Type:	Package
Title:	Routines for L1 Estimation
Version:	0.60
Date:	2025-07-08
Maintainer:	Felipe Osorio <faosorios.stat@gmail.com>
Description:	L1 estimation for linear regression using Barrodale and Roberts' method <doi:10.1145/355616.361024> and the EM algorithm <doi:10.1023/A:1020759012226>. Estimation of mean and covariance matrix using the multivariate Laplace distribution, density, distribution function, quantile function and random number generation for univariate and multivariate Laplace distribution <doi:10.1080/03610929808832115>. Implementation of Naik and Plungpongpun <doi:10.1007/0-8176-4487-3_7> for the Generalized spatial median estimator is included.
Depends:	R(≥ 3.5.0), fastmatrix
LinkingTo:	fastmatrix
Imports:	stats, grDevices, graphics
License:	GPL-3
URL:	https://github.com/faosorios/L1pack
NeedsCompilation:	yes
LazyLoad:	yes
Packaged:	2025-07-08 19:41:28 UTC; root
Author:	Felipe Osorio [aut, cre], Tymoteusz Wolodzko [aut]
Repository:	CRAN
Date/Publication:	2025-07-08 20:10:02 UTC

The symmetric Laplace distribution

Description

Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter location and scale parameter scale.

Usage

dlaplace(x, location = 0, scale = 1, log = FALSE)
plaplace(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qlaplace(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rlaplace(n, location = 0, scale = 1)

Arguments

Details

If location or scale are not specified, they assume the default values of 0 and 1 respectively.

The Laplace distribution with location \mu and scale \phi has density

f(x) = \frac{1}{\sqrt{2}\phi} \exp(-\sqrt{2}|x-\mu|/\phi),

where -\infty < y < \infty, -\infty < \mu < \infty and \phi > 0. The mean is \mu and the variance is \phi^2.

The cumulative distribution function, assumes the form

F(x) = \left\{\begin{array}{ll} \frac{1}{2} \exp(\sqrt{2}(x - \mu)/\phi) & x < \mu, \\ 1 - \frac{1}{2} \exp(-\sqrt{2}(x - \mu)/\phi) & x \geq \mu. \end{array}\right.

The quantile function, is given by

F^{-1}(p) = \left\{\begin{array}{ll} \mu + \frac{\phi}{\sqrt{2}} \log(2p) & p < 0.5, \\ \mu - \frac{\phi}{\sqrt{2}} \log(2(1-p)) & p \geq 0.5. \end{array}\right.

Value

dlaplace, plaplace, and qlaplace are respectively the density, distribution function and quantile function of the Laplace distribution. rlaplace generates random deviates drawn from the Laplace distribution, the length of the result is determined by n.

Author(s)

Felipe Osorio and Tymoteusz Wolodzko

References

Kotz, S., Kozubowski, T.J., Podgorski, K. (2001). The Laplace Distributions and Generalizations. Birkhauser, Boston.

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications, 2nd Ed. Chapman & Hall, Boca Raton.

See Also

Distributions for other standard distributions and rmLaplace for the random generation from the multivariate Laplace distribution.

Examples

x <- rlaplace(1000)
## QQ-plot for Laplace data against true theoretical distribution:
qqplot(qlaplace(ppoints(1000)), x, main = "Laplace QQ-plot",
  xlab = "Theoretical quantiles", ylab = "Sample quantiles")
abline(c(0,1), col = "red", lwd = 2)

Estimation of location and scatter using the multivariate Laplace distribution

Description

Estimates the location vector and scatter matrix assuming the data came from a multivariate Laplace distribution.

Usage

LaplaceFit(x, data, subset, na.action, tol = 1e-6, maxiter = 200)

Arguments

Value

A list with class 'LaplaceFit' containing the following components:

Generic function print show the results of the fit.

References

Yavuz, F.G., Arslan, O. (2018). Linear mixed model with Laplace distribution (LLMM). Statistical Papers 59, 271-289.

See Also

cov

Examples

fit <- LaplaceFit(stack.x)
fit

# covariance matrix
p <- fit$dims[2]
Sigma <- (4 * (p + 1)) * fit$Scatter
Sigma

Wilson-Hilferty transformation

Description

Returns the Wilson-Hilferty transformation for multivariate Laplace deviates.

Usage

WH.Laplace(x, center, Scatter)

Arguments

Details

Let T = D/(2p) be a Gamma distributed random variable, where D^2 denotes the squared Mahalanobis distance defined as

D^2 = (\bold{x} - \bold{\mu})^T \bold{\Sigma}^{-1} (\bold{x} - \bold{\mu}).

