Density distribution

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.

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 $\mu_4 / \mu_2^2 - 3$.

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
#> <distribution[1]>
#> [1] density[801]
mean(dist)
#> [1] -1.820392e-16
variance(dist)
#> [1] 0.9989458
skewness(dist)
#> [1] -2.336246e-15
kurtosis(dist)
#> [1] -0.01395399
density(dist, 0)
#> [1] 0.3989676
cdf(dist, 1.96)
#> [1] 0.9750312
quantile(dist, 0.975)
#> [1] 1.959466

# The density needn't be standardised, it is scaled to integrate to one
dist_density(list(c(0, 1)), list(c(2, 2)))
#> Warning: The density integrates to 2, standardising it to integrate to 1.
#> <distribution[1]>
#> [1] density[2]

# A kernel density estimate
kd <- density(rnorm(100))
dist_density(list(kd$x), list(kd$y))
#> <distribution[1]>
#> [1] density[512]

Arguments

Details

See also

Examples