lqnorm {VGAM} | R Documentation |
Minimizes the L-q norm of residuals in a linear model.
lqnorm(qpower=2, link="identity", earg=list(), method.init=1, imu=NULL, shrinkage.init=0.95)
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, earg |
Link function applied to the `mean' mu,
and extra argument optionally used by the link function.
See Links for more details.
|
method.init |
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 method.init=1 .
|
shrinkage.init |
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 method.init=3 .
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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.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
Convergence failure is common, therefore the user is advised to be cautious and monitor convergence!
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.
Thomas W. Yee
Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.
set.seed(123) d = data.frame(x = sort(runif(n <- 100))) realfun = function(x) 4 + 5*x d = transform(d, y = realfun(x) + rnorm(n, sd=exp(1))) d$y[1] = 4 * d$y[1] # Outlier d$x[1] = -1 # Outlier fit = vglm(y ~ x, fam = lqnorm(qpower=1.2), data=d) coef(fit, matrix=TRUE) fitted(fit)[1:4,] fit@misc$qpower fit@misc$objectiveFunction ## Not run: # Graphical check with(d, plot(x, y, main=paste("LS=red, lqnorm=blue (qpower = ", fit@misc$qpower, "), truth=black", sep=""), col="blue")) it = lm(y ~ x, data=d) with(d, lines(x, fitted(fit), col="blue")) with(d, lines(x, it$fitted, col="red")) with(d, lines(x, realfun(x), col="black")) ## End(Not run)