notation.md

Global Notation and Conventions

Technical reference index | Parameterization crosswalk | Documentation contract

Typographic conventions

Scalars are italic lower-case letters, vectors are bold lower-case letters, and matrices are bold upper-case letters. Random variables are upper-case where the distinction is material; realized observations are lower-case. A hat denotes an estimate, a tilde a transformed or approximated quantity, and a superscript in parentheses denotes an indexed realization rather than a power when stated.

All logarithms are natural logarithms unless log10, $\log_{10}$, or a distribution chapter explicitly says otherwise. Products and sums over empty sets follow the usual conventions 1 and 0. double.NegativeInfinity represents log density $\log 0$.

Core symbols

Symbol	Definition
$y_i$	Observed or latent hydrologic magnitude for logical record $i$
$t_i$	Integer time/index attached to record $i$
$x_{ij}$	Covariate $j$ for record or site $i$
$n$	Number of observations or time points, defined locally
$m$	Number of logical pointwise likelihood units, defined locally
$p$	Number of fitted parameters; also a probability when locally qualified
$\theta$	Natural model/distribution parameter vector
$\beta$	Trend, regression, or link-space coefficient vector
$\Theta$	Admissible parameter space
$f(y\mid\theta)$	PDF or PMF of $Y$ conditional on $\theta$
$F(y\mid\theta)$	CDF $\Pr(Y\le y\mid\theta)$
$S(y\mid\theta)$	Survival function $1-F(y\mid\theta)$
$F^{-1}(p\mid\theta)$	Left-continuous quantile $\inf{y:F(y\mid\theta)\ge p}$
$\alpha$	Exceedance probability unless a family chapter defines it as a parameter
$q_\alpha$	AEP quantile $F^{-1}(1-\alpha)$
$T=1/\alpha$	Return period under the stated stationary/event-rate assumptions
$L(y\mid\theta)$	Data likelihood
$\ell_D(\theta;y)$	Data log-likelihood
$\pi(\theta)$	Prior density
$\ell_P(\theta)$	Log-prior
$p(\theta\mid y)$	Posterior density
$h(x)$, $h^{-1}(\eta)$	Link and inverse link
$J$, $D$	Jacobian matrix or determinant, defined locally

Estimating-Equation and GMM Symbols

Symbol	Definition
$\mathbf g_i(\theta)$	pointwise estimating-function vector for logical record $i$
$\overline{\mathbf g}_n(\theta)$	sample mean of estimating functions
$\mathbf S(\theta)$	covariance of pointwise estimating functions
$\mathbf W$	GMM weighting matrix, ordinarily $\mathbf S^{-1}$ after updating
$\mathbf D$	Jacobian $\partial\overline{\mathbf g}_n/\partial\theta^\mathsf T$
$\mathbf H$	Hessian of an external-information penalty
$\mathbf B$	penalized GMM bread matrix $\mathbf D^\mathsf T\mathbf W\mathbf D+\mathbf H$
WEDS	weighted error-direction score used by Bulletin 17C linked-MVN heuristics

Estimation and Diagnostic Symbols

Symbol	Definition
$\widehat{\theta}_{\mathrm{MLE}}$	maximizer of data log likelihood over the implemented bounded parameter space
$\widehat{\theta}_{\mathrm{MAP}}$	maximizer of the implemented data-plus-prior log target
$Q(\theta)$	GMM objective under the weighting and penalty convention stated locally
$\widehat R$	maximum of split rank-normalized and folded rank-normalized R-hat
$N_{\mathrm{eff}}$	rank-normalized bulk effective sample size; tail ESS is reported separately
$\mathrm{lppd}$	log pointwise posterior predictive density
$p_{\mathrm{WAIC}}$	sum of posterior variances of pointwise log likelihoods
$\mathrm{WAIC}$	$-2(\mathrm{lppd}-p_{\mathrm{WAIC}})$
$\mathrm{elpd}_{\mathrm{loo}}$	expected log predictive density under leave-one-out prediction
$k$	generalized-Pareto tail shape used by PSIS; compare it with the draw-count reliability limit
$T(y)$	discrepancy statistic in a predictive check
$p_B$	one-sided predictive tail proportion $\Pr{T(\widetilde y)\ge T(y)}$

Time-Series and Rating-Curve Symbols

Symbol	Definition
$B$	backshift operator, $By_t=y_{t-1}$
$\Delta^d$	$d$-fold difference operator $(1-B)^d$
$g(y_t)$	configured response transform
$p,d,q$	AR order, integration/difference order, and MA order
$b$	maximum exogenous-covariate lag in ARIMAX
$\phi_j$	autoregressive coefficient at lag $j$
$\theta_j$	moving-average coefficient at lag $j$
$\varepsilon_t$	Gaussian innovation on the fitted model scale
$\sigma$	innovation SD; log10-discharge residual SD in the rating curve
$h_k$	rating-curve zero-flow/activation stage for control $k$
$a_k$	fitted $\log_{10}$ rating coefficient; $\alpha_k=10^{a_k}$
$\beta_k$	rating-curve power exponent for control $k$
$q(h;\theta)$	median discharge predicted at stage $h$