Thus, the Wilson-Hilferty transformation is given by

z = \frac{T^{1/3} - (1 - \frac{1}{9p})}{(\frac{1}{9p})^{1/2}}%

and z is approximately distributed as a standard normal distribution. This is useful, for instance, in the construction of QQ-plots.

References

Osorio, F., Galea, M., Henriquez, C., Arellano-Valle, R. (2023). Addressing non-normality in multivariate analysis using the t-distribution. AStA Advances in Statistical Analysis 107, 785-813.

Terrell, G.R. (2003). The Wilson-Hilferty transformation is locally saddlepoint. Biometrika 90, 445-453.

Wilson, E.B., Hilferty, M.M. (1931). The distribution of chi-square. Proceedings of the National Academy of Sciences of the United States of America 17, 684-688.

Examples

Scatter <- matrix(c(1,.5,.5,1), ncol = 2)
Scatter

# generate the sample
y <- rmLaplace(n = 500, Scatter = Scatter)
fit <- LaplaceFit(y)
z <- WH.Laplace(fit)
par(pty = "s")
qqnorm(z, main = "Transformed distances QQ-plot")
abline(c(0,1), col = "red", lwd = 2)

Confidence intervals from lad models

Description

Computes confidence intervals for one or more parameters in a fitted model associated to a lad object.

Usage

## S3 method for class 'lad'
confint(object, parm, level = 0.95, ...)

Arguments

Details

confint is a generic function. Confidence intervals associated to lad objects are asymptotic, and needs suitable coef and vcov methods to be available.

Value

A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labelled as (1-level)/2 and 1 - (1-level)/2 in % (by default 2.5% and 97.5%).

See Also

confint.glm and confint.nls in package MASS.

Examples

fm <- lad(stack.loss ~ ., data = stackloss, method = "BR")
confint(fm) # based on asymptotic normality

QQ-plot with simulated envelopes

Description

Constructs a normal QQ-plot using a Wilson-Hilferty transformation for the estimated Mahalanobis distances obtained from the LaplaceFit procedure.

Usage

envelope.Laplace(object, reps = 50, conf = 0.95, plot.it = TRUE)

Arguments

Value

A list with the following components :

References

Atkinson, A.C. (1985). Plots, Transformations and Regression. Oxford University Press, Oxford.

Osorio, F., Galea, M., Henriquez, C., Arellano-Valle, R. (2023). Addressing non-normality in multivariate analysis using the t-distribution. AStA Advances in Statistical Analysis 107 785-813.

Vallejos, R., Osorio, F., Ferrer, C. (2025+). A new coefficient to measure agreement between continuous variables. Working paper.

Excess returns for Martin Marietta and American Can companies

Description

Data from the Martin Marietta and American Can companies collected over a period of 5 years on a monthly basis.

Usage

data(ereturns)

Format

A data frame with 60 observations on the following 4 variables.

Date

the month in which the observations were collected.

am.can

excess returns from the American Can company.

m.marietta

excess returns from the Martin Marietta company.

CRSP

an index for the excess rate returns for the New York stock exchange.

Source

Butler, R.J., McDonald, J.B., Nelson, R.D., and White, S.B. (1990). Robust and partially adaptive estimation of regression models. The Review of Economics and Statistics 72, 321-327.

L1 concordance correlation coefficient

Description

Calculates L1 concordance correlation coefficient for evaluating the degree of agreement between measurements generated by two different methods.

Usage

l1ccc(x, data, equal.means = FALSE, boots = TRUE, nsamples = 1000, subset, na.action)

Arguments

Value

A list with class 'l1ccc' containing the following named components:

References

King, T.S., Chinchilli, V.M. (2001). A generalized concordance correlation coefficient for continuous and categorical data. Statistics in Medicine 20, 2131-2147.

King, T.S., Chinchilli, V.M. (2001). Robust estimators of the concordance correlation coefficient. Journal of Biopharmaceutical Statistics 11, 83-105.

Lin, L. (1989). A concordance correlation coefficient to evaluate reproducibility. Biometrics 45, 255-268.

Vallejos, R., Osorio, F., Ferrer, C. (2025+). A new coefficient to measure agreement between two continuous variables. Working paper.

