# Optimising Fuzzy Neural Network Architecture for Dissolved Oxygen Prediction and Risk Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Data-Driven Models and ANN

#### 1.2. Fuzzy Artificial Neural Networks for Risk Assessment

#### 1.3. Objectives

## 2. Materials and Methods

#### 2.1. Data Collection

^{2}in area) in southern Alberta, Canada. The Bow River is 645 km long and averages a 0.4% slope over its length [10]. The headwaters of the river are located at Bow Lake, in the Rocky Mountains, from where it flows south-easterly to Calgary (drainage area of 7870 km

^{2}), meeting the Oldman River and eventually draining into Hudson Bay [9,41]. The river is supplied by snowmelt from the Rocky Mountains, rainfall and runoff, and discharge from groundwater. The River has an average annual discharge of 90 m

^{3}/s, and an average width and depth of 100 m and 1.5 m, respectively [42].

^{3}/s for the selected period.

#### 2.2. ANN and Uncertainty Analysis

#### 2.2.1. Network Architecture

_{H}), and the amount of data used for training, validation and testing (known as data-division) for the early-stopping procedure described above. There is no consistent method used in the literature for each of these factors [30,46,47]. Typically, an ad hoc, or trial-and-error method is used to select the number of neurons [17,23,29,31]. The number of neurons selected must balance the complexity and generalisation of the final model; too many neurons increase the complexity and hence the processing speed, while reducing the transparency of the model. Not enough neurons risk reducing model performance and forgoing the ability of modelling non-linear systems. Similarly, the issue of data-division, which can have significant impacts on final model structure, is also predominantly conducted in an ad hoc and trial-and-error basis [31]. Generally speaking, two broad methods are available: In the first, each subset should have data that are statistically similar, including similar patterns or trends. Conversely, a method can be selected where each subset is based on some physical property, such as grouping the subsets chronologically.

_{H}and data-division for the ANN model described. The smallest number of neurons and the least amount of data for training is targeted. The first is to reduce computational effort. The second is to prevent the risk of over-fitting to the training data, and to have a larger dataset for testing for more robust statistical inference of that dataset.

_{H}and amount of training data) can be objectively selected. In doing so, both the processing time (i.e., the number of epochs, the amount of training data and n

_{H}) and the complexity (n

_{H}) of the system is accounted for in the final model architecture.

#### 2.2.2. Network Coefficients

#### 2.3. Risk Analysis

_{L}and A

_{R}, respectively:

_{L}+ A

_{R},

_{L}represents the cumulative probability between a and x which is assumed to equal the probability P(X < x), since the fuzzy number defines any values to less than a to be impossible (i.e., μ = 0). Given the fact that the fuzzy number is not symmetrical, the lengths of the two intervals [a, x] and [x', b] can be used to establish a relationship between A

_{L}and A

_{R}. Then, A

_{L}can be estimated as:

_{L}= P(X < x) = μ(x)/{1 + [(b − x')/(x − a)]}.

^{3}/s and 220 m

^{3}/s (at 2 m

^{3}/s intervals), and T ranged between 0 and 25 °C (at 0.2 °C intervals). For each prediction, the risk of low DO to be below either 5, 6.5 or 9.5 mg/L threshold was calculated using the previously described defuzzification technique. These intervals of Q and T were selected to reflect typical conditions in the Bow River. The thresholds correspond to the minimum acceptable DO concentration for the protection of aquatic life for 1-day (at 5 mg/L) [11], and for the protection of aquatic life in cold, freshwater for early-life stages (at 9.5 mg/L) and other-life stages (6.5 mg/L) [3].

## 3. Results and Discussion

#### 3.1. Network Architecture

_{H}and the percentage of data for each of the training, validation and testing subsets. Sample results of the proposed method are shown in Figure 2. Figure 2a shows the mean MSE (solid black line) of the test dataset for the initial 50%:25%:25% data-division scenario, with n

_{H}varying between 1 and 20 neurons. This simulation was repeated 100 times to account for the random selection of data and the upper and lower limits of MSE for each of these simulations are shown in grey. This figure demonstrates that the number of neurons did not have a noticeable impact on the MSE for this configuration. The most significant outcome of this process is that the variability (the difference between the upper and lower limits) of the performance seems to decrease after n

_{H}= 6 and increases again after n

_{H}= 12, with the lowest MSE at n

_{H}= 10. This result has two important implications: first, increasing the model complexity results in limited improvement of model performance, suggesting that a simpler model structure may be more suitable to describe the system. Second, the variability in performance indicates that the initial selection of data in each training subset can highly influence the performance of the test dataset, especially at the lower (i.e., n

_{H}< 6) and higher (i.e., n

_{H}> 12) ends of the spectrum of the proposed number of neurons. This suggests that an optimum selection of hidden neurons lies within this range (6 < n

_{H}< 12).

