R中具有等式和不等式约束系数的回归 [英] Regression with equality and inequality constrained coefficients in R

问题描述

我正在尝试使用RSS获得估计的约束系数. Beta系数被限制在[0,1]之间且总和为1.此外，我的第三个参数被限制在(-1,1)之间.利用以下内容，我可以使用模拟变量获得一个不错的解决方案，但是在实际数据集上实施该方法时，我总是得出一个非唯一的解决方案.反过来，我想知道是否有一种数值上更稳定的方法来获取我的估计参数.

I am trying to obtain estimated constrained coefficients using RSS. The beta coefficients are constrained between [0,1] and sum to 1. Additionally, my third parameter is constrained between (-1,1). Utilizing the below I can obtain a nice solution using simulated variables but when implementing the methodology on my real data set I keep arriving at a non-unique solution. In turn, I'm wondering if there is a more numerically stable way to obtain my estimated parameters.

set.seed(234)
k = 2
a = diff(c(0, sort(runif(k-1)), 1))
n = 1e4
x = matrix(rnorm(k*n), nc = k)
a2 = -0.5
y = a2 * (x %*% a) + rnorm(n)
f = function(u){sum((y - u[3] * (x %*% u[1:2]))^2)}
g = function(v){

v1 = v[1]
v2 = v[2]
u = vector(mode = "double", length = 3)

# ensure in (0,1)
v1 = 1 / (1 + exp(-v1))

# ensure add up to 1
u[1:2] = c(v1, 1 - sum(v1))

# ensure between [-1,1]
u[3] = (v2^2 - 1) / (v2^2 + 1)
u
}

res = optim(rnorm(2), function(v) f(g(v)), hessian = TRUE, method = "BFGS")
eigen(res$hessian)$values
res$convergence
rbind(Est = res$par, SE = sqrt(diag(solve(res$hessian))))
rbind(g(res$par),c(a,a2))

前往 http://zoonek.free.fr/blosxom/R/2012-06-01_Optimization.html

推荐答案

最简单的解决具有相等和不等式约束的优化问题的方法很可能是通过增强的拉格朗日"方法.例如，在R中，这是在 alabama 包中实现的.

The simplest way to solve optimization problems with equality and inequality constraints will most likely be through the "augmented Lagrangian" approach. In R this is, for example, realized in the alabama package.

# function and gradient
fn = function(u){sum((y - u[3] * (x %*% u[1:2]))^2)}
gr = function(u) numDeriv::grad(fn, u)

# constraint sum(u) == 1
heq = function(u) sum(u) - 1
# constraints 0 <= u[1],u[2] <= 1; -1 <= u[3] <= 1
hin = function(u) c(u[1], u[2], 1-u[1], 1-u[2], u[3]+1, 1-u[3])

sol_a = alabama::auglag(c(0.5, 0.5, 0), fn, gr, hin=hin, heq=heq)
sol_a
## $par
## [1]  1.0000000  0.3642904 -0.3642904
## $value
## [1] 10094.74
## ...
## $hessian
##             [,1]        [,2]        [,3]
## [1,] 15009565054  9999999977  9999992926
## [2,]  9999999977 10000002578  9999997167
## [3,]  9999992926  9999997167 10000022569

对于其他包含拉格朗日增强法"过程的软件包，请参见CRAN任务视图中的优化.

For other packages containing an "augmented Lagrangian" procedure see the CRAN Task View on optimization.

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