# A DFA-based bivariate regression model for estimating the dependence of PM2.5 among neighbouring cities

## TL;DR

This paper introduces a DFA-based bivariate regression model that estimates multi-scale dependence between variables, demonstrated through PM2.5 pollution data among neighboring cities, revealing seasonal and scale-dependent influences.

## Contribution

The paper presents a novel DFA-based regression model that captures multi-scale dependence, providing new estimators and indices for analyzing complex variable relationships.

## Key findings

- The model accurately depicts dependence between PM2.5 variables.
- Dependence varies across seasons and time scales.
- New indices effectively evaluate scale-dependent relationships.

## Abstract

On the basis of detrended fluctuation analysis (DFA), we propose a new bivariate linear regression model. This new model provides estimators of multi-scale regression coefficients to measure the dependence between variables and corresponding variables of interest with multi-scales. Numerical tests are performed to illustrate that the proposed DFA-based regression estimators are capable of accurately depicting the dependence between the variables of interest and can be used to identify different dependence at different time scales. We apply this model to analyze the PM2.5 series of three adjacent cities (Beijing, Tianjin, and Baoding) in Northern China. The estimated regression coefficients confirmed the dependence of PM2.5 among the three cities and illustrated that each city has different influence on the others at different seasons and at different time scales. Two statistics based on the scale-dependent $t$-statistic and the partial detrended cross-correlation coefficient are used to demonstrate the significance of the dependence. Three new scale-dependent evaluation indices show that the new DFA-based bivariate regression model can provide rich information on studied variables.