Evaluating type-B uncertainty

The shorter name tb has been defined as an alias for type_b, to resolve the names of objects in this module.

Least-squares regression

line_fit() implements an ordinary least-squares straight-line regression calculation that accepts uncertain real numbers for the independent and dependent variables.

line_fit_wls() implements a weighted least-squares straight-line regression calculation. It accepts uncertain real numbers for the independent and dependent variables. It is also possible to specify weights for the regression.

line_fit_wtls() implements a total least-squares algorithm for a straight-line fitting that can perform a weighted least-squares regression when both y and x data are uncertain real numbers, it also handles correlation between (x,y) data pairs.

Probability distributions: real-valued problems

Functions that convert the half-width of a one-dimensional distribution to a standard uncertainty:

uniform()

triangular()

u_shaped()

arcsine()

Probability distributions: complex-valued problems

Functions that convert information about two-dimensional distributions into standard uncertainties:

uniform_ring()

uniform_disk()

unknown_phase_product()

A table of distributions

The mapping distribution is provided so that the functions above can be selected by name. For example,
>>> a = 1.5
>>> ureal( 1, type_b.distribution['gaussian'](a) )
ureal(1.0,1.5,inf)
>>> ureal( 1, type_b.distribution['uniform'](a) )
ureal(1.0,0.8660254037844387,inf)
>>> ureal( 1, type_b.distribution['arcsine'](a) )
ureal(1.0,1.0606601717798212,inf)
Keys to distribution are (case-sensitive):

gaussian

uniform

triangular

arcsine

u_shaped

uniform_ring

uniform_disk

Module contents

mean(seq, *args, **kwargs)

Return the arithmetic mean of data in seq

Parameters:

seq – a sequence, ndarray, or iterable, of numbers or uncertain numbers
args – optional arguments when seq is an ndarray
kwargs – optional keyword arguments when seq is an ndarray

An uncertain number is returned if seq contains uncertain numbers.

Example

>>> seq = [ ureal(1,1), ureal(2,1), ureal(3,1) ]
>>> function.mean(seq)
ureal(2.0,0.5773502691896257,inf)

Note

When seq is an empty ndarray or a ndarray containing any NaN elements NaN is returned.

In other cases, a ZeroDivisionError is raised when there are no elements in seq.

line_fit(x, y)

Least-squares fit intercept and slope

New in version 1.2.

Parameters:

x – sequence of independent variable data
y – sequence of dependent variable data

Return type:

a LineFitOLS

y must be a sequence of uncertain real numbers.

Performs an ordinary least-squares regression.

Note

Uncertainty in the parameter estimates is found by propagation through the regression formulae. This does not take residuals into account.

The function type_a.line_fit() performs a regression analysis that evaluates uncertainty in the parameter estimates using the residuals.

If appropriate, the results from both type-A and type-B analyses can be merged (see type_a.merge()).

Example:

>>> a0 =10
>>> b0 = -3
>>> u0 = .2

>>> x = [ float(x_i) for x_i in xrange(10) ]
>>> y = [ ureal(b0*x_i + a0,u0) for x_i in x ]

>>> a,b = tb.line_fit(x,y).a_b
>>> a
ureal(10.0,0.1175507627290...,inf)
>>> b
ureal(-3.0,0.02201927530252...,inf)

line_fit_wls(x, y, u_y=None)

Weighted least-squares linear regression

New in version 1.2.

Parameters:

x – sequence of independent variable data
y – sequence of dependent variable data
u_y – sequence of uncertainties in y

Return type:

LineFitWLS

y must be a sequence of uncertain real numbers.

Performs a weighted least-squares regression.

Weights are calculated from the uncertainty of the y elements unless the sequence u_y is provided.

Note

The uncertainty in the parameter estimates is found by propagation of uncertainty through the regression formulae. This does not take account of the residuals.

The function type_a.line_fit_wls() can be used to carry out a regression analysis that obtains uncertainty in the parameter estimates due to the residuals.

