def polyfit(x, y, deg, rcond=None, full=False):
"""%s
Notes
-----
Any masked values in x is propagated in y, and vice-versa.
"""
order = int(deg) + 1
x = asarray(x)
mx = getmask(x)
y = asarray(y)
if y.ndim == 1:
m = mask_or(mx, getmask(y))
elif y.ndim == 2:
y = mask_rows(y)
my = getmask(y)
if my is not nomask:
m = mask_or(mx, my[:,0])
else:
m = mx
else:
raise TypeError,"Expected a 1D or 2D array for y!"
if m is not nomask:
x[m] = y[m] = masked
# Set rcond
if rcond is None :
if x.dtype in (np.single, np.csingle):
rcond = len(x)*_single_eps
else :
rcond = len(x)*_double_eps
# Scale x to improve condition number
scale = abs(x).max()
if scale != 0 :
x = x / scale
# solve least squares equation for powers of x
v = vander(x, order)
c, resids, rank, s = _lstsq(v, y.filled(0), rcond)
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
warnings.warn("Polyfit may be poorly conditioned", np.RankWarning)
# scale returned coefficients
if scale != 0 :
if c.ndim == 1 :
c /= np.vander([scale], order)[0]
else :
c /= np.vander([scale], order).T
if full :
return c, resids, rank, s, rcond
else :
return c
def assert_array_compare(comparison,
x,
y,
err_msg='',
verbose=True,
header='',
fill_value=True):
"""Asserts that a comparison relation between two masked arrays is satisfied
elementwise."""
# Fill the data first
# xf = filled(x)
# yf = filled(y)
# Allocate a common mask and refill
m = mask_or(getmask(x), getmask(y))
x = masked_array(x, copy=False, mask=m, keep_mask=False, subok=False)
y = masked_array(y, copy=False, mask=m, keep_mask=False, subok=False)
if ((x is masked) and not (y is masked)) or \
((y is masked) and not (x is masked)):
msg = build_err_msg([x, y],
err_msg=err_msg,
verbose=verbose,
header=header,
names=('x', 'y'))
raise ValueError(msg)
# OK, now run the basic tests on filled versions
return utils.assert_array_compare(comparison,
x.filled(fill_value),
y.filled(fill_value),
err_msg=err_msg,
verbose=verbose,
header=header)
def assert_array_compare(comparison, x, y, err_msg='', header='',
fill_value=True):
"""Asserts that a comparison relation between two masked arrays is satisfied
elementwise."""
xf = filled(x)
yf = filled(y)
m = mask_or(getmask(x), getmask(y))
x = masked_array(xf, copy=False, subok=False, mask=m).filled(fill_value)
y = masked_array(yf, copy=False, subok=False, mask=m).filled(fill_value)
if ((x is masked) and not (y is masked)) or \
((y is masked) and not (x is masked)):
msg = build_err_msg([x, y], err_msg, header=header, names=('x', 'y'))
raise ValueError(msg)
if (x.dtype.char != "O") and (x.dtype.char != "S"):
x = x.astype(float_)
if isinstance(x, N.ndarray) and x.size > 1:
x[N.isnan(x)] = 0
elif N.isnan(x):
x = 0
if (y.dtype.char != "O") and (y.dtype.char != "S"):
y = y.astype(float_)
if isinstance(y, N.ndarray) and y.size > 1:
y[N.isnan(y)] = 0
elif N.isnan(y):
y = 0
try:
cond = (x.shape==() or y.shape==()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg
+ '\n(shapes %s, %s mismatch)' % (x.shape,
y.shape),
header=header,
names=('x', 'y'))
assert cond, msg
val = comparison(x,y)
if m is not nomask and fill_value:
val = masked_array(val, mask=m, copy=False)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100-100.0*reduced.count(1)/len(reduced)
msg = build_err_msg([x, y],
err_msg
+ '\n(mismatch %s%%)' % (match,),
header=header,
names=('x', 'y'))
assert cond, msg
except ValueError:
msg = build_err_msg([x, y], err_msg, header=header, names=('x', 'y'))
raise ValueError(msg)
def almost(a, b, decimal=6, fill_value=True):
"""Returns True if a and b are equal up to decimal places.
