ln-normal.md

Ln-Normal Distribution

Distribution index · Base-10 Log-Normal · Parameterization crosswalk

Purpose and Public Parameterization

Numerics.Distributions.LnNormal is a natural-lognormal family whose public constructor and GetParameters use the natural-space mean and standard deviation, $(m,s)$, not the mean and standard deviation of $\ln X$. Both are in observation units and must be positive for a meaningful lognormal model. This differs intentionally from LogNormal, whose public parameters are log-space moments.

The constructor converts $(m,s)$ to internal Normal parameters

$$ \sigma_{\ln}^2=\log\left[1+(s/m)^2\right],\qquad \mu_{\ln}=\log m-\frac12\sigma_{\ln}^2. \tag{LN.1} $$

The public Mu and Sigma properties expose these internal log-space values, while GetParameters, ParametersToString, Mean, and StandardDeviation expose or return natural-space moments. Code that mixes these surfaces can silently change parameterization.

The physical-parameter cache preserves the exact serialized natural-space mean and standard deviation through clone and round-trip operations. Persisted coordinates and logarithmic bases are unchanged.

Distribution

For $x>0$,

$$ f(x)=\frac{1}{x\sigma_{\ln}\sqrt{2\pi}} \exp\left[-\frac{(\ln x-\mu_{\ln})^2}{2\sigma_{\ln}^2}\right],\qquad F(x)=\Phi\left(\frac{\ln x-\mu_{\ln}}{\sigma_{\ln}}\right). \tag{LN.2} $$

The survival is $1-F$ and

$$ Q(p)=\exp[\mu_{\ln}+\sigma_{\ln}\Phi^{-1}(p)]. \tag{LN.3} $$

All positive moments exist, with $E[X^r]=\exp(r\mu_{\ln}+r^2\sigma_{\ln}^2/2)$. Median and mode are $e^{\mu_{\ln}}$ and $e^{\mu_{\ln}-\sigma_{\ln}^2}$. The upper tail is unbounded and heavier than an exponential tail but lighter than a power law.

Likelihood, Priors, and Estimation

For exact positive values,

$$ \ell=-n\log\sigma_{\ln}-\sum_i\log x_i-\frac n2\log(2\pi) -\frac{1}{2\sigma_{\ln}^2}\sum_i(\ln x_i-\mu_{\ln})^2. \tag{LN.4} $$

RMC.BestFit uses natural-space $(m,s)$ as model coefficients and passes them through the constructor conversion before evaluating exact, censored, uncertain, interval, and threshold likelihood components. Quantile priors are evaluated from (LN.3). Sample-dependent starts and working bounds come from the Numerics constraint routine. The direct-moment conversion guards only nonpositive standard deviation explicitly; nonpositive or near-zero mean can lead to non-finite internal values and must be rejected during data/model validation.

When parameters vary with covariates, the trends predict natural-space mean and standard deviation, not $(\mu_{\ln},\sigma_{\ln})$. This distinction changes both interpretation and the nonlinear response to covariates.

Compile-Checked API Example

private static (double Density, double Cdf, double Quantile) EvaluateLnNormal()
{
    var distribution = new global::Numerics.Distributions.LnNormal(1500.0, 500.0);
    return Evaluate(distribution, value: 2200.0, nonExceedanceProbability: 0.99);
}

Here 1,500 and 500 are the arithmetic mean and standard deviation in discharge units. Passing log-space values would define a different and usually nonsensical model.

Validation and Limitations

Required evidence includes round-trip recovery of $(m,s)$ after conversion, density normalization, CDF/quantile inversion, theoretical natural-space moments, and comparison with a natural-lognormal reference. The compiled fixture guards current constructor semantics. Positive support can be appropriate for discharge, but lognormal tail extrapolation remains a substantive assumption and may underrepresent heavy flood-generating mechanisms.

The distribution verification matrix identifies the current independent formula and fitted-objective comparisons, retained Bayesian/MLE recovery designs, and their distinct acceptance rules. The compiled example guards API compatibility; it does not run an estimator or establish scientific accuracy by itself.

References

[1] E. L. Crow and K. Shimizu, eds., Lognormal Distributions: Theory and Applications. Marcel Dekker, 1988.