Struct LogNormal

pub struct LogNormal { /* private fields */ }

The Log Normal Distribution

Description

Density, distribution function, quantile function and random generation for the log normal distribution whose logarithm has mean equal to meanlog and standard deviation equal to sdlog.

Arguments

• meanlog, sdlog: mean and standard deviation of the distribution on the log scale with default values of 0 and 1 respectively.

Details

The log normal distribution has density

$ f(x) = 1/(\sqrt(2 \pi) \sigma x) e^-((log x - \mu)^2 / (2 \mu^2)) $

where $ \mu $ and $ \sigma $ are the mean and standard deviation of the logarithm. The mean is $ E(X) = exp(\mu + 1/2 \sigma^2) $, the median is $ med(X) = exp(\mu) $, and the variance $ Var(X) = exp(2*\mu + \sigma^2)*(exp(\sigma^2) - 1) $ and hence the coefficient of variation is $ \sqrt(exp(\sigma^2) - 1) $ which is approximately $ \sigma $ when that is small (e.g., $ \sigma < 1/2 $).

Density Plot

let lnorm = LogNormalBuilder::new().build();
let x = <[f64]>::sequence(-1.0, 8.0, 1000);
let y = x
    .iter()
    .map(|x| lnorm.density(x).unwrap())
    .collect::<Vec<_>>();

let root = SVGBackend::new("density.svg", (1024, 768)).into_drawing_area();
Plot::new()
    .with_options(PlotOptions {
        x_axis_label: "x".to_string(),
        y_axis_label: "density".to_string(),
        ..Default::default()
    })
    .with_plottable(Line {
        x,
        y,
        color: BLACK,
        ..Default::default()
    })
    .plot(&root)
    .unwrap();

Note

The cumulative hazard $ H(t) = - \log(1 - F(t)) $ is $ -plnorm(t, r, lower = FALSE, log = TRUE) $.

Source

dlnorm is calculated from the definition (in ‘Details’). [pqr]lnorm are based on the relationship to the normal.

Consequently, they model a single point mass at exp(meanlog) for the boundary case sdlog = 0.

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 1, chapter 14. Wiley, New York.

See Also

Distributions for other standard distributions, including dnorm for the normal distribution.

Examples

let norm = NormalBuilder::new().build();
println!("{}", norm.density(0));
let lnorm = LogNormalBuilder::new().build();
println!("{}", lnorm.density(1));

Trait Implementations

impl Distribution for LogNormal

fn density<R>(&self, x: R) -> NonNan<f64>

where R: Into<NonNan<f64>>,

The density of the values at a given point

fn log_density<R>(&self, x: R) -> NonNan<f64>

where R: Into<NonNan<f64>>,

The logarithmic density of the values at a given point

fn probability<R>( &self, q: R, lower_tail: bool, ) -> GreaterThanEqualZero<LessThanEqualOne<NonNan<f64>>>

where R: Into<NonNan<f64>>,

PDF; The probability that a value is found in a distribution (inverse of quantile)

fn log_probability<R>( &self, q: R, lower_tail: bool, ) -> LessThanEqualZero<NonNan<f64>>

where R: Into<NonNan<f64>>,

log(PDF); The logarithmic probability that a value is found in a distribution (inverse of quantile)

fn quantile<P>(&self, p: P, lower_tail: bool) -> NonNan<f64>

where P: Into<GreaterThanEqualZero<LessThanEqualOne<NonNan<f64>>>>,

The value in the distribution that is associated with a probability (inverse of probability)

fn log_quantile<LP>(&self, p: LP, lower_tail: bool) -> NonNan<f64>

where LP: Into<LessThanEqualZero<NonNan<f64>>>,

The logarithmic value in the distribution that is associated with a probability (inverse of probability)

fn random_sample<R>(&self, rng: &mut R) -> NonNan<f64>

where R: RNG,

Generates a random sample from the distribution