Detrending-moving-average-based multivariate regression model for nonstationary series

Dependency between a response variable and the explanatory variables is a relationship of universal concern in various real-world problems. Multivariate linear regression (MLR) is a well-known method to focus on this issue. However, it is limited to dealing with stationary variables. In this work, we develop a MLR framework based on detrending moving average (DMA) analysis to reveal the actual dependency among variables with nonstationary measures. The DMA-based MLR can generate multiscale regression coefficients, which characterize different dependent behavior at different timescales. Artificial tests show that the DMA-MLR model can successfully resist the impact of trends on the studied series and produce more accurate regression coefficients with multiscale. Furthermore, some scale-dependent statistics are developed to deduce some important relationships in three typical DMA-based MLR models, which help us to deeply understand the DMA-MLR models in theory. The application of the proposed DMA-MLR framework to Beijing's air quality index system demonstrates that fine particulate matter with diameter $\le 2.5\mu \text{m}$ $\left({\mathrm{PM}}_{2.5}\right)$ is the dominant pollutant affecting the air quality of Beijing in recent years.

Figure 1

PDFs of regression coefficients of the quaternary linear regression model ($y=-x_1-0.5x_2+0.5x_3+x_4+\epsilon$) based on the DMA method and the traditional method. The four subplots are for the four regression coefficients in order. In each case, three scales $s=10$, 40, and 70 are considered for the DMA-based regression model.

Figure 2

Effect of the four types of trends on regression estimators. The left panel is for $\left\{x_i\left(t\right)\right\}$ ($i=1$, 2, 3, 4) adding linear trend (a), quadratic trend (c), cubic trend (e), and sinusoidal trend (g) with the other variables unchanged. The right panel is for the corresponding $\widehat{{\beta }_i}$ estimated by DMA-based (solid line) and traditional OSL-based (dashed line) methods. Dotted line locates the expected result, ${\beta }_i=1$. It demonstrates that the DMA-based estimators are close to the true values, but the estimations of the traditional OLS-based method deviate seriously.

Figure 3

Estimated DMA-based and OLS-based regression coefficients for the quaternary regression model $y=0.8x_1+0.3x_2-0.2x_3-0.7x_4+\epsilon$. Error bars indicate the standard deviations calculated from 100 independent realizations of the corresponding processes.

Figure 4

The performance of semipartial cross-correlation coefficient for ARFIMA series involving trends. (a) The multiscale SPCC obtained by DMA (blue squares) and DFA (violet triangles). The red dashed line denotes the traditional SPCC. (b) More cases with higher-order trends are shown. We take second-order polynomial fitting for the DFA method. The coefficient of each case is actually averaged over all the timescales in (a).

Figure 5

DMA-MLR modeling for four explanatory variables $\left\{x_i\left(t\right)\right\}$ ($i=1$, 2, 3, 4) and an explained variable $\left\{y\right(t\left)\right\}$ generated by BMFs model. (a) DMXA coefficient of every two $\left\{x_i\left(t\right)\right\}$. (b) The DMA-based SPCC between series $\left\{y\right(t\left)\right\}$ and each $\left\{x_i\left(t\right)\right\}$. The black dashed line represents zero.

Figure 6

Examination of the relationship between $R^2\left(s\right)$ and $R_{yi}^2\left(s\right)$, which is the coefficient of determination of the quaternary and ternary linear regression models based on the DMA framework. Subplots (a), (b), (c), and (d) are for Model C established by three $x_i$ with the exclusion of $x_1$, $x_2$, $x_3$, and $x_4$, respectively. The dashed line in the four subplots is $R^2\left(s\right)$ of Model A (all of them are identical). The hollow symbols denote the quantities determined by the right side of Eq. (16). The perfect agreement suggests that the relationship (16) holds.

Figure 7

Examination of the relationship between the DMA-based PCC and SPCC. The four subplots correspond to DMA-based PCCs between the series $\left\{y\right(t\left)\right\}$ and one of $\left\{x_i\left(t\right)\right\}$. The hollow symbols and the dashed line are the quantities determined by the left-hand side and right-hand side of Eq. (17), respectively. The perfect agreement suggests that the relationship (17) holds.

Figure 8

Examination of the relationship between the DMA-based SPCC and DMA-based regression coefficient. The four subplots are the DMA-based SPCC between series $\left\{y\right(t\left)\right\}$ and one of $\left\{x_i\left(t\right)\right\}$. The hollow symbols and the dashed line are the quantities determined by the left-hand side and right-hand side of Eq. (18), respectively. The perfect coincidence suggests that the relationship (18) holds.

Figure 9

The quaternary linear regression model of Beijing's AQI for the years 2014–2019. Panels (a) and (b) are the estimated scale-dependent regression coefficient and scale-dependent SPCC based on the DMA-MLR framework, respectively. The shaded zones around $\widehat{\beta }\left(s\right)$ denote the 95% confidence intervals.

Figure 10

Annual indicators of the quaternary linear regression model of Beijing's AQI. (a–d) The averaged SPCC, averaged PCC, averaged coefficient of determination, and averaged tolerance over timescales of 7 to 56 days based on the DMA-MLR framework, respectively.