dist_transformed: Modify a distribution with a transformation

In distributional: Vectorised Probability Distributions

Modify a distribution with a transformation

Description

A transformed distribution applies a monotonic transformation to an existing distribution. This is useful for creating derived distributions such as log-normal (exponential transformation of normal), or other custom transformations of base distributions.

The density(), mean(), and variance() methods are approximate as they are based on numerical derivatives.

Usage

dist_transformed(dist, transform, inverse)

Arguments

dist	A univariate distribution vector.
transform	A function used to transform the distribution. This transformation should be monotonic over appropriate domain.
inverse	The inverse of the transform function.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_transformed.html

Let Y = g(X) where X is the base distribution with transformation function transform = g and inverse = g^{-1}. The transformation g must be monotonic over the support of X.

Support: g(S_X) where S_X is the support of X

Mean: Approximated numerically using a second-order Taylor expansion:

E(Y) \approx g(\mu_X) + \frac{1}{2}g''(\mu_X)\sigma_X^2

where \mu_X and \sigma_X^2 are the mean and variance of the base distribution X, and g'' is the second derivative of the transformation. The derivative is computed numerically using numDeriv::hessian().

Variance: Approximated numerically using the delta method:

\mathrm{Var}(Y) \approx [g'(\mu_X)]^2\sigma_X^2 + \frac{1}{2}[g''(\mu_X)\sigma_X^2]^2

where g' is the first derivative (Jacobian) computed numerically using numDeriv::jacobian().

Probability density function (p.d.f): Using the change of variables formula:

f_Y(y) = f_X(g^{-1}(y)) \left|\frac{d}{dy}g^{-1}(y)\right|

where f_X is the p.d.f. of the base distribution and the Jacobian |d/dy \, g^{-1}(y)| is computed numerically using numDeriv::jacobian().

Cumulative distribution function (c.d.f):

For monotonically increasing g:

F_Y(y) = F_X(g^{-1}(y))

For monotonically decreasing g:

F_Y(y) = 1 - F_X(g^{-1}(y))

where F_X is the c.d.f. of the base distribution.

Quantile function: The inverse of the c.d.f.

For monotonically increasing g:

Q_Y(p) = g(Q_X(p))

For monotonically decreasing g:

Q_Y(p) = g(Q_X(1-p))

where Q_X is the quantile function of the base distribution.

See Also

numDeriv::jacobian(), numDeriv::hessian()

Examples

# Create a log normal distribution
dist <- dist_transformed(dist_normal(0, 0.5), exp, log)
density(dist, 1) # dlnorm(1, 0, 0.5)
cdf(dist, 4) # plnorm(4, 0, 0.5)
quantile(dist, 0.1) # qlnorm(0.1, 0, 0.5)
generate(dist, 10) # rlnorm(10, 0, 0.5)

distributional documentation built on June 11, 2026, 9:07 a.m.