If fill_value is True, masked values considered equal. Otherwise, masked values
are considered unequal.
"""
m = mask_or(getmask(a), getmask(b))
d1 = filled(a)
d2 = filled(b)
if d1.dtype.char == "O" or d2.dtype.char == "O":
return np.equal(d1, d2).ravel()
x = filled(masked_array(d1, copy=False, mask=m), fill_value).astype(float_)
y = filled(masked_array(d2, copy=False, mask=m), 1).astype(float_)
d = np.around(np.abs(x - y), decimal) <= 10.0**(-decimal)
return d.ravel()
def almost(a, b, decimal=6, fill_value=True):
"""Returns True if a and b are equal up to decimal places.
If fill_value is True, masked values considered equal. Otherwise, masked values
are considered unequal.
"""
m = mask_or(getmask(a), getmask(b))
d1 = filled(a)
d2 = filled(b)
if d1.dtype.char == "O" or d2.dtype.char == "O":
return N.equal(d1,d2).ravel()
x = filled(masked_array(d1, copy=False, mask=m), fill_value).astype(float_)
y = filled(masked_array(d2, copy=False, mask=m), 1).astype(float_)
d = N.around(N.abs(x-y),decimal) <= 10.0**(-decimal)
return d.ravel()
def approx (a, b, fill_value=1, rtol=1.e-5, atol=1.e-8):
"""Returns true if all components of a and b are equal subject to given tolerances.
If fill_value is 1, masked values considered equal.
If fill_value is 0, masked values considered unequal.
The relative error rtol should be positive and << 1.0
The absolute error atol comes into play for those elements of b that are
very small or zero; it says how small a must be also.
"""
m = mask_or(getmask(a), getmask(b))
d1 = filled(a)
d2 = filled(b)
if d1.dtype.char == "O" or d2.dtype.char == "O":
return N.equal(d1,d2).ravel()
x = filled(masked_array(d1, copy=False, mask=m), fill_value).astype(float_)
y = filled(masked_array(d2, copy=False, mask=m), 1).astype(float_)
d = N.less_equal(umath.absolute(x-y), atol + rtol * umath.absolute(y))
return d.ravel()
def assert_array_compare(comparison, x, y, err_msg="", verbose=True, header="", fill_value=True):
"""Asserts that a comparison relation between two masked arrays is satisfied
elementwise."""
# Fill the data first
# xf = filled(x)
# yf = filled(y)
# Allocate a common mask and refill
m = mask_or(getmask(x), getmask(y))
x = masked_array(x, copy=False, mask=m, keep_mask=False, subok=False)
y = masked_array(y, copy=False, mask=m, keep_mask=False, subok=False)
if ((x is masked) and not (y is masked)) or ((y is masked) and not (x is masked)):
msg = build_err_msg([x, y], err_msg=err_msg, verbose=verbose, header=header, names=("x", "y"))
raise ValueError(msg)
# OK, now run the basic tests on filled versions
return utils.assert_array_compare(
comparison, x.filled(fill_value), y.filled(fill_value), err_msg=err_msg, verbose=verbose, header=header
)
def approx(a, b, fill_value=True, rtol=1.e-5, atol=1.e-8):
"""Returns true if all components of a and b are equal subject to given tolerances.
If fill_value is True, masked values considered equal. Otherwise, masked values
are considered unequal.
The relative error rtol should be positive and << 1.0
The absolute error atol comes into play for those elements of b that are very
small or zero; it says how small a must be also.
"""
m = mask_or(getmask(a), getmask(b))
d1 = filled(a)
d2 = filled(b)
if d1.dtype.char == "O" or d2.dtype.char == "O":
return np.equal(d1, d2).ravel()
x = filled(masked_array(d1, copy=False, mask=m), fill_value).astype(float_)
y = filled(masked_array(d2, copy=False, mask=m), 1).astype(float_)
d = np.less_equal(umath.absolute(x - y), atol + rtol * umath.absolute(y))
return d.ravel()
def polyfit(x, y, deg, rcond=None, full=False):
"""
Least squares polynomial fit.
