def test_with_dof_2(self):
"""
Same test as above, but now ensure that the
ensemble calculation can deal with influences
that are not in sequence
"""
TOL = 1E-10
# The dummy variables mean that those included
# in the ensemble will not be a consecutive
# series of IDs, provided we include the
# dummies in the equations, as is done below
# with zero weight.
v = ureal(4.999, 0.0032, 5, independent=False)
dummy_1 = ureal(1, 1)
i = ureal(0.019661, 0.0000095, 5, independent=False)
dummy_2 = ureal(1, 1)
phi = ureal(1.04446, 0.00075, 5, independent=False)
dummy_3 = ureal(1, 1)
real_ensemble([v, i, phi], 5)
set_correlation(-0.36, v, i)
set_correlation(0.86, v, phi)
set_correlation(-0.65, i, phi)
r = v * cos(phi) / i + (0 * dummy_1)
x = v * sin(phi) / i + (0 * dummy_2)
z = v / i + (0 * dummy_3)
# The presence of finite DoF should not alter previous results
equivalent(uncertainty(r), 0.0699787279884, TOL)
equivalent(uncertainty(x), 0.295716826846, TOL)
equivalent(uncertainty(z), 0.236602971835, TOL)
equivalent(math.sqrt(welch_satterthwaite(r)[0]), uncertainty(r), TOL)
equivalent(math.sqrt(welch_satterthwaite(x)[0]), uncertainty(x), TOL)
equivalent(math.sqrt(welch_satterthwaite(z)[0]), uncertainty(z), TOL)
equivalent(get_correlation(r, x), -0.591484610819, TOL)
equivalent(get_correlation(x, z), 0.992797472722, TOL)
equivalent(get_correlation(r, z), -0.490623905441, TOL)
equivalent(dof(r), 5, TOL)
equivalent(dof(x), 5, TOL)
equivalent(dof(z), 5, TOL)
def line_fit_wtls(x, y, u_x, u_y, a0_b0=None, r_xy=None, label=None):
"""Return a total least-squares straight-line fit
.. versionadded:: 1.2
:arg x: sequence of independent-variable data
:arg y: sequence of dependent-variable data
:arg u_x: sequence of uncertainties in ``x``
:arg u_y: sequence of uncertainties in ``y``
:arg a0_b0: a pair of initial estimates for the intercept and slope
:arg r_xy: correlation between x-y pairs
:arg label: suffix labeling the uncertain numbers `a` and `b`
:returns: an object containing the fitting results
:rtype: :class:`.LineFitWTLS`
The optional argument ``a_b`` can be used to provide a pair
of initial estimates for the intercept and slope.
Based on paper by M Krystek and M Anton,
*Meas. Sci. Technol.* **22** (2011) 035101 (9pp)
**Example**::
# Pearson-York test data see, e.g.,
# Lybanon, M. in Am. J. Phys 52 (1) 1984
>>> x=[0.0,0.9,1.8,2.6,3.3,4.4,5.2,6.1,6.5,7.4]
>>> wx=[1000.0,1000.0,500.0,800.0,200.0,80.0,60.0,20.0,1.8,1.0]
>>> y=[5.9,5.4,4.4,4.6,3.5,3.7,2.8,2.8,2.4,1.5]
>>> wy=[1.0,1.8,4.0,8.0,20.0,20.0,70.0,70.0,100.0,500.0]
# standard uncertainties required for weighting
>>> ux=[1./math.sqrt(wx_i) for wx_i in wx ]
>>> uy=[1./math.sqrt(wy_i) for wy_i in wy ]
>>> result = ta.line_fit_wtls(x,y,ux,uy)
>>> intercept, slope = result.a_b
>>> intercept
ureal(5.47991018...,0.29193349...,8)
>>> slope
ureal(-0.48053339...,0.057616740...,8)
"""
N = len(x)
df = N - 2
if df <= 0 or N != len(y):
raise RuntimeError("Invalid sequences: len({!r}), len({!r})".format(
x, y))
if N != len(u_x) or N != len(u_y):
raise RuntimeError("Invalid sequences: len({!r}), len({!r})".format(
u_x, u_y))
independent = r_xy is None
x_u = [
ureal(value(x_i), u_i, inf, None, independent=independent)
for x_i, u_i in izip(x, u_x)
]
y_u = [
ureal(value(y_i), u_i, inf, None, independent=independent)
for y_i, u_i in izip(y, u_y)
]
if not independent:
for x_i, y_i, r_i in izip(x_u, y_u, r_xy):
x_i.set_correlation(r_i, y_i)
result = type_b.line_fit_wtls(x_u, y_u, a_b=a0_b0)
a, b = result.a_b
N = result.N
ssr = result.ssr
r_ab = a.get_correlation(b)
a = ureal(a.x,
a.u,
df,
label='a_{}'.format(label) if label is not None else None,
independent=False)
b = ureal(b.x,
b.u,
df,
label='b_{}'.format(label) if label is not None else None,
independent=False)
real_ensemble((a, b), df)
a.set_correlation(r_ab, b)
return LineFitWTLS(a, b, ssr, N)
def line_fit_rwls(x, y, s_y, label=None):
"""Return a relative weighted least-squares straight-line fit
.. versionadded:: 1.2
The ``s_y`` values are used to scale variability in the ``y`` data.
