Least-squares approximation
===========================

The following description is taken from scikit-learn version 0.18.2 [1]_.

The Least Squares method fits a linear model with coefficients :math:`{\bf \beta} = \left(\beta_0, \beta_1,\dotsc,\beta_{nx}\right)` to minimize the residual sum of squares between the observed responses in the dataset, and the responses predicted by the linear approximation.
Mathematically it solves a problem of the form:

.. math ::
  \min_\limits{{\bf \beta}}||{\bf X\beta-y}||_2^2,

where :math:`{\bf X} = \left(1,{{\bf x}^{(1)}}^T,\dots,{{\bf x}^{(nt)}}^T\right)^T` with dimensions (:math:`nt\times nx+1`).

.. [1] http://scikit-learn.org/stable/modules/linear_model.html

Usage
-----

.. code-block:: python

  import numpy as np
  import matplotlib.pyplot as plt
  
  from smt.surrogate_models import LS
  
  xt = np.array([0.0, 1.0, 2.0, 3.0, 4.0])
  yt = np.array([0.0, 1.0, 1.5, 0.9, 1.0])
  
  sm = LS()
  sm.set_training_values(xt, yt)
  sm.train()
  
  num = 100
  x = np.linspace(0.0, 4.0, num)
  y = sm.predict_values(x)
  
  plt.plot(xt, yt, "o")
  plt.plot(x, y)
  plt.xlabel("x")
  plt.ylabel("y")
  plt.legend(["Training data", "Prediction"])
  plt.show()
  
::

  ___________________________________________________________________________
     
                                      LS
  ___________________________________________________________________________
     
   Problem size
     
        # training points.        : 5
     
  ___________________________________________________________________________
     
   Training
     
     Training ...
     Training - done. Time (sec):  0.0004039
  ___________________________________________________________________________
     
   Evaluation
     
        # eval points. : 100
     
     Predicting ...
     Predicting - done. Time (sec):  0.0000441
     
     Prediction time/pt. (sec) :  0.0000004
     
  
.. figure:: ls_Test_test_ls.png
  :scale: 80 %
  :align: center

Options
-------

.. list-table:: List of options
  :header-rows: 1
  :widths: 15, 10, 20, 20, 30
  :stub-columns: 0

  *  -  Option
     -  Default
     -  Acceptable values
     -  Acceptable types
     -  Description
  *  -  print_global
     -  True
     -  None
     -  ['bool']
     -  Global print toggle. If False, all printing is suppressed
  *  -  print_training
     -  True
     -  None
     -  ['bool']
     -  Whether to print training information
  *  -  print_prediction
     -  True
     -  None
     -  ['bool']
     -  Whether to print prediction information
  *  -  print_problem
     -  True
     -  None
     -  ['bool']
     -  Whether to print problem information
  *  -  print_solver
     -  True
     -  None
     -  ['bool']
     -  Whether to print solver information
  *  -  data_dir
     -  None
     -  None
     -  ['str']
     -  Directory for loading / saving cached data; None means do not save or load