scipy.linalg.lstsq#

scipy.linalg.lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False, check_finite=True, lapack_driver=None)[source]#

Compute least-squares solution to equation Ax = b.

Compute a vector x such that the 2-norm |b - A x| is minimized.

Parameters:

a(M, N) array_like: Left-hand side array
b(M,) or (M, K) array_like: Right hand side array
condfloat, optional: Cutoff for ‘small’ singular values; used to determine effective rank of a. Singular values smaller than cond * largest_singular_value are considered zero.
overwrite_abool, optional: Discard data in a (may enhance performance). Default is False.
overwrite_bbool, optional: Discard data in b (may enhance performance). Default is False.
check_finitebool, optional: Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
lapack_driverstr, optional: Which LAPACK driver is used to solve the least-squares problem. Options are 'gelsd', 'gelsy', 'gelss'. Default ('gelsd') is a good choice. However, 'gelsy' can be slightly faster on many problems. 'gelss' was used historically. It is generally slow but uses less memory.

Added in version 0.17.0.

Returns:

x(N,) or (N, K) ndarray: Least-squares solution.
residues(K,) ndarray or float: Square of the 2-norm for each column in b - a x, if M > N and rank(A) == n (returns a scalar if b is 1-D). Otherwise a (0,)-shaped array is returned.
rankint: Effective rank of a.
s(min(M, N),) ndarray or None: Singular values of a. The condition number of a is s[0] / s[-1].

Raises:

LinAlgError: If computation does not converge.
ValueError: When parameters are not compatible.

See also

scipy.optimize.nnls: linear least squares with non-negativity constraint

Notes

When 'gelsy' is used as a driver, residues is set to a (0,)-shaped array and s is always None.

Examples

>>> import numpy as np
>>> from scipy.linalg import lstsq
>>> import matplotlib.pyplot as plt

Suppose we have the following data:

>>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
>>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])

We want to fit a quadratic polynomial of the form y = a + b*x**2 to this data. We first form the “design matrix” M, with a constant column of 1s and a column containing x**2:

>>> M = x[:, np.newaxis]**[0, 2]
>>> M
array([[  1.  ,   1.  ],
       [  1.  ,   6.25],
       [  1.  ,  12.25],
       [  1.  ,  16.  ],
       [  1.  ,  25.  ],
       [  1.  ,  49.  ],
       [  1.  ,  72.25]])

We want to find the least-squares solution to M.dot(p) = y, where p is a vector with length 2 that holds the parameters a and b.

>>> p, res, rnk, s = lstsq(M, y)
>>> p
array([ 0.20925829,  0.12013861])

Plot the data and the fitted curve.

>>> plt.plot(x, y, 'o', label='data')
>>> xx = np.linspace(0, 9, 101)
>>> yy = p[0] + p[1]*xx**2
>>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$')
>>> plt.xlabel('x')
>>> plt.ylabel('y')
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.grid(alpha=0.25)
>>> plt.show()