10. Fitting

Here are some exercises to do together: we put these equations into sympy to do curve fitting.

We can precede this with a discussion of fitting and how that gives us a system of linear equations.

10.1. Fitting a straight line

\[\begin{split}m x_1 + b = y_1 \\ m x_2 + b = y_2\end{split}\]

which is equivalent to:

\[\begin{split}m x_1 + b - y_1 = 0 \\ m x_2 + b - y_2 = 0\end{split}\]

from sympy import *
init_printing(use_unicode=True)
m, b = symbols('m b')
x1, x2, y1, y2 = symbols('x1 x2 y1 y2')
solve([m*x1 + b - y1, m*x2 + b - y2], m, b)
# HIT ENTER HERE - there will be a long result

# now specialize it to specific points
x1 = 1
y1 = 7
x2 = 2
y2 = 3
solve([m*x1 + b - y1, m*x2 + b - y2], m, b)
result: {b: 11, m: -3}

10.2. Fitting a parabola

\[\begin{split}a {x_1}^2 + b x_1 + c = y_1 \\ a {x_2}^2 + b x_2 + c = y_2 \\ a {x_3}^2 + b x_3 + c = y_3\end{split}\]

which is equivalent to:

\[\begin{split}a {x_1}^2 + b x_1 + c - y_1 = 0\\ a {x_2}^2 + b x_2 + c - y_2 = 0 \\ a {x_3}^2 + b x_3 + c - y_3 = 0\end{split}\]

from sympy import *
init_printing(use_unicode=True)
a, b, c = symbols('a b c')
x1, x2, x3, y1, y2, y3 = symbols('x1 x2 x3 y1 y2 y3')
solve([a*x1**2 + b*x1 + c - y1, a*x2**2 + b*x2 + c - y2, a*x3**2 + b*x3 + c - y3], a, b, c)
# HIT ENTER HERE - there will be a long result

# now specialize it to specific points
x1 = 1
y1 = 1
x2 = 2
y2 = 2
x3 = 3
y3 = 7
solve([a*x1**2 + b*x1 + c - y1, a*x2**2 + b*x2 + c - y2, a*x3**2 + b*x3 + c - y3], a, b, c)
result: {a: 2, b: -5, c: 4}

10.3. Fitting a cubic

\[\begin{split}a {x_1}^3 + b {x_1}^2 + c x_1 + d = y_1 \\ a {x_2}^3 + b {x_2}^2 + c x_2 + d = y_2 \\ a {x_3}^3 + b {x_3}^2 + c x_3 + d = y_3 \\ a {x_4}^3 + b {x_4}^2 + c x_4 + d = y_4\end{split}\]

which is equivalent to:

\[\begin{split}a {x_1}^3 + b {x_1}^2 + c x_1 + d - y_1 = 0\\ a {x_2}^3 + b {x_2}^2 + c x_2 + d - y_2 = 0 \\ a {x_3}^3 + b {x_3}^2 + c x_3 + d - y_3 = 0 \\ a {x_4}^3 + b {x_4}^2 + c x_4 + d - y_4 = 0\end{split}\]

from sympy import *
init_printing(use_unicode=True)
a, b, c, d = symbols('a b c d')
x1, x2, x3, x4, y1, y2, y3, y4 = symbols('x1 x2 x3 x4 y1 y2 y3 y4')
solve([a*x1**3 + b*x1**2 + c*x1 + d - y1,
       a*x2**3 + b*x2**2 + c*x2 + d - y2,
       a*x3**3 + b*x3**2 + c*x3 + d - y3,
       a*x4**3 + b*x4**2 + c*x4 + d - y4],
       a, b, c, d)
# long result
x1 = -4
y1 = -3
x2 = -2
y2 = 0
x3 = 2
y3 = 0
x4 = 4
y4 = 2
solve([a*x1**3 + b*x1**2 + c*x1 + d - y1,
       a*x2**3 + b*x2**2 + c*x2 + d - y2,
       a*x3**3 + b*x3**2 + c*x3 + d - y3,
       a*x4**3 + b*x4**2 + c*x4 + d - y4],
       a, b, c, d)