Multidimensional Latin Hypercube Design and Optimization

1. Introduction

Multidimensional Latin Hypercube Design and Optimization

2. Software environment

2.1 matlab

3. Main process

3.1 Latin hypercube design

A Latin hypercube design is a method used in experimental design and optimization to ensure that sample points are evenly distributed at different levels of multiple variables. Its basic idea is to divide the sample space into uniform sub-regions, and then select a sample point in each sub-region.

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 生成二维拉丁超立方设计
design = lhsdesign(numSamples, 2);

% 将设计映射到指定的变量范围
samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);

% 绘制二维图
scatter(samplesX, samplesY, 'filled');
xlabel('X');
ylabel('Y');
title('Latin Hypercube Design (2D)');

3.2 Stochastic Optimal Latin Hypercube Design ROLHD

Randomized Optimization of Latin Hypercube Design (ROLHD) is a method for optimizing Latin Hypercube Design. It improves the original Latin hypercube design by introducing randomness and stochastic optimization algorithm to increase the uniformity and discreteness of sample points.

1 The basic idea of ​​the ROLHD method is as follows:

2 Generate an initial Latin hypercube design.

3 The initial design is improved using a stochastic optimization algorithm.

4 In each optimization iteration, a random perturbation or optimization operation is performed on the current design.

5 Evaluate the quality of the new design according to a certain evaluation index (such as maximum and minimum distance, variance, etc.).

6 If the new design is better than the current design, then accept the new design as the current optimal design; otherwise, keep the current design.

Repeat steps 3-5 until the specified number of iterations is reached or the stopping condition is met.

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 迭代改进型拉丁超立方设计 (ROLHD)
numIterations = 100; % 迭代次数
bestDesign = lhsdesign(numSamples, 2); % 初始的拉丁超立方设计
bestCriterion = computeCriterion(bestDesign); % 计算初始设计的准则值

for iteration = 1:numIterations
    % 生成随机变换
    transformedDesign = applyRandomTransform(bestDesign);
    
    % 计算变换后的设计的准则值
    transformedCriterion = computeCriterion(transformedDesign);
    
    % 比较新旧设计的准则值，更新当前最优设计
    if transformedCriterion < bestCriterion
        bestDesign = transformedDesign;
        bestCriterion = transformedCriterion;
    end
end

% 将设计映射到指定的变量范围
samplesX = bestDesign(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
samplesY = bestDesign(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);

% 绘制二维图
scatter(samplesX, samplesY, 'filled');
xlabel('X');
ylabel('Y');
title('Optimized Latin Hypercube Design');

% 计算设计的准则值（示例中使用随机值作为准则值，实际应根据具体问题定义准则）
function criterion = computeCriterion(design)
    criterion = rand(); % 使用随机值作为准则值，实际应根据具体问题定义准则
end

% 应用随机变换（示例中使用随机排序作为变换，实际应根据具体问题定义变换）
function transformedDesign = applyRandomTransform(design)
    transformedDesign = design(randperm(size(design, 1)), :);
end

3.3 Maximum and minimum distance generation optimization

Maximum Minimum Distance is an optimization method for improving Latin hypercube designs. Its goal is to maximize the minimum distance between sample points to increase the discreteness and uniformity between sample points.

The following is the basic idea of ​​​​the generation optimization of the maximum and minimum distance method:

1 First, generate an initial Latin hypercube design.

2 Calculate the minimum distance between sample points in the initial design.

3 Optimization iterations are performed, each iteration generating a new design and computing the minimum distance.

4 In each iteration, for each sample point, a new location is generated by random reassignment. This can be done by swapping the positions of the two sample points or by randomly choosing a new position within the range of sample points.

5 Calculate the minimum distance between sample points in the new design.

6 If the minimum distance of the new design is better than the previous design, accept this design as the current optimal design; otherwise, keep the previous design.

Repeat steps 4-6 until the specified number of iterations is reached or the stopping condition is met.