Examples

## data from Bland and Altman (1986). The Lancet 327, 307-310.
x <- list(Large = c(494,395,516,434,476,557,413,442,650,433,
          417,656,267,478,178,423,427),
          Mini  = c(512,430,520,428,500,600,364,380,658,445,
          432,626,260,477,259,350,451))
x <- as.data.frame(x)

plot(Mini ~ Large, data = x, xlim = c(100,800), ylim = c(100,800),
     xlab = "PERF by Large meter", ylab = "PERF by Mini meter")
abline(c(0,1), col = "gray", lwd = 2)

## estimating L1 concordance coefficient
z <- l1ccc(~ Mini + Large, data = x, boots = FALSE)
z
## output:
# Call:
# l1ccc(x = ~ Mini + Large, data = x, boots = FALSE)
#
# L1 coefficients using:
# Laplace  Gaussian  U-statistic 
#  0.7456    0.7607       0.7642 
#
# Lin's coefficients:
# estimate  accuracy precision 
#   0.9395    0.9974    0.9419 

Minimum absolute residual (L1) regression

Description

Performs an L1 regression on a matrix of explanatory variables and a vector of responses.

Usage

l1fit(x, y, intercept = TRUE, tolerance = 1e-07, print.it = TRUE)

Arguments

Details

The Barrodale-Roberts algorithm, which is a specialized linear programming algorithm, is used.

Value

list defining the regression (compare with function lsfit).

References

Barrodale, I., Roberts, F.D.K. (1973). An improved algorithm for discrete L1 linear approximations. SIAM Journal of Numerical Analysis 10, 839-848.

Barrodale, I., Roberts, F.D.K. (1974). Solution of an overdetermined system of equations in the L1 norm. Communications of the ACM 17, 319-320.

Bloomfield, P., Steiger, W.L. (1983). Least Absolute Deviations: Theory, Applications, and Algorithms. Birkhauser, Boston, Mass.

Examples

l1fit(stack.x, stack.loss)

Least absolute deviations regression

Description

This function is used to fit linear models considering Laplace errors.

Usage

lad(formula, data, subset, na.action, method = "BR", tol = 1e-7, maxiter = 200,
  x = FALSE, y = FALSE, contrasts = NULL)

Arguments

Value

An object of class lad representing the linear model fit. Generic function print, show the results of the fit.

The functions print and summary are used to obtain and print a summary of the results. The generic accessor functions coefficients, fitted.values and residuals extract various useful features of the value returned by lad.

Author(s)

The design was inspired by the R function lm.

References

Barrodale, I., Roberts, F.D.K. (1974). Solution of an overdetermined system of equations in the L1 norm. Communications of the ACM 17, 319-320.

Phillips, R.F. (2002). Least absolute deviations estimation via the EM algorithm. Statistics and Computing 12, 281-285.

Examples

fm <- lad(stack.loss ~ ., data = stackloss, method = "BR")
summary(fm)

data(ereturns)
fm <- lad(m.marietta ~ CRSP, data = ereturns, method = "EM")
summary(fm)
# basic observations
fm$basic

Fitter functions for least absolute deviation (LAD) regression

Description

This function is a switcher among various numerical fitting functions (lad.fit.BR, and lad.fit.EM). The argument method does the switching: "BR" for lad.fit.BR, etc. This should usually not be used directly unless by experienced users.

Usage

lad.fit(x, y, method = "BR", tol = 1e-7, maxiter = 200)

Arguments

Value

a list with components:

See Also

lad.fit.BR, lad.fit.EM.

Examples

x <- cbind(1, stack.x)
fm <- lad.fit(x, stack.loss, method = "BR")
fm

Fit a least absolute deviation (LAD) regression model

Description

Fits a linear model using LAD methods, returning the bare minimum computations.

Usage

lad.fit.BR(x, y, tol = 1e-7)
lad.fit.EM(x, y, tol = 1e-7, maxiter = 200)

Arguments

Value

The bare bones of a lad object: the coefficients, residuals, fitted values, and some information used by summary.lad.

See Also

lad, lad.fit, lm

Examples

x <- cbind(1, stack.x)
z <- lad.fit.BR(x, stack.loss)
z

Multivariate Laplace distribution

Description

These functions provide the density and random number generation from the multivariate Laplace distribution.

Usage

dmLaplace(x, center = rep(0, nrow(Scatter)), Scatter = diag(length(center)), log = FALSE)
rmLaplace(n = 1, center = rep(0, nrow(Scatter)), Scatter = diag(length(center)))

Arguments

Details

The multivariate Laplace distribution is a multidimensional extension of the univariate symmetric Laplace distribution. There are multiple forms of the multivariate Laplace distribution. Here, a particular case of the multivarite power exponential distribution introduced by Gomez et al. (1998) is considered.

