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numpy.polynomial.laguerre.lagfit

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Least squares fit of Laguerre series to data.

Return the coefficients of a Laguerre series of degree deg that is the
least squares fit to the data values y given at points x. If y is
1-D the returned coefficients will also be 1-D. If y is 2-D multiple
fits are done, one for each column of y, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form

.. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),(more...)


        def lagfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Laguerre series to data.

Return the coefficients of a Laguerre series of degree deg that is the
least squares fit to the data values y given at points x. If y is
1-D the returned coefficients will also be 1-D. If y is 2-D multiple
fits are done, one for each column of y, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form

.. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),

where n is deg.

Since numpy version 1.7.0, lagfit also supports NA. If any of the
elements of x, y, or w are NA, then the corresponding rows of the
linear least squares problem (see Notes) are set to 0. If y is 2-D,
then an NA in any row of y invalidates that whole row.

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. If not None, the contribution of each point
(x[i],y[i]) to the fit is weighted by w[i]. Ideally the
weights are chosen so that the errors of the products w[i]*y[i]
all have the same variance.  The default value is None.

Returns
-------
coef : ndarray, shape (M,) or (M, K)
Laguerre coefficients ordered from low to high. If y was 2-D,
the coefficients for the data in column k  of y are in column
k.

[residuals, rank, singular_values, rcond] : present when full = True
Residuals of the least-squares fit, the effective rank of the
scaled Vandermonde matrix and its singular values, and the
specified value of rcond. For more details, see linalg.lstsq.

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', RankWarning)

--------
chebfit, legfit, polyfit, hermfit, hermefit
lagval : Evaluates a Laguerre series.
lagvander : pseudo Vandermonde matrix of Laguerre series.
lagweight : Laguerre weight function.
linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.

Notes
-----
The solution is the coefficients of the Laguerre series p that
minimizes the sum of the weighted squared errors

.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,

where the :math:w_j are the weights. This problem is solved by
setting up as the (typically) overdetermined matrix equation

.. math:: V(x) * c = w * y,

where V is the weighted pseudo Vandermonde matrix of x, c are the
coefficients to be solved for, w are the weights, and y are the
observed values.  This equation is then solved using the singular value
decomposition of V.

If some of the singular values of V are so small that they are
neglected, then a RankWarning will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning.  The rcond parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.

Fits using Laguerre series are probably most useful when the data can
be approximated by sqrt(w(x)) * p(x), where w(x) is the Laguerre
weight. In that case the weight sqrt(w(x[i]) should be used
together with data values y[i]/sqrt(w(x[i]). The weight function is
available as lagweight.

References
----------
.. [1] Wikipedia, "Curve fitting",
http://en.wikipedia.org/wiki/Curve_fitting

Examples
--------
>>> from numpy.polynomial.laguerre import lagfit, lagval
>>> x = np.linspace(0, 10)
>>> err = np.random.randn(len(x))/10
>>> y = lagval(x, [1, 2, 3]) + err
>>> lagfit(x, y, 2)
array([ 0.96971004,  2.00193749,  3.00288744])

"""
order = int(deg) + 1
x = np.asarray(x) + 0.0
y = np.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 len(x) != len(y):
raise TypeError("expected x and y to have same length")

# set up the least squares matrices in transposed form
lhs = lagvander(x, deg).T
rhs = y.T
if w is not None:
w = np.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected 1D vector for w")
if len(x) != len(w):
raise TypeError("expected x and w to have same length")
# apply weights. Don't use inplace operations as they
# can cause problems with NA.
lhs = lhs * w
rhs = rhs * w

# set rcond
if rcond is None :
rcond = len(x)*np.finfo(x.dtype).eps

# Determine the norms of the design matrix columns.
if issubclass(lhs.dtype.type, np.complexfloating):
scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1))
else:
scl = np.sqrt(np.square(lhs).sum(1))
scl[scl == 0] = 1

# Solve the least squares problem.
c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)
c = (c.T/scl).T

# warn on rank reduction
if rank != order and not full:
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)

if full :
return c, [resids, rank, s, rcond]
else :
return c


        y = f(x)
#
coef3 = lag.lagfit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(lag.lagval(x, coef3), y)
#
coef4 = lag.lagfit(x, y, 4)

        assert_almost_equal(lag.lagval(x, coef4), y)
#
coef2d = lag.lagfit(x, np.array([y,y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3,coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = lag.lagfit(x, yw, 3, w=w)

        assert_almost_equal(wcoef3, coef3)
#
wcoef2d = lag.lagfit(x, np.array([yw,yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3,coef3]).T)
# test scaling with complex values x points whose square


        y = f(x)
#
coef3 = lag.lagfit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(lag.lagval(x, coef3), y)
#
coef4 = lag.lagfit(x, y, 4)

        assert_almost_equal(lag.lagval(x, coef4), y)
#
coef2d = lag.lagfit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = lag.lagfit(x, yw, 3, w=w)

        assert_almost_equal(wcoef3, coef3)
#
wcoef2d = lag.lagfit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
# test scaling with complex values x points whose square