拉丁超立方体抽样（LHS）研究（Matlab代码实现）

article/2025/10/25 13:17:22

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📋📋📋本文目录如下：🎁🎁🎁

目录

💥1 概述

📚2 运行结果

🌈3 Matlab代码实现

🎉4 参考文献

💥1 概述

拉丁超立方体抽样（LHS）是一种分层随机过程，提供了一种从变量分布中抽样变量的有效方法（Iman和Conover，1980）。LHS涉及从k个变量X1、X2、…Xk中的每一个的规定分布中采样ns值。每个变量的累积分布被划分为N个等概率区间。从每个间隔中随机选择一个值（图1）。为每个变量获得的N个值与其他变量随机配对。与简单的随机抽样不同，该方法通过最大程度地分层每个边际分布，确保了每个变量范围的完全覆盖。

详细文章讲解见第3部分。

📚2 运行结果

部分代码：

% Coefficients in rational approximations.
a = [ -3.969683028665376e+01 2.209460984245205e+02 ...
-2.759285104469687e+02 1.383577518672690e+02 ...
-3.066479806614716e+01 2.506628277459239e+00 ];
b = [ -5.447609879822406e+01 1.615858368580409e+02 ...
-1.556989798598866e+02 6.680131188771972e+01 ...
-1.328068155288572e+01 ];
c = [ -7.784894002430293e-03 -3.223964580411365e-01 ...
-2.400758277161838e+00 -2.549732539343734e+00 ...
4.374664141464968e+00 2.938163982698783e+00 ];
d = [ 7.784695709041462e-03 3.224671290700398e-01 ...
2.445134137142996e+00 3.754408661907416e+00 ];

% Define break-points.
plow = 0.02425;
phigh = 1 - plow;

% Initialize output array.
z = zeros(size(p));

% Rational approximation for central region:
k = plow <= p & p <= phigh;
if any(k(:))
q = p(k) - 0.5;
r = q.*q;
z(k) = (((((a(1)*r+a(2)).*r+a(3)).*r+a(4)).*r+a(5)).*r+a(6)).*q ./ ...
(((((b(1)*r+b(2)).*r+b(3)).*r+b(4)).*r+b(5)).*r+1);
end

% Rational approximation for lower region:
k = 0 < p & p < plow;
if any(k(:))
q = sqrt(-2*log(p(k)));
z(k) = (((((c(1)*q+c(2)).*q+c(3)).*q+c(4)).*q+c(5)).*q+c(6)) ./ ...
((((d(1)*q+d(2)).*q+d(3)).*q+d(4)).*q+1);
end

% Rational approximation for upper region:
k = phigh < p & p < 1;
if any(k(:))
q = sqrt(-2*log(1-p(k)));
z(k) = -(((((c(1)*q+c(2)).*q+c(3)).*q+c(4)).*q+c(5)).*q+c(6)) ./ ...
((((d(1)*q+d(2)).*q+d(3)).*q+d(4)).*q+1);
end

% Case when P = 0:
z(p == 0) = -Inf;

% Case when P = 1:
z(p == 1) = Inf;

% Cases when output will be NaN:
k = p < 0 | p > 1 | isnan(p);
if any(k(:))
z(k) = NaN;
end

% The relative error of the approximation has absolute value less
% than 1.15e-9. One iteration of Halley's rational method (third
% order) gives full machine precision.
k = 0 < p & p < 1;
if any(k(:))
e = 0.5*erfc(-z(k)/sqrt(2)) - p(k); % error
u = e * sqrt(2*pi) .* exp(z(k).^2/2); % f(z)/df(z)
%z(k) = z(k) - u; % Newton's method
z(k) = z(k) - u./( 1 + z(k).*u/2 ); % Halley's method
end