在回归曲面Z中，当Z小于预定义值时，如何求出最大X值？

Question

我已经从一个回归模型中创建了一个尺寸为i x i的X，Y网格和相应的表面Z。如何提取X的最大值的坐标，其中Z小于给定的阈值？

输入数据

x = np.array( )
y = np.array( )
z = np.array( )

data = np.column_stack((x, y, z))

# regular grid covering the domain of the data
X, Y = np.meshgrid(x, y)

XX = X.flatten()
YY = Y.flatten()

# best-fit quadratic curve
A = np.c_[np.ones(data.shape[0]), data[:, :2], np.prod(data[:, :2], axis=1), data[:, :2]**2]
C, _, _, _ = scipy.linalg.lstsq(A, data[:, 2])

# evaluate it on a grid
Z = np.dot(np.c_[np.ones(XX.shape), XX, YY, XX * YY, XX**2, YY**2],C).reshape(X.shape)

# Desired output

max(X, where Z < a) = [x1, y1, z1]

具有最大化X值的坐标的列表，其中Z小于a。

Answer 1

这里是一个示例3D曲面拟合程序，它根据数据点从拟合中预测的Z值是否低于指定的阈值来提取数据子集。此示例还具有三维散点图、三维曲面图和等高线图。这里，使用curve_fit而不是linalg.lstsq()进行拟合。

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


# this is threshhold value for Z subset
zThreshhold = 5.0

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 thaere 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 thaere 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 thaere can be memory and process problems


# fulle 3D quadratic
def func(data, a, b, c, d, e, f):
    X = data[0]
    Y = data[1]
    return a + b*X + c*Y + d*X*Y + e*X*X + f*Y*Y


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([1.0, 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]

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

    # 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)


zSubset = []
for i in range(len(modelPredictions)):
    if modelPredictions[i] < zThreshhold:
        zSubset.append([xData[i], yData[i], zData[i], modelPredictions[i]])

print("X, Y, Z, and Predicted values below Z threshold of", zThreshhold)
for i in range(len(zSubset)):
    print(zSubset[i])