.. _chap-fitting:

*********
 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.

Fitting a straight line
=======================

.. math::

   m x_1 + b = y_1 \\
   m x_2 + b = y_2

which is equivalent to:


.. math::

   m x_1 + b - y_1 = 0 \\
   m x_2 + b - y_2 = 0

::

   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}

Fitting a parabola
==================

.. math::

   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

which is equivalent to:


.. math::

   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

::

   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}


Fitting a cubic
===============

.. math::

   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

which is equivalent to:


.. math::


   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

::

   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)