Selection of the data time interval for the prediction of maximum ozone concentrations

Abstract

This paper highlights the problem of step-length selection for the one-step-ahead prediction of ozone called the data time interval. This is done using a case study-based comparison of two approaches for predicting the maximum daily values of tropospheric ozone. The first approach is the 1-day-ahead prediction and the second is the prediction of the maximum values based on a multi-step-ahead iteration of 1-h predictions. Gaussian process modelling is utilised for this comparison. In particular, evolving Gaussian-process models are used that update on-line with the incoming measurement data. These sorts of models have been successfully used in the past for the prediction of ozone pollution. This paper contributes an assessment of the way that the maximum ozone values are predicted. A comparison of the daily maximum ozone values forecasted by a model based on 1-day-ahead predictions with those obtained by iterated 1-h-ahead predictions of the ozone with predictions at predetermined hours of the day is given. The forecast results are in favour of the on-line model based on hourly predictions when approaching closer to the real maximum values of ozone, and in favour of the daily predictions when they are made on a daily basis.

Appendix: Performance measures

The following are performance measures used in the study.

• The root-mean-square error—RMSE:

  $$\begin{aligned} \mathrm {RMSE} = \sqrt{\frac{1}{N}\sum _{i=1}^N (E(\hat{y}_i)-y_i)^2}, \end{aligned}$$

  (9)

  where $y_i$ and $\hat{y}_i$ are the observation and the prediction in the i-th step, respectively, $E(\cdot )$ denotes the expectation, i.e., the mean value, of the random variable, and N is the number of used observations.

• The standardised mean-squared error—SMSE

  $$\begin{aligned} \mathrm {SMSE}=\frac{1}{N}\frac{\sum _{i=1}^N(E(\hat{y}_i)-y_i)^2}{\sigma _y^2}, \end{aligned}$$

  (10)

  where $\sigma _y^2$ is the variance of the observations.

• The Pearson’s correlation coefficient—PCC:

  $$\begin{aligned} \mathrm {PCC}=\frac{\sum _{i=1}^N(E(\hat{y}_i)-E(\hat{{\mathbf {y}}})) (y_i-E({\mathbf {y}}))}{N\sigma _y\sigma _{\hat{y}}}, \end{aligned}$$

  (11)

  where $E(\hat{{\mathbf {y}}})$ is the expectation, i.e., the mean value, of the vector of predictions, and $\sigma _y$, $\sigma _{\hat{y}}$ are the standard deviations of the observations and the predictions, respectively.

• The mean fractional bias—MFB:

  $$\begin{aligned} \mathrm {MFB}=\frac{1}{N}\sum _{i=1}^N\frac{E(\hat{y}_i)-y_i}{\frac{1}{2}(E(\hat{y}_i)+y_i)}. \end{aligned}$$

  (12)

• The factor of the modelled values within a factor of two of the observations—FAC2:

  $$\begin{aligned} \mathrm {FAC2}=\frac{1}{N}\sum _{i=1}^Nn_i\ \mathrm {with}\ n_i= {\left\{ \begin{array}{ll} 1 &{} \mathrm {for} \,\,\,0.5\le |\frac{E(\hat{y}_i)}{y_i}|\le 2,\\ 0 &{} \mathrm {else}. \end{array}\right. } \end{aligned}$$

  (13)

RMSE and SMSE are frequently used measures for the accuracy of the predictions’ mean values, which are 0 in the case of a perfect model. SMSE is the standardised measure with values between 0 and 1. PCC is a measure of the associativity and is not sensitive to bias. Its value is between $-\,1$ and $+\,1$, with ideally linearly correlated values resulting in a value 1. MFB is the measure that bounds the maximum bias and gives additional weight to underestimations and less weight to overestimations. Its value is between $-\,2$ and $+\,2$, with the value 0 in the case of a perfect model. FAC2 indicates the fraction of the data that satisfies the condition from Eq. (13). Its value is between 0 and 1, with the perfect model resulting in a value of 1.