Calculate root mean square of 3D deviation after surface fitting in python

Question

My goal is to determine the 3D deviation (and its RMS) between a set of 3D data points and a fitted paraboloid in Python.

Starting from this: Paraboloid (3D parabola) surface fitting python, I can compute the RMS. If I understand correctly, the error and the RMS are computed along the Z-axis. Is it right?

I tried (without success) to determine the 3D deviation and RMS between the fitted surface and the data points, but I cannot get their. Does anyone have some advices to solve this, please?

import numpy as np
from scipy.optimize import curve_fit

# Initial guess parameters
p0 = [1.5,0.4,1.5,0.4,1]

# INPUT DATA
x = [0.4,0.165,0.165,0.585,0.585]
y = [.45, .22, .63, .22, .63]
z = np.array([1, .99, .98,.97,.96])

# FIT
def paraBolEqn(data,a,b,c,d,e):
    x,y = data
    return -(((x-b)/a)**2+((y-d)/c)**2)+e

data = np.vstack((x,y))
popt, _ = curve_fit(paraBolEqn,data,z,p0)

# Deviation and RMS along Z axis
modelPredictions = paraBolEqn(data, *popt) 
absError = modelPredictions - z
RMSE = np.sqrt(np.mean(np.square(absError))) # Root Mean Squared Error along Z axis
print('RMSE (along Z axis):', RMSE)

# Deviation and RMS in 3D
# ??

Answer 1

Here is a graphical Python surface fitter using your data and equation that draws a 3D scatter plot, a 3D surface plot, and a contour plot. You should be able to click-drag with the mouse and rotate the 3D plots in 3-space for visual inspection. Note that you have 5 data points and 5 equation parameters, so you get what is in effect a perfect fit - the RMSE is effectively zero, the R-squared is 1.0, and the scipy code gives a warning when calculating the covariance matrix.

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

x = [0.4,0.165,0.165,0.585,0.585]
y = [.45, .22, .63, .22, .63]
z = [1, .99, .98,.97,.96]

# alias data to match previous example
xData = numpy.array(x, dtype=float)
yData = numpy.array(y, dtype=float)
zData = numpy.array(z, dtype=float)

# place the data in a single list
data = [xData, yData, zData]


def func(data,a,b,c,d,e):

    # extract data from the single list
    x = data[0]
    y = data[1]
    return -(((x-b)/a)**2+((y-d)/c)**2)+e


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

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

    # extract data from the single list
    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)

    # extract data from the single list
    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)

    # extract data from the single list
    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



if __name__ == "__main__":    
    initialParameters = [1.5,0.4,1.5,0.4,1] # from the posting

    # 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 parameters', 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)