Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

View source: R/family.normal.R

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

1 2 |

`qpower` |
A single numeric, must be greater than one, called |

`link` |
Link function applied to the ‘mean’ |

`imethod` |
Must be 1, 2 or 3.
See |

`imu` |
Numeric, optional initial values used for the fitted values.
The default is to use |

`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 |

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
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 = paste("LS = red, lqnorm = blue (qpower = ",
fit@misc$qpower, "), truth = black", sep = ""), col = "blue"))
lmfit <- lm(y ~ x, data = ldata)
with(ldata, lines(x, fitted(fit), col = "blue"))
with(ldata, lines(x, lmfit$fitted, col = "red"))
with(ldata, lines(x, realfun(x), col = "black"))
## End(Not run)
``` |

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