_{H}increases from 1 to 20. While the mean value does not show a notable change, the variability of the time needed for training (i.e., the number of epochs) drastically decreases as n

_{H}increases from 1 to 5. This means that a simpler model structure may require more time to train, and the performance of these simpler architectures (n

_{H}= 1 to 5) is more variable. This is likely because the initial dataset selection has a higher impact on the final model performance for less complex models. The lowest number of mean epochs for this analysis occurred at n

_{H}= 19, with 26 epochs. However, note that the variability of MSE at n

_{H}= 19 (in Figure 2a) is high, and that n

_{H}= 19 falls outside the range 6 < n

_{H}< 12, identified above.

_{H}= 10 scenario, which was the best performing scenario, i.e., had the lowest mean MSE for each data-division scenarios when compared to other n

_{H}values. However, the subplot illustrates that the MSE for the test dataset does not show a major trend as the amount of data for training is increased from the initial 50%. This means that for this scenario (n

_{H}= 10) increasing the amount of data used for training has minimal impact on model performance, indicating that using the least amount of data for training (and thus having a higher fraction available for validating and testing) would be ideal. Note that the mean MSE values were generally higher for all data-division scenarios when the selected n

_{H}was between 1 and 5 (follow the example shown in Figure 2a).

_{H}= 10 case, which demonstrates that the mean and the variability of the number of epochs does not demonstrate a clear trend, as the amount of data used for training is increased. The significance of this analysis is that the amount of computational effort (or time) does not necessarily decrease as a larger fraction of data is used for training. Given this result, the least amount of training data (50%) is the preferred choice for the number of neurons that result in the lowest MSE, which is n

_{H}= 10 as described above. For the n

_{H}= 10 case, the overall mean number of epochs for each data-division scenario is low ranging between 24 and 40 epochs.

_{H}= 10 with a 50%:25%:25% data-division was selected as the optimum architecture for this research. The fact that the mean and the variability of MSE was the lowest at n

_{H}= 10 makes it a preferred option over the n

_{H}= 19 case, which as a lower number of mean epochs but had higher variability in MSE. In other words, higher model performance was selected over model training speed (mean epochs at n

_{H}= 10 ranged between 9 and 122 for the 50%:25%:25% data-division scenario). Secondly no significant trend was seen as the amount of data used for training, validation and testing was altered, however lower MSE values were seen at n

_{H}= 10 compared to other at n

_{H}values. Thus, the option that guarantees the largest amount of independent data for validation and training is preferred. Given the fact that the mean MSE for the testing dataset does not show a significant change as the per cent of training data is increased from 50% to 75%, the initial 50%:25%:25% division is maintained as the final selection.

#### 3.2. Network Coefficients

^{®}Xeon microprocessor (with 4 GB RAM). The results of this optimisation are summarised in Table 2, which shows the amount of data captured within the resulting α-cut intervals after each optimisation.

^{L}or 0

^{R}) at different times steps are joined together creating bands representative of the predicted fuzzy numbers calculated at each time step. Note that the superscripts L and R define the lower bound and upper bounds of the interval at a membership level of 0, respectively. In doing so, it is apparent that all the observed values fall within the μ = 0 interval for the years 2006, 2007 and 2010, and nearly all observations in 2004. This difference in 2004 is because the per cent of data included within the μ = 0 interval was selected to be 99.5% rather than 100% to prevent over-fitting; the optimisation algorithm was designed to eliminate the outliers first to minimise the predicted interval. The low DO values in 2004 are the lowest of the study period, and thus are not captured by the FNN.