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 最大最小距离法生成拉丁超立方设计
design = maxminLHD(numSamples, 2);

% 将设计映射到指定的变量范围
samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);

% 绘制二维图
scatter(samplesX, samplesY, 'filled');
xlabel('X');
ylabel('Y');
title('Optimized Latin Hypercube Design using Max-Min Distance');

% 最大最小距离法生成优化的拉丁超立方设计函数
function design = maxminLHD(numSamples, numVariables)
    design = lhsdesign(numSamples, numVariables);
    
    % 进行最大最小距离优化
    for i = 1:numVariables
        col = design(:, i);
        col = (col - min(col)) / (max(col) - min(col));
        design(:, i) = col;
    end
end

3.4 Maximum entropy optimization

The maximum entropy method finds the optimal sample point layout through iterative optimization to maximize the uncertainty of the sample point distribution. By continuously adjusting the position of the sample points and keeping the constraints, the maximum entropy method can improve the Latin hypercube design, so that the sample points are more evenly distributed and cover the entire sample space.

1 Determine the range and constraints of the variables, such as minimum and maximum values ​​for each variable.

2 Generate the initial Latin hypercube design.

3 Calculate the entropy of the initial design as the current maximum entropy.

4 Iterations of optimization are performed, and each iteration generates a new design and calculates its entropy.

5 In each iteration, for each sample point, a new location is generated by random reassignment. This can be done by swapping the positions of the two sample points or by randomly choosing a new position within the range of sample points.

6 Adjust the sample point locations in the new design according to the constraints to ensure that it complies with the range and constraints of the variables.

7 Calculate the entropy of the new design.

8 If the entropy of the new design is greater than the current maximum entropy, accept the new design as the current optimal design; otherwise, keep the previous design.

9 Repeat steps 4-8 until the specified number of iterations is reached or the stopping condition is met.

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 最大熵法生成拉丁超立方设计
design = maxEntropyLHD(numSamples, 2);

% 将设计映射到指定的变量范围
samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);

% 绘制二维图
scatter(samplesX, samplesY, 'filled');
xlabel('X');
ylabel('Y');
title('Optimized Latin Hypercube Design using Maximum Entropy');

% 最大熵法生成优化的拉丁超立方设计函数
function design = maxEntropyLHD(numSamples, numVariables)
    design = lhsdesign(numSamples, numVariables);
    
    % 进行最大熵法优化
    for i = 1:numVariables
        col = design(:, i);
        col = -log(1 - col);
        col = (col - min(col)) / (max(col) - min(col));
        design(:, i) = col;
    end
end

3.5 Simulated annealing algorithm optimization

Through the iterative optimization process of the simulated annealing algorithm, the quality of the Latin hypercube design can be gradually improved. As the temperature decreases, the algorithm tends to accept solutions with smaller differences, thereby gradually converging to a better design.
1 Initialize the initial solution: Generate an initial Latin hypercube design as the current solution.

2. Set the initial temperature and annealing strategy: determine the initial temperature and annealing strategy, for example, the initial temperature is higher, and the temperature gradually decreases during the annealing process.

3. Iterative optimization process: In each iteration, an adjacent solution is selected according to the current temperature. Adjacent solutions can be obtained by randomly perturbing the sample point positions of the current solution or performing optimization operations, such as exchanging the positions of two sample points.

4 Calculate the difference or cost function between the current solution and adjacent solutions: Calculate the difference or cost function value between the current solution and adjacent solutions. The difference can be a measure of distance between sample points, linear correlation, etc.

5 Calculate the probability of accepting the difference based on the difference and the current temperature: Calculate the probability of accepting an adjacent solution based on the difference and the current temperature. In general, the probability of accepting an adjacent solution is higher if the difference is smaller or if the current temperature is higher.

6 Accept or reject adjacent solutions based on probabilities: Accept or reject adjacent solutions based on probabilities. If the adjacent solution is accepted, make it the current solution; otherwise keep the current solution unchanged.

