,
Julie Zhou [aut, ctb],
Jason Hou-Liu [ctb]| Title: | Optimal Regression Design under the Second-Order Least Squares Estimator |
|---|---|
| Description: | With given inputs that include number of points, discrete design space, a measure of skewness, models and parameter value, this package calculates the objective value, optimal designs and plot the equivalence theory under A- and D-optimal criteria under the second-order Least squares estimator. This package is based on the paper "Properties of optimal regression designs under the second-order least squares estimator" by Chi-Kuang Yeh and Julie Zhou (2021) <doi:10.1007/s00362-018-01076-6>. |
| Authors: | Chi-Kuang Yeh [aut, cre] ,
Julie Zhou [aut, ctb],
Jason Hou-Liu [ctb] |
| Maintainer: | Chi-Kuang Yeh <[email protected]> |
| License: | GPL-3 |
| Version: | 0.0.3 |
| Built: | 2024-11-04 05:51:29 UTC |
| Source: | https://github.com/chikuang/slsedesign |
Calculate the A-optimal design under the second-order Least squares estimator
Aopt(N, u, tt, FUN, theta, num_iter = 1000)Aopt(N, u, tt, FUN, theta, num_iter = 1000)
N |
The number of sample points in the design space. |
u |
The discretized design space. |
tt |
The level of skewness between 0 to 1 (inclusive). When tt=0, it is equivalent to compute the A-optimal design under the ordinary least squares estimator. |
FUN |
The function to calculate the derivative of the given model. |
theta |
The parameter value of the model. |
num_iter |
Maximum number of iteration. |
This function calculates the A-optimal design and the loss function under the A-optimality. The loss function under A-optimality is defined as the trace of the inverse of the Fisher information matrix
A list that contains 1. Value of the objective function at solution. 2. Status. 3. Optimal design
poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } Npt <- 101 my_design <- Aopt(N = Npt, u = seq(-1, +1, length.out = Npt), tt = 0, FUN = poly3, theta = rep(0,4), num_iter = 2000) round(my_design$design, 3) my_design$valpoly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } Npt <- 101 my_design <- Aopt(N = Npt, u = seq(-1, +1, length.out = Npt), tt = 0, FUN = poly3, theta = rep(0,4), num_iter = 2000) round(my_design$design, 3) my_design$val
Calculate the loss function of the A-optimal design
calc_phiA(design, theta, FUN, tt, A)calc_phiA(design, theta, FUN, tt, A)
design |
The resulted design that contains the design points and the associated weights |
theta |
The parameter value of the model |
FUN |
The function to calculate the derivative of the given model. |
tt |
The level of skewness |
A |
The calculated covariance matrix |
This function calculates the loss function of the design problem under the A-optimality. The loss function under A-optimality is defined as the trace of the inverse of the Fisher information matrix
The loss of the model at each design points
my_design <- data.frame(location = c(0, 180), weight = c(1/2, 1/2)) theta <- c(0.05, 0.5) peleg <- function(xi, theta){ deno <- (theta[1] + xi * theta[2])^2 rbind(-xi/deno, -xi^2/deno) } A <- matrix(c(1, 0, 0, 0, 0.2116, 1.3116, 0, 1.3116, 15.462521), byrow = TRUE, ncol = 3) res <- calc_phiA(my_design, theta, peleg, 0, A) resmy_design <- data.frame(location = c(0, 180), weight = c(1/2, 1/2)) theta <- c(0.05, 0.5) peleg <- function(xi, theta){ deno <- (theta[1] + xi * theta[2])^2 rbind(-xi/deno, -xi^2/deno) } A <- matrix(c(1, 0, 0, 0, 0.2116, 1.3116, 0, 1.3116, 15.462521), byrow = TRUE, ncol = 3) res <- calc_phiA(my_design, theta, peleg, 0, A) res
Calculate the loss function of the D-optimal design
calc_phiD(design, theta, FUN, tt, A)calc_phiD(design, theta, FUN, tt, A)
design |
The resulted design that contains the design points and the associated weights |
theta |
The parameter value of the model |
FUN |
The function to calculate the derivative of the given model. |
tt |
The level of skewness |
A |
The calculated covariance matrix |
This function calculates the loss function of the design problem under the D-optimality. The loss function under D-optimality is defined as the log determinant of the inverse of the Fisher information matrix
The loss of the model at each design points
my_design <- data.frame(location = c(0, 180), weight = c(1/2, 1/2)) theta <- c(0.05, 0.5) peleg <- function(xi, theta){ deno <- (theta[1] + xi * theta[2])^2 rbind(-xi/deno, -xi^2/deno) } A <- matrix(c(1, 0, 0, 0, 0.2116, 1.3116, 0, 1.3116, 15.462521), byrow = TRUE, ncol = 3) res <- calc_phiA(my_design, theta, peleg, 0, A) resmy_design <- data.frame(location = c(0, 180), weight = c(1/2, 1/2)) theta <- c(0.05, 0.5) peleg <- function(xi, theta){ deno <- (theta[1] + xi * theta[2])^2 rbind(-xi/deno, -xi^2/deno) } A <- matrix(c(1, 0, 0, 0, 0.2116, 1.3116, 0, 1.3116, 15.462521), byrow = TRUE, ncol = 3) res <- calc_phiA(my_design, theta, peleg, 0, A) res
Calculate the D-optimal design under the SLSE
Dopt(N, u, tt, FUN, theta, num_iter = 1000)Dopt(N, u, tt, FUN, theta, num_iter = 1000)
N |
The number of sample points in the design space. |
u |
The discretized design space. |
tt |
The level of skewness. When tt=0, it is equivalent to compute the D-optimal design under the ordinary least squares estimator. |
FUN |
The function to calculate the derivative of the given model. |
theta |
The parameter value of the model. |
num_iter |
Maximum number of iteration. |
This function calculates the D-optimal design and the loss function under the D-optimality. The loss function under D-optimality is defined as the log determinant of the inverse of the Fisher information matrix.