Do a best fit polynomial of degree 'deg' of 'x' to 'y'. Return value is a
vector of polynomial coefficients [pk ... p1 p0]. Eg, for ``deg = 2``::
p2*x0^2 + p1*x0 + p0 = y1
p2*x1^2 + p1*x1 + p0 = y1
p2*x2^2 + p1*x2 + p0 = y2
.....
p2*xk^2 + p1*xk + p0 = yk
Parameters
----------
x : array_like
1D vector of sample points.
y : array_like
1D vector or 2D array of values to fit. The values should run down the
columns in the 2D case.
deg : integer
Degree of the fitting polynomial
rcond: {None, float}, optional
Relative condition number of the fit. Singular values smaller than this
relative to the largest singular value will be ignored. The defaul value
is len(x)*eps, where eps is the relative precision of the float type,
about 2e-16 in most cases.
full : {False, boolean}, optional
Switch determining nature of return value. When it is False just the
coefficients are returned, when True diagnostic information from the
singular value decomposition is also returned.
Returns
-------
coefficients, [residuals, rank, singular_values, rcond] : variable
When full=False, only the coefficients are returned, running down the
appropriate colume when y is a 2D array. When full=True, the rank of the
scaled Vandermonde matrix, its effective rank in light of the rcond
value, its singular values, and the specified value of rcond are also
returned.
Warns
-----
RankWarning : if rank is reduced and not full output
The warnings can be turned off by:
>>> import warnings
>>> warnings.simplefilter('ignore',np.RankWarning)
See Also
--------
polyval : computes polynomial values.
Notes
-----
If X is a the Vandermonde Matrix computed from x (see
http://mathworld.wolfram.com/VandermondeMatrix.html), then the
polynomial least squares solution is given by the 'p' in
X*p = y
where X.shape is a matrix of dimensions (len(x), deg + 1), p is a vector of
dimensions (deg + 1, 1), and y is a vector of dimensions (len(x), 1).
This equation can be solved as
p = (XT*X)^-1 * XT * y
where XT is the transpose of X and -1 denotes the inverse. However, this
method is susceptible to rounding errors and generally the singular value
decomposition of the matrix X is preferred and that is what is done here.
The singular value method takes a paramenter, 'rcond', which sets a limit on
the relative size of the smallest singular value to be used in solving the
equation. This may result in lowering the rank of the Vandermonde matrix, in
which case a RankWarning is issued. If polyfit issues a RankWarning, try a
fit of lower degree or replace x by x - x.mean(), both of which will
generally improve the condition number. The routine already normalizes the
vector x by its maximum absolute value to help in this regard. The rcond
parameter can be set to a value smaller than its default, but the resulting
fit may be spurious. The current default value of rcond is len(x)*eps, where
eps is the relative precision of the floating type being used, generally
around 1e-7 and 2e-16 for IEEE single and double precision respectively.
This value of rcond is fairly conservative but works pretty well when x -
x.mean() is used in place of x.
DISCLAIMER: Power series fits are full of pitfalls for the unwary once the
degree of the fit becomes large or the interval of sample points is badly
centered. The problem is that the powers x**n are generally a poor basis for
the polynomial functions on the sample interval, resulting in a Vandermonde
matrix is ill conditioned and coefficients sensitive to rounding erros. The
computation of the polynomial values will also sensitive to rounding errors.
Consequently, the quality of the polynomial fit should be checked against
the data whenever the condition number is large. The quality of polynomial
fits *can not* be taken for granted. If all you want to do is draw a smooth
curve through the y values and polyfit is not doing the job, try centering
the sample range or look into scipy.interpolate, which includes some nice
spline fitting functions that may be of use.
For more info, see
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
but note that the k's and n's in the superscripts and subscripts
on that page. The linear algebra is correct, however.
Notes
-----
Any masked values in x is propagated in y, and vice-versa.
