to a family function or a character string naming the family. Every family function has a link argument allowing to specify the link function to be applied on the response variable. If not specified, default links are used. For details see `family`. By default, a linear gaussian model `gaussian()` is applied. - `lx`: Vector of lower bounds for the predictors. Should be in the same order as `predvars`. - `ux`: Vector of upper bounds for the predictors. Should be in the same order as `predvars`. - `iter`: Maximum number of iterations. - `k`: Number of design points. Must be at least equal to the number of model parameters to avoid singularity of the FIM. - `inipars`: A vector of initial estimates for the unknown parameters. It must match `parvars` or the argument `param` of the function `fimfunc`, when provided. - `fimfunc`: A function. Returns the FIM as a `matrix`. Required when `formula` is missing. See 'Details' of `minimax`. - `ICA.control`: ICA control parameters. For details, see `ICA.control`. - `sens.control`: Control Parameters for Calculating the ELB. For details, see `sens.control`. - `initial`: A matrix of the initial design points and weights that will be inserted into the initial solutions (countries) of the algorithm. Every row is a design, i.e. a concatenation of `x` and `w`. Will be coerced to a `matrix` if necessary. See 'Details' of `minimax`. - `npar`: Number of model parameters. Used when `fimfunc` is given instead of `formula` to specify the number of model parameters. If not given, the sensitivity plot may be shifted below the y-axis. When `NULL`, it is set to `length(inipars)`. - `plot_3d`: Which package should be used to plot the sensitivity (derivative) function for two-dimensional design space. Defaults to `"lattice"`. - `x`: A vector of candidate design (support) points. When is not set to `NULL` (default), the algorithm only finds the optimal weights for the candidate points in `x`. Should be set when the user has a finite number of candidate design points and the purpose is to find the optimal weight for each of them (when zero, they will be excluded from the design). For design points with more than one dimension, see 'Details' of `sensminimax`. - `crtfunc`: (Optional) a function that specifies an arbitrary criterion. It must have especial arguments and output. See 'Details' of `minimax`. - `sensfunc`: (Optional) a function that specifies the sensitivity function for `crtfunc`. See 'Details' of `minimax`. ## Returns an object of class `minimax` that is a list including three sub-lists: - **`arg`**: A list of design and algorithm parameters. - **`evol`**: A list of length equal to the number of iterations that stores the information about the best design (design with least criterion value) of each iteration. `evol[[iter]]` contains: |||| |:--|:--|:--| |`iter`||Iteration number.| |`x`||Design points.| |`w`||Design weights.| |`min_cost`||Value of the criterion for the best imperialist (design).| |`mean_cost`||Mean of the criterion values of all the imperialists.| |`sens`||An object of class `'sensminimax'` . See below.| |`param`||Vector of parameters.| - **`empires`**: A list of all the empires of the last iteration. - **`alg`**: A list with following information: |||| |:--|:--|:--| |`nfeval`||Number of function evaluations. It does not count the function evaluations from checking the general equivalence theorem.| |`nlocal`||Number of successful local searches.| |`nrevol`||Number of successful revolutions.| |`nimprove`||Number of successful movements toward the imperialists in the assimilation step.| |`convergence`||Stopped by `'maxiter'` or `'equivalence'` ?| - **`method`**: A type of optimal designs used. - **`design`**: Design points and weights at the final iteration. - **`out`**: A data frame of design points, weights, value of the criterion for the best imperialist (min_cost), and Mean of the criterion values of all the imperialistsat each iteration (mean_cost). The list `sens` contains information about the design verification by the general equivalence theorem. See `sensminimax` for more details. It is given every `ICA.control$checkfreq` iterations and also the last iteration if `ICA.control$checkfreq >= 0`. Otherwise, `NULL`. `param` is a vector of parameters that is the global minimum of the minimax criterion or the global maximum of the standardized maximin criterion over the parameter space, given the current `x`, `w`. ## Details Let $M(\xi, \theta_0)$ be the Fisher information matrix (FIM) of a $k-$point design $\xi$ and $\theta_0$ be the vector of the initial estimates for the unknown parameters. A locally D-optimal design $\xi*$ minimizes over $\Xi$ $$ -\log|M(\xi, \theta_0)|.