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Statistical methods such as multiple linear regression (MLR) and classification and regression tree (CART) analysis were used to build prediction models for the levels of pollutant concentrations in Macao using meteorological and air quality historical data to three periods: (i) from 2013 to 2016, (ii) from 2015 to 2018, and (iii) from 2013 to 2018. The variables retained by the models were identical for nitrogen dioxide (NO2), particulate matter (PM10), PM2.5, but not for ozone (O3) Air pollution data from 2019 was used for validation purposes. The model for the 2013 to 2018 period was the one that performed best in prediction of the next-day concentrations levels in 2019, with high coefficient of determination (R2), between predicted and observed daily average concentrations (between 0.78 and 0.89 for all pollutants), and low root mean square error (RMSE), mean absolute error (MAE), and biases (BIAS). To understand if the prediction model was robust to extreme variations in pollutants concentration, a test was performed under the circumstances of a high pollution episode for PM2.5 and O3 during 2019, and the low pollution episode during the period of implementation of the preventive measures for COVID-19 pandemic. Regarding the high pollution episode, the period of the Chinese National Holiday of 2019 was selected, in which high concentration levels were identified for PM2.5 and O3, with peaks of daily concentration exceeding 55 μg/m3 and 400 μg/m3, respectively. The 2013 to 2018 model successfully predicted this high pollution episode with high coefficients of determination (of 0.92 for PM2.5 and 0.82 for O3). The low pollution episode for PM2.5 and O3 was identified during the 2020 COVID-19 pandemic period, with a low record of daily concentration for PM2.5 levels at 2 μg/m3 and O3 levels at 50 μg/m3, respectively. The 2013 to 2018 model successfully predicted the low pollution episode for PM2.5 and O3 with a high coefficient of determination (0.86 and 0.84, respectively). Overall, the results demonstrate that the statistical forecast model is robust and able to correctly reproduce extreme air pollution events of both high and low concentration levels.
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It is known that the probability is not a conserved quantity in the stock market, given the fact that it corresponds to an open system. In this paper we analyze the flow of probability in this system by expressing the ideal Black-Scholes equation in the Hamiltonian form. We then analyze how the non-conservation of probability affects the stability of the prices of the Stocks. Finally, we find the conditions under which the probability might be conserved in the market, challenging in this way the non-Hermitian nature of the Black-Scholes Hamiltonian.
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