Find optimised function parameters in python for function with two variables

i am new here and also new to Python. I realized that Python can be a powerful tool to optimize curves and fit data, so here I am. I have some experience with VBA, so i have some general knowledge about programming. I have a problem and I will try to describe it to you as good as possible.

I have a system of straight lines, which have different slopes and different intercepts. Each line represents a condition at which a second parameter is constant.

So for example:

• x1[0,1,2,3,4] y1[1,2,3,4,5] and z1[1,1,1,1,1] for the first line,
• x2[0,1,2,3,4] , y2[2,4,6,8,10] and z2[2,2,2,2,2] for the second line and
• x3[0,1,2,3,4], y3[3,6,9,12,15] and z3[3,3,3,3,3] for the third line.

To express these variables in one function I can express the slope and the intercept in terms of z, so that the combined function for all three lines would be:

y=(a1*z+b1)*x+(a2*z+b2) with a1=1, b1=0, a2=1, b2=0

Is it possible to get the values for a1, b1, a2 and b2 from a Python program?

I ask with this very easy example, because in reality I have measured data and the data therefore normally fluctuate a bit.

I want to play around with the function to describe the slope and intercepts, To optimise for fitting the whole set of data best. I know I want to describe the lines of constant z as linear and i want to have the best fitting general function to describe my data. The function to describe the slopes can be linear or anything else.

I want to test, which fits best. If I change the slope function for example one set of data points at z1=const. will not fit so well, while another set with z2=const. will fit better and so on, so that I have the smallest error possible in the end for the whole data set. I hope that makes sense to you. I would be happy if you could tell me if this is possible at all and how I need to build my code.

Is there a limit of how many lines I can fit? Sometimes I have 5, sometimes 15.

Thank you in advance!

I tried to get the optimised version in Excel but I think it will take ages! Also the visual analysis is not as good as mathematical anaylsis and reducing the error in multiple lines is difficult for me.

This is a quite simple linear algebra problem:

import numpy as np
from numpy.linalg import lstsq

'''
x1[0,1,2,3,4], y1[1,2,3,4,5]   and z1[1,1,1,1,1] for the first line,
x2[0,1,2,3,4], y2[2,4,6,8,10]  and z2[2,2,2,2,2] for the second line and
x3[0,1,2,3,4], y3[3,6,9,12,15] and z3[3,3,3,3,3]

y=(a1*z + b1)*x + (a2*z + b2) with a1=1, b1=0, a2=1, b2=0

[xz z x 1][a1] = [y]
[xz z x 1][a2]   [y]
[xz z x 1][b1]   [y]
[xz z x 1][b2]   [y]
[xz z x 1]       [y]
[...     ]       [...]
'''

x = np.array((0, 1, 2, 3, 4, 0, 1, 2, 3,  4, 0, 1, 2,  3,  4))
y = np.array((1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15))
z = np.array((1, 1, 1, 1, 1, 2, 2, 2, 2,  2, 3, 3, 3,  3,  3))

a = np.stack((
    x*z, z, x, np.ones_like(x),
), axis=1)

params, *_ = lstsq(a, y)
print(params)
print(a @ params)

[ 1.00000000e+00  1.00000000e+00 -8.11406854e-16  3.35250086e-15]
[ 1.  2.  3.  4.  5.  2.  4.  6.  8. 10.  3.  6.  9. 12. 15.]