-log|M(\xi, \theta_0)|. $$ One can adjust the tuning parameters in `ICA.control` to set a stopping rule based on the general equivalence theorem. See "Examples" below. ## Examples ```r ################################# # Exponential growth model ################################ # See how we set stopping rule by adjusting 'stop_rule', 'checkfreq' and 'stoptol' # It calls the 'senslocally' function every checkfreq = 50 iterations to # calculate the ELB. if ELB is greater than stoptol = .95, then the algoithm stops. # initializing by one iteration res1 <- locally(formula = ~a + exp(-b*x), predvars = "x", parvars = c("a", "b"), lx = 0, ux = 1, inipars = c(1, 10), iter = 1, k = 2, ICA.control= ICA.control(rseed = 100, stop_rule = "equivalence", checkfreq = 20, stoptol = .95)) # update the algorithm res1 <- update(res1, 150) #stops at iteration 21 because ELB is greater than .95 ### fixed x, lx and ux are only required for equivalence theorem res1.1 <- locally(formula = ~a + exp(-b*x), predvars = "x", parvars = c("a", "b"), lx = 0, ux = 1, inipars = c(1, 10), iter = 100, x = c(.25, .5, .75), ICA.control= ICA.control(rseed = 100)) plot(res1.1) # we can not have an optimal design using this x ################################ ## two parameter logistic model ################################ res2 <- locally(formula = ~1/(1 + exp(-b *(x - a))), predvars = "x", parvars = c("a", "b"), family = binomial(), lx = -3, ux = 3, inipars = c(1, 3), iter = 1, k = 2, ICA.control= list(rseed = 100, stop_rule = "equivalence", checkfreq = 50, stoptol = .95)) res2 <- update(res2, 100) # stops at iteration 51 ################################ # A model with two predictors ################################ # mixed inhibition model res3 <- locally(formula = ~ V*S/(Km * (1 + I/Kic)+ S * (1 + I/Kiu)), predvars = c("S", "I"), parvars = c("V", "Km", "Kic", "Kiu"), family = gaussian(), lx = c(0, 0), ux = c(30, 60), k = 4, iter = 300, inipars = c(1.5, 5.2, 3.4, 5.6), ICA.control= list(rseed = 100, stop_rule = "equivalence", checkfreq = 50, stoptol = .95)) # stops at iteration 100 # fixed x res3.1 <- locally(formula = ~ V*S/(Km * (1 + I/Kic)+ S * (1 + I/Kiu)), predvars = c("S", "I"), parvars = c("V", "Km", "Kic", "Kiu"), family = gaussian(), lx = c(0, 0), ux = c(30, 60), iter = 100, x = c(20, 4, 20, 4, 10, 0, 0, 30, 3, 2), inipars = c(1.5, 5.2, 3.4, 5.6), ICA.control= list(rseed = 100)) ################################### # user-defined optimality criterion ################################## # When the model is defined by the formula interface # A-optimal design for the 2PL model. # the criterion function must have argument x, w fimfunc and the parameters defined in 'parvars'. # use 'fimfunc' as a function of the design points x, design weights w and # the 'parvars' parameters whenever needed. Aopt <-function(x, w, a, b, fimfunc){ sum(diag(solve(fimfunc(x = x, w = w, a = a, b = b)))) } ## the sensitivtiy function # xi_x is a design that put all its mass on x in the definition of the sensitivity function # x is a vector of design points Aopt_sens <- function(xi_x, x, w, a, b, fimfunc){ fim <- fimfunc(x = x, w = w, a = a, b = b) M_inv <- solve(fim) M_x <- fimfunc(x = xi_x, w = 1, a = a, b = b) sum(diag(M_inv %*% M_x %*% M_inv)) - sum(diag(M_inv)) } res4 <- locally(formula = ~1/(1 + exp(-b * (x-a))), predvars = "x", parvars = c("a", "b"), family = "binomial", lx = -3, ux = 3, inipars = c(1, 1.25), iter = 1, k = 2, crtfunc = Aopt, sensfunc = Aopt_sens, ICA.control = list(checkfreq = Inf)) res4 <- update(res4, 50) # When the FIM of the model is defined directly via the argument 'fimfunc' # the criterion function must have argument x, w fimfunc and param. # use 'fimfunc' as a function of the design points x, design weights w # and param whenever needed. Aopt2 <-function(x, w, param, fimfunc){ sum(diag(solve(fimfunc(x = x, w = w, param = param)))) } ## the sensitivtiy function # xi_x is a design that put all its mass on x in the definition of the sensitivity function # x is a vector of design points Aopt_sens2 <- function(xi_x, x, w, param, fimfunc){ fim <- fimfunc(x = x, w = w, param = param) M_inv <- solve(fim) M_x <- fimfunc(x = xi_x, w = 1, param = param) sum(diag(M_inv %*% M_x %*% M_inv)) - sum(diag(M_inv)) } res4.1 <- locally(fimfunc = FIM_logistic, lx = -3, ux = 3, inipars = c(1, 1.25), iter = 1, k = 2, crtfunc = Aopt2, sensfunc = Aopt_sens2, ICA.control = list(checkfreq = Inf)) res4.1 <- update(res4.1, 50) plot(res4.1) # locally c-optimal design # example from Chaloner and Larntz (1989) Figure 3 c_opt <-function(x, w, a, b, fimfunc){ gam <- log(.95/(1-.95)) M <- fimfunc(x = x, w = w, a = a, b = b) c <- matrix(c(1, -gam * b^(-2)), nrow = 1) B <- t(c) %*% c sum(diag(B %*% solve(M))) } c_sens <- function(xi_x, x, w, a, b, fimfunc){ gam <- log(.95/(1-.95)) M <- fimfunc(x = x, w = w, a = a, b = b) M_inv <- solve(M) M_x <- fimfunc(x = xi_x, w = 1, a = a, b = b) c <- matrix(c(1, -gam * b^(-2)), nrow = 1) B <- t(c) %*% c sum(diag(B %*% M_inv %*% M_x %*% M_inv)) - sum(diag(B %*% M_inv)) } res4.2 <- locally(formula = ~1/(1 + exp(-b * (x-a))), predvars = "x", parvars = c("a", "b"), family = "binomial", lx = -1, ux = 1, inipars = c(0, 7), iter = 1, k = 2, crtfunc = c_opt, sensfunc = c_sens, ICA.control = list(rseed = 1, checkfreq = Inf)) res4.2 <- update(res4.2, 100) ``` ## References Masoudi E, Holling H, Wong W.K. (2017). Application of Imperialist Competitive Algorithm to Find Minimax and Standardized Maximin Optimal Designs. Computational Statistics and Data Analysis, 113, 330-345. ## See Also `senslocally`

locally function