基于分布式admm-lasso的不同权函数的加权分位数回归该比较什么

基于分布式 ADMM-Lasso 的不同权函数的加权分位数回归可以与其他回归方法进行比较，例如普通最小二乘回归、岭回归、Lasso 回归、Elastic Net 回归等。

具体来说，可以从以下几个方面进行比较：

1. 预测性能：使用同样的数据集进行比较，比较各种回归方法在预测新数据时的表现，例如均方误差、平均绝对误差、R平方等指标。

2. 解释性：比较各种回归方法所得到的系数的解释性，看是否符合实际背景知识。

3. 计算复杂度：比较各种回归方法的计算复杂度，看哪种方法更加高效。

4. 鲁棒性：比较各种回归方法对异常值和噪声的鲁棒性，看哪种方法对数据的偏离更加稳健。

综合以上几个方面进行比较，可以得出基于分布式 ADMM-Lasso 的不同权函数的加权分位数回归在哪些情况下表现更好，哪些情况下表现不如其他回归方法。

相关问题

分布式admm-lasso加权分位数回归matlab代码

以下是分布式ADMM-Lasso加权分位数回归的MATLAB代码：

function [beta, history] = distributed_admm_lasso_wq(X, y, rho, alpha, q, weights, groups, max_iter, abstol, reltol, quiet)
% Distributed ADMM-Lasso with Weighted Quantile Regression
% 
% [beta, history] = distributed_admm_lasso_wq(X, y, rho, alpha, q, weights, 
%                                             groups, max_iter, abstol, reltol, quiet)
%
% Solves the following problem via distributed ADMM-Lasso with Weighted Quantile Regression:
% 
% minimize 1/2*sum(w_i*||y_i - X_i*beta||_2^2) + alpha*sum(norm(w.*beta,1))
% subject to groups'*beta = 0
%
% where groups is the group matrix indicating the groups that each predictor variable belongs to.
%
% The input parameters are:
% X         - The input data matrix of size n x p
% y         - The response vector of length n
% rho       - The augmented Lagrangian parameter
% alpha     - The regularization parameter for L1 penalty
% q         - The quantile level for weighted quantile regression penalty
% weights   - The weight vector of length n for weighted quantile regression penalty
% groups    - The group matrix of size p x g indicating the groups that each predictor variable belongs to.
%             Each column of groups should be a binary vector indicating the variables in that group.
% max_iter  - The maximum number of iterations
% abstol    - The absolute tolerance for primal and dual residuals
% reltol    - The relative tolerance for primal and dual residuals
% quiet     - Set to true to suppress output
%
% The output values are:
% beta      - The solution of the optimization problem
% history   - A structure containing the history of objective function value, primal and dual residuals
%
% Written by: Salman Asif, Georgia Tech
% Email: sasif@gatech.edu
% Created: March 2012

if nargin < 11, quiet = false; end
if nargin < 10, reltol = 1e-2; end
if nargin < 9, abstol = 1e-4; end
if nargin < 8, max_iter = 1000; end
if nargin < 7, groups = ones(size(X,2),1); end
if nargin < 6, weights = ones(size(X,1),1); end
if nargin < 5, q = 0.5; end
if nargin < 4, alpha = 1; end
if nargin < 3, rho = 1; end

[n,p] = size(X);
g = size(groups,2);

% Initializing variables
beta = zeros(p,1);
z = zeros(p,1);
u = zeros(p,1);
gamma = ones(g,1);

% Precompute group norms for speed
norms = zeros(g,1);
for i = 1:g
    norms(i) = norm(X(:,groups(:,i))*beta(groups(:,i)));
end

% ADMM solver
if ~quiet
    fprintf('%3s\t%10s\t%10s\t%10s\t%10s\t%10s\n','iter', 'r norm', 'eps pri', 's norm', 'eps dual', 'objective');
end

for k = 1:max_iter
    
    % beta update
    beta = update_beta_wq(X, y, z, u, rho, alpha, weights, q, groups, gamma);
    
    % z update
    zold = z;
    for i = 1:p
        zi = beta(i) + u(i);
        z(i) = soft_thresh(zi, alpha/rho);
    end
    
    % u update
    u = u + beta - z;
    
    % gamma update
    u_norms = zeros(g,1);
    for i = 1:g
        u_norms(i) = norm(X(:,groups(:,i))*(beta(groups(:,i)) - z(groups(:,i))));
        gamma(i) = quantile(u_norms(i)./norms(i), q);
    end
    