It is assumed that the standard deviation of each ``y`` value is
proportional to the corresponding ``s_y`` scale factor.
The unknown common factor in the uncertainties is estimated from the residuals.
:arg x: sequence of stimulus data (independent-variable)
:arg y: sequence of response data (dependent-variable)
:arg s_y: sequence of scale factors
:arg label: suffix to label the uncertain numbers `a` and `b`
:returns: an object containing regression results
:rtype: :class:`.LineFitRWLS`
**Example**::
>>> x = [1,2,3,4,5,6]
>>> y = [3.014,5.225,7.004,9.061,11.201,12.762]
>>> s_y = [0.2,0.2,0.2,0.4,0.4,0.4]
>>> fit = type_a.line_fit_rwls(x,y,s_y)
>>> a, b = fit.a_b
>>>
>>> print(fit)
<BLANKLINE>
Relative Weighted Least-Squares Results:
<BLANKLINE>
Intercept: 1.14(12)
Slope: 1.973(41)
Correlation: -0.87
Sum of the squared residuals: 1.3395217958...
Number of points: 6
<BLANKLINE>
"""
N = len(x)
df = N - 2
if df <= 0 or N != len(y) or N != len(s_y):
raise RuntimeError(
"Invalid sequences: len({!r}), len({!r}), len({!r})".format(
x, y, s_y))
x = value_seq(x)
y = value_seq(y)
a_, b_, siga, sigb, r_ab, ssr, N = _line_fit_wls(x, y, s_y)
sigma_hat = math.sqrt(ssr / df)
siga *= sigma_hat
sigb *= sigma_hat
a = ureal(a_,
siga,
df,
label='a_{}'.format(label) if label is not None else None,
independent=False)
b = ureal(b_,
sigb,
df,
label='b_{}'.format(label) if label is not None else None,
independent=False)
real_ensemble((a, b), df)
a.set_correlation(r_ab, b)
return LineFitRWLS(a, b, ssr, N)
def line_fit(x, y, label=None):
"""Return a least-squares straight-line fit to the data
.. versionadded:: 1.2
:arg x: sequence of stimulus data (independent-variable)
:arg y: sequence of response data (dependent-variable)
:arg label: suffix to label the uncertain numbers `a` and `b`
:returns: an object containing regression results
:rtype: :class:`.LineFitOLS`
Performs an ordinary least-squares regression of ``y`` to ``x``.
**Example**::
>>> x = [1,2,3,4,5,6,7,8,9]
>>> y = [15.6,17.5,36.6,43.8,58.2,61.6,64.2,70.4,98.8]
>>> result = type_a.line_fit(x,y)
>>> a,b = result.a_b
>>> a
ureal(4.8138888888888...,4.8862063121833...,7)
>>> b
ureal(9.4083333333333...,0.8683016476563...,7)
>>> y_p = a + b*5.5
>>> dof(y_p)
7.0
"""
N = len(x)
df = N - 2
if df <= 0 or N != len(y):
raise RuntimeError("Invalid sequences: len({!r}), len({!r})".format(
x, y))
x = value_seq(x)
y = value_seq(y)
S_x = math.fsum(x)
S_y = math.fsum(y)
k = S_x / N
t = [(x_i - k) for x_i in x]
S_tt = math.fsum(t_i * t_i for t_i in t)
b_ = math.fsum(t_i * y_i / S_tt for t_i, y_i in izip(t, y))
a_ = (S_y - b_ * S_x) / N
siga = math.sqrt((1.0 + S_x * S_x / (N * S_tt)) / N)
sigb = math.sqrt(1.0 / S_tt)
r_ab = -S_x / (N * S_tt * siga * sigb)
# Sum of squared residuals needed to correctly calculate parameter uncertainties
f = lambda x_i, y_i: (y_i - a_ - b_ * x_i)**2
ssr = math.fsum(f(x_i, y_i) for x_i, y_i in izip(x, y))
data_u = math.sqrt(ssr / df)
siga *= data_u
sigb *= data_u
a = ureal(a_,
siga,
df=df,
label='a_{}'.format(label) if label is not None else None,
independent=False)
b = ureal(b_,
sigb,
df=df,
label='b_{}'.format(label) if label is not None else None,
independent=False)
real_ensemble((a, b), df)
a.set_correlation(r_ab, b)
return LineFitOLS(a, b, ssr, N)
def multi_estimate_real(seq_of_seq, labels=None):
"""Return a sequence of uncertain real numbers
:arg seq_of_seq: a sequence of sequences of data
:arg labels: a sequence of `str` labels
:rtype: seq of :class:`~lib.UncertainReal`
The sequences in ``seq_of_seq`` must all be the same length.
Each sequence contains
a sample of data associated with a particular quantity.
