Quantile distribution — dist_quantile • distributional

The Quantile distribution is a non-parametric distribution defined by a set of values at specified quantile probabilities. This distribution is useful for representing empirical distributions or elicited expert knowledge when only quantile information is available. The distribution uses linear interpolation between quantiles and can be used to approximate complex distributions that may not have simple parametric forms.

dist_quantile(x, quantile)

dist_percentile(x, percentile)

Arguments

x: A list of values
quantile: A list of quantile probabilities (between 0 and 1)
percentile: A list of percentiles (between 0 and 100)

Details

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

In the following, let $X$ be a Quantile random variable defined by values $x_1, x_2, \ldots, x_n$ at quantile probabilities $q_1, q_2, \ldots, q_n$ where $0 \le q_i \le 1$.

Support: $[\min(x_i), \max(x_i)]$ if $\min(q_i) > 0$ or $\max(q_i) < 1$, otherwise support is approximated from the specified quantiles.

Mean: Approximated numerically using spline interpolation and numerical integration:

$$ E(X) \approx \int_0^1 Q(u) du $$

where $Q(u)$ is a spline function interpolating the quantile values.

Variance: Approximated numerically.

Probability density function (p.d.f): Approximated numerically using kernel density estimation from generated samples.

Cumulative distribution function (c.d.f): Defined by linear interpolation:

$$ F(t) = \begin{cases} q_1 & \text{if } t < x_1 \\ q_i + \frac{(t - x_i)(q_{i+1} - q_i)}{x_{i+1} - x_i} & \text{if } x_i \le t < x_{i+1} \\ q_n & \text{if } t \ge x_n \end{cases} $$

Quantile function: Defined by linear interpolation:

$$ Q(u) = x_i + \frac{(u - q_i)(x_{i+1} - x_i)}{q_{i+1} - q_i} $$

for $q_i \le u \le q_{i+1}$.

Examples

dist <- dist_normal()
probs <- seq(0.01, 0.99, by = 0.01)
x <- vapply(probs, quantile, double(1L), x = dist)
dist_quantile(list(x), list(probs))
#> <distribution[1]>
#> [1] quantile[99]
dist_percentile(list(x), list(probs * 100))
#> <distribution[1]>
#> [1] quantile[99]