math-style-guide.md

Mathematical Style Guide

Consistent notation is used across IRIDIUM documentation and, when applicable, in a future preprint. Use this guide when adding or editing formulas.

Typesetting

• In Markdown, use inline math with $...$ and display math with $$...$$.
• Symbols must match the symbol glossary. The same symbol should not denote different things in different documents.

Conventions

Concept	Notation	Notes
Graph	$G = (V, E)$ or $G = (V, E, W)$	$V$ nodes, $E$ edges, $W$ optional weights.
Node set size	$	V
Feature matrix (time $t$)	$X_t \in \mathbb{R}^{N \times F}$	$F$ features per node.
Time index	$t$, $t+1$, $t-T+1:t$	Subscript or range; avoid reusing $t$ for other meanings in the same section.
Forecast horizon	$H$	Number of future steps.
District index	$d$ or $k$	For equity; $m$ for indicator index.
Route	$r$	A path or itinerary.
Model parameters	$\theta$	Global or local model weights.
Loss	$\mathcal{L}$, $\mathcal{L}_k$	$k$ for client or district.
Weights (routing/equity)	$\alpha, \beta, \gamma, \delta$ or $w_m$	Defined in context.

Federated Learning

• Local objective: $\min_\theta \sum_{k=1}^{K} p_k \mathcal{L}_k(\theta)$.
• Client update: $\theta_k^{(t)}$; aggregation: $\theta^{(t+1)} = \sum_k \frac{n_k}{\sum_j n_j} \theta_k^{(t)}$ (FedAvg-style). Use $n_k$ for local sample count, $p_k$ for weight.

Forecasting

• Graph: $G = (V, E, W)$. Input window: $X_{t-T+1:t}$. Forecast: $\hat{Y}{t+1:t+H} = f\theta(X_{t-T+1:t}, G)$. Loss: MAE, RMSE, or MAPE as defined in symbol-glossary.

Routing

• Route utility: $J(r) = \alpha T(r) + \beta C(r) + \gamma E(r) + \delta P(r)$ with $T$ travel time, $C$ cost, $E$ emissions proxy, $P$ penalty. Define symbols on first use.

Equity

• District $d$, indicators $m$. Normalised value: $z_{d,m}$. Composite: $MES_d = \sum_m w_m z_{d,m}$. State that MES is a derived analytic index, not an official statistic.

Anomaly

• Deviation: $z_t = (x_t - \mu_{t,w}) / \sigma_{t,w}$ (rolling window $w$). Severity or score: $S_t$ with defined components. Define $\mu_{t,w}$, $\sigma_{t,w}$ in text.

Avoid

• Em dash or en dash in math or text (use hyphen).
• Conflicting use of $k$ (e.g. both client index and time step).
• Undefined symbols. When in doubt, add a short phrase: "where $N$ is the number of nodes."