An uncertain number will be created for the quantity
from sample statistics. The covariance
between the different quantities will also be evaluated.
A sequence of elementary uncertain numbers is returned. These uncertain numbers
are considered to be related, allowing a degrees-of-freedom calculations
to be performed on derived quantities.
**Example**::
# From Appendix H2 in the GUM
>>> V = [5.007,4.994,5.005,4.990,4.999]
>>> I = [19.663E-3,19.639E-3,19.640E-3,19.685E-3,19.678E-3]
>>> phi = [1.0456,1.0438,1.0468,1.0428,1.0433]
>>> v,i,p = type_a.multi_estimate_real((V,I,phi),labels=('V','I','phi'))
>>> v
ureal(4.999000...,0.0032093613071761...,4, label='V')
>>> i
ureal(0.019661,9.471008394041335...e-06,4, label='I')
>>> p
ureal(1.044460...,0.0007520638270785...,4, label='phi')
>>> r = v/i*cos(p)
>>> r
ureal(127.732169928102...,0.071071407396995...,4.0)
"""
M = len(seq_of_seq)
N = len(seq_of_seq[0])
if labels is not None and len(labels) != M:
raise RuntimeError("Incorrect number of labels: '{!r}'".format(labels))
# Calculate the deviations from the mean for each sequence
means = []
dev = []
for i, seq_i in enumerate(seq_of_seq):
if len(seq_i) != N:
raise RuntimeError("{:d}th sequence length inconsistent".format(i))
mu_i = value(sum(seq_i) / N)
means.append(mu_i)
dev.append(tuple(value(x_j) - mu_i for x_j in seq_i))
# calculate the covariance matrix
N_N_1 = N * (N - 1)
u = []
cv = [] # M elements of len M-1, M-2, ...
for i, seq_i in enumerate(dev):
u_i = math.sqrt(math.fsum(d_i**2 for d_i in seq_i) / N_N_1)
u.append(u_i)
cv.append([])
for seq_j in dev[i + 1:]:
cv[i].append(
math.fsum(d_i * d_j
for d_i, d_j in izip(seq_i, seq_j)) / N_N_1)
# Create a list of elementary uncertain numbers
# to return a list of standard uncertainties
# to normalise the CV matrix.
df = N - 1
rtn = []
for i in xrange(M):
mu_i = means[i]
u_i = u[i]
l_i = labels[i] if labels is not None else ""
rtn.append(ureal(mu_i, u_i, df, l_i, independent=False))
# Create the list of ensemble id's,
# assign it to the register in the context,
# set the correlation between nodes
real_ensemble(rtn, df)
for i in xrange(M):
u_i = u[i]
un_i = rtn[i]
for j in xrange(M - 1 - i):
cv_ij = cv[i][j]
if cv_ij != 0.0:
r = cv_ij / (u_i * u[i + j + 1])
un_j = rtn[i + j + 1]
set_correlation_real(un_i, un_j, r)
return rtn
def multiple_ureal(x_seq, u_seq, df, label_seq=None):
"""Return a sequence of related elementary uncertain real numbers
:arg x_seq: a sequence of values (estimates)
:arg u_seq: a sequence of standard uncertainties
:arg df: the degrees-of-freedom
:arg label_seq: a sequence of labels
:rtype: a sequence of :class:`~lib.UncertainReal`
Defines an set of uncertain real numbers with
the same number of degrees-of-freedom.
Correlation between any pairs of this set of uncertain
numbers defined will not invalidate degrees-of-freedom
calculations.
(see: R Willink, *Metrologia* 44 (2007) 340-349, Sec. 4.1)
**Example**::
# Example from GUM-H2
>>> x = [4.999,19.661E-3,1.04446]
>>> u = [3.2E-3,9.5E-6,7.5E-4]
>>> labels = ['V','I','phi']
>>> v,i,phi = multiple_ureal(x,u,4,labels)
>>> set_correlation(-0.36,v,i)
>>> set_correlation(0.86,v,phi)
>>> set_correlation(-0.65,i,phi)
>>> r = v/i*cos(phi)
>>> r
ureal(127.732169928102...,0.0699787279883717...,4.0)
"""
if len(x_seq) != len(u_seq):
raise RuntimeError("unequal length sequences: x={!r} u={!r}".format(
x_seq, u_seq))
if label_seq is None:
label_seq = [None] * len(x_seq)
elif is_sequence(label_seq):
if len(x_seq) != len(label_seq):
raise RuntimeError(
"unequal length sequences: x={!r} label_seq={!r}".format(
x_seq, u_seq))
else:
raise RuntimeError("invalid `label_seq`: {!r}".format(label_seq))
rtn = [
# NB `ureal` creates constant objects when u == 0
ureal(x_i, u_i, df, label=l_i, independent=False)
for x_i, u_i, l_i in izip(x_seq, u_seq, label_seq)
]
# Only non-constant UNs can be collected in an ensemble
lib.real_ensemble(
[un_i for un_i in rtn if not lib._is_uncertain_real_constant(un_i)],
df)
# All uncertain numbers are returned, including the constants
return rtn
