Fixing brokern math symbols, updating JOSS paper.

Author: HadenSmith

CITATION.cff

@@ -24,7 +24,22 @@ keywords:
   - C#
 authors:
   - family-names: "Smith"
-    given-names: "Haden"
+    given-names: "C. Haden"
     email: "cole.h.smith@usace.army.mil"
     affiliation: "U.S. Army Corps of Engineers, Risk Management Center"
     orcid: "https://orcid.org/0000-0001-7881-5814"
+  - family-names: "Fields"
+    given-names: "Woodrow L."
+    affiliation: "U.S. Army Corps of Engineers, Risk Management Center"
+  - family-names: "Gonzalez"
+    given-names: "Julian"
+    affiliation: "U.S. Army Corps of Engineers, Risk Management Center"
+  - family-names: "Niblett"
+    given-names: "Sadie"
+    affiliation: "U.S. Army Corps of Engineers, Risk Management Center"
+  - family-names: "Beam"
+    given-names: "Brennan"
+    affiliation: "U.S. Army Corps of Engineers, Risk Management Center"
+  - family-names: "Skahill"
+    given-names: "Brian"
+    affiliation: "U.S. Army Corps of Engineers, Risk Management Center"

docs/distributions/copulas.md

@@ -145,7 +145,7 @@ The Frank copula has **no tail dependence** and produces a symmetric dependence
 |----------|-------|
 | Generator | $\varphi(t) = -\ln\!\left(\frac{e^{-\theta t} - 1}{e^{-\theta} - 1}\right)$ |
 | CDF | $C(u,v) = -\frac{1}{\theta}\ln\!\left(1 + \frac{(e^{-\theta u}-1)(e^{-\theta v}-1)}{e^{-\theta}-1}\right)$ |
-| Parameter range | $\theta \in (-\infty, \infty) \setminus \{0\}$ |
+| Parameter range | $\theta \in (-\infty, \infty) \setminus \lbrace 0\rbrace$ |
 | Tail dependence | $\lambda_L = \lambda_U = 0$ |
 
 The Frank copula is the only Archimedean copula that allows both positive and negative dependence ($\theta > 0$ for positive, $\theta < 0$ for negative).
@@ -195,7 +195,7 @@ var amhCopula = new AMHCopula(0.5);
 | Student-t | Symmetric | $\rho \in [-1, 1]$, $\nu > 2$ | Heavy-tailed joint extremes |
 | Clayton | Lower tail | $\theta \in (0, \infty)$ | Joint low extremes (droughts) |
 | Gumbel | Upper tail | $\theta \in [1, \infty)$ | Joint high extremes (floods) |
-| Frank | None | $\theta \in \mathbb{R} \setminus \{0\}$ | Moderate symmetric dependence |
+| Frank | None | $\theta \in \mathbb{R} \setminus \lbrace 0\rbrace$ | Moderate symmetric dependence |
 | Joe | Upper tail | $\theta \in [1, \infty)$ | Strong upper tail dependence |
 | AMH | None | $\theta \in [-1, 1]$ | Weak dependence structures |
 

docs/distributions/multivariate.md

@@ -653,7 +653,7 @@ A random vector $\mathbf{X} = (X_1, X_2, \ldots, X_k)$ follows a multinomial dis
 P(X_1 = x_1, \ldots, X_k = x_k) = \frac{n!}{\prod_{i=1}^{k} x_i!} \prod_{i=1}^{k} p_i^{x_i}
 ```
 
-with constraints $x_i \in \{0, 1, \ldots, n\}$, $\sum_{i=1}^{k} x_i = n$, $p_i \geq 0$, and $\sum_{i=1}^{k} p_i = 1$.
+with constraints $x_i \in \lbrace 0, 1, \ldots, n\rbrace$, $\sum_{i=1}^{k} x_i = n$, $p_i \geq 0$, and $\sum_{i=1}^{k} p_i = 1$.
 
 The multinomial coefficient $n! / \prod x_i!$ counts the number of ways to arrange $n$ trials into $k$ categories with the specified counts.
 

docs/distributions/uncertainty-analysis.md

@@ -28,7 +28,7 @@ x^{*}_{b,1}, x^{*}_{b,2}, \ldots, x^{*}_{b,n} \sim \hat{F}(\hat{\theta}), \quad
 \hat{Q}^{*}_b = \hat{F}^{-1}(p \mid \hat{\theta}^{*}_b), \quad b = 1, 2, \ldots, B
 ```
 
-**Step 4.** The empirical distribution of $\{\hat{Q}^{*}_1, \hat{Q}^{*}_2, \ldots, \hat{Q}^{*}_B\}$ approximates the sampling distribution of the quantile estimator $\hat{Q}$.
+**Step 4.** The empirical distribution of $\lbrace\hat{Q}^{*}_1, \hat{Q}^{*}_2, \ldots, \hat{Q}^{*}_B\rbrace$ approximates the sampling distribution of the quantile estimator $\hat{Q}$.
 
