def test_basic(self):
dts = list(
zip(['f2', 'f4', 'f8', 'c8', 'c16'], [
np.float16, np.float32, np.float64, np.complex64, np.complex128
]))
for dt1, dt2 in dts:
for attr in ('bits', 'eps', 'epsneg', 'iexp', 'machar', 'machep',
'max', 'maxexp', 'min', 'minexp', 'negep', 'nexp',
'nmant', 'precision', 'resolution', 'tiny'):
assert_equal(getattr(finfo(dt1), attr),
getattr(finfo(dt2), attr), attr)
assert_raises(ValueError, finfo, 'i4')
def test_instances():
iinfo(10)
finfo(3.0)
def test_singleton(self):
ftype = finfo(half)
ftype2 = finfo(half)
assert_equal(id(ftype), id(ftype2))
def test_singleton(self):
ftype = finfo(longdouble)
ftype2 = finfo(longdouble)
assert_equal(id(ftype), id(ftype2))
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
"""
Least squares polynomial fit.
Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg`
to points `(x, y)`. Returns a vector of coefficients `p` that minimises
the squared error.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int
Degree of the fitting polynomial
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (M,), optional
Weights to apply to the y-coordinates of the sample points. For
gaussian uncertainties, use 1/sigma (not 1/sigma**2).
cov : bool, optional
Return the estimate and the covariance matrix of the estimate
If full is True, then cov is not returned.
Returns
-------
p : ndarray, shape (deg + 1,) or (deg + 1, K)
Polynomial coefficients, highest power first. If `y` was 2-D, the
coefficients for `k`-th data set are in ``p[:,k]``.
residuals, rank, singular_values, rcond
Present only if `full` = True. Residuals of the least-squares fit,
the effective rank of the scaled Vandermonde coefficient matrix,
its singular values, and the specified value of `rcond`. For more
details, see `linalg.lstsq`.
V : ndarray, shape (M,M) or (M,M,K)
Present only if `full` = False and `cov`=True. The covariance
matrix of the polynomial coefficient estimates. The diagonal of
this matrix are the variance estimates for each coefficient. If y
is a 2-D array, then the covariance matrix for the `k`-th data set
are in ``V[:,:,k]``
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if `full` = False.
The warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
polyval : Compute polynomial values.
linalg.lstsq : Computes a least-squares fit.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution minimizes the squared error
.. math ::
E = \\sum_{j=0}^k |p(x_j) - y_j|^2
in the equations::
x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0]
x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1]
...
x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]
The coefficient matrix of the coefficients `p` is a Vandermonde matrix.
`polyfit` issues a `RankWarning` when the least-squares fit is badly
conditioned. This implies that the best fit is not well-defined due
to numerical error. The results may be improved by lowering the polynomial
degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter
can also be set to a value smaller than its default, but the resulting
fit may be spurious: including contributions from the small singular
values can add numerical noise to the result.
Note that fitting polynomial coefficients is inherently badly conditioned
when the degree of the polynomial is large or the interval of sample points
is badly centered. The quality of the fit should always be checked in these
cases. When polynomial fits are not satisfactory, splines may be a good
alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
http://en.wikipedia.org/wiki/Curve_fitting
.. [2] Wikipedia, "Polynomial interpolation",
http://en.wikipedia.org/wiki/Polynomial_interpolation
Examples
--------
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> z = np.polyfit(x, y, 3)
>>> z
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
It is convenient to use `poly1d` objects for dealing with polynomials:
>>> p = np.poly1d(z)
>>> p(0.5)
0.6143849206349179
>>> p(3.5)
-0.34732142857143039
>>> p(10)
22.579365079365115
High-order polynomials may oscillate wildly:
>>> p30 = np.poly1d(np.polyfit(x, y, 30))
/... RankWarning: Polyfit may be poorly conditioned...
>>> p30(4)
-0.80000000000000204
>>> p30(5)
-0.99999999999999445
>>> p30(4.5)
-0.10547061179440398
Illustration:
>>> import matplotlib.pyplot as plt
>>> xp = np.linspace(-2, 6, 100)
>>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--')
>>> plt.ylim(-2,2)
(-2, 2)
>>> plt.show()
"""
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0]:
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None:
rcond = len(x) * finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected a 1-d array for weights")
if w.shape[0] != y.shape[0]:
raise TypeError("expected w and y to have the same length")
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs * lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T / scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning, stacklevel=2)
if full:
return c, resids, rank, s, rcond
elif cov:
Vbase = inv(dot(lhs.T, lhs))
Vbase /= NX.outer(scale, scale)
# Some literature ignores the extra -2.0 factor in the denominator, but
# it is included here because the covariance of Multivariate Student-T
# (which is implied by a Bayesian uncertainty analysis) includes it.
# Plus, it gives a slightly more conservative estimate of uncertainty.
if len(x) <= order + 2:
raise ValueError("the number of data points must exceed order + 2 "
"for Bayesian estimate the covariance matrix")
fac = resids / (len(x) - order - 2.0)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:, :, NX.newaxis] * fac
else:
return c
