A Wavelet-based Neural Network Model to Predict Ambient Air Pollutants’ Concentration

Abstract

The present paper proposes a wavelet based recurrent neural network model to forecast one step ahead hourly, daily mean and daily maximum concentrations of ambient CO, NO₂, NO, O₃, SO₂ and PM_2.5 — the most prevalent air pollutants in urban atmosphere. The time series of each air pollutant has been decomposed into different time-scale components using maximum overlap wavelet transform (MODWT). These time-scale components were made to pass through Elman network. The number of nodes in the network was decided on the basis of the strength (power) of the corresponding input signals. The wavelet network model was then used to obtain one-step ahead forecasts for a period extending from January 2009 to June 2010. The model results for out of sample forecast are reasonably good in terms of model performance parameters such as mean absolute error (MAE), mean absolute percentage error (MAPE), root mean square error (RMSE), normalized mean absolute error (NMSE), index of agreement (IOA) and standard average error (SAE). The MAPE values for daily maximum concentrations of CO, NO₂, NO, O₃, SO₂ and PM2.5 were found to be 9.5%, 17.37%, 21.20%, 13.79%, 17.77% and 11.94%, respectively, at ITO, Delhi, India. Bearing in mind that the forecasts are for daily maximum concentrations tested over a long validation period, the forecast performance of the model may be considered as reasonably good. The model results demonstrate that a judicious selection of wavelet network design may be employed successfully for air quality forecasting.

Appendix -1

$$ {\text{Observed Mean}} = \frac{1}{\text{n}}\left( {\sum {{x_o}} } \right) $$

$$ {\text{Predicted Mean}} = \frac{1}{\text{n}}\left( {\sum {{x_p}} } \right) $$

$$ {\text{Observed Standard Deviation}} = \sqrt {{\frac{{\sum { \left( {x{ _o} - \bar{x_o}} \right)^2} }}{{n - 1}}}} $$

$$ {\text{Predicted Standard Deviation}} = \sqrt {{\frac{{\sum { \left( {x{ _p} - \bar{x_p}} \right)^2} }}{{n - 1}}}} $$

$$ {\text{Mean Absolute Error }}\left( {\text{MSE}} \right) = \frac{{\sum {\left| {\left. {{x_o} - {x_p}} \right|} \right.} }}{n} $$

$$ {\text{Mean Absolute Percentage Error}} = \left( {\frac{1}{n}\sum {\left| {\left. {\frac{{{x_o} - {x_p}}}{{{x_o}}}} \right|} \right.} } \right) \times 100 $$

$$ {\text{Root Mean Square Error}} = \sqrt {{\left( {\frac{1}{n}\sum {{{\left( {{x_o} - {x_p}} \right)}^2}} } \right)}} $$

$$ {\text{Normalized Mean Square Error}} = \frac{{\left( {\frac{1}{n}\sum {{{\left( {{x_o} - {x_p}} \right)}^2}} } \right)}}{{{{\bar{x}}_o} \times {{\bar{x}}_p}}} $$

$$ {\text{Correlation Coefficient}} = \frac{{\frac{1}{n}\sum {\left( {{x_{{oi}}} - {{\bar{x}}_o}} \right)\left( {{x_{{pi}}} - {{\bar{x}}_p}} \right)} }}{{\sqrt {{\left[ {\frac{1}{n}\sum {{{\left( {{x_{{oi}}} - {{\bar{x}}_o}} \right)}^2} \cdot \frac{1}{n}\sum {{{\left( {{x_{{pi}}} - {{\bar{x}}_p}} \right)}^2}} } } \right]}} }} $$

$$ {\text{Index of Agreement}} = 1 - \frac{{\sum {{{\left( {{x_{{oi}}} - {x_p}} \right)}^2}} }}{{{{\left( {\left| {\left. {{x_p} - {{\bar{x}}_o}} \right|} \right. + \left| {\left. {{x_o} - {{\bar{x}}_o}} \right|} \right.} \right)}^2}}} $$

Where:

x _o :

Observed values

x _o :

Predicted values

${\bar{x}_o}$ :

Observed mean

${\bar{x}_p}$ :

Predicted mean

n :

Number of observation

i :

Vary from 1 to n