如何使用数据集拟合3D曲面？

Question

我正在尝试将X，Y，Z数据集拟合到未知曲面。

不幸的是，线性拟合不足以显示表面数据。我认为多项式拟合可能适合这种情况。另外，问题在于我不知道如何构建多项式拟合函数来完成曲面拟合。

任何帮助都会很棒。

谢谢

import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

X = [[2, 2, 2], [1.5, 1.5, 1.5], [0.5, 0.5, 0.5]]
Y = [[3, 2, 1], [3, 2, 1], [3, 2, 1]]
Z = [[2.4, 2.5, 2.2], [2.4, 3, 2.5], [4, 3.3, 8]]

# ================= Plot figure =================  ##
Fontsize_set = {'size': 20}
fig = plt.figure(figsize=[8, 5], dpi=140, facecolor='w')
ax = fig.gca(projection='3d')
ax.grid(color='y', linestyle='--', linewidth=0.5)
ax.tick_params(labelsize=20)
ax.set_xlim3d(0, 3)
ax.set_ylim3d(0, 6)
ax.set_zlim3d(0, 10)
ax.view_init(30, 45)
ax.scatter(X, Y, Z, s=50, color='k', marker='o', linewidth=None, alpha=1)
# ax.plot_surface(X, Y, Z)
fig.tight_layout()
plt.show()

Answer 1

你在这里

= ^ .. ^ =

代码说明：

import numpy as np
from scipy.optimize import curve_fit
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt


# test function
def function(data, a, b, c):
    x = data[0]
    y = data[1]
    return a * (x**b) * (y**c)

# setup test data
raw_data = [2.0, 2.0, 2.0], [1.5, 1.5, 1.5], [0.5, 0.5, 0.5],[3.0, 2.0, 1.0], [3.0, 2.0, 1.0],\
       [3.0, 2.0, 1.0], [2.4, 2.5, 2.2], [2.4, 3.0, 2.5], [4.0, 3.3, 8.0]

# convert data into proper format
x_data = []
y_data = []
z_data = []
for item in raw_data:
    x_data.append(item[0])
    y_data.append(item[1])
    z_data.append(item[2])

# get fit parameters from scipy curve fit
parameters, covariance = curve_fit(function, [x_data, y_data], z_data)

# create surface function model
# setup data points for calculating surface model
model_x_data = np.linspace(min(x_data), max(x_data), 30)
model_y_data = np.linspace(min(y_data), max(y_data), 30)
# create coordinate arrays for vectorized evaluations
X, Y = np.meshgrid(model_x_data, model_y_data)
# calculate Z coordinate array
Z = function(np.array([X, Y]), *parameters)

# setup figure object
fig = plt.figure()
# setup 3d object
ax = Axes3D(fig)
# plot surface
ax.plot_surface(X, Y, Z)
# plot input data
ax.scatter(x_data, y_data, z_data, color='red')
# set plot descriptions
ax.set_xlabel('X data')
ax.set_ylabel('Y data')
ax.set_zlabel('Z data')

plt.show()

Answer 2

这里是带有散点图，表面图和轮廓图的其他图形示例。您应该能够按住鼠标按钮并旋转3D图。

import numpy, scipy, scipy.optimize
import matplotlib
from mpl_toolkits.mplot3d import  Axes3D
from matplotlib import cm # to colormap 3D surfaces from blue to red
import matplotlib.pyplot as plt

graphWidth = 800 # units are pixels
graphHeight = 600 # units are pixels

# 3D contour plot lines
numberOfContourLines = 16


def SurfacePlot(func, data, fittedParameters):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

    matplotlib.pyplot.grid(True)
    axes = Axes3D(f)

    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    xModel = numpy.linspace(min(x_data), max(x_data), 20)
    yModel = numpy.linspace(min(y_data), max(y_data), 20)
    X, Y = numpy.meshgrid(xModel, yModel)

    Z = func(numpy.array([X, Y]), *fittedParameters)

    axes.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=1, antialiased=True)

    axes.scatter(x_data, y_data, z_data) # show data along with plotted surface

    axes.set_title('Surface Plot (click-drag with mouse)') # add a title for surface plot
    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label
    axes.set_zlabel('Z Data') # Z axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot or else there can be memory and process problems


def ContourPlot(func, data, fittedParameters):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    xModel = numpy.linspace(min(x_data), max(x_data), 20)
    yModel = numpy.linspace(min(y_data), max(y_data), 20)
    X, Y = numpy.meshgrid(xModel, yModel)

    Z = func(numpy.array([X, Y]), *fittedParameters)

    axes.plot(x_data, y_data, 'o')

    axes.set_title('Contour Plot') # add a title for contour plot
    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    CS = matplotlib.pyplot.contour(X, Y, Z, numberOfContourLines, colors='k')
    matplotlib.pyplot.clabel(CS, inline=1, fontsize=10) # labels for contours

    plt.show()
    plt.close('all') # clean up after using pyplot or else there can be memory and process problems


def ScatterPlot(data):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)

    matplotlib.pyplot.grid(True)
    axes = Axes3D(f)
    x_data = data[0]
    y_data = data[1]
    z_data = data[2]

    axes.scatter(x_data, y_data, z_data)

    axes.set_title('Scatter Plot (click-drag with mouse)')
    axes.set_xlabel('X Data')
    axes.set_ylabel('Y Data')
    axes.set_zlabel('Z Data')

    plt.show()
    plt.close('all') # clean up after using pyplot or else there can be memory and process problems


def func(data, a, b, c):
    x = data[0]
    y = data[1]
    return (a * x) + (y * b) + c


if __name__ == "__main__":
    xData = numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0])
    yData = numpy.array([11.0, 12.1, 13.0, 14.1, 15.0, 16.1, 17.0, 18.1, 90.0])
    zData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.5, 6.6, 7.7, 8.0, 9.9])

    data = [xData, yData, zData]

    initialParameters = [1.0, 1.0, 1.0] # these are the same as scipy default values in this example

    # here a non-linear surface fit is made with scipy's curve_fit()
    fittedParameters, pcov = scipy.optimize.curve_fit(func, [xData, yData], zData, p0 = initialParameters)

    ScatterPlot(data)
    SurfacePlot(func, data, fittedParameters)
    ContourPlot(func, data, fittedParameters)

    print('fitted prameters', fittedParameters)

    modelPredictions = func(data, *fittedParameters) 

    absError = modelPredictions - zData

    SE = numpy.square(absError) # squared errors
    MSE = numpy.mean(SE) # mean squared errors
    RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
    Rsquared = 1.0 - (numpy.var(absError) / numpy.var(zData))
    print('RMSE:', RMSE)
    print('R-squared:', Rsquared)