If necessary, the results of both type-A and type-B analyses can be merged (see type_a.merge()).

Example:

>>> x = [1,2,3,4,5,6]
>>> y = [3.2, 4.3, 7.6, 8.6, 11.7, 12.8]
>>> u_y = [0.5,0.5,0.5,1.0,1.0,1.0]
>>> y = [ ureal(y_i,u_y_i) for y_i, u_y_i in zip(y,u_y) ]

>>> fit = type_b.line_fit_wls(x,y)
>>> a, b = fit.a_b
>>> a
ureal(0.8852320675105...,0.5297081435088...,inf)
>>> b
ureal(2.0569620253164...,0.1778920167412...,inf)

line_fit_wtls(x, y, u_x=None, u_y=None, a_b=None, r_xy=None)

Perform straight-line regression with uncertainty in x and y

New in version 1.2.

Parameters:

x – list of uncertain real numbers for the independent variable
y – list of uncertain real numbers for the dependent variable
u_x – a sequence of uncertainties for the x data
u_y – a sequence of uncertainties for the y data
a_b – a pair of initial estimates for the intercept and slope
r_xy – correlation between x-y pairs [default: 0]

Returns a LineFitWTLS object

The elements of x and y must be uncertain numbers with non-zero uncertainties. If specified, the optional arguments u_x and u_y will be used uncertainties to weight the data for the regression, otherwise the uncertainties of the uncertain numbers in the sequences are used.

The optional argument a_b can be used to provide a pair of initial estimates for the intercept and slope. Otherwise, initial estimates will be obtained by calling line_fit_wls.

Implements a Weighted Total Least Squares algorithm that allows for correlation between x-y pairs. See reference:

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), January 1984
>>> xin=[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]
>>> yin=[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]

# Convert weights to standard uncertainties
>>> uxin=[1./math.sqrt(wx_i) for wx_i in wx ]
>>> uyin=[1./math.sqrt(wy_i) for wy_i in wy ]

# Define uncertain numbers
>>> x = [ ureal(xin_i,uxin_i) for xin_i,uxin_i in zip(xin,uxin) ]
>>> y = [ ureal(yin_i,uyin_i) for yin_i,uyin_i in zip(yin,uyin) ]

# TLS returns uncertain numbers
>>> a,b = type_b.line_fit_wtls(x,y).a_b
>>> a
ureal(5.47991018...,0.29193349...,inf)
>>> b
ureal(-0.48053339...,0.057616740...,inf)

class LineFitOLS(a, b, ssr, N)

Class to hold results from an ordinary linear regression to data.

New in version 1.2.

property N: The number of points in the sample

property a_b: Return the intercept a and slope b as a tuple of uncertain numbers

property intercept: Return the intercept as an uncertain number.

property slope: Return the slope as an uncertain number.

property ssr

Sum of the squared residuals

The sum of the squared deviations between values predicted by the model and the actual data.

If weights are used during the fit, the squares of weighted deviations are summed.

x_from_y(yseq, x_label=None)

Estimate the stimulus x corresponding to the responses in yseq

Parameters:

yseq – a sequence of further observations of y
x_label – a label for the return uncertain number x

The items in yseq must be uncertain real numbers.

Note

When x_label is defined, the uncertain number returned will be declared an intermediate result (using result())

y_from_x(x, y_label=None)

Return an uncertain number y that predicts the response to x

Parameters:

x – an uncertain real number
y_label – a label for the return uncertain number y

This is a prediction of a single future response y to a stimulus x

Note

When y_label is defined, the uncertain number returned will be declared an intermediate result (using result())

class LineFitWLS(a, b, ssr, N)

This object holds results from a weighted LS linear regression to data.

New in version 1.2.

property N: The number of points in the sample

property a_b: Return the intercept a and slope b as a tuple of uncertain numbers

property intercept: Return the intercept as an uncertain number.

property slope: Return the slope as an uncertain number.

property ssr

Sum of the squared residuals

The sum of the squared deviations between values predicted by the model and the actual data.

