lqnorm: Minimizing the L-q norm Family Function

In VGAM: Vector Generalized Linear and Additive Models

Minimizing the L-q norm Family Function

Description

Minimizes the L-q norm of residuals in a linear model.

Usage

lqnorm(qpower = 2, link = "identitylink",
       imethod = 1, imu = NULL, ishrinkage = 0.95)

Arguments

qpower	A single numeric, must be greater than one, called q below. The absolute value of residuals are raised to the power of this argument, and then summed. This quantity is minimized with respect to the regression coefficients.
link	Link function applied to the ‘mean’ \mu. See Links for more details.
imethod	Must be 1, 2 or 3. See CommonVGAMffArguments for more information. Ignored if imu is specified.
imu	Numeric, optional initial values used for the fitted values. The default is to use imethod = 1.
ishrinkage	How much shrinkage is used when initializing the fitted values. The value must be between 0 and 1 inclusive, and a value of 0 means the individual response values are used, and a value of 1 means the median or mean is used. This argument is used in conjunction with imethod = 3.

Details

This function minimizes the objective function

\sum_{i=1}^n \; w_i (|y_i - \mu_i|)^q

where q is the argument qpower, \eta_i = g(\mu_i) where g is the link function, and \eta_i is the vector of linear/additive predictors. The prior weights w_i can be inputted using the weights argument of vlm/vglm/vgam etc.; it should be just a vector here since this function handles only a single vector or one-column response.

Numerical problem will occur when q is too close to one. Probably reasonable values range from 1.5 and up, say. The value q=2 corresponds to ordinary least squares while q=1 corresponds to the MLE of a double exponential (Laplace) distibution. The procedure becomes more sensitive to outliers the larger the value of q.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Convergence failure is common, therefore the user is advised to be cautious and monitor convergence!

Note

This VGAM family function is an initial attempt to provide a more robust alternative for regression and/or offer a little more flexibility than least squares. The @misc slot of the fitted object contains a list component called objectiveFunction which is the value of the objective function at the final iteration.

Author(s)

Thomas W. Yee

References

Yee, T. W. and Wild, C. J. (1996). Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

See Also

uninormal.

Examples

set.seed(123)
ldata <- data.frame(x = sort(runif(nn <- 10 )))
realfun <- function(x) 4 + 5*x
ldata <- transform(ldata, y = realfun(x) + rnorm(nn, sd = exp(-1)))
# Make the first observation an outlier
ldata <- transform(ldata, y = c(4*y[1], y[-1]), x = c(-1, x[-1]))
fit <- vglm(y ~ x, lqnorm(qpower = 1.2), data = ldata)
coef(fit, matrix = TRUE)
head(fitted(fit))
fit@misc$qpower
fit@misc$objectiveFunction

## Not run: 
# Graphical check
with(ldata, plot(x, y,
     main = paste0("LS = red, lqnorm = blue (qpower = ",
     fit@misc$qpower, "), truth = black"), col = "blue"))
lmfit <- lm(y ~ x, data = ldata)
with(ldata, lines(x,  fitted(fit), col = "blue"))
with(ldata, lines(x, lmfit$fitted, col = "red"))
with(ldata, lines(x,  realfun(x),  col = "black")) 
## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.