Abstract
In the water management literature both the normal and log-normal distribution are commonly used to model stochastic water pollution. The normality assumption is usually motivated by the central limit theorem, while the log-normality assumption is often motivated by the need to avoid the possibility of negative pollution loads. We utilize the truncated normal distribution as an alternative to these distributions. Using probabilistic constraints in a cost-minimization model for the Baltic Sea, we show that the distribution assumption bias is between 1% and 60%. Simulations show that a greater difference is to be expected for data with a higher degree of truncation. Using the normal distribution instead of the truncated normal distribution leads to an underestimation of the true cost. On the contrary, the difference in cost when using the normal versus the log-normal can be positive as well as negative.
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Notes
Kampas and Adamidis [18] also develop a deterministic equivalent for normal–log-normal distribution.
With shape we imply varying values of skewness and kurtosis. Skewness is a measure of symmetry, i.e. how a variable is distributed to the left and right of the center point. Kurtosis measures the peakedness of a distribution and it reflects the probability of extreme events. Data sets with high kurtosis have heavy tails (the uniform distribution is an extreme example).
Xu et al. [28] investigate the normal and log-normal distribution assumptions for sediment yields that affect returns from farming. Gren et al. [13] adopted a similar approach but in the context of water pollution. Kampas and White [17] compare the normal and log-normal distributional assumptions to a non-parametric assumption.
For the percent point function, we start with the probability and compute the corresponding Q for the cumulative distribution function. Hence, note that Q is defined by the left-hand-side of Eq. 7.
Hence, the mean load of the log-normally distributed pollution will be characterized by Q*. The generality of Eqs. 10–13 makes it equally applicable for the normal as well as the log-normal distribution. Hence, \( \hat Q \) from Eq. 10 equals \( Q* - {K_\alpha }{\left[ {{\text{Var}}(Q)} \right]^{0.5}} \)from Eq. 7, assuming normal distribution.
The choice between log-normal and truncated normal depends on the shape of the stochastic load data.
Either one of Eqs. 13 and 15 can be used to obtain the percent point function. Giving a numerical example; suppose that v = 0, E(Q) = 0, Var(Q) = 1, and α = 0.5, than \( {K_\alpha } = {\Phi^{ - 1}}\left( {0.75} \right) = 0.67 \) applying Eq. 15. Applying Eq. 13 we have \( {K_\alpha } = F_{\text{TN}}^{ - 1}\left( {0.5} \right) = 0.67 \).
The marginal cost of wetland restoration is assumed to be 5,050 SEK/ha for all countries besides Poland, Russia, Latvia, Lithuania, and Estonia, where a 60% lower cost is assumed, as suggested by Gren et al. [11]. Moreover, it has been assumed that 5% of arable land can be converted to wetland. Finally, each hectare of wetland is assumed to reduce the amount of nitrogen by 0.150 tonnes.
In principle, it would be interesting to compare the implications of different distributional assumptions for nitrogen to those for phosphorus, as both nutrients contribute to eutrophication of the Baltic Sea. Therefore, we carried out similar calculations for phosphorus, but could establish that the probability of erroneously predicting negative phosphorus loads using the normal distribution was also zero. Therefore, for phosphorus, the difference between a normal and truncated normal distribution would also be zero.
The average stochastic pollution can be reduced to zero at most. For sizable standard deviations compared to Q*, this could restrict the possibility to state that \( P\left( {Q \leqslant {Q^*}} \right) \geqslant \alpha \) with a high level of reliability.
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Acknowledgments
The authors want to thank Clas Eriksson, Karin Larsen, Monica Campos, Yves Surry, and Erik Ansink for valuable comments. The usual disclaimer applies. Funding from Baltic Nest Institute, Aarhus University, Denmark, and the BONUS program is gratefully acknowledged.
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Kataria, M., Elofsson, K. & Hasler, B. Distributional Assumptions in Chance-Constrained Programming Models of Stochastic Water Pollution. Environ Model Assess 15, 273–281 (2010). https://doi.org/10.1007/s10666-009-9205-7
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DOI: https://doi.org/10.1007/s10666-009-9205-7