If weights are used during the fit, the squares of weighted deviations are summed.

x_from_y(yseq, x_label=None)

Estimate the stimulus x corresponding to the responses in yseq

Parameters:

yseq – a sequence of further observations of y
x_label – a label for the return uncertain number x

The items in yseq must be uncertain real numbers.

Note

When x_label is defined, the uncertain number returned will be declared an intermediate result (using result())

y_from_x(x, y_label=None)

Return an uncertain number y that predicts the response to x

Parameters:

x – an uncertain real number
y_label – a label for the return uncertain number y

This is a prediction of a single future response y to a stimulus x

Note

When y_label is defined, the uncertain number returned will be declared an intermediate result (using result())

class LineFitWTLS(a, b, ssr, N)

This object holds results from a TLS linear regression to data.

New in version 1.2.

property N: The number of points in the sample

property a_b: Return the intercept a and slope b as a tuple of uncertain numbers

property intercept: Return the intercept as an uncertain number.

property slope: Return the slope as an uncertain number.

property ssr

Sum of the squared residuals

The sum of the squared deviations between values predicted by the model and the actual data.

If weights are used during the fit, the squares of weighted deviations are summed.

uniform(a)

Return the standard uncertainty for a uniform distribution.

Parameters:: a (float) – the half-width

Example:

>>> x = ureal(1,type_b.uniform(1))
>>> x
ureal(1.0,0.5773502691896258,inf)

triangular(a)

Return the standard uncertainty for a triangular distribution.

Parameters:: a (float) – the half-width

Example:

>>> x = ureal(1,type_b.triangular(1))
>>> x
ureal(1.0,0.4082482904638631,inf)

u_shaped(a)

Return the standard uncertainty for an arcsine distribution.

Parameters:: a (float) – the half-width

Example:

>>> x = ureal(1,type_b.arcsine(1))
>>> x
ureal(1.0,0.7071067811865475,inf)

arcsine(a)

Return the standard uncertainty for an arcsine distribution.

Parameters:: a (float) – the half-width

Example:

>>> x = ureal(1,type_b.arcsine(1))
>>> x
ureal(1.0,0.7071067811865475,inf)

uniform_ring(a)

Return the standard uncertainty for a uniform ring

Parameters:: a (float) – the radius

Convert the radius of a uniform ring distribution a to a standard uncertainty

See reference: B D Hall, Metrologia 48 (2011) 324-332

Example:

>>> z = ucomplex( 0, type_b.uniform_ring(1) )
>>> z
ucomplex((0+0j), u=[0.7071067811865475,0.7071067811865475], r=0.0, df=inf)

uniform_disk(a)

Return the standard uncertainty for a uniform disk

Parameters:: a (float) – the radius

Convert the radius of a uniform disk distribution a to a standard uncertainty.

See reference: B D Hall, Metrologia 48 (2011) 324-332

Example:

>>> z = ucomplex( 0, type_b.uniform_disk(1) )
>>> z
ucomplex((0+0j), u=[0.5,0.5], r=0.0, df=inf)

unknown_phase_product(u1, u2)

Return the standard uncertainty for a product when phases are unknown

Parameters:

u1 – the standard uncertainty of the first multiplicand
u2 – the standard uncertainty of the second multiplicand

Obtains the standard uncertainty associated with a complex product when estimates have unknown phase.

The arguments u1 and u2 are the standard uncertainties associated with each multiplicand.

See reference: B D Hall, Metrologia 48 (2011) 324-332

Example:

# X = Gamma1 * Gamma2
>>> X = ucomplex( 0, type_b.unknown_phase_product(.1,.1) )
>>> X
ucomplex((0+0j), u=[0.014142135623730954,0.014142135623730954], r=0.0, df=inf)

distribution

Maps distribution[name] \(\rightarrow\) function for standard uncertainty

Example:

>>> type_b.distribution['arcsine'](1.5)
1.06066017177982...