四阶Hermite 多项式4维函数回归预测 matlab示例

以下是使用四阶Hermite 多项式进行4维函数回归预测的matlab示例代码：

%% 生成训练数据
x1 = linspace(-1, 1, 100)';
x2 = linspace(-1, 1, 100)';
x3 = linspace(-1, 1, 100)';
x4 = linspace(-1, 1, 100)';
[X1, X2, X3, X4] = ndgrid(x1, x2, x3, x4);
Y = sin(pi*X1).*cos(pi*X2).*exp(X3).*log(1+X4.^2) + 0.1*randn(size(X1));

%% 建立模型
X = [ones(size(X1(:))), X1(:), X2(:), X3(:), X4(:), ...
X1(:).^2, X2(:).^2, X3(:).^2, X4(:).^2, ...
X1(:).*X2(:), X1(:).*X3(:), X1(:).*X4(:), X2(:).*X3(:), X2(:).*X4(:), X3(:).*X4(:), ...
X1(:).^3, X2(:).^3, X3(:).^3, X4(:).^3, ...
(X1(:).^2).*X2(:), (X1(:).^2).*X3(:), (X1(:).^2).*X4(:), ...
X1(:).*(X2(:).^2), X2(:).*(X3(:).^2), X3(:).*(X4(:).^2), ...
X1(:).^4, X2(:).^4, X3(:).^4, X4(:).^4, ...
(X1(:).^3).*X2(:), (X1(:).^3).*X3(:), (X1(:).^3).*X4(:), ...
(X2(:).^3).*X1(:), (X2(:).^3).*X3(:), (X2(:).^3).*X4(:), ...
(X3(:).^3).*X1(:), (X3(:).^3).*X2(:), (X3(:).^3).*X4(:), ...
(X4(:).^3).*X1(:), (X4(:).^3).*X2(:), (X4(:).^3).*X3(:), ...
X1(:).^2.*X2(:).^2, X1(:).^2.*X3(:).^2, X1(:).^2.*X4(:).^2, ...
X2(:).^2.*X3(:).^2, X2(:).^2.*X4(:).^2, ...
X3(:).^2.*X4(:).^2, ...
X1(:).*X2(:).*X3(:), X1(:).*X2(:).*X4(:), X1(:).*X3(:).*X4(:), X2(:).*X3(:).*X4(:)];

beta = X \ Y(:); % 回归系数

%% 预测
x1_new = linspace(-1, 1, 50)';
x2_new = linspace(-1, 1, 50)';
x3_new = linspace(-1, 1, 50)';
x4_new = linspace(-1, 1, 50)';
[X1_new, X2_new, X3_new, X4_new] = ndgrid(x1_new, x2_new, x3_new, x4_new);
X_new = [ones(size(X1_new(:))), X1_new(:), X2_new(:), X3_new(:), X4_new(:), ...
X1_new(:).^2, X2_new(:).^2, X3_new(:).^2, X4_new(:).^2, ...
X1_new(:).*X2_new(:), X1_new(:).*X3_new(:), X1_new(:).*X4_new(:), X2_new(:).*X3_new(:), X2_new(:).*X4_new(:), X3_new(:).*X4_new(:), ...
X1_new(:).^3, X2_new(:).^3, X3_new(:).^3, X4_new(:).^3, ...
(X1_new(:).^2).*X2_new(:), (X1_new(:).^2).*X3_new(:), (X1_new(:).^2).*X4_new(:), ...
X1_new(:).*(X2_new(:).^2), X2_new(:).*(X3_new(:).^2), X3_new(:).*(X4_new(:).^2), ...
X1_new(:).^4, X2_new(:).^4, X3_new(:).^4, X4_new(:).^4, ...
(X1_new(:).^3).*X2_new(:), (X1_new(:).^3).*X3_new(:), (X1_new(:).^3).*X4_new(:), ...
(X2_new(:).^3).*X1_new(:), (X2_new(:).^3).*X3_new(:), (X2_new(:).^3).*X4_new(:), ...
(X3_new(:).^3).*X1_new(:), (X3_new(:).^3).*X2_new(:), (X3_new(:).^3).*X4_new(:), ...
(X4_new(:).^3).*X1_new(:), (X4_new(:).^3).*X2_new(:), (X4_new(:).^3).*X3_new(:), ...
X1_new(:).^2.*X2_new(:).^2, X1_new(:).^2.*X3_new(:).^2, X1_new(:).^2.*X4_new(:).^2, ...
X2_new(:).^2.*X3_new(:).^2, X2_new(:).^2.*X4_new(:).^2, ...
X3_new(:).^2.*X4_new(:).^2, ...
X1_new(:).*X2_new(:).*X3_new(:), X1_new(:).*X2_new(:).*X4_new(:), X1_new(:).*X3_new(:).*X4_new(:), X2_new(:).*X3_new(:).*X4_new(:)];

Y_pred = reshape(X_new * beta, size(X1_new)); % 预测结果

%% 可视化
figure
subplot(1,2,1)
scatter3(X1(:), X2(:), X3(:), 20, Y(:), 'filled')
xlabel('x_1'), ylabel('x_2'), zlabel('x_3')
title('训练数据')

subplot(1,2,2)
surf(X1_new, X2_new, X3_new, Y_pred)
xlabel('x_1'), ylabel('x_2'), zlabel('x_3')
title('预测结果')