Zephyrus

Created by Shaunik Musukula, Aarush Gupta, and Ayush Iyengar

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📚 Table of Contents

• ⚙️ Introduction
• 📐 The Math / Structure
• 📂 Directory Structure
• 📊 Benchmarks

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⚙️ Introduction

Zephyrus uses model archutectures that use a Geospatial Neural Network (GNN), Finite Difference Method (FDM), and Guassion Process Regression (GPR). Additionally, it experiments with the use of a log-cosh loss function and the Huber loss function. The ultimate goal is to create a more accurate representation of air quality in socioeconomically disadvantaged areas.

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📐 The Math / Structure

Geospatial Neural Network (GNN)

FDM (Finite Difference Method)

We used FDM to create a dense grid with the sparse data points via spatial interpolation. Euler's method used in this project is defined as:

$$\vec{y}_{n+1} = \vec{y}_n + h \vec{F}(\vec{y}_n, t_n).$$

Using FDM, we compute AQI values over a grid defined by lat and long. For each grid point $$(i, j)$$:

1. Define the grid as:

$$ \text{\Large Latitude grid: } {x_i \mid i = 1, \dots, n}, \quad \text{\Large Longitude grid: } {y_j \mid j = 1, \dots, m} $$

2. We calculate weights based on the inverse distance:

$$W_{i,j} = \frac{1}{\sqrt{(x - x_i)^2 + (y - y_j)^2} + \epsilon},$$

where $$\epsilon = 1 \times 10^{-5}$$ to prevent division by zero.

3. Aggregation of AQI values:

$$AQI_{i,j} = \frac{\sum_{k=1}^{n} W_{i,j} \cdot AQI_k}{\sum_{k=1}^{n} W_{i,j}}$$

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📊 Benchmarks

Summary of Results

Methodology	Loss
GNN w/tan-cosh Loss	0.0115
Smaller GNN w/ Huber Loss	0.0119