7 Lowering the temperature: lowering the temperature according to the annealing strategy, such as reducing the proportion of the temperature or adjusting the temperature according to the number of iterations.

Repeat steps 3-7 until a stopping condition is met (e.g. the maximum number of iterations is reached or the temperature is below a certain threshold).

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 模拟退火算法生成优化的拉丁超立方设计
design = simulatedAnnealingLHD(numSamples, 2, rangeX, rangeY);

% 绘制二维图
scatter(design(:,1), design(:,2), 'filled');
xlabel('X');
ylabel('Y');
title('Optimized Latin Hypercube Design using Simulated Annealing');

% 模拟退火算法生成优化的拉丁超立方设计函数
function design = simulatedAnnealingLHD(numSamples, numVariables, rangeX, rangeY)
    % 初始化设计
    design = lhsdesign(numSamples, numVariables);
    
    % 定义能量函数
    function energy = energyFunction(design)
        % 将设计映射到变量范围
        samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
        samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);
        
        % 计算设计的能量（距离的总和）
        distances = pdist([samplesX, samplesY]);
        energy = sum(distances);
    end

    % 进行模拟退火优化
    initialEnergy = energyFunction(design);
    currentDesign = design;
    currentEnergy = initialEnergy;
    
    temperature = 1;
    coolingRate = 0.95;
    
    while temperature > 1e-8
        % 随机选择两个变量进行交换
        i = randi([1, numSamples]);
        j = randi([1, numSamples]);
        
        newDesign = currentDesign;
        newDesign([i j], :) = newDesign([j i], :);
        newEnergy = energyFunction(newDesign);
        
        % 根据 Metropolis 准则接受或拒绝新设计
        if newEnergy < currentEnergy
            currentDesign = newDesign;
            currentEnergy = newEnergy;
        else
            acceptanceProbability = exp((currentEnergy - newEnergy) / temperature);
            if rand < acceptanceProbability
                currentDesign = newDesign;
                currentEnergy = newEnergy;
            end
        end
        
        % 降低温度
        temperature = temperature * coolingRate;
    end
    
    % 返回优化后的设计
    design = currentDesign;
end

3.6 Mean Optimization for Reducing Linear Dependency

RLCMO) is an optimization method for improving the uniformity and correlation of Latin hypercube designs. Its goal is to achieve a better uniform distribution by adjusting the position of the sample points in the Latin hypercube design so that the linear correlation between the sample points in the design is as small as possible.
The specific steps of RLCMO are as follows:

1 First, generate an initial Latin hypercube design.

2 Calculate the condition number of the design matrix, which measures the linear dependence of the current design.

3 Perform optimization iterations, each iteration generating a new design and computing its condition number.

4 In each iteration, for each sample point, a new location is generated by random reassignment. This can be done by swapping the positions of the two sample points or by randomly choosing a new position within the range of sample points.

5. Adjust the sample point location in the new design according to the constraints to ensure that it complies with the range and constraints of the variables.

6 Calculate the condition number of the new design.

7 If the condition number of the new design is less than the current condition number, accept the new design as the current optimal design; otherwise, accept the new design according to a certain probability.

8 Repeat steps 3-7 until the specified number of iterations is reached or the stopping condition is met.

% 设置变量范围和样本数量
rangeX = [0 1]; % X 变量范围
rangeY = [0 1]; % Y 变量范围
numSamples = 20; % 样本数量

% 减小线性相关性的均值优化生成拉丁超立方设计
design = reduceCorrelationLHD(numSamples, 2);

% 将设计映射到指定的变量范围
samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);

% 绘制二维图
scatter(samplesX, samplesY, 'filled');
xlabel('X');
ylabel('Y');
title('Optimized Latin Hypercube Design with Reduced Correlation');

% 减小线性相关性的均值优化生成拉丁超立方设计函数
function design = reduceCorrelationLHD(numSamples, numVariables)
    design = lhsdesign(numSamples, numVariables);
    
    % 减小线性相关性的均值优化
    for i = 1:numVariables
        col = design(:, i);
        col = col - mean(col);
        design(:, i) = col;
    end
end

3.7 Optimization method of greedy strategy

The greedy strategy is an optimization method based on the choice of local optimum, which can be used to optimize Latin hypercube designs. The method selects and adjusts the location of sample points step by step to maximize the uniformity and correlation of the design.