"""
order = int(deg) + 1
x = asarray(x)
mx = getmask(x)
y = asarray(y)
if y.ndim == 1:
m = mask_or(mx, getmask(y))
elif y.ndim == 2:
y = mask_rows(y)
my = getmask(y)
if my is not nomask:
m = mask_or(mx, my[:, 0])
else:
m = mx
else:
raise TypeError, "Expected a 1D or 2D array for y!"
if m is not nomask:
x[m] = y[m] = masked
# Set rcond
if rcond is None:
rcond = len(x) * np.finfo(x.dtype).eps
# Scale x to improve condition number
scale = abs(x).max()
if scale != 0:
x = x / scale
# solve least squares equation for powers of x
v = vander(x, order)
c, resids, rank, s = _lstsq(v, y.filled(0), rcond)
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
warnings.warn("Polyfit may be poorly conditioned", np.RankWarning)
# scale returned coefficients
if scale != 0:
if c.ndim == 1:
c /= np.vander([scale], order)[0]
else:
c /= np.vander([scale], order).T
if full:
return c, resids, rank, s, rcond
else:
return c
def polyfit(x, y, deg, rcond=None, full=False):
"""
Least squares polynomial fit.
Do a best fit polynomial of degree 'deg' of 'x' to 'y'. Return value is a
vector of polynomial coefficients [pk ... p1 p0]. Eg, for ``deg = 2``::
p2*x0^2 + p1*x0 + p0 = y1
p2*x1^2 + p1*x1 + p0 = y1
p2*x2^2 + p1*x2 + p0 = y2
.....
p2*xk^2 + p1*xk + p0 = yk
Parameters
----------
x : array_like
1D vector of sample points.
y : array_like
1D vector or 2D array of values to fit. The values should run down the
columns in the 2D case.
deg : integer
Degree of the fitting polynomial
rcond: {None, float}, optional
Relative condition number of the fit. Singular values smaller than this
relative to the largest singular value will be ignored. The defaul value
is len(x)*eps, where eps is the relative precision of the float type,
about 2e-16 in most cases.
full : {False, boolean}, optional
Switch determining nature of return value. When it is False just the
coefficients are returned, when True diagnostic information from the
singular value decomposition is also returned.
Returns
-------
coefficients, [residuals, rank, singular_values, rcond] : variable
When full=False, only the coefficients are returned, running down the
appropriate colume when y is a 2D array. When full=True, the rank of the
scaled Vandermonde matrix, its effective rank in light of the rcond
value, its singular values, and the specified value of rcond are also
returned.
Warns
-----
RankWarning : if rank is reduced and not full output
The warnings can be turned off by:
>>> import warnings
>>> warnings.simplefilter('ignore',np.RankWarning)
See Also
--------
polyval : computes polynomial values.
Notes
-----
If X is a the Vandermonde Matrix computed from x (see
http://mathworld.wolfram.com/VandermondeMatrix.html), then the
polynomial least squares solution is given by the 'p' in
X*p = y
where X.shape is a matrix of dimensions (len(x), deg + 1), p is a vector of
dimensions (deg + 1, 1), and y is a vector of dimensions (len(x), 1).
This equation can be solved as
p = (XT*X)^-1 * XT * y
where XT is the transpose of X and -1 denotes the inverse. However, this
method is susceptible to rounding errors and generally the singular value
decomposition of the matrix X is preferred and that is what is done here.
The singular value method takes a paramenter, 'rcond', which sets a limit on
the relative size of the smallest singular value to be used in solving the
equation. This may result in lowering the rank of the Vandermonde matrix, in
which case a RankWarning is issued. If polyfit issues a RankWarning, try a
fit of lower degree or replace x by x - x.mean(), both of which will
generally improve the condition number. The routine already normalizes the
vector x by its maximum absolute value to help in this regard. The rcond
parameter can be set to a value smaller than its default, but the resulting
fit may be spurious. The current default value of rcond is len(x)*eps, where
eps is the relative precision of the floating type being used, generally
around 1e-7 and 2e-16 for IEEE single and double precision respectively.
This value of rcond is fairly conservative but works pretty well when x -
x.mean() is used in place of x.
