lstsq#

rlinalg.lstsq(a: Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], b: Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | bool | int | float | complex | str | bytes | _NestedSequence[bool | int | float | complex | str | bytes], tol: float = 1e-07, overwrite_a: bool = False, check_finite: bool = True) → Tuple[ndarray, ndarray, int]#

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

  • tol (float) – The absolute tolerance to use for QR decomposition.

  • overwrite_a (bool, optional) – Discard data in a (may enhance performance). Default is False.

  • check_finite (bool, 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.

Returns:

  • x ((N,) or (N, K) ndarray) – Least-squares solution. For underdetermined systems, x[i] will be zero for all i >= rank(A).

  • 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.

  • rank (int) – Effective rank of a.

Raises:

ValueError – When parameters are not compatible.

Notes

This function uses a QR decomposition rather than a SVD decomposition in the case of numpy.linalg.lstsq or scipy.linalg.lstsq. This is the same behaviour as lm.fit.

See also

numpy.linalg.lstsq

The NumPy implementation using LAPACK.

scipy.linalg.lstsq

The NumPy implementation using LAPACK.

lm.fit

Documentation of the equivalent R function lm.fit.

Examples

>>> import numpy
>>> from rlinalg import lstsq

Suppose we have the following data:

>>> x = numpy.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
>>> y = numpy.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[:, numpy.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, rank = lstsq(M, y)
>>> p
array([0.20925829, 0.12013861])

References

Chambers, J. M. (1992) Linear models. Chapter 4 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

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