The basic idea of ​​the greedy strategy is to start from the initial design and make optimization adjustments one by one to improve the characteristics of the design. The specific steps are as follows:
In summary, distance minimization is a common method for optimizing Latin hypercube designs, and can produce satisfactory uniform distribution results in most cases.
1 Generate an initial Latin hypercube design.

2 For each sample point, select an optimal position for adjustment according to certain rules. Here, an evaluation index needs to be defined to measure the pros and cons of each location, for example, a distance index or a correlation index can be used.

3 For the selected optimal position, move the current sample point to this position, and update the design matrix.

4 Repeat steps 2-3 until all sample points are optimally adjusted.

5 Evaluate the final optimized design to check for improved homogeneity and correlation. If further optimization is required, you can start over from step 2, or use other optimization methods for iterative improvement.

% 贪心策略生成优化的拉丁超立方设计函数
function design=lhs(rangeX,rangeY,rangeZ,rangeW,numSamples,iterCount)
% 贪心策略生成优化的拉丁超立方设计
design = greedyOptimizationLHD(numSamples, 4, rangeX, rangeY, rangeZ, rangeW, iterCount);

% 打印优化后的设计样本点
disp(design);
end


% 贪心策略生成优化的拉丁超立方设计函数
function design = greedyOptimizationLHD(numSamples, numVariables, rangeX, rangeY, rangeZ, rangeW, iterCount)
    % 生成初始的拉丁超立方设计
    design = lhsdesign(numSamples, numVariables);
    
    % 将设计映射到变量范围
    samplesX = design(:, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
    samplesY = design(:, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);
    samplesZ = design(:, 3) * (rangeZ(2) - rangeZ(1)) + rangeZ(1);
    samplesW = design(:, 4) * (rangeW(2) - rangeW(1)) + rangeW(1);
    
    % 计算初始设计的适应度
    initialFitness = calculateFitness(samplesX, samplesY, samplesZ, samplesW);
    
    % 迭代改进设计
    for iter = 1:iterCount
        % 随机选择一个样本点
        index = randi(numSamples);

        % 保存上一轮迭代的样本点
        prevX = samplesX(index);
        prevY = samplesY(index);
        prevZ = samplesZ(index);
        prevW = samplesW(index);
        
        % 随机生成新的样本点
        newX = rand * (rangeX(2) - rangeX(1)) + rangeX(1);
        newY = rand * (rangeY(2) - rangeY(1)) + rangeY(1);
        newZ = rand * (rangeZ(2) - rangeZ(1)) + rangeZ(1);
        newW = rand * (rangeW(2) - rangeW(1)) + rangeW(1);
        
        % 替换选中的样本点
        samplesX(index) = newX;
        samplesY(index) = newY;
        samplesZ(index) = newZ;
        samplesW(index) = newW;
        
        % 计算新设计的适应度
        newFitness = calculateFitness(samplesX, samplesY, samplesZ, samplesW);
        
        % 如果新设计的适应度更好，则接受新设计
        if newFitness > initialFitness
            initialFitness = newFitness;
        else
            % 恢复原来的样本点
            samplesX(index) = design(index, 1) * (rangeX(2) - rangeX(1)) + rangeX(1);
            samplesY(index) = design(index, 2) * (rangeY(2) - rangeY(1)) + rangeY(1);
            samplesZ(index) = design(index, 3) * (rangeZ(2) - rangeZ(1)) + rangeZ(1);
            samplesW(index) = design(index, 4) * (rangeW(2) - rangeW(1)) + rangeW(1);

            samplesX(index) = prevX;
            samplesY(index) = prevY;
            samplesZ(index) = prevZ;
            samplesW(index) = prevW;
        end
    end
    
