IsoplotR (version 6.5)

titterington: Linear regression of X,Y,Z-variables with correlated errors

Description

Implements the maximum likelihood algorithm of Ludwig and Titterington (1994) for linear regression of three dimensional data with correlated uncertainties.

Usage

titterington(x)

Value

A four-element list of vectors containing:

par

4-element vector c(a,b,A,B) where a is the intercept of the X-Y regression, b is the slope of the X-Y regression, A is the intercept of the X-Z regression, and B is the slope of the X-Z regression.

cov

[4x4]-element covariance matrix of par

mswd

the mean square of the residuals (a.k.a `reduced Chi-square') statistic

p.value

p-value of a Chi-square test for linearity

df

the number of degrees of freedom for the Chi-square test (2\(n\)-4)

tfact

the \(100(1-\alpha/2)\%\) percentile of the t-distribution with \((n-2k+1)\) degrees of freedom

Arguments

x

an [nx9] matrix with the following columns: X, sX, Y, sY, Z, sZ, rXY, rXZ, rYZ.

Details

Ludwig and Titterington (1994)'s 3-dimensional linear regression algorithm for data with correlated uncertainties is an extension of the 2-dimensional algorithm by Titterington and Halliday (1979), which itself is equivalent to the algorithm of York et al. (2004). Given \(n\) triplets of (approximately) collinear measurements \(X_i\), \(Y_i\) and \(Z_i\) (for \(1 \leq i \leq n\)), their uncertainties \(s[X_i]\), \(s[Y_i]\) and \(s[Z_i]\), and their covariances cov[\(X_i,Y_i\)], cov[\(X_i,Z_i\)] and cov[\(Y_i,Z_i\)], the titterington function fits two slopes and intercepts with their uncertainties. It computes the MSWD as a measure of under/overdispersion. Overdispersed datasets (MSWD>1) can be dealt with in the same three ways that are described in the documentation of the isochron function.

References

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.

Titterington, D.M. and Halliday, A.N., 1979. On the fitting of parallel isochrons and the method of maximum likelihood. Chemical Geology, 26(3), pp.183-195.

York, D., Evensen, N.M., Martinez, M.L. and De Basebe Delgado, J., 2004. Unified equations for the slope, intercept, and standard errors of the best straight line. American Journal of Physics, 72(3), pp.367-375.

See Also

york, isochron, ludwig

Examples

d <- matrix(c(0.1677,0.0047,1.105,0.014,0.782,0.015,0.24,0.51,0.33,
              0.2820,0.0064,1.081,0.013,0.798,0.015,0.26,0.63,0.32,
              0.3699,0.0076,1.038,0.011,0.819,0.015,0.27,0.69,0.30,
              0.4473,0.0087,1.051,0.011,0.812,0.015,0.27,0.73,0.30,
              0.5065,0.0095,1.049,0.010,0.842,0.015,0.27,0.76,0.29,
              0.5520,0.0100,1.039,0.010,0.862,0.015,0.27,0.78,0.28),
            nrow=6,ncol=9)
colnames(d) <- c('X','sX','Y','sY','Z','sZ','rXY','rXZ','rYZ')
titterington(d)