# KLopt.lnorm: Calculation of KL-optimal discriminating design for lognormal... In rodd: Optimal Discriminating Designs

## Description

Calculates an approximation xi^{**} of the KL-optimal design (in case of lognormal errors) xi^* for discrimination between a given list of error densities {f_i(x,theta_i), i = 1,…,nu}. This procedure is based on the work [8]. This function mimics `tpopt` almost entirely. It is planed to combine `tpopt` and `KLopt.lnorm` in the future. See `tpopt` for the detailed description of the arguments marked with “-//-”.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```KLopt.lnorm( x, w = rep(1, length(x)) / length(x), eta, sq.var, theta.fix, theta.var = NULL, p, x.lb = min(x), x.rb = max(x), opt = list()) ```

## Arguments

 `x` -//- `w` -//- `eta` a list of means for the error densities {f_i(x,theta_i), i = 1,…,nu} between which proposed optimization should be performed. Every function from this list should be defined in the form of eta_i(x,theta_i), where x is one dimensional variable from X and θ_i is a vector of corresponding model parameters. We will refer to length of this list as nu. `sq.var` a list of variances for the error densities {f_i(x,theta_i), i = 1,…,nu} between which proposed optimization should be performed. Every function from this list should be defined in the form of v^2_i(x,theta_i). This list also has the length equal to nu. `theta.fix` -//- `theta.var` -//- `p` -//- `x.lb` -//- `x.rb` -//- `opt` -//-

## Value

Object of class “KLopt.lnorm” which contains the following fields:

x, w, efficiency, functional

-//-

eta

a list of means from the input.

sq.var

a list of variances from the input.

