distributional (version 0.9.0)

dist_density: Density distribution

Description

The density distribution is a non-parametric distribution defined by the values of its probability density function at a set of points. This is useful for representing a distribution which is known only through its density, such as a kernel density estimate, a posterior evaluated over a grid, or a density obtained numerically.

The density is linearly interpolated between the given values, and is zero beyond them. All other properties of the distribution are computed exactly from that interpolation.

Usage

dist_density(x, density)

Arguments

x

A list of values

density

A list of density values at x

Details

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

In the following, let \(X\) be a random variable with density \(f_1, f_2, \ldots, f_n\) given at the values \(x_1 < x_2 < \cdots < x_n\). Write \(w_i = x_{i+1} - x_i\) for the width of the \(i\)th interval, and \(s_i = (f_{i+1} - f_i) / w_i\) for the slope of the density over it.

Support: \([x_1, x_n]\)

Probability density function (p.d.f): Linear interpolation of the given values, standardised to integrate to one:

$$ f(t) = f_i + (t - x_i) s_i \quad \text{for } x_i \le t \le x_{i+1} $$

and \(f(t) = 0\) for \(t < x_1\) or \(t > x_n\).

Cumulative distribution function (c.d.f): The exact integral of the interpolated density, which is piecewise quadratic. With \(h = t - x_i\) and \(F_i = F(x_i)\),

$$ F(t) = F_i + f_i h + \frac{s_i h^2}{2} $$

Quantile function: The inverse of the above, obtained by solving the quadratic within the interval containing the requested probability.

Mean: Computed exactly from the interpolated density,

$$ E(X) = \sum_{i=1}^{n-1} \left[ x_i m_i + w_i^2 \left( \frac{f_i}{6} + \frac{f_{i+1}}{3} \right) \right] $$

where \(m_i = w_i (f_i + f_{i+1}) / 2\) is the probability of the \(i\)th interval.

Variance, skewness, and excess kurtosis: Computed from the central moments \(\mu_k = E[(X - E(X))^k]\). Writing \(c_i = x_i - E(X)\),

$$ \mu_2 = \sum_{i=1}^{n-1} \left[ c_i^2 m_i + \frac{c_i w_i^2 (f_i + 2 f_{i+1})}{3} + \frac{w_i^3 (f_i + 3 f_{i+1})}{12} \right] $$

$$ \mu_3 = \sum_{i=1}^{n-1} \left[ c_i^3 m_i + \frac{c_i^2 w_i^2 (f_i + 2 f_{i+1})}{2} + \frac{c_i w_i^3 (f_i + 3 f_{i+1})}{4} + \frac{w_i^4 (f_i + 4 f_{i+1})}{20} \right] $$

$$ \mu_4 = \sum_{i=1}^{n-1} \left[ c_i^4 m_i + \frac{2 c_i^3 w_i^2 (f_i + 2 f_{i+1})}{3} + \frac{c_i^2 w_i^3 (f_i + 3 f_{i+1})}{2} + \frac{c_i w_i^4 (f_i + 4 f_{i+1})}{5} + \frac{w_i^5 (f_i + 5 f_{i+1})}{30} \right] $$

Variance is \(\mu_2\), skewness is \(\mu_3 / \mu_2^{3/2}\), and excess kurtosis is \(\mu_4 / \mu_2^2 - 3\).

See Also

dist_quantile(), stats::density()

Examples

# A distribution given by its density over a grid
at <- seq(-4, 4, by = 0.01)
dist <- dist_density(list(at), list(dnorm(at)))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
density(dist, 0)
cdf(dist, 1.96)
quantile(dist, 0.975)

# The density needn't be standardised, it is scaled to integrate to one
dist_density(list(c(0, 1)), list(c(2, 2)))

# A kernel density estimate
kd <- density(rnorm(100))
dist_density(list(kd$x), list(kd$y))