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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.
dist_transformed(dist, transform, inverse)Arguments
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
Examples
# Create a log normal distribution
dist <- dist_transformed(dist_normal(0, 0.5), exp, log)
density(dist, 1) # dlnorm(1, 0, 0.5)
#> [1] 0.7978846
cdf(dist, 4) # plnorm(4, 0, 0.5)
#> [1] 0.9972194
quantile(dist, 0.1) # qlnorm(0.1, 0, 0.5)
#> [1] 0.5268835
generate(dist, 10) # rlnorm(10, 0, 0.5)
#> [[1]]
#> [1] 1.7645954 1.0285990 1.0192352 0.8064466 0.7581369 0.4255272 0.7577866
#> [8] 1.4767394 1.2341949 0.7469440
#>