DISCLAIMER: Power series fits are full of pitfalls for the unwary once the
degree of the fit becomes large or the interval of sample points is badly
centered. The problem is that the powers x**n are generally a poor basis for
the polynomial functions on the sample interval, resulting in a Vandermonde
matrix is ill conditioned and coefficients sensitive to rounding erros. The
computation of the polynomial values will also sensitive to rounding errors.
Consequently, the quality of the polynomial fit should be checked against
the data whenever the condition number is large. The quality of polynomial
fits *can not* be taken for granted. If all you want to do is draw a smooth
curve through the y values and polyfit is not doing the job, try centering
the sample range or look into scipy.interpolate, which includes some nice
spline fitting functions that may be of use.
For more info, see
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
but note that the k's and n's in the superscripts and subscripts
on that page. The linear algebra is correct, however.
Notes
-----
Any masked values in x is propagated in y, and vice-versa.
"""
order = int(deg) + 1
x = asarray(x)
mx = getmask(x)
y = asarray(y)
if y.ndim == 1:
m = mask_or(mx, getmask(y))
elif y.ndim == 2:
y = mask_rows(y)
my = getmask(y)
if my is not nomask:
m = mask_or(mx, my[:,0])
else:
m = mx
else:
raise TypeError,"Expected a 1D or 2D array for y!"
if m is not nomask:
x[m] = y[m] = masked
# Set rcond
if rcond is None :
rcond = len(x)*np.finfo(x.dtype).eps
# Scale x to improve condition number
scale = abs(x).max()
if scale != 0 :
x = x / scale
# solve least squares equation for powers of x
v = vander(x, order)
c, resids, rank, s = _lstsq(v, y.filled(0), rcond)
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
warnings.warn("Polyfit may be poorly conditioned", np.RankWarning)
# scale returned coefficients
if scale != 0 :
if c.ndim == 1 :
c /= np.vander([scale], order)[0]
else :
c /= np.vander([scale], order).T
if full :
return c, resids, rank, s, rcond
else :
return c
def assert_array_compare(comparison,
x,
y,
err_msg='',
header='',
fill_value=True):
"""Asserts that a comparison relation between two masked arrays is satisfied
elementwise."""
xf = filled(x)
yf = filled(y)
m = mask_or(getmask(x), getmask(y))
x = masked_array(xf, copy=False, subok=False, mask=m).filled(fill_value)
y = masked_array(yf, copy=False, subok=False, mask=m).filled(fill_value)
if ((x is masked) and not (y is masked)) or \
((y is masked) and not (x is masked)):
msg = build_err_msg([x, y], err_msg, header=header, names=('x', 'y'))
raise ValueError(msg)
if (x.dtype.char != "O") and (x.dtype.char != "S"):
x = x.astype(float_)
if isinstance(x, N.ndarray) and x.size > 1:
x[N.isnan(x)] = 0
elif N.isnan(x):
x = 0
if (y.dtype.char != "O") and (y.dtype.char != "S"):
y = y.astype(float_)
if isinstance(y, N.ndarray) and y.size > 1:
y[N.isnan(y)] = 0
elif N.isnan(y):
y = 0
try:
cond = (x.shape == () or y.shape == ()) or x.shape == y.shape
if not cond:
msg = build_err_msg([x, y],
err_msg + '\n(shapes %s, %s mismatch)' %
(x.shape, y.shape),
header=header,
names=('x', 'y'))
assert cond, msg
val = comparison(x, y)
if m is not nomask and fill_value:
val = masked_array(val, mask=m, copy=False)
if isinstance(val, bool):
cond = val
reduced = [0]
else:
reduced = val.ravel()
cond = reduced.all()
reduced = reduced.tolist()
if not cond:
match = 100 - 100.0 * reduced.count(1) / len(reduced)
msg = build_err_msg([x, y],
err_msg + '\n(mismatch %s%%)' % (match, ),
header=header,
names=('x', 'y'))
assert cond, msg
except ValueError:
msg = build_err_msg([x, y], err_msg, header=header, names=('x', 'y'))
raise ValueError(msg)