Categorical: Categorical distribution

Description

Probability mass function, distribution function, quantile function and random generation for the categorical distribution.

Usage

dcat(x, prob, log = FALSE)
pcat(q, prob, lower.tail = TRUE, log.p = FALSE)
qcat(p, prob, lower.tail = TRUE, log.p = FALSE, labels)
rcat(n, prob, labels)
rcatlp(n, log_prob, labels)

Arguments

x, q: vector of quantiles.
prob, log_prob: vector of length $m$, or $m$-column matrix of non-negative weights (or their logarithms in log_prob).
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
p: vector of probabilities.
labels: if provided, labeled factor vector is returned. Number of labels needs to be the same as number of categories (number of columns in prob).
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

$$ \Pr(X = k) = \frac{w_k}{\sum_{j=1}^m w_j} $$

Cumulative distribution function $$ \Pr(X \le k) = \frac{\sum_{i=1}^k w_i}{\sum_{j=1}^m w_j} $$

It is possible to sample from categorical distribution parametrized by vector of unnormalized log-probabilities $\alpha_1,\dots,\alpha_m$ without leaving the log space by employing the Gumbel-max trick (Maddison, Tarlow and Minka, 2014). If $g_1,\dots,g_m$ are samples from Gumbel distribution with cumulative distribution function $F(g) = \exp(-\exp(-g))$, then $k = \mathrm{arg\,max}_i \{g_i + \alpha_i\}$ is a draw from categorical distribution parametrized by vector of probabilities $p_1,\dots,p_m$, such that $p_i = \exp(\alpha_i) / [\sum_{j=1}^m \exp(\alpha_j)]$. This is implemented in rcatlp function parametrized by vector of log-probabilities log_prob.

References

Maddison, C. J., Tarlow, D., & Minka, T. (2014). A* sampling. [In:] Advances in Neural Information Processing Systems (pp. 3086-3094). https://arxiv.org/abs/1411.0030

Examples

Run this code


# Generating 10 random draws from categorical distribution
# with k=3 categories occuring with equal probabilities
# parametrized using a vector

rcat(10, c(1/3, 1/3, 1/3))

# or with k=5 categories parametrized using a matrix of probabilities
# (generated from Dirichlet distribution)

p <- rdirichlet(10, c(1, 1, 1, 1, 1))
rcat(10, p)

x <- rcat(1e5, c(0.2, 0.4, 0.3, 0.1))
plot(prop.table(table(x)), type = "h")
lines(0:5, dcat(0:5, c(0.2, 0.4, 0.3, 0.1)), col = "red")

p <- rdirichlet(1, rep(1, 20))
x <- rcat(1e5, matrix(rep(p, 2), nrow = 2, byrow = TRUE))
xx <- 0:21
plot(prop.table(table(x)))
lines(xx, dcat(xx, p), col = "red")

xx <- seq(0, 21, by = 0.01)
plot(ecdf(x))
lines(xx, pcat(xx, p), col = "red", lwd = 2)

pp <- seq(0, 1, by = 0.001)
plot(ecdf(x))
lines(qcat(pp, p), pp, col = "red", lwd = 2)

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