jax.numpy.polyfit

jax.numpy.polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False)

Least squares polynomial fit to data.

Jax implementation of numpy.polyfit().

Given a set of data points (x, y) and degree of polynomial deg, the function finds a polynomial equation of the form:

\[y = p(x) = p[0] x^{deg} + p[1] x^{deg - 1} + ... + p[deg]\]

Parameters:

• x (ArrayLike) – Array of data points of shape (M,).

• y (ArrayLike) – Array of data points of shape (M,) or (M, K).

• deg (int) – Degree of the polynomials. It must be specified statically.

• rcond (float | None) – Relative condition number of the fit. Default value is len(x) * eps. It must be specified statically.

• full (bool) – Switch that controls the return value. Default is False which restricts the return value to the array of polynomail coefficients p. If True, the function returns a tuple (p, resids, rank, s, rcond). It must be specified statically.

• w (ArrayLike | None) – Array of weights of shape (M,). If None, all data points are considered to have equal weight. If not None, the weight \(w_i\) is applied to the unsquared residual of \(y_i - \widehat{y}_i\) at \(x_i\), where \(\widehat{y}_i\) is the fitted value of \(y_i\). Default is None.

• cov (bool) – Boolean or string. If True, returns the covariance matrix scaled by resids/(M-deg-1) along with ploynomial coefficients. If cov='unscaled', returns the unscaaled version of covariance matrix. Default is False. cov is ignored if full=True. It must be specified statically.

Returns:

• An array polynomial coefficients p if full=False and cov=False.

• A tuple of arrays (p, resids, rank, s, rcond) if full=True. Where

  • p is an array of shape (M,) or (M, K) containing the polynomial coefficients.

  • resids is the sum of squared residual of shape () or (K,).

  • rank is the rank of the matrix x.

  • s is the singular values of the matrix x.

  • rcond as the array.

• A tuple of arrays (p, C) if full=False and cov=True. Where

  • p is an array of shape (M,) or (M, K) containing the polynomial coefficients.

  • C is the covariance matrix of polynomial coefficients of shape (deg + 1, deg + 1) or (deg + 1, deg + 1, 1).

Return type:

Array | tuple[Array, …]

Note

Unlike numpy.polyfit() implementation of polyfit, jax.numpy.polyfit() will not warn on rank reduction, which indicates an ill conditioned matrix.

See also

• jax.numpy.poly(): Finds the polynomial coefficients of the given sequence of roots.

• jax.numpy.polyval(): Evaluate a polynomial at specific values.

• jax.numpy.roots(): Computes the roots of a polynomial for given coefficients.

Examples

>>> x = jnp.array([3., 6., 9., 4.])
>>> y = jnp.array([[0, 1, 2],
...                [2, 5, 7],
...                [8, 4, 9],
...                [1, 6, 3]])
>>> p = jnp.polyfit(x, y, 2)
>>> with jnp.printoptions(precision=2, suppress=True):
...   print(p)
[[ 0.2  -0.35 -0.14]
 [-1.17  4.47  2.96]
 [ 1.95 -8.21 -5.93]]

If full=True, returns a tuple of arrays as follows:

>>> p, resids, rank, s, rcond = jnp.polyfit(x, y, 2, full=True)
>>> with jnp.printoptions(precision=2, suppress=True):
...   print("Polynomial Coefficients:", "\n", p, "\n",
...         "Residuals:", resids, "\n",
...         "Rank:", rank, "\n",
...         "s:", s, "\n",
...         "rcond:", rcond)
Polynomial Coefficients:
[[ 0.2  -0.35 -0.14]
[-1.17  4.47  2.96]
[ 1.95 -8.21 -5.93]]
Residuals: [0.37 5.94 0.61]
Rank: 3
s: [1.67 0.47 0.04]
rcond: 4.7683716e-07

If cov=True and full=False, returns a tuple of arrays having polynomial coefficients and covariance matrix.

>>> p, C = jnp.polyfit(x, y, 2, cov=True)
>>> p.shape, C.shape
((3, 3), (3, 3, 1))