# Asymmetric Least Squares Quantile Regression Asymmetric least squares, a special case of maximizing an asymmetric likelihood function of a normal distribution. This allows for expectile/quantile regression using asymmetric least squares error loss. ```r amlnormal(w.aml = 1, parallel = FALSE, lexpectile = "identitylink", iexpectile = NULL, imethod = 1, digw = 4) ``` ## Arguments - `w.aml`: Numeric, a vector of positive constants controlling the percentiles. The larger the value the larger the fitted percentile value (the proportion of points below the ``w-regression plane''). The default value of unity results in the ordinary least squares (OLS) solution. - `parallel`: If `w.aml` has more than one value then this argument allows the quantile curves to differ by the same amount as a function of the covariates. Setting this to be `TRUE` should force the quantile curves to not cross (although they may not cross anyway). See `CommonVGAMffArguments` for more information. - `lexpectile, iexpectile`: See `CommonVGAMffArguments` for more information. - `imethod`: Integer, either 1 or 2 or 3. Initialization method. Choose another value if convergence fails. - `digw`: Passed into `Round` as the `digits` argument for the `w.aml` values; used cosmetically for labelling. ## Details This is an implementation of Efron (1991) and full details can be obtained there. Equation numbers below refer to that article. The model is essentially a linear model (see `lm`), however, the asymmetric squared error loss function for a residual $r$ is $r^2$ if $r <= 0$ and $w*r^2$ if $r > 0$. The solution is the set of regression coefficients that minimize the sum of these over the data set, weighted by the `weights` argument (so that it can contain frequencies). Newton-Raphson estimation is used here. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## References Efron, B. (1991). Regression percentiles using asymmetric squared error loss. **Statistica Sinica**, 1 , 93--125. ## Author(s) Thomas W. Yee ## Note On fitting, the `extra` slot has list components `"w.aml"` and `"percentile"`. The latter is the percent of observations below the ``w-regression plane'', which is the fitted values. One difficulty is finding the `w.aml` value giving a specified percentile. One solution is to fit the model within a root finding function such as `uniroot`; see the example below. For `amlnormal` objects, methods functions for the generic functions `qtplot` and `cdf` have not been written yet. See the note in `amlpoisson` on the jargon, including **expectiles** and **regression quantiles**. The `deviance` slot computes the total asymmetric squared error loss (2.5). If `w.aml` has more than one value then the value returned by the slot is the sum taken over all the `w.aml` values. This `VGAM` family function could well be renamed `amlnormal()` instead, given the other function names `amlpoisson`, `amlbinomial`, etc. In this documentation the word **quantile** can often be interchangeably replaced by **expectile** (things are informal here). ## See Also `amlpoisson`, `amlbinomial`, `amlexponential`, `bmi.nz`, `extlogF1`, `alaplace1`, `denorm`, `lms.bcn` and similar variants are alternative methods for quantile regression. ## Examples ```r ## Not run: # Example 1 ooo <- with(bmi.nz, order(age)) bmi.nz <- bmi.nz[ooo, ] # Sort by age (fit <- vglm(BMI ~ sm.bs(age), amlnormal(w.aml = 0.1), bmi.nz)) fit@extra # Gives the w value and the percentile coef(fit, matrix = TRUE) # Quantile plot with(bmi.nz, plot(age, BMI, col = "blue", main = paste(round(fit@extra$percentile, digits = 1), "expectile-percentile curve"))) with(bmi.nz, lines(age, c(fitted(fit)), col = "black")) # Example 2 # Find the w values that give the 25, 50 and 75 percentiles find.w <- function(w, percentile = 50) { fit2 <- vglm(BMI ~ sm.bs(age), amlnormal(w = w), data = bmi.nz) fit2@extra$percentile - percentile } # Quantile plot with(bmi.nz, plot(age, BMI, col = "blue", las = 1, main = "25, 50 and 75 expectile-percentile curves")) for (myp in c(25, 50, 75)) { # Note: uniroot() can only find one root at a time bestw <- uniroot(f = find.w, interval = c(1/10^4, 10^4), percentile = myp) fit2 <- vglm(BMI ~ sm.bs(age), amlnormal(w = bestw$root), bmi.nz) with(bmi.nz, lines(age, c(fitted(fit2)), col = "orange")) } # Example 3; this is Example 1 but with smoothing splines and # a vector w and a parallelism assumption. ooo <- with(bmi.nz, order(age)) bmi.nz <- bmi.nz[ooo, ] # Sort by age fit3 <- vgam(BMI ~ s(age, df = 4), data = bmi.nz, trace = TRUE, amlnormal(w = c(0.1, 1, 10), parallel = TRUE)) fit3@extra # The w values, percentiles and weighted deviances # The linear components of the fit; not for human consumption: coef(fit3, matrix = TRUE) # Quantile plot with(bmi.nz, plot(age, BMI, col="blue", main = paste(paste(round(fit3@extra$percentile, digits = 1), collapse = ", "), "expectile-percentile curves"))) with(bmi.nz, matlines(age, fitted(fit3), col = 1:fit3@extra$M, lwd = 2)) with(bmi.nz, lines(age, c(fitted(fit )), col = "black")) # For comparison ## End(Not run) ```

amlnormal function