How to do two-dimensional regression analysis in P

Firstly, I am not familiar with Python and I still barely understand the mechanism of Python code. But I need to do some statistical analysis through Python.

I have tried many many ways to figure out but I failed.

Basically, I have 3 arrays of data (assume these arrays are X, Y, Z).
I did some analysis with (X, Y) and (Z, Y) by making the scatter plot and put the best fit with the data to see the correlation.
№1 and №2 are quite easy enough.
Now I need to see the edge on view from the graph which is the one with combined X and Z. So, I made the equation (see below).

import pylab as pl
import numpy as np
from pylab import *
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt


s = np.loadtxt('New_list3.txt')
s = s.T

x = s[1]
y = s[2]
z = s[4]

upper_error = s[5]
lower_error = s[6]
asymmetric_error = [lower_error, upper_error]

def func(X, a1, a2, a3):
    x1, y1 = X

    return a1 * x1 + a2 * y1 + a3

popt, pcov = curve_fit(func,(x,y),z)
new_x=func((x,y),popt[0],popt[1],0)
new_y=z
new_z = np.polyfit(new_x,new_y,1)

p = np.poly1d(new_z)

plt.plot(func((x, y), popt[0], popt[1], 0), z, '.k')
pl.plot(new_x, p(new_x), "r-")
plt.errorbar(new_x, z ,yerr=asymmetric_error, ecolor='b', capsize=3, marker ='o', fmt='none')

print popt    
plt.show()

Now I cannot figure out to find the errors from that equation which is y = a1 * x + a2 * z + a3. I found the best values of a1, a2, a3. However, not the errors.

How can I find the "errors" of a1, a2, a3?

Here is example Python code using scipy.optimize.curve_fit to fit a surface, and it makes a 3D scatterplot of the raw data, a 3D scatterplot of the errors, a surface plot, and a contour plot. Change this to use your own data and function, and you should be done.

import numpy, scipy
import 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(equationFunc, data, params):
    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 = equationFunc(numpy.array([X, Y]), *params)

    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(equationFunc, data, params):
    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 = equationFunc(numpy.array([X, Y]), *params)

    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, title):
    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, depthshade=False, color='k')

    axes.set_title(title)
    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


def EquationFunc(data, *params):
    p0 = params[0]
    p1 = params[1]
    return p0 + numpy.sqrt(data[0]) + numpy.cos(data[1] / p1)


if __name__ == "__main__":

    # raw data
    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])

    pInitial = (1.0, 1.0)
    popt, pcov = scipy.optimize.curve_fit(EquationFunc,(xData,yData),zData, p0=pInitial)

    dataForPlotting = [xData, yData, zData]

    ScatterPlot([xData, yData, zData], 'Data Scatter Plot (click-drag with mouse)')
    SurfacePlot(EquationFunc, [xData, yData, zData], popt)
    ContourPlot(EquationFunc, [xData, yData, zData], popt)

    absError = zData - EquationFunc((xData,yData), *popt)
    ScatterPlot([xData, yData, absError], 'Error Scatter Plot (click-drag with mouse)')