def y_from_x(self, x, s_label=None, y_label=None):
"""Return an uncertain number ``y`` that predicts the response to ``x``
:arg x: a real number, or an uncertain real number
:arg s_label: a label for an elementary uncertain number associated with observation variability
:arg y_label: a label for the return uncertain number `y`
This is a prediction of a single future response ``y`` to a stimulus ``x``
The variability in observations is based on residuals obtained during regression.
An uncertain real number can be used for ``x``, in which
case the associated uncertainty will also be propagated into ``y``.
.. note::
When ``y_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
a, b = self._a_b
df = self._N - 2
u = math.sqrt(self._ssr / df)
noise = ureal(0, u, df, label=s_label, independent=False)
append_real_ensemble(a, noise)
if y_label is None:
y = a + b * x + noise
else:
y = result(a + b * x + noise, label=y_label)
return y
def y_from_x(self, x, s_y, s_label=None, y_label=None):
"""Return an uncertain number ``y`` that predicts the response to ``x``
:arg x: a real number, or an uncertain real number
:arg s_y: response variability uncertainty
:arg s_label: a label for an elementary uncertain number associated with response variability
:arg y_label: a label for the return uncertain number `y`
Returns a single future response ``y`` predicted for a stimulus ``x``.
The standard uncertainty ``s_y`` is used to create an additive
component of uncertainty associated with variability in the ``y`` value.
An uncertain real number can be used for ``x``, in which
case the associated uncertainty is also propagated into ``y``.
.. note::
When ``y_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
a, b = self._a_b
df = self.N - 2
noise = ureal(0, s_y, df, label=s_label)
append_real_ensemble(a, noise)
if y_label is None:
y = a + b * x + noise
else:
y = result(a + b * x + noise, label=y_label)
return y
def x_from_y(self, yseq, u_yseq, x_label=None, y_label=None):
"""Estimate the stimulus ``x`` corresponding to the responses in ``yseq``
:arg yseq: a sequence of further observations of ``y``
:arg u_yseq: the standard uncertainty of the ``yseq`` data
:arg x_label: a label for the return uncertain number `x`
:arg y_label: a label for the estimate of `y` based on ``yseq``
The variations in ``yseq`` values are assumed to result from
independent random effects.
..note::
When ``x_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
a, b = self._a_b
p = len(yseq)
y = math.fsum(yseq) / p
y = ureal(y,
u_yseq / math.sqrt(p),
inf,
label=y_label,
independent=False)
append_real_ensemble(a, y)
if x_label is None:
x = (y - a) / b
else:
x = result((y - a) / b, label=x_label)
return x
def x_from_y(self, yseq, s_y, x_label=None, y_label=None):
"""Estimate the stimulus ``x`` corresponding to the responses in ``yseq``
:arg yseq: a sequence of further observations of ``y``
:arg s_y: a scale factor for the uncertainty of the ``yseq``
:arg x_label: a label for the return uncertain number `x`
:arg y_label: a label for the estimate of `y` based on ``yseq``
..note::
When ``x_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
df = self._N - 2
a, b = self._a_b
p = len(yseq)
y = math.fsum(yseq) / p
y = ureal(y,
s_y * math.sqrt(self._ssr / df / p),
df,
label=y_label,
independent=False)
append_real_ensemble(a, y)
if x_label is None:
x = (y - a) / b
else:
x = result((y - a) / b, label=x_label)
return x
def x_from_y(self, yseq, x_label=None, y_label=None):
"""Estimate the stimulus ``x`` corresponding to the responses in ``yseq``
:arg yseq: a sequence of ``y`` observations
:arg x_label: a label for the return uncertain number `x`
:arg y_label: a label for the estimate of `y` based on ``yseq``
.. note::
When ``x_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
**Example** ::
>>> x_data = [0.1, 0.1, 0.1, 0.3, 0.3, 0.3, 0.5, 0.5, 0.5,
... 0.7, 0.7, 0.7, 0.9, 0.9, 0.9]
>>> y_data = [0.028, 0.029, 0.029, 0.084, 0.083, 0.081, 0.135, 0.131,
... 0.133, 0.180, 0.181, 0.183, 0.215, 0.230, 0.216]
>>> fit = type_a.line_fit(x_data,y_data)
>>> x0 = fit.x_from_y( [0.0712, 0.0716] )
>>> x0
ureal(0.2601659751037...,0.01784461112558...,13.0)
"""
df = self._N - 2
a, b = self._a_b
p = len(yseq)
y = math.fsum(yseq) / p
y = ureal(y,
math.sqrt(self._ssr / df / p),
df,
label=y_label,
independent=False)
append_real_ensemble(a, y)
if abs(b) < 1E-15:
# When b == 0, the best-fit line is horizontal
# so no use of new y data is made
x = a
else:
x = (y - a) / b
if x_label is not None:
x = result(x, label=x_label)
return x
def x_from_y(self, y_data, u_y_data, x_label=None, y_label=None):
"""Estimate the stimulus ``x`` corresponding to the responses in ``y_data``
:arg y_data: a sequence of further observations of ``y``
:arg u_y_data: the standard uncertainty of the ``y_data`` elements
:arg x_label: a label for the return uncertain number `x`
:arg y_label: a label for the estimate of `y` based on ``y_data``
The variations in ``y_data`` values are assumed to result from
independent random effects.
.. note::
When ``x_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
a, b = self._a_b
p = len(y_data)
y = math.fsum(y_data) / p
y = ureal(y,
u_y_data / math.sqrt(p),
inf,
label=y_label,
independent=False)
append_real_ensemble(a, y)
if abs(b) < 1E-15:
# When b == 0, the best-fit line is horizontal
# so no use of new y data is made
x = a
else:
x = (y - a) / b
if x_label is not None:
x = result(x, label=x_label)
return x
def y_from_x(self, x, s_y, s_label=None, y_label=None):
"""Return an uncertain number ``y`` that predicts the response to ``x``
:arg x: a real number, or an uncertain real number
:arg s_y: a scale factor for the response uncertainty
:arg s_label: a label for an elementary uncertain number associated with observation variability
:arg y_label: a label for the return uncertain number `y`
Returns a single future response ``y`` predicted for a stimulus ``x``.
Because there is different variability in
the response to different stimuli, the
scale factor ``s_y`` is required. It is assumed
that the standard deviation in the ``y`` value is
proportional to ``s_y``.
An uncertain real number can be used for ``x``, in which
case the associated uncertainty is also propagated into ``y``.
.. note::
When ``y_label`` is defined, the uncertain number returned will be
declared an intermediate result (using :func:`~.result`)
"""
a, b = self._a_b
df = self._N - 2
u = math.sqrt(s_y * self._ssr / df)
noise = ureal(0, u, df, label=s_label)
append_real_ensemble(a, noise)
if y_label is None:
y = a + b * x + noise
else:
y = result(a + b * x + noise, label=y_label)
return y