IsoplotR (version 1.0)

ludwig: Linear regression of U-Pb data with correlated errors, taking into account decay constant uncertainties.

Description

Implements the maximum likelihood algorithm for Total-Pb/U isochron regression of Ludwig (1998)

Usage

ludwig(x, ...)
# S3 method for default
ludwig(x, ...)
# S3 method for UPb
ludwig(x, exterr = FALSE, alpha = 0.05, model = 1, ...)

Arguments

x

an object of class UPb

...

optional arguments

exterr

propagate external sources of uncertainty (e.g., decay constants)?

alpha

cutoff value for confidence intervals

model

one of three regression models:

1: fit a discordia line through the data using the maximum likelihood algorithm of Ludwig (1998), which assumes that the scatter of the data is solely due to the analytical uncertainties. In this case, IsoplotR will either calculate an upper and lower intercept age (for Wetherill concordia), or a lower intercept age and common (\(^{207}\)Pb/\(^{206}\)Pb)\(_\circ\)-ratio intercept (for Tera-Wasserburg). If \(MSWD\)>0, then the analytical uncertainties are augmented by a factor \(\sqrt{MSWD}\).

2: fit a discordia line ignoring the analytical uncertainties

3: fit a discordia line using a modified maximum likelihood algorithm that includes accounts for any overdispersion by adding a geological (co)variance term.

Value

par

a two-element vector with the lower concordia intercept and initial \(^{207}\)Pb/\(^{206}\)Pb-ratio.

cov

the covariance matrix of par

df

the degrees of freedom of the model fit (\(3n-3\), where \(n\) is the number of aliquots).

mswd

the mean square of weighted deviates (a.k.a. reduced Chi-square statistic) for the fit.

p.value

p-value of a Chi-square test for the linear fit

w

the overdispersion, i.e., a two-element vector with the estimated standard deviation of the (assumedly) Normal distribution that underlies the true isochron; and the \(100(1-\alpha)\%\) confidence interval (only relevant if model = 3).

Details

The 3-dimensional regression algorithm of Ludwig and Titterington (1994) was modified by Ludwig (1998) to fit so-called `Total Pb-U isochrons'. These are constrained to a radiogenic endmember composition that falls on the concordia line. In its most sophisticated form, this algorithm does not only allow for correlated errors between variables, but also between aliquots. IsoplotR currently uses this algorithm to propagate decay constant uncertainties in the total Pb-U isochron ages. Future versions of the program will generalise this approach to other chronometers as well.

References

Ludwig, K.R., 1998. On the treatment of concordant uranium-lead ages. Geochimica et Cosmochimica Acta, 62(4), pp.665-676.

Ludwig, K.R. and Titterington, D.M., 1994. Calculation of \(^{230}\)Th/U isochrons, ages, and errors. Geochimica et Cosmochimica Acta, 58(22), pp.5031-5042.

See Also

concordia, titterington, isochron

Examples

# NOT RUN {
f <- system.file("UPb4.csv",package="IsoplotR")
d <- read.data(f,method="U-Pb",format=4)
fit <- ludwig(d)
# }