    % 更新最终的设计,是有每轮更新的,这种是做了归一化处理的
    design(:, 1) = (samplesX - rangeX(1)) / (rangeX(2) - rangeX(1));
    design(:, 2) = (samplesY - rangeY(1)) / (rangeY(2) - rangeY(1));
    design(:, 3) = (samplesZ - rangeZ(1)) / (rangeZ(2) - rangeZ(1));
    design(:, 4) = (samplesW - rangeW(1)) / (rangeW(2) - rangeW(1));

    design(:, 1) = samplesX;
    design(:, 2) = samplesY;
    design(:, 3) = samplesZ;
    design(:, 4) = samplesW;
end

% 计算设计的适应度
function fitness = calculateFitness(samplesX, samplesY, samplesZ, samplesW)
    % 计算设计的适应度（距离的总和的倒数）
    distances = pdist([samplesX, samplesY, samplesZ, samplesW]);
    fitness = 1 / sum(distances);
end

This code implements an optimized Latin hypercube design using a greedy strategy generation. Here is the explanation and function of the code:

rangeX and rangeY are the ranges for the X and Y variables.
numSamples is the number of samples to generate.
design = greedyOptimizationLHD(numSamples, 2, rangeX, rangeY); Call the greedyOptimizationLHD function to generate an optimized Latin hypercube design.
scatter(design(:,1), design(:,2), 'filled'); Draws a 2D plot showing the resulting design points.
The greedyOptimizationLHD function implements the process of generating an optimized Latin hypercube design by a greedy strategy.
First, it generates an initial Latin hypercube design using the lhsdesign function.
Then, map the design to the specified variable ranges.
Next, calculate the fitness of the initial design, where fitness is defined as the inverse of the sum of distances.
Iteratively improve the design, each iteration randomly selects a sample point, randomly generates a new sample point, and replaces the selected sample point. If the fitness of the new design is better, the new design is accepted; otherwise, the original sample points are restored.
Finally, update the final design and return.
The calculateFitness function calculates the fitness of a design. Here fitness is defined as the reciprocal of the sum of distances, and distance minimization is a common method for optimizing Latin hypercube designs.
Please note that the number of iterations in the code is set to 100, which is an adjustable parameter and can be adjusted according to the specific situation. In addition, the designed fitness calculation method can also be modified as needed.

Yes, in this code, the calculation of fitness is based on the sum of distances between sample points. Therefore, the smaller the sum of the distances, the higher the fitness, which means that the distribution of sample points is more uniform.

By minimizing the sum of the distances between sample points, the sample points can be evenly distributed over the entire variable range. This is one of the common methods for optimizing Latin hypercube designs and produces satisfactory results in most cases.

When the distance between sample points is small, they are closer to a uniform distribution. This ensures overall coverage of the design space and reduces redundancy among sample points. Such designs allow for better representation of latent features over the range of variables and for exploration of the solution space of experimental or optimization problems.

If your problem domain has different ranges in different dimensions, and those ranges vary widely, I recommend using the normalized design results. Normalization can unify the range of different dimensions to the range of [0, 1], so that the changes between different dimensions have a relatively consistent scale.

Using normalized design results has the following advantages:

Unified scale: Normalization can eliminate scale differences between different dimensions, ensuring that they have relatively consistent ranges. This way, you can more easily compare and weigh the effects of different dimensions.
Reusability: Normalized design results can be more easily integrated with other models, algorithms or methods. You can import the design into other models using the same normalized range without additional scaling.
Interpretability: Normalized design results are easier to understand and interpret because they are in a uniform range. You can use the normalized values ​​directly, regardless of the actual extent of each dimension.
So if your problem involves multiple dimensions with different ranges, I recommend using the normalized design results. This can better handle the scale difference between different dimensions, and make subsequent analysis and application more convenient.