A list that contains 1. Value of the objective function at solution. 2. Status. 3. Optimal design
poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } Npt <- 101 my_design <- Dopt(N = Npt, u = seq(-1, +1, length.out = Npt), tt = 0, FUN = poly3, theta = rep(0,4), num_iter = 2000) round(my_design$design, 3) my_design$valpoly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } Npt <- 101 my_design <- Dopt(N = Npt, u = seq(-1, +1, length.out = Npt), tt = 0, FUN = poly3, theta = rep(0,4), num_iter = 2000) round(my_design$design, 3) my_design$val
Verify the optimality condition for the A-optimal design
plot_direction_Aopt(u, design, tt, FUN, theta)plot_direction_Aopt(u, design, tt, FUN, theta)
u |
The discretized design points. |
design |
The A-optimal design that contains the design points and the associated weights |
tt |
The level of skewness. |
FUN |
The function to calculate the derivative of the given model. |
theta |
The parameter value of the model. |
This function produces the figure for the directional derivative of the given A-optimal design of the compact supports. According to the general equivalence theorem, for an optimal design, all the negative value of the directional derivative should be below zero line.
The plot of the negative value of the directional derivative of an A-optimal design
poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } design = data.frame(location = c(-1, -0.464, 0.464, 1), weight = c(0.151, 0.349, 0.349, 0.151)) u = seq(-1, 1, length.out = 201) plot_direction_Aopt(u=u, design=design, tt=0, FUN = poly3, theta = rep(0,4))poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } design = data.frame(location = c(-1, -0.464, 0.464, 1), weight = c(0.151, 0.349, 0.349, 0.151)) u = seq(-1, 1, length.out = 201) plot_direction_Aopt(u=u, design=design, tt=0, FUN = poly3, theta = rep(0,4))
Verify the optimality condition for the D-optimal design
plot_direction_Dopt(u, design, tt, FUN, theta)plot_direction_Dopt(u, design, tt, FUN, theta)
u |
The discretized design points. |
design |
The D-optimal design that contains the design points and the associated weights. |
tt |
The level of skewness. |
FUN |
The function to calculate the derivative of the given model. |
theta |
The parameter value of the model. |
This function produces the figure for the negative value of the directional derivative of the given D-optimal design of the compact supports. According to the general equivalence theorem, for an optimal design, all the directional derivative should be below zero line.
The plot of the negative value of the directional derivative of a D-optimal design
poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } design = data.frame(location = c(-1, -0.447, 0.447, 1), weight = rep(0.25, 4)) u = seq(-1, 1, length.out = 201) plot_direction_Dopt(u, design, tt=0, FUN = poly3, theta = rep(0, 4))poly3 <- function(xi, theta){ matrix(c(1, xi, xi^2, xi^3), ncol = 1) } design = data.frame(location = c(-1, -0.447, 0.447, 1), weight = rep(0.25, 4)) u = seq(-1, 1, length.out = 201) plot_direction_Dopt(u, design, tt=0, FUN = poly3, theta = rep(0, 4))
Plot the weight distribution of the optimal design for univaraite regression model
plot_weight(design)plot_weight(design)
design |
The resulted design that contains the design points and the associated weights |
This functions produce a figure that contains the location and their associated weights of the resulted optimal design measures.
The plot that shows the given optimal design
Des = list(location = c(-1, +1), weight = c(0.5, 0.5)) plot_weight(Des)Des = list(location = c(-1, +1), weight = c(0.5, 0.5)) plot_weight(Des)