## Student Perspectives: Change in the air: Tackling Bristol’s nitrogen oxide problem

A post by Dom Owens, PhD student on the Compass programme.

“Air pollution kills an estimated seven million people worldwide every year” – World Health Organisation

Many particulates and chemicals are present in the air in urban areas like Bristol, and this poses a serious risk to our respiratory health. It is difficult to model how these concentrations behave over time due to the complex physical, environmental, and economic factors they depend on, but identifying if and when abrupt changes occur is crucial for designing and evaluating public policy measures, as outlined in the local Air Quality Annual Status Report.  Using a novel change point detection procedure to account for dependence in time and space, we provide an interpretable model for nitrogen oxide (NOx) levels in Bristol, telling us when these structural changes occur and describing the dynamics driving them in between.

## Model and Change Point Detection

We model the data with a piecewise-stationary vector autoregression (VAR) model:

In between change points the time series $\boldsymbol{Y}_{t}$, a $d$-dimensional vector, depends on itself linearly over $p \geq 1$ previous time steps through parameter matrices $\boldsymbol{A}_i^{(j)}, i=1, \dots, p$ with intercepts $\boldsymbol{\mu}^{(j)}$, but at unknown change points $k_j, j = 1, \dots, q$ the parameters switch abruptly. $\{ \boldsymbol{\varepsilon}_{t} \in \mathbb{R}^d : t \geq 1 \}$ are white noise errors, and we have $n$ observations.