torch.distributions.LogNormal

class LogNormal extends Distribution

new LogNormal(loc: number | Tensor, scale: number | Tensor, options?: DistributionOptions)

readonlyloc(Tensor): – Mean of the underlying normal distribution.
readonlyscale(Tensor): – Standard deviation of the underlying normal distribution.
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Log-Normal distribution: continuous distribution for positive values.

If X ~ LogNormal(μ, σ), then log(X) ~ Normal(μ, σ). Distribution of log-transformed normal. Essential for:

Modeling positive-valued skewed data (sizes, durations, concentrations)
Income and wealth distributions
Particle sizes in physical systems
Biological measurements (enzyme concentrations, organism sizes)
Financial returns and stock prices (under geometric Brownian motion)
Multiplicative processes and growth phenomena
Right-skewed data with long tail

If log(X) ~ Normal(μ, σ), then X ~ LogNormal(μ, σ) where μ = loc (mean of log(X)) and σ = scale (std of log(X))

\begin{aligned} \text{PDF: } f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right) \quad x > 0 \\ \text{Mean: } \mathbb{E}[X] = \exp\left(\mu + \frac{\sigma^2}{2}\right) \\ \text{Median: } \exp(\mu) \\ \text{Mode: } \exp(\mu - \sigma^2) \\ \text{Variance: } \text{Var}(X) = \left(\exp(\sigma^2) - 1\right) \exp(2\mu + \sigma^2) \\ \text{Entropy: } H(X) = \frac{1}{2}\ln(2\pi e\sigma^2) + \mu \end{aligned}

Log-transformation: log(X) has normal distribution N(loc, scale)
Geometric mean: Median = exp(loc), related to underlying normal mean
Right-skewed: Mean Median Mode (positive skew)
Variance relationship: More spread in log-space → more spread in original
Multiplicative process: Product of independent lognormals stays lognormal
Heavy tail: Extreme values more likely than in normal distribution

Positive support: Always produces strictly positive values (never zero)
Right-skewed: Mean is larger than median; central tendency not intuitive
Variance can be large: High scale causes exponentially larger variance
Parameter interpretation: loc and scale are on log-scale, not original scale

Examples

// Basic log-normal with underlying N(0, 1)
const lognorm = new torch.distributions.LogNormal(0, 1);
lognorm.sample();  // log(X) ~ N(0, 1), so X ~ LogNormal(0, 1)

// Income distribution: right-skewed with long tail
const loc = 10.5;  // log-scale location
const scale = 0.8;  // log-scale spread
const income_dist = new torch.distributions.LogNormal(loc, scale);
const incomes = income_dist.sample([10000]);  // 10000 income samples

// Particle size distribution
const mean_log_size = 2;  // log(μm)
const std_log_size = 0.5;
const size_dist = new torch.distributions.LogNormal(mean_log_size, std_log_size);
const sizes = size_dist.sample([1000]);

// Geometric growth process
// Multiplicative process: X_t = X_0 * exp(Y_1 + Y_2 + ... + Y_t)
// where Y_i are normal increments → X_t is lognormal
const initial = 100;
const daily_log_mean = 0.001;  // ~0.1% daily growth
const daily_log_std = 0.02;  // 2% volatility
const stock_price_log = torch.linspace(-5, 5, 100);  // Log-prices
const dist = new torch.distributions.LogNormal(daily_log_mean, daily_log_std);

// Financial modeling: log-returns
// Stock return: S_t = S_0 * exp(R) where R ~ N(μ, σ)
const mu = 0.08;  // 8% expected return (annual)
const sigma = 0.15;  // 15% volatility
const return_dist = new torch.distributions.LogNormal(mu, sigma);
const stock_prices = return_dist.sample([252]);  // 252 trading days

// Batched distributions with different parameters
const locs = torch.tensor([0, 1, 2]);
const scales = torch.tensor([0.5, 1, 1.5]);
const dist = new torch.distributions.LogNormal(locs, scales);
const samples = dist.sample();