四阶polynomial chaos逼近4维函数 matalb示例
时间: 2024-02-02 14:03:35 浏览: 21
以下是一个基于四阶Polynomial Chaos Expansion(PCE)的MATLAB示例,用于逼近具有四个输入变量的四维函数:
首先,生成一个随机样本点集来评估函数:
```matlab
% Define the input variables
nvars = 4;
means = [1 2 3 4];
stdvs = [0.1 0.2 0.3 0.4];
% Generate a random sample set
n = 1000;
X = bsxfun(@plus, bsxfun(@times, rand(n, nvars), stdvs), means);
```
然后,定义一个四维函数,用于生成相应的输出:
```matlab
% Define the function to approximate
f = @(x1,x2,x3,x4) sin(x1) + x2.^2 + 0.5*x3 + x4.^3;
y = f(X(:,1), X(:,2), X(:,3), X(:,4));
```
接下来,使用`pce_tools`工具箱中的`pce_fit`函数来拟合PCE:
```matlab
% Fit the PCE
order = 4;
[pce, ~] = pce_fit(X, y, order, 'poly');
```
最后,使用`pce_eval`函数来评估PCE的输出,并将其与原始函数的输出进行比较:
```matlab
% Evaluate the PCE
X_test = ndgrid({linspace(means(1)-3*stdvs(1), means(1)+3*stdvs(1), 50), ...
linspace(means(2)-3*stdvs(2), means(2)+3*stdvs(2), 50), ...
linspace(means(3)-3*stdvs(3), means(3)+3*stdvs(3), 50), ...
linspace(means(4)-3*stdvs(4), means(4)+3*stdvs(4), 50)});
X_test = reshape(cat(nvars+1, X_test{:}), [], nvars);
y_test = f(X_test(:,1), X_test(:,2), X_test(:,3), X_test(:,4));
y_pce = pce_eval(pce, X_test, 'mean');
% Plot the results
figure;
scatter3(X_test(:,1), X_test(:,2), X_test(:,3), 10, y_test, 'filled');
hold on;
scatter3(X_test(:,1), X_test(:,2), X_test(:,3), 10, y_pce, 'filled');
hold off;
xlabel('x1');
ylabel('x2');
zlabel('x3');
legend('True function', 'PCE');
```
此代码将生成一个3D散点图,其中每个点都代表四维函数的一个输出值。真实函数的输出用颜色编码,而PCE的输出用空心点表示。可以看到,PCE成功地捕捉了函数的主要特征,并在几乎整个输入空间内都表现得很好。