#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Regression Analysis
===================
Defines various objects to perform statistical regression analysis:
- :func:`linear_regression`: Implements multiple linear regression.
References
----------
.. [1] Wikipedia. (n.d.). Regression analysis. Retrieved May 24, 2014, from
http://en.wikipedia.org/wiki/Regression_analysis
"""
from __future__ import division, unicode_literals
import numpy as np
from colour.algebra import as_array
__author__ = 'Colour Developers'
__copyright__ = 'Copyright (C) 2013 - 2014 - Colour Developers'
__license__ = 'New BSD License - http://opensource.org/licenses/BSD-3-Clause'
__maintainer__ = 'Colour Developers'
__email__ = 'colour-science@googlegroups.com'
__status__ = 'Production'
__all__ = ['linear_regression']
[docs]def linear_regression(y, x=None, additional_statistics=False):
"""
Performs the statistics computation about the ideal trend line from given
data using the *least-squares* method.
The equation of the line is :math:`y=b+mx` or
:math:`y=b+m1x1+m1x2+...+mnxn` where the dependent variable :math:`y` value
is a function of the independent variable :math:`x` values.
Parameters
----------
y : array_like
Dependent and already known :math:`y` variable values used to curve
fit an ideal trend line.
x : array_like, optional
Independent :math:`x` variable(s) values corresponding with :math:`y`
variable.
additional_statistics : ndarray
Output additional regression statistics, by default only the :math:`b`
variable and :math:`m` coefficients are returned.
Returns
-------
ndarray, ({{mn, mn-1, ..., b}, {sum_of_squares_residual}})
Regression statistics.
Raises
------
ValueError
If :math:`y` and :math:`x` variables have incompatible dimensions.
References
----------
.. [2] Wikipedia. (n.d.). Simple linear regression. Retrieved May 24,
2014, from http://en.wikipedia.org/wiki/Simple_linear_regression
Examples
--------
Linear regression with the dependent and already known :math:`y` variable:
>>> y = np.array([1, 2, 1, 3, 2, 3, 3, 4, 4, 3])
>>> linear_regression(y) # doctest: +ELLIPSIS
array([ 0.2909090..., 1. ])
Linear regression with the dependent :math:`y` variable and independent
:math:`x` variable:
>>> x1 = np.array([40, 45, 38, 50, 48, 55, 53, 55, 58, 40])
>>> linear_regression(y, x1) # doctest: +ELLIPSIS
array([ 0.1225194..., -3.3054357...])
Multiple linear regression with the dependent :math:`y` variable and
multiple independent :math:`x_i` variables:
>>> x2 = np.array([25, 20, 30, 30, 28, 30, 34, 36, 32, 34])
>>> linear_regression(y, tuple(zip(x1, x2))) # doctest: +ELLIPSIS
array([ 0.0998002..., 0.0876257..., -4.8303807...])
Multiple linear regression with additional statistics:
>>> linear_regression(y, tuple(zip(x1, x2)), True) # doctest: +ELLIPSIS
(array([ 0.0998002..., 0.0876257..., -4.8303807...]), array([ 2.1376249...]))
"""
y = as_array(y)
if x is None:
x = np.arange(1, len(y) + 1)
else:
x = as_array(x)
if len(x) != len(y):
raise ValueError(
'"y" and "x" variables have incompatible dimensions!')
x = np.vstack([np.array(x).T, np.ones(len(x))]).T
result = np.linalg.lstsq(x, y)
if additional_statistics:
return result[0:2]
else:
return result[0]
Colour 0.3