VGAMextra (version 0.0-6)

uninormalQlink: Quantile regression: Link function for the quantiles of the normal distribution.

Description

Computes the uninormalQlink transformation, its inverse and the first two derivatives.

Usage

uninormalQlink(theta, percentile = stop("Enter percentiles."),
                    sd = NULL, wrt.param = NULL,
                    bvalue = NULL, inverse = FALSE,
                    deriv = 0, short = TRUE, tag = FALSE)

Value

For deriv = 0, the uninormalQlink transformation of

theta, i.e. \(\mu\), when inverse = FALSE. If inverse = TRUE, then \(\theta\) becomes \(\eta\), and the inverse, \(\eta - \sigma \Phi^{-1}(perc)\),

for given

\(\sigma\), is returned.

When deriv = 1

theta becomes

\(\theta = (\mu, \sigma)= (\theta_1, \theta_2)\), and

\(\eta = (\eta_1, \eta_2)\) with

\(\eta_2 = \log~\sigma\), and the argument wrt.param must be considered:

A) If inverse = FALSE, then

\(d\)

eta1 / \(d\)

\(\mu\) is returned when

wrt.param = 1, and

\(d\)

eta1 / \(d\)

\(\sigma\) if

wrt.param = 2.

B) For inverse = TRUE, this link returns

\(d\)

\(\mu\) / \(d\)

eta1 and

\(d\)

\(\sigma\) / \(d\)

eta1 conformably arranged in a matrix, if wrt.param = 1, as a function of \(\theta_i\), \(i = 1, 2\). When wrt.param = 2, then

\(d\mu\) / \(d\)

eta2 and

\(d\sigma\) / \(d\)

eta2

is returned.

For deriv = 2, the second derivatives in terms of theta are similarly returned.

Arguments

theta

Numeric or character. This is \(\theta\) ('mean' parameter) but iy may be \(\eta\) depending on the other parameters. See below for further details.

percentile

Numeric. A vector of percentiles of interest, denoted as perc.

sd

Numeric, positive. The 'standard deviation' parameter (required), denoted as \(\sigma\).

wrt.param

Positive integer, either \(1\) or \(2\). The partial derivatives are computed with respect to one of the two linear predictors involved with this link. Further details listed below.

bvalue, inverse, deriv, short, tag

See Links.

Author

V. Miranda and Thomas W. Yee.

Details

A 2-parameter link for the quantiles of the normal distribution. It can only be used within uninormalff as the first linear predictor. It is defined as $$ \tt{uninormalQlink}(\mu; \sigma) = \eta_1(\mu; \sigma) = \mu + \sigma \cdot \Phi^{-1}(perc), $$ where \(\Phi\) is the error function (see, e.g., erf), \(\mu in (-\infty, \infty)\), and \(\sigma > 0\). This link is expressly a function of \(\theta = \mu\), therefore \(sigma\) must be entered at every call. Numerical values of \(\sigma\) out of range may result in Inf, -Inf, NA or NaN.

See Also

uninormalff, uninormal, Links.

Examples

    eta <- seq(-3, 3, by = 0.1) # this is eta = log(Normal - Quantiles).
    sigma  <- exp(1)    # 'sigma' argument.
    percentile <- c(25, 50, 75, 95)  # some percentiles of interest.
 
 ## E1. Get 'mean' values.
   theta <- uninormalQlink(theta = eta, percentile = percentile,
                          sd = sigma, inverse = TRUE)  # Mu
   
 if (FALSE) {
 ## E2. Plot theta vs. eta, 'shape' fixed, for different percentiles.
plot(theta[, 1], eta, type = "l", las = 1, lty = 2, lwd = 3,
     ylim = c(-10, 10), xlim = c(-10, 10),
     main = "uninormalQlink(theta; shape), fixed 'shape'.",
     xlab = "Theta (scale)", ylab = "uninormalQlink")
abline(v = 0, h = 0, col = "red")
lines(theta[, 2], eta, lty = 2, lwd = 3, col = "blue")
lines(theta[, 3], eta, lty = 2, lwd = 3, col = "orange")
lines(theta[, 4], eta, lty = 2, lwd = 3, col = "red")
legend("bottomright", c("25th Perc", "50th Perc", "75th Perc", "95th Perc"),
      col = c("black", "blue", "orange", "red"), lty = c(2, 2, 2, 2),
      lwd = rep(3, 4))

 }
 
 ## E3. uninormalQlink() and its inverse ##
    etabis  <- uninormalQlink(theta = theta, percentile = percentile,
                              sd = sigma, inverse = FALSE)
    my.diff <- eta - etabis
    summary(my.diff)     # Zero