3D曲线拟合 [英] 3D curvefitting

问题描述

我有a,b个点的离散规则网格及其对应的c值，并对其进行进一步插值以获得平滑曲线.现在，从插值数据中，我还想创建一个用于曲线拟合的多项式方程.如何在多项式中拟合3D图?

I have discrete regular grid of a,b points and their corresponding c values and I interpolate it further to get a smooth curve. Now from interpolation data, I further want to create a polynomial equation for curve fitting. How to fit 3D plot in polynomial?

我尝试在MATLAB中执行此操作.我使用了MATLAB(r2010a)中的Surface拟合工具箱来对3维数据进行曲线拟合.但是，在MATLAB/MAPLE或任何其他软件中，如何找到最适合数据的公式.有什么建议吗?另外最有用的是一些实际的代码示例，例如PDF文件，网络等.

I try to do this in MATLAB. I used Surface fitting toolbox in MATLAB (r2010a) to curve fit 3-dimensional data. But, how does one find a formula that fits a set of data to the best advantage in MATLAB/MAPLE or any other software. Any advice? Also most useful would be some real code examples to look at, PDF files, on the web etc.

这只是我数据的一小部分.

This is just a small portion of my data.

a = [ 0.001 .. 0.011];

b = [1, .. 10];

c = [ -.304860225, .. .379710865]; 

谢谢.

推荐答案

要将曲线拟合到一组点上，我们可以使用普通最小二乘回归. MathWorks提供了一个解决方案页面来描述该过程.

To fit a curve onto a set of points, we can use ordinary least-squares regression. There is a solution page by MathWorks describing the process.

作为一个例子，让我们从一些随机数据开始:

As an example, let's start with some random data:

% some 3d points
data = mvnrnd([0 0 0], [1 -0.5 0.8; -0.5 1.1 0; 0.8 0 1], 50);

如 @BasSwinckels 所示，通过构造所需的 pinv 到解决超定系统表示为Ax=b:

As @BasSwinckels showed, by constructing the desired design matrix, you can use mldivide or pinv to solve the overdetermined system expressed as Ax=b:

% best-fit plane
C = [data(:,1) data(:,2) ones(size(data,1),1)] \ data(:,3);    % coefficients

% evaluate it on a regular grid covering the domain of the data
[xx,yy] = meshgrid(-3:.5:3, -3:.5:3);
zz = C(1)*xx + C(2)*yy + C(3);

% or expressed using matrix/vector product
%zz = reshape([xx(:) yy(:) ones(numel(xx),1)] * C, size(xx));

接下来，我们将结果可视化:

Next we visualize the result:

% plot points and surface
figure('Renderer','opengl')
line(data(:,1), data(:,2), data(:,3), 'LineStyle','none', ...
    'Marker','.', 'MarkerSize',25, 'Color','r')
surface(xx, yy, zz, ...
    'FaceColor','interp', 'EdgeColor','b', 'FaceAlpha',0.2)
grid on; axis tight equal;
view(9,9);
xlabel x; ylabel y; zlabel z;
colormap(cool(64))

如前所述，我们可以通过向自变量矩阵(Ax=b中的A)添加更多项来获得高阶多项式拟合.

As was mentioned, we can get higher-order polynomial fitting by adding more terms to the independent variables matrix (the A in Ax=b).

说我们想用一个常数，线性，相互作用和平方项(1，x，y，xy，x ^ 2，y ^ 2)拟合二次模型.我们可以手动执行此操作:

Say we want to fit a quadratic model with constant, linear, interaction, and squared terms (1, x, y, xy, x^2, y^2). We can do this manually:

% best-fit quadratic curve
C = [ones(50,1) data(:,1:2) prod(data(:,1:2),2) data(:,1:2).^2] \ data(:,3);
zz = [ones(numel(xx),1) xx(:) yy(:) xx(:).*yy(:) xx(:).^2 yy(:).^2] * C;
zz = reshape(zz, size(xx));

统计信息"工具箱中还有一个帮助器功能 x2fx ，帮助建立几个模型订单的设计矩阵:

There is also a helper function x2fx in the Statistics Toolbox that helps in building the design matrix for a couple of model orders:

C = x2fx(data(:,1:2), 'quadratic') \ data(:,3);
zz = x2fx([xx(:) yy(:)], 'quadratic') * C;
zz = reshape(zz, size(xx));

最后，在文件交换上，有一个出色的功能 polyfitn John D'Errico，可让您指定所涉及的各种多项式阶数和项:

Finally there is an excellent function polyfitn on the File Exchange by John D'Errico that allows you to specify all kinds of polynomial orders and terms involved:

model = polyfitn(data(:,1:2), data(:,3), 2);
zz = polyvaln(model, [xx(:) yy(:)]);
zz = reshape(zz, size(xx));

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