Functions for estimating measurement uncertainty from standard uncertainties and either sensitivity coefficients or (for some methods) expressions or functions. Correlation is supported via either a correlation or covariance matrix.
uncert(obj, …) # S3 method for default
uncert(obj, c, method = c("GUM", "MC"), cor,
cov, distrib=NULL, distrib.pars=NULL, B=200, x=NULL, keep.x = TRUE,
u=obj, …)
# S3 method for expression
uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"),
cor, cov, distrib=NULL, distrib.pars=NULL,
B=200, delta=0.01, keep.x = TRUE, …)
# S3 method for function
uncert(obj, x, u, method=c("NUM", "kragten", "k2", "MC"),
cor, cov, distrib=NULL, distrib.pars=NULL,
B=200, delta=0.01, keep.x = TRUE, …)
# S3 method for formula
uncert(obj, x, u, method=c("GUM", "NUM", "kragten", "k2", "MC"),
cor, cov, distrib=NULL, distrib.pars=NULL,
B=200, delta=0.01, keep.x = TRUE, …)
An R object used for method dispatch; see below. Methods currently exist for numeric vector, expression, function, or formula
For the default method, a numeric vector of standard uncertainties. For
the formula or expression methods, a named list of standard uncertainties.Note that
for the default method, u is set to the value of obj, allowing
specification of either as the first argument
A numeric vector of senstivity coefficients.
For the expression or formula methods, an R object which can be used
as an environment by eval. For the function method, a list of
parameters supplied to FUN via do.call.
Method of uncertainty evaluation. The current list of methods is:
First-order error propagation (the “law of propagation of uncertainty”) as implemented by the GUM.
Numerical differentiation using a simple small step size.
Numerical estimation of uncertainty following Kragten (Kragten (1994).
A symmetric modification of Kragten's approach described by Ellison (Ellison (2005)).
Monte Carlo simulation.
A (square, symmetric) correlation or covariance matrix, respectively.
If neither is specified, cor is set to the identity matrix.
For method="MC", a character vector of length
length(x) or a named list of names of distribution functions associated
with u. See Details for defaults. The list format may include
user-specified functions. Silently ignored for other methods.
For method="MC", a list of
lists of parameters describing the distributions associated with u
to be passed to the relevant distribution function. If distrib is present
but distrib.pars is not, neither are included in the return value
unless method="MC". See Details for defaults when method="MC".
Silently ignored for other methods.
Number of Monte Carlo replicates.
Step size for numerical differentiation.
For method="MC", if keepx=TRUE, the simulated replicates of
x are included in the return object.
Additional parameters to be passed to a function (for the function method) or used in an expression (for expression or formula method).
An object of class ‘uncert’ or, for method="MC" of class ‘uncertMC’.
See uncert-class and uncertMC-class for details.
The default “GUM” method applies first-order error propagation principles to estimate
a combined standard uncertainty from a set of sensitivity coefficients and either a
set of standard uncertainties and a correlation matrix (which defaults to an identity matrix)
or a covariance matrix. Both options use the same calculation, which is simply
(t(c) %*% cov) %*% c ; standard uncertainties are first combined with
the correlation matrix provided to form the covariance matrix. Since the correlation
matrix defaults to the identity matrix, the default is combination without
correlation.
The default method takes obj as a vector of uncertainty contributions unless u
is specified, in which case u is used. It is not necessary to specify both.
The expression method requires obj to be a differentiable R expression which can
be evaluated in the environment x to provide a numeric value.
For the function method, obj must be an R function which takes parameters from x and
returns a numeric value.
For the formula method, obj must be a formula with no left-hand side (e.g. ~a*x+b*x^2)
which can be evaluated in the environment x to provide a numeric value.
The formula and expression methods first calculate derivatives for the expression or formula,
evaluate them using the supplied values of x and then pass the resulting sensitivity
coefficients, with supplied u, cor or cov to uncert.default.
The derivatives for the “GUM” method (formula and expression methods only) are algorithmic
derivatives (that is, algebraic or analytical derivatives) obtained using deriv
applied to expr and formula.
Numerical derivatives are computed in different ways depending on the method specified:
- For method="NUM", the derivatives are calculated as
\((f(x+delta*u)-f(x-delta*u))/(2*delta*u)\).
- For method="kragten", derivatives are calculated as
\((f(x+u*sign(delta))-f(x))/u\).
- For method="k2", derivatives are calculated as
\((f(x+u)-f(x-u))/(2*u)\).
"NUM" is likely to give a close approximation to analytical differentiation provided that
delta is appreciably less than 1 but not so small as to give step sizes near machine
precision. "k2" is equivalent to "NUM" with delta=1.0. Both will give zero coefficients
at stationary points (e.g minima), leading to under-estimation of uncertainty if
the curvature is large. "kragten" uses a deliberately one-sided (and large) step to
avoid this problem; as a result, "kragten" is a poorer (sometimes much poorer) estimate of
the analytical differential but likely a better approximation to the truth.
Since these methods rely on u, if u is unspecified and cov is
provided, u is extracted from cov (using sqrt(diag(cov))). It is
assumed that the row and column order in cov is identical to the order of named
parameters in x.
Derivatives (and uncertainty contributions) are computed for all parameters in
x. Additional parameters used in FUN, expr or formula
may be included in …; these will be treated as constants in the
uncertainty calculation.
If distrib is missing, or if it is a list with some members missing, the distribution
is assumed Normal and distrib$name is set to "norm". Similarly, if distrib.pars
or a member of it is missing, the default parameters for x$name are
list(mean=x$name, sd=u$name). If the list is not named, names(x) are used
(so the list must be in order of names(x)).
If method="MC", uncert calls uncertMC. Distributions and
distribution parameters are required and B must be present and >1. See uncertMC
for details of distribution specification.
For other evaluation methods, the distributions are silently ignored.
JCGM 100 (2008) Evaluation of measurement data - Guide to the expression of uncertainty in measurement. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf. (JCGM 100:2008 is a public domain copy of ISO/IEC Guide to the expression of uncertainty in measurement (1995) ).
Kragten, J. (1994) Calculating standard deviations and confidence intervals with a universally applicable spreadsheet technique, Analyst, 119, 2161-2166.
Ellison, S. L. R. (2005) Including correlation effects in an improved spreadsheet calculation of combined standard uncertainties, Accred. Qual. Assur. 10, 338-343.
For method="MC" see uncertMC and uncertMC-class.
# NOT RUN {
expr <- expression(a+b*2+c*3+d/2)
x <- list(a=1, b=3, c=2, d=11)
u <- lapply(x, function(x) x/10)
u.expr<-uncert(expr, x, u, method="NUM")
u.expr
#Compare with default:
uncert(u=c(0.1, 0.3, 0.2, 1.1), c=c(1.0, 2.0, 3.0, 0.5))
#... or with function method
f <- function(a,b,c,d) a+b*2+c*3+d/2
u.fun<-uncert(f, x, u, method="NUM")
u.fun
#.. or with the formula method
u.form<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM")
u.form
#An example with correlation
u.cor<-diag(1,4)
u.cor[3,4]<-u.cor[4,3]<-0.5
u.formc<-uncert(~a+b*2+c*3+d/2, x, u, method="NUM", cor=u.cor)
u.formc
#A Monte Carlo example
#See uncertMC for a less linear example
u.formc.MC<-uncert(~a+b*2+c*3+d/2, x, u, method="MC", cor=u.cor, B=200)
u.formc.MC
# }
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