Bivariate and Spatial Symbols

Symbol	Definition
$U=F_X(X),V=F_Y(Y)$	probability-integral transforms of bivariate marginals
$C_\psi(u,v)$, $c_\psi(u,v)$	copula CDF and density with dependence parameters $\psi$
$\lambda_L,\lambda_U$	lower- and upper-tail dependence coefficients
$S$	number of sites in a spatial analysis
$\mathbf s_j$	site coordinate: projected Cartesian pair or latitude/longitude pair, according to the selected metric
$\mathbf x_j$	site-covariate vector in the order supplied to GeneralLinearFunction
$\xi_j,\alpha_j,\kappa_j$	Numerics GEV location, scale, and shape at site $j$
$\boldsymbol\epsilon_a$	latent spatial regression-error vector for parameter field $a$
$\sigma_a,r_a,p_a$	spatial-error SD, correlation range, and powered-exponential exponent
$\mathbf R_C$	Gaussian-copula correlation matrix for within-row site dependence
$w_j$	multiplier on site $j$'s marginal spatial log density
$h_{jk}$	Cartesian separation in projected units or great-circle separation in kilometres

Copula dependence describes association among observations within the same row. Spatial regression errors describe persistent spatial departures of GEV parameter surfaces. These two structures are not interchangeable.

Probability convention

In frequency analysis, $\alpha$ denotes annual exceedance probability (AEP) and $p=1-\alpha$ denotes nonexceedance probability. Thus

$$ q_\alpha=F^{-1}(1-\alpha). \tag{1} $$

Under a stationary annual-maximum model with one independent maximum per year, the conventional return period is $T=1/\alpha$. It is an average recurrence interval, not a schedule. The probability of at least one exceedance in $N$ independent years is

$$ 1-(1-\alpha)^N. \tag{2} $$

Peaks-over-threshold and nonstationary chapters state their event-rate/time-varying conventions separately; Equation (2) must not be transferred without checking those assumptions.

Units policy

RMC.BestFit stores numeric values without a runtime unit type. Every analysis must define units in its input provenance and output report.

• Location and quantiles have the same units as the modeled magnitude.
• Scale parameters generally have magnitude units, subject to the exact Numerics family parameterization.
• Shape, skewness, correlations, copula parameters, probabilities, and standardized diagnostics are generally dimensionless.
• A linear time slope has parameter units per index unit; polynomial coefficients have parameter units per index-unit power.
• A regression coefficient has response-parameter units divided by its covariate units.
• Logarithms of dimensional quantities require a stated reference scale in a rigorous physical interpretation. Software commonly applies them numerically; reports should state the input units so a change of units is not mistaken for the same parameterization.

No implicit conversion occurs between cubic feet per second and cubic metres per second, feet and metres, calendar and water years, or daily and annual rates.

Index and ordering conventions

An API double[] parameters uses implementation order, not alphabetical order. The authoritative order is the corresponding model's Parameters collection and the family-specific crosswalk. C# indexes are zero-based; equations use one-based mathematical subscripts unless stated.

DataFrame threshold start and end indexes are inclusive. The generic ExactSeries(IList<double>) constructor assigns integer indexes beginning at zero. Nonstationary hydrologic work should use explicit time indexes and a documented origin.

Numerical conventions

• Impossible parameter/data combinations return negative infinity in log space.
• Probabilities, scales, denominators, exponents, and integrations may use documented finite floors or clamps. Each model chapter identifies its exact safeguards.
• Equality at a continuous censoring boundary has probability zero, so inclusive/exclusive endpoint notation does not alter a continuous likelihood. It does matter for discrete models.
• A finite calculation is not by itself evidence of identifiability, convergence, adequate fit, or safe extrapolation.
• Deterministic examples state inputs and dependency versions. Stochastic examples state seed, chain configuration, diagnostic criteria, and tolerances.

Abbreviations

Abbreviation	Meaning
AEP	Annual exceedance probability
AR, MA, ARIMA, ARIMAX	Autoregressive; moving average; integrated ARMA; ARIMA with exogenous covariates
B17C	Bulletin 17C
CDF, PDF, PMF	Cumulative distribution, probability density, probability mass function
DEMCz, DEMCzs	Differential-evolution Markov chain variants implemented in RMC.Numerics
DIC	Deviance information criterion
EMA	Expected Moments Algorithm
ESS	Effective sample size
GEV, GPD	Generalized extreme-value, generalized Pareto distribution
GMM	Generalized method of moments
HMC	Hamiltonian Monte Carlo; present in Numerics but not exposed by BestFit's sampler enum
LOO, LOOIC	Leave-one-out cross-validation and its information criterion
MAP, MLE	Maximum a posteriori, maximum likelihood estimate
MCMC	Markov chain Monte Carlo
MGBT	Multiple Grubbs-Beck Test
NUTS	No-U-Turn Sampler; BestFit's gradient-based sampler option
POT	Peaks over threshold
PSIS	Pareto-smoothed importance sampling
R-hat	Between/within-chain convergence diagnostic
WAIC	Widely applicable information criterion
WEDS	Weighted error-direction score

Technical reference index | Parameterization crosswalk | Documentation contract