The multivariate Laplace distribution with location \bold{\mu} = center and \bold{\Sigma} = Scatter has density

f(\bold{x}) = \frac{\Gamma(k/2)}{\pi^{k/2}\Gamma(k)2^{k+1}}|\bold{\Sigma}|^{-1/2} \exp\left\{-\frac{1}{2}[(\bold{x} - \bold{\mu})^T \bold{\Sigma}^{-1} (\bold{x} - \bold{\mu})]^{1/2}\right\}.

The function rmLaplace is an interface to C routines, which make calls to subroutines from LAPACK. The matrix decomposition is internally done using the Cholesky decomposition. If Scatter is not non-negative definite then there will be a warning message.

Value

If x is a matrix with n rows, then dmLaplace returns a n\times 1 vector considering each row of x as a copy from the multivariate Laplace.

If n = 1, then rmLaplace returns a vector of the same length as center, otherwise a matrix of n rows of random vectors.

References

Fang, K.T., Kotz, S., Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Chapman & Hall, London.

Gomez, E., Gomez-Villegas, M.A., Marin, J.M. (1998). A multivariate generalization of the power exponential family of distributions. Communications in Statistics - Theory and Methods 27, 589-600.

Examples

# dispersion parameters
Scatter <- matrix(c(1,.5,.5,1), ncol = 2)
Scatter

# generate the sample
y <- rmLaplace(n = 2000, Scatter = Scatter)

# scatterplot of a random bivariate Laplace sample with center
# vector zero and scale matrix 'Scatter'
par(pty = "s")
plot(y, xlab = "", ylab = "")
title("bivariate Laplace sample", font.main = 1)

Simulate responses from lad models

Description

Simulate one or more responses from the distribution corresponding to a fitted lad object.

Usage

## S3 method for class 'lad'
simulate(object, nsim = 1, seed = NULL, ...)

Arguments

Value

For the "lad" method, the result is a data frame with an attribute "seed". If argument seed is NULL, the attribute is the value of .Random.seed before the simulation was started.

Author(s)

Tymoteusz Wolodzko and Felipe Osorio

Examples

fm <- lad(stack.loss ~ ., data = stackloss)
sm <- simulate(fm, nsim = 4)

Computation of the generalized spatial median

Description

Computation of the generalized spatial median estimator as defined by Rao (1988).

Usage

spatial.median(x, data, subset, na.action, tol = 1e-6, maxiter = 200)

Arguments

Details

An interesting fact is that the generalized spatial median estimator proposed by Rao (1988) is the maximum likelihood estimator under the Kotz-type distribution discussed by Naik and Plungpongpun (2006). The generalized spatial median estimators are defined as \hat{\bold{\mu}} and \hat{\bold{\Sigma}} which minimize

\frac{n}{2}\log|\bold{\Sigma}| + \sum\limits_{i=1}^n \sqrt{(\bold{x} - \bold{\mu})^T \bold{\Sigma}^{-1} (\bold{x} - \bold{\mu})},

simultaneously with respect to \bold{\mu} and \bold{\Sigma}.

The function spatial.median follows the iterative reweighting algorithm of Naik and Plungpongpun (2006).

Value

A list with class 'spatial.median' containing the following components:

Generic function print show the results of the fit.

References

Naik, D.N., Plungpongpun, K. (2006). A Kotz-type distribution for multivariate statistical inference. In: Balakrishnan, N., Sarabia, J.M., Castillo, E. (Eds) Advances in Distribution Theory, Order Statistics, and Inference. Birkhauser Boston, pp. 111-124.

Rao, C.R. (1988). Methodology based on the L1-norm in statistical inference. Sankhya, Series A 50, 289-313.

See Also

cov, LaplaceFit

Examples

z <- spatial.median(stack.x)
z

Calculate variance-covariance matrix from lad models

Description

Returns the variance-covariance matrix of the main parameters of a fitted model for lad objects. The “main” parameters of model correspond to those returned by coef, and typically do not contain the nuisance scale parameter.

Usage

## S3 method for class 'lad'
vcov(object, ...)

Arguments

Value

A matrix of the estimated covariances between the parameter estimates in the linear regression model. This should have row and column names corresponding to the parameter names given by the coef method.