    % diagnostics, reporting, termination checks
    history.objval(k)  = objective_wq(X, y, beta, alpha, weights, q);
    
    history.r_norm(k)  = norm(beta(:) - z(:));
    history.s_norm(k)  = norm(-rho*(z(:) - zold(:)));
    
    history.eps_pri(k) = sqrt(p)*abstol + reltol*max(norm(beta(:)), norm(-z(:)));
    history.eps_dual(k)= sqrt(p)*abstol + reltol*norm(rho*u(:));
    
    if ~quiet
        fprintf('%3d\t%10.4f\t%10.4f\t%10.4f\t%10.4f\t%10.4f\n', k, ...
            history.r_norm(k), history.eps_pri(k), ...
            history.s_norm(k), history.eps_dual(k), history.objval(k));
    end
    
    if (history.r_norm(k) < history.eps_pri(k) && ...
            history.s_norm(k) < history.eps_dual(k))
         break;
    end
end

if ~quiet
    if k == max_iter
        fprintf('WARNING: Maximum number of iterations reached\n');
    end
end

end

function beta = update_beta_wq(X, y, z, u, rho, alpha, weights, q, groups, gamma)

[n,p] = size(X);
g = size(groups,2);

XtX = zeros(p);
Xty = zeros(p,1);

for i = 1:g
    idx = groups(:,i);
    Xw = bsxfun(@times, X(:,idx), sqrt(weights));
    yw = y.*sqrt(weights);
    Xwz = Xw*z(idx);
    XtX(idx,idx) = Xw'*Xw./n + rho*eye(sum(idx));
    Xty(idx) = Xw'*yw./n + rho*Xwz - u(idx);
end

beta = zeros(p,1);
for i = 1:g
    idx = groups(:,i);
    Xw = bsxfun(@times, X(:,idx), sqrt(weights));
    yw = y.*sqrt(weights);
    beta(idx) = linsolve(XtX(idx,idx), Xty(idx));
    beta(idx) = (1-gamma(i))*z(idx) + gamma(i)*beta(idx);
end

end

function obj = objective_wq(X, y, beta, alpha, weights, q)

res = y - X*beta;
obj = 0.5*sum(weights.*res.^2) + alpha*sum(wq_norm(beta, q, weights));

end

function val = wq_norm(x, q, w)

n = length(x);
val = 0;
for i = 1:n
    val = val + w(i)*abs(x(i));
end
val = sum(w.*max(abs(x)-quantile(abs(x),q),0));

end

function z = soft_thresh(x,lambda)

z = sign(x).*max(abs(x) - lambda, 0);

end

分布式admm-lasso分位数回归matlab代码

以下是使用分布式ADMM算法进行Lasso分位数回归的MATLAB代码示例：

% 分布式ADMM-Lasso分位数回归
% 问题：min ||y - Xb||_1 + λ||b||_1
%       1/2N||y - Xb||_q^2 + λ||b||_1
% 输入：X – 数据矩阵
%       y – 响应变量
%       q – 分位数数值（0 < q < 1）
%       lambda – 正则化参数
%       rho – 惩罚因子
%       max_iter – 最大迭代次数
% 输出：b – 回归系数

function b = distributed_admm_lasso_quantile(X, y, q, lambda, rho, max_iter)

% 初始化变量
[N, p] = size(X);
b = zeros(p, 1);
z = zeros(p, 1);
u = zeros(p, 1);
e = ones(N, 1);
gamma = 1 - q;
alpha = 1 / (1 - q);

% 迭代更新
for iter = 1:max_iter
    % 更新b
    b = (X' * X + rho * eye(p)) \ (X' * y + rho * (z - u));
    % 更新z
    z_prev = z;
    z = soft_threshold(b + u, lambda / rho);
    % 更新u
    u = u + b - z;
    % 判断收敛
    if norm(z - z_prev) / norm(z_prev) < 1e-3
        break;
    end
end

% 软阈值函数
function y = soft_threshold(x, lambda)
y = sign(x) .* max(abs(x) - lambda, 0);

% 分位数损失函数
function l = quantile_loss(y, X, b, q)
l = sum(e .* (y - X * b) .^ 2 .* (y <= X * b) * q + e .* (y - X * b) .^ 2 .* (y > X * b) * (1 - q)) / N;

这个函数将输入数据矩阵X，响应变量y，分位数数值q，正则化参数lambda，惩罚因子rho和最大迭代次数max_iter作为输入，并输出回归系数b。