 From this bootstrap distribution, we can compute several useful summaries. The **bootstrap standard error** is the sample standard deviation of the bootstrap replicates:
 
@@ -258,7 +258,7 @@ The Percentile method [[1]](#1) is the simplest bootstrap confidence interval. I
 CI_{1-\alpha} = \left[\hat{Q}^{*}_{(\alpha/2)},\;\hat{Q}^{*}_{(1-\alpha/2)}\right]
 ```
 
-where $\hat{Q}^{*}_{(p)}$ denotes the $p$-th percentile of the bootstrap distribution $\{\hat{Q}^{*}_1, \ldots, \hat{Q}^{*}_B\}$. For a 90% confidence interval ($\alpha = 0.1$), this takes the 5th and 95th percentiles of the bootstrap replicates.
+where $\hat{Q}^{*}_{(p)}$ denotes the $p$-th percentile of the bootstrap distribution $\lbrace\hat{Q}^{*}_1, \ldots, \hat{Q}^{*}_B\rbrace$. For a 90% confidence interval ($\alpha = 0.1$), this takes the 5th and 95th percentiles of the bootstrap replicates.
 
 The Percentile method is intuitive and easy to implement, but it does **not** correct for bias or skewness in the bootstrap distribution. It works well when the bootstrap distribution is approximately symmetric and the estimator is approximately unbiased. This is the default method used by the `Estimate()` method.
 
@@ -405,7 +405,7 @@ where $\tilde{Q} = \hat{Q}^{1/3}$ is the transformed original estimate. The conf
 CI_{1-\alpha} = \left[\left(\tilde{Q} + t^{*}_{(\alpha/2)}\cdot\widetilde{SE}\right)^3,\;\left(\tilde{Q} + t^{*}_{(1-\alpha/2)}\cdot\widetilde{SE}\right)^3\right]
 ```
 
-where $t^{*}_{(p)}$ is the $p$-th percentile of $\{t^{*}_1, \ldots, t^{*}_B\}$, and $\widetilde{SE}$ is the standard deviation of the transformed bootstrap replicates $\{\tilde{Q}^{*}_1, \ldots, \tilde{Q}^{*}_B\}$.
+where $t^{*}_{(p)}$ is the $p$-th percentile of $\lbrace t^{*}_1, \ldots, t^{*}_B\rbrace$, and $\widetilde{SE}$ is the standard deviation of the transformed bootstrap replicates $\lbrace\tilde{Q}^{*}_1, \ldots, \tilde{Q}^{*}_B\rbrace$.
 
 The Bootstrap-t method is the most computationally expensive method because it requires a **double bootstrap**: each of the $B$ outer replications requires an inner bootstrap (300 replications by default) to estimate the standard error. However, it can provide the most accurate coverage probabilities for location parameters and is second-order accurate.
 

docs/sampling/convergence-diagnostics.md

@@ -209,7 +209,7 @@ where $N$ is the number of samples, $\rho_k$ is the autocorrelation at lag $k$,
 
 ### Mathematical Derivation
 
-The ESS formula arises from analyzing the variance of the sample mean of a correlated sequence. For a stationary process $\{\theta_1, \theta_2, \ldots, \theta_N\}$ with marginal variance $\sigma^2$ and autocorrelation function $\rho_k = \text{Corr}(\theta_t, \theta_{t+k})$, the variance of the sample mean $\bar{\theta} = \frac{1}{N}\sum_{t=1}^{N}\theta_t$ is:
+The ESS formula arises from analyzing the variance of the sample mean of a correlated sequence. For a stationary process $\lbrace\theta_1, \theta_2, \ldots, \theta_N\rbrace$ with marginal variance $\sigma^2$ and autocorrelation function $\rho_k = \text{Corr}(\theta_t, \theta_{t+k})$, the variance of the sample mean $\bar{\theta} = \frac{1}{N}\sum_{t=1}^{N}\theta_t$ is:
 
 ```math
 \text{Var}(\bar{\theta}) = \frac{\sigma^2}{N}\left(1 + 2\sum_{k=1}^{N-1}\left(1 - \frac{k}{N}\right)\rho_k\right)

docs/sampling/mcmc.md

@@ -320,7 +320,7 @@ where:
 - $\gamma = 2.38 / \sqrt{2d}$ is the default jump rate (`Jump` property), with $d$ the number of parameters
 - $z_{R_1}$ and $z_{R_2}$ are two randomly selected states from the **population matrix** (a memory of past states from all chains)
 - $e \sim \mathcal{N}(0, b^2)$ is a small noise perturbation with default $b = 10^{-3}$ (`Noise` property)
-- $R_1$ and $R_2$ are drawn uniformly without replacement from $\{1, 2, \ldots, M\}$, where $M$ is the current size of the population matrix
+- $R_1$ and $R_2$ are drawn uniformly without replacement from $\lbrace 1, 2, \ldots, M\rbrace$, where $M$ is the current size of the population matrix
 
 The proposal is accepted using the standard Metropolis ratio in log space:
 

docs/sampling/random-generation.md

@@ -222,7 +222,7 @@ Latin Hypercube Sampling divides each dimension's range $[0, 1)$ into $n$ equal
 x_{ij} = \frac{\pi_j(i) + U_{ij}}{n}, \quad i = 0, \ldots, n-1
 ```
 
-where $\pi_j$ is a random permutation of $\{0, 1, \ldots, n-1\}$ (independent for each dimension) and $U_{ij} \sim \text{Uniform}(0, 1)$. The library's `Median` variant replaces $U_{ij}$ with $0.5$, placing each point at the stratum center.
+where $\pi_j$ is a random permutation of $\lbrace 0, 1, \ldots, n-1\rbrace$ (independent for each dimension) and $U_{ij} \sim \text{Uniform}(0, 1)$. The library's `Median` variant replaces $U_{ij}$ with $0.5$, placing each point at the stratum center.
 
 The random permutations are generated using the **Fisher-Yates shuffle**, and each dimension uses an independent Mersenne Twister seeded from a master RNG.