Logistic {stats}

R Documentation

The Logistic Distribution

Description

Density, distribution function, quantile function and random generation for the logistic distribution with parameters location and scale.

Usage

dlogis(x, location = 0, scale = 1, log = FALSE)
plogis(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qlogis(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rlogis(n, location = 0, scale = 1)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
location, scale	location and scale parameters.
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].

Details

If location or scale are omitted, they assume the default values of 0 and 1 respectively.

The Logistic distribution with location = \mu and scale = \sigma has distribution function

F(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}%

and density

f(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}%

It is a long-tailed distribution with mean \mu and variance \pi^2/3 \sigma^2.

Value

dlogis gives the density, plogis gives the distribution function, qlogis gives the quantile function, and rlogis generates random deviates.

Note

qlogis(p) is the same as the well known ‘logit’ function, logit(p) = \log(p/(1-p)), and plogis(x) has consequently been called the ‘inverse logit’.

The distribution function is a rescaled hyperbolic tangent, plogis(x) == (1+ tanh(x/2))/2, and it is called a sigmoid function in contexts such as neural networks.

Source

[dpr]logis are calculated directly from the definitions.

rlogis uses inversion.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapter 23. Wiley, New York.

Examples

var(rlogis(4000, 0, scale = 5))# approximately (+/- 3)
pi^2/3 * 5^2

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