MATLAB中[ ˜Q, ˜R] = qr(H);代码解释

此代码使用MATLAB中的qr函数计算输入参数H的QR分解。QR分解是一种将矩阵分解为一个正交矩阵Q乘以一个上三角矩阵R的技术。

该函数返回的是矩阵H的QR分解。其中，计算得到的正交矩阵Q存储在变量˜Q中，上三角矩阵R存储在变量˜R中。用法示例如下：

[ ˜Q, ˜R ] = qr( H );
% H为待分解的矩阵，˜Q为计算得到的正交矩阵，˜R为上三角矩阵

注意，如果矩阵H不是方阵，则QR分解将会使用一些特殊技巧来分解。在此情况下，上三角矩阵R的列数将小于矩阵的行数和列数。

相关问题

生成一段matlab编程：先验分布为α ∼ N (0, 5) ， β˜ ∼ U(0, 10)， ˜σ ∼ U(0, 30)， ˜θ = [ ˜α β˜ ˜σ]′ ,Yi = α + β Xi + εi , εi i.i.d ∼ N (0, σ2 ),Calculate the posterior mean and variance of θ ˜, and plot the histogram for each elements of θ ˜

Sure, here is the MATLAB code for calculating the posterior mean and variance of θ ˜ and plotting the histogram for each element of θ ˜:

% Define the prior distributions
prior_alpha = makedist('Normal','mu',0,'sigma',sqrt(5));
prior_beta = makedist('Uniform','lower',0,'upper',10);
prior_sigma = makedist('Uniform','lower',0,'upper',30);

% Generate some simulated data
N = 100;
X = rand(N,1)*10;
alpha_true = 2;
beta_true = 5;
sigma_true = 2;
Y = alpha_true + beta_true*X + randn(N,1)*sigma_true;

% Define the likelihood function
likelihood = @(theta) -sum(log(normpdf(Y,theta(1)+theta(2)*X,sqrt(theta(3)^2))));

% Define the posterior distribution
posterior = @(theta) likelihood(theta) + log(pdf(prior_alpha,theta(1))) + log(pdf(prior_beta,theta(2))) + log(pdf(prior_sigma,theta(3)));

% Use MCMC to sample from the posterior distribution
n_samples = 10000;
theta_samples = zeros(n_samples,3);
theta_samples(1,:) = [0,0,0];
for i=2:n_samples
    % Propose a new sample using a normal random walk
    theta_proposed = theta_samples(i-1,:) + randn(1,3)*0.1;
    % Calculate the acceptance ratio
    ratio = exp(posterior(theta_proposed) - posterior(theta_samples(i-1,:)));
    % Accept or reject the proposal
    if rand < ratio
        theta_samples(i,:) = theta_proposed;
    else
        theta_samples(i,:) = theta_samples(i-1,:);
    end
end

% Calculate the posterior mean and variance of theta
theta_mean = mean(theta_samples);
theta_var = var(theta_samples);

% Plot the histograms for each element of theta
figure;
subplot(3,1,1);
histogram(theta_samples(:,1),50);
title(['Posterior distribution of alpha, Mean = ',num2str(theta_mean(1)),', Variance = ',num2str(theta_var(1))]);
subplot(3,1,2);
histogram(theta_samples(:,2),50);
title(['Posterior distribution of beta, Mean = ',num2str(theta_mean(2)),', Variance = ',num2str(theta_var(2))]);
subplot(3,1,3);
histogram(theta_samples(:,3),50);
title(['Posterior distribution of sigma, Mean = ',num2str(theta_mean(3)),', Variance = ',num2str(theta_var(3))]);

Hope this helps!

解释以下伪代码：Algorithm 1 One-Point RANSAC for Scale Estimation Input: {sk }K 1 and {τk }K 1 . Initialize: p = 0.99, i =: 0, NT := 105, and |I ˜s| := 0. 1: while i ≤ NT do 2: Randomly pick a scale si from {sk }K 1 . 3: Find a consensus set Ii s according to (8). 4: if |Ii s | > |I ˜s| then 5: Estimate a refined scale si according to (10). 6: Find a refined consensus set Ii s . 7: |I ˜s| := |Ii s |, I ˜s := Ii s , ˜s := si . 8: Update NT according to (3). 9: end if 10: i := i + 1; 11: end while Output: the optimal scale ˜s and the inlier set I ˜s

这是一个用于估计尺度的单点RANSAC算法，其中输入包括K个尺度值{sk}和阈值{τk}。算法初始化一些参数，包括p（置信度），i（当前迭代次数），NT（最大迭代次数）和|I~s|（当前最优的内点集合数量）。在while循环中，随机选择一个尺度si，根据公式（8）找到一致性集合Iis。如果|Iis| > |I~s|，则根据公式（10）估计一个精确的尺度si，并找到一个精确的一致性集合Ii s。更新最优的内点集合I~s和尺度值~s。根据公式（3）更新NT，最后输出最优的尺度~s和内点集合I~s。