theta.fix, theta.var, p, x.lb, x.rb, max.iter, done.iter, des.eff, time

-//-

`plot.KLopt.lnorm`, `summary.KLopt.lnorm`, `print.KLopt.lnorm`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347``` ```## Not run: ### Examples from [8] ### Cases 1 and 3 are presented here; case 2 can be computed using the ### function tpopt (see the description of this function for exact example) library(mvtnorm) ### Example 1 from [8]; EMAX vs MM #List of models eta.1 <- function(x, theta.1) theta.1[1] * x + theta.1[2] * x / (x + theta.1[3]) eta.2 <- function(x, theta.2) theta.2[1] * x / (x + theta.2[2]) eta <- list(eta.1, eta.2) #List of fixed parameters theta.1 <- c(1, 1, 1) theta.2 <- c(1, 1) theta.fix <- list(theta.1, theta.2) #Comparison table p <- matrix( c( 0,1, 0,0 ), c(length(eta), length(eta)), byrow = TRUE) ### Case 1 #List of variances sq.var.1 <- function(x, theta.1) 1 sq.var.2 <- function(x, theta.2) 1 sq.var <- list(sq.var.1, sq.var.2) #Case 1, method 1 res <- KLopt.lnorm( x = seq(0.1, 5, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 1) ) plot(res) summary(res) #Case 1, method 2 res <- KLopt.lnorm( x = seq(0.1, 5, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 2) ) plot(res) summary(res) ### case 3 #List of variances sq.var.1 <- function(x, theta.1) exp(eta.1(x, theta.1)) sq.var.2 <- function(x, theta.2) exp(eta.2(x, theta.2)) sq.var <- list(sq.var.1, sq.var.2) #Case 3, method 1 res <- KLopt.lnorm( x = seq(0.1, 5, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 1) ) plot(res) summary(res) #Case 3, method 2 res <- KLopt.lnorm( x = seq(0.1, 5, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 2) ) plot(res) summary(res) ### Example 2 from [8]; sigmoidal #List of models eta.1 = function(x, theta.1) theta.1[1] - theta.1[2] * exp(-theta.1[3] * x ^ theta.1[4]) eta.2 <- function(x, theta.2) theta.2[1] - theta.2[2] * exp(-theta.2[3] * x) #List of fixed parameters theta.1.mean <- c(2, 1, 0.8, 1.5) sigma <- 0.3 theta.1.sigma <- matrix( c( sigma,0, 0,sigma ), c(2, 2), byrow = TRUE) grid <- expand.grid( theta.1.mean[1], theta.1.mean[2], seq(theta.1.mean[3] - sqrt(sigma), theta.1.mean[3] + sqrt(sigma), length.out = 5), seq(theta.1.mean[4] - sqrt(sigma), theta.1.mean[4] + sqrt(sigma), length.out = 5) ) theta.2 <- c(2,1,1) theta.fix <- list() for(i in 1:length(grid[,1])) theta.fix[[length(theta.fix)+1]] <- as.numeric(grid[i,]) theta.fix[[length(theta.fix)+1]] <- theta.2 density.on.grid <- dmvnorm(grid[,3:4], mean = theta.1.mean[3:4], sigma = theta.1.sigma) density.on.grid <- density.on.grid / sum(density.on.grid) eta <- list() for(i in 1:length(grid[,1])) eta <- c(eta, eta.1) eta <- c(eta, eta.2) #Comparison table p <- rep(0,length(eta)) for(i in 1:length(grid[,1])) p <- rbind(p, c(rep(0,length(eta)-1), density.on.grid[i])) p <- rbind(p, rep(0,length(eta))) p <- p[-1,] ### Case 1 sq.var.1 <- function(x, theta.1) 1 sq.var.2 <- function(x, theta.2) 1 sq.var <- list() for(i in 1:length(grid[,1])) sq.var <- c(sq.var, sq.var.1) sq.var <- c(sq.var, sq.var.2) #Case 1, method 1 res <- KLopt.lnorm( x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 1) ) plot(res) summary(res) #Case 1, method 2 res <- KLopt.lnorm( x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 2) ) plot(res) summary(res) ### Case 3 sq.var.1 <- function(x, theta.1) exp(eta.1(x, theta.1)) sq.var.2 <- function(x, theta.2) exp(eta.2(x, theta.2)) sq.var <- list() for(i in 1:length(grid[,1])) sq.var <- c(sq.var, sq.var.1) sq.var <- c(sq.var, sq.var.2) #Case 3, method 1 res <- KLopt.lnorm( x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 1) ) plot(res) summary(res) #Case 3, method 2 res <- KLopt.lnorm( x = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(method = 2) ) plot(res) summary(res) ### Example 3 from [8]; dose response #List of models eta.1 <- function(x, theta.1) theta.1[1] + theta.1[2] * x eta.2 <- function(x, theta.2) theta.2[1] + theta.2[2] * x * (theta.2[3] - x) eta.3 <- function(x, theta.3) theta.3[1] + theta.3[2] * x / (theta.3[3] + x) eta.4 <- function(x, theta.4) theta.4[1] + theta.4[2] / (1 + exp((theta.4[3] - x) / theta.4[4])) #List of fixed parameters theta.1 <- c(60, 0.56) theta.2 <- c(60, 7 / 2250, 600) theta.3 <- c(60, 294, 25) theta.4.mean <- c(49.62, 290.51, 150, 45.51) a <- 45 b <- 20 grid <- expand.grid( c(theta.4.mean[1] - b, theta.4.mean[1], theta.4.mean[1] + a), c(theta.4.mean[2] - b, theta.4.mean[2], theta.4.mean[2] + a), c(theta.4.mean[3] - b, theta.4.mean[3], theta.4.mean[3] + a), c(theta.4.mean[4] - b, theta.4.mean[4], theta.4.mean[4] + a) ) eta <- list() eta <- c(eta, eta.1, eta.2, eta.3) for(i in 1:length(grid[,1])) eta <- c(eta, eta.4) theta.fix <- list(theta.1, theta.2, theta.3) for(i in 1:length(grid[,1])) theta.fix[[length(theta.fix) + 1]] <- as.numeric(grid[i,]) density.on.grid <- rep(1,length(grid[,1])) density.on.grid <- density.on.grid / sum(density.on.grid) #Comparison table p <- rep(0, length(eta)) p <- rbind(p, c(1, rep(0, length(eta) - 1))) p <- rbind(p, c(1, 1, rep(0,length(eta) - 2))) for(i in 1:length(grid[,1])) p <- rbind(p, c(rep(density.on.grid[i], 3), rep(0, length(eta) - 3))) ### Case 1 #List of variances sq.var.1 <- function(x, theta.1) 1 sq.var.2 <- function(x, theta.2) 1 sq.var.3 <- function(x, theta.3) 1 sq.var.4 <- function(x, theta.4) 1 sq.var <- list() sq.var <- c(sq.var, sq.var.1, sq.var.2, sq.var.3) for(i in 1:length(grid[,1])) sq.var <- c(sq.var, sq.var.4) #Case 1, method 1 #Design estimation res <- KLopt.lnorm( x = seq(0, 500, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(max.iter = 10) ) plot(res) summary(res) #Case 1, method 2 #Design estimation res <- KLopt.lnorm( x = seq(0, 500, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list( method = 2, max.iter = 10, weights.evaluation.max.iter = 50, support.epsilon = 1e-4 ) ) plot(res) summary(res) ### Case 3 #List of variances sq.var.1 <- function(x, theta.1) exp(1e-2 * eta.1(x,theta.1)) sq.var.2 <- function(x, theta.2) exp(1e-2 * eta.2(x,theta.2)) sq.var.3 <- function(x, theta.3) exp(1e-2 * eta.3(x,theta.3)) sq.var.4 <- function(x, theta.4) exp(1e-2 * eta.4(x,theta.4)) sq.var <- list() sq.var <- c(sq.var, sq.var.1, sq.var.2, sq.var.3) for(i in 1:length(grid[,1])) sq.var <- c(sq.var, sq.var.4) #Case 3, method 1 #Design estimation res <- KLopt.lnorm( x = seq(0, 500, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(max.iter = 10) ) plot(res) summary(res) #Case 3, method 2 eta.2 <- function(x, theta.2) theta.2[1] + theta.2[2] * x - theta.2[3] * x * x theta.2 <- c(60, 7 * 600 / 2250, 7 / 2250) eta <- list() eta <- c(eta, eta.1, eta.2, eta.3) for(i in 1:length(grid[,1])) eta <- c(eta, eta.4) theta.fix <- list(theta.1, theta.2, theta.3) for(i in 1:length(grid[,1])) theta.fix[[length(theta.fix) + 1]] <- as.numeric(grid[i,]) #Design estimation res <- KLopt.lnorm( x = seq(0, 500, length.out = 10), eta = eta, sq.var = sq.var, theta.fix = theta.fix, p = p, opt = list(max.iter = 6, method = 2) ) plot(res) summary(res) ## End(Not run) ```