The nlj package—named in honor of British statistician Norman Lloyd Johnson—provides specialized tools for working with the unbounded Johnson SU distribution. In addition to functions for density evaluation, distribution and quantile computation, and random variate generation, the package offers automatic normalization routines and a generalized transformation-based regression model that leverages the inverse hyperbolic sine transformation.
Install the development version of nlj from GitHub using devtools:
# install.packages("devtools")
devtools::install_github("shanedrabing/nlj")nlj provides a suite of tools for the Johnson SU distribution, automatic data normalization, and advanced regression modeling. The following examples demonstrate the package’s core functionality.
The Johnson-SU distribution functions—djohnson, pjohnson, qjohnson, and rjohnson—provide density evaluation, cumulative distribution computation, quantile inversion, and random sample generation. They support the full Johnson-SU parameterization for flexible control of skewness and kurtosis.
djohnson computes the Johnson-SU density for a specified set of input values.
x <- seq(-pi, pi, length.out = 300)
d <- nlj::djohnson(x, gamma = -1.7, delta = 2.1, xi = -1.45, lambda = 1.45)
plot(x, d, type = "l",
main = "Johnson-SU Density",
xlab = "X", ylab = "Density")
pjohnson returns the cumulative distribution function (CDF) values for specified quantiles under the Johnson-SU model.
q <- seq(-pi, pi, length.out = 300)
p <- nlj::pjohnson(q, gamma = -1.7, delta = 2.1, xi = -1.45, lambda = 1.45)
plot(q, p, type = "l",
main = "Johnson-SU Cumulative Distribution",
xlab = "Quantile", ylab = "Probability")
qjohnson obtains the quantile (inverse CDF) corresponding to specified probability levels, facilitating probability-based analyses within the Johnson-SU framework.
p <- seq(0, 1, length.out = 300)
q <- nlj::qjohnson(p, gamma = -1.7, delta = 2.1, xi = -1.45, lambda = 1.45)
plot(p, q, type = "l",
main = "Johnson-SU Quantile",
xlab = "Probability", ylab = "Quantile")
rjohnson generates random samples from the Johnson-SU distribution, useful for simulation, bootstrapping, or Monte Carlo studies.
nlj::rjohnson(5, gamma = -1.7, delta = 2.1, xi = -1.45, lambda = 1.45)
#> [1] 0.1163819 2.4153221 -0.9112831 -0.1996155 0.4708389The package provides znorm and zjohnson, automated normalization functions that compute and apply data transformations, track transformation parameters, and support denormalization. These utilities streamline the process of preparing data in standardized or distribution-adapted form.
zjohnson performs Johnson-SU normalization by estimating distribution parameters that align the input data to a standard normal reference. It adapts to skewness and heavy tails, producing a transformation that accommodates non-Gaussian data characteristics.
# Example data: Wind speed
x <- sort(airquality$Wind)
d <- density(x)
# Apply Johnson-SU normalization
zj <- nlj::zjohnson(x)
# Extract the density estimates
xd <- d$x
yd <- d$y
yj <- zj$fd(xd)
# Plot the density comparison
plot(range(xd), range(c(yd, yj)), type = "n",
main = "Wind Speed Density Estimation",
xlab = "Wind Speed", ylab = "Density")
lines(xd, yd, col = "black") # Original density
lines(xd, yj, col = "red") # Johnson-SU normalized density
legend(min(xd), max(c(yd, yj)),
c("Kernel", "Johnson-SU"),
col = c("black", "red"),
lwd = 1, bg = "white")
lm.gat implements the Generalized Asinh Transformation Model, fitting regression models with a parameterized inverse hyperbolic sine transform on predictors. This approach can enhance model fit in the presence of nonlinearity, particularly when linear models are insufficient.
The following example illustrates lm.gat in action. We predict miles per gallon (mpg) from engine horsepower (hp) using both a standard linear model and a GATM model. The comparison highlights how the GATM transformation captures non-linear patterns that a simple linear fit does not.
# Example data: Horsepower and miles per gallon (MPG)
i <- order(mtcars$hp)
x <- mtcars$hp[i]
y <- mtcars$mpg[i]
# Fit simple regression
m <- lm(y ~ x)
# Fit non-linear model
gat <- nlj::lm.gat(y ~ x, iterations = 3)
# Plot
plot(x, y, pch = 16,
main = "Horsepower (HP) vs Miles Per Gallon (MPG)",
xlab = "HP", ylab = "MPG")
lines(x, m$fitted.values, col = "red")
lines(x, gat$z, col = "blue")
legend(min(x), max(y),
c("Simple", "GATM"),
col = c("red", "blue"),
lwd = 1, bg = "white")
The next example extends lm.gat to a model with multiple interaction terms, demonstrating its ability to manage high-dimensional, non-linear relationships. We illustrate this by modeling quarter-mile time (qsec) as a function of hp, mpg, and wt, including all two-way and three-way interactions.
First, we fit a standard linear model with these terms and examine its summary. Then, we apply lm.gat to the same formula, showcasing how the parameterized asinh transformation can yield a superior fit by accommodating complex variable interactions.
# Fit simple regression
m <- lm(qsec ~ hp * mpg * wt, mtcars)
summary(m)
#>
#> Call:
#> lm(formula = qsec ~ hp * mpg * wt, data = mtcars)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.5219 -0.5882 -0.0978 0.4013 3.1959
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 14.0426771 10.9005197 1.288 0.210
#> hp 0.0017784 0.0590665 0.030 0.976
#> mpg 0.0249053 0.3921269 0.064 0.950
#> wt 0.2427188 3.4981471 0.069 0.945
#> hp:mpg -0.0005924 0.0026322 -0.225 0.824
#> hp:wt 0.0016601 0.0179826 0.092 0.927
#> mpg:wt 0.1050975 0.1451373 0.724 0.476
#> hp:mpg:wt -0.0003813 0.0008658 -0.440 0.664
#>
#> Residual standard error: 1.102 on 24 degrees of freedom
#> Multiple R-squared: 0.7055, Adjusted R-squared: 0.6196
#> F-statistic: 8.215 on 7 and 24 DF, p-value: 4.052e-05# Fit complex regression
gat <- nlj::lm.gat(qsec ~ hp * mpg * wt, mtcars,
iterations = 6, penalty = 1e-9)
summary(gat$fit)
#>
#> Call:
#> stats::lm(formula = formula, data = mut)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.506e-03 -7.267e-04 -2.282e-05 6.394e-04 2.601e-03
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.130e+00 1.880e-03 1133.138 < 2e-16 ***
#> hp -4.703e-04 2.501e-04 -1.881 0.07220 .
#> mpg -2.400e-03 2.681e-04 -8.952 4.08e-09 ***
#> wt -7.029e-03 6.350e-04 -11.069 6.52e-11 ***
#> hp:mpg 1.682e-06 4.890e-05 0.034 0.97284
#> hp:wt -3.599e-04 1.176e-04 -3.061 0.00537 **
#> mpg:wt -7.182e-04 9.360e-05 -7.673 6.55e-08 ***
#> hp:mpg:wt -8.171e-05 1.108e-05 -7.371 1.30e-07 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.001326 on 24 degrees of freedom
#> Multiple R-squared: 0.9224, Adjusted R-squared: 0.8997
#> F-statistic: 40.74 on 7 and 24 DF, p-value: 8.352e-12