^{L}and 0.2

^{L}levels) compared to the higher DO events. However, it should be noted that compared to a crisp ANN, the proposed method provides some possibility of low DO, whereas the former only predicts a crisp result without a possibility of low DO. Thus, the ability to capture the full array of minimum DO within different intervals is an advantaged of the proposed method over existing methods.

#### 3.3. Risk Analysis

^{3}/s at 2 m

^{3}/s intervals, and water temperature was between 0 and 25 °C at 0.2 °C intervals. For each combination of Q and T, the fuzzy DO was calculated using the FNN. The inverse transformation was used to calculate the probability of predicted DO to be below 5 mg/L for each combination of inputs.

^{3}/s: the probability of low DO is more than 90%. The risk of low DO decreases with higher flow rate and lower temperature. The utility of this method is that a water-resource manager can use forecasted water temperature data and expected flow rates to quantify the risk of low DO events in the Bow River, and can plan accordingly. For example, if the risk of low DO reaches a specified numerical threshold or trigger, different actions or strategies (e.g., increasing flow rate in the river by controlled release from the upstream dams) can be implemented. The quantification of the risk to specific probabilities means that the severity of the response can be tuned to the severity of the calculated risk.

^{3}/s, respectively. For these conditions, the probability of DO to be less than 9.5 mg/L ranges between ~50% to more than 90% (based on results presented in Figure 11). This risk increases in the summer months where the average daily water temperature in the Bow River is usually above 10 °C; under this condition there is a high risk of low DO even at high flow rates, as seen in Figure 11. In contrast to this, Figure 10 shows that there is a relatively higher risk of low DO (below 6.5 mg/L) in the spring (April and May) and late summer (September), when flow rate can be as low as 50 m

^{3}/s, and the water temperature varies between 10 to 15 °C, resulting in a risk of low DO of about ~60%. These examples are meant to illustrate the potential utility of the data-driven and abiotic input parameter DO model, that can be used to assess the risk of low DO. Given that it is a data-driven approach, the model can be continually updated as more data become available, further refining the predictions.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**An aerial view Calgary, Canada showing the locations of: (

**a**) Water Survey of Canada flow monitoring site “Bow River at Calgary (ID: 05BH004); (

**b**) Bonnybrook; (

**c**) Fish Creek; and (

**d**) Pine Creek wastewater treatment plants; and two water quality sampling sites: (

**e**) Stier’s Ranch; and (

**f**) Highwood.

**Figure 2.**Sample results of the coupled method to determine the optimum number of neurons in the hidden layer and percentage of data for training, validation and testing subsets; the mean (solid black line) and upper and lower limits (in grey) of: (

**a**) the Mean Squared Error for the test dataset for each number of hidden neurons; (

**b**) the number of epochs for training; (

**c**) the Mean Squared Error for a range of training data size; and (

**d**) the number of epochs for 10 hidden neurons.

**Figure 3.**Sample results of the Fuzzy Neural Network optimisation algorithm to estimate the fuzzy number values of selected weights and biases in the FNN model.

**Figure 4.**A comparison of the observed and predicted crisp (black dots) and fuzzy intervals at μ = 0 (grey lines) for minimum Dissolved Oxygen in the Bow River for the training, validation and testing datasets.

**Figure 5.**Time-series comparison of the observations and Fuzzy Neural Network minimum Dissolved Oxygen for 2004 and 2006.

**Figure 6.**Time-series comparison of the observations and Fuzzy Neural Network minimum Dissolved Oxygen for 2007 and 2010.

**Figure 7.**Detailed view of time series for minimum observed Dissolved Oxygen and predicted fuzzy Dissolved Oxygen for 2004, 2006, 2007 and 2010, corresponding to days with low Dissolved Oxygen events.

**Figure 8.**Membership functions of the predicted minimum Dissolved Oxygen (dashed line) and observed minimum Dissolved Oxygen corresponding to the lowest Dissolved Oxygen observation for each year (solid black line) between 2004 and 2012.

**Figure 9.**Results of the low Dissolved Oxygen identification and risk analyses tool for Dissolved Oxygen less than 5 mg/L.

**Figure 10.**Results of the low Dissolved Oxygen identification and risk analyses tool for Dissolved Oxygen less than 6.5 mg/L.

**Figure 11.**Results of the low Dissolved Oxygen identification and risk analyses tool for Dissolved Oxygen less than 9.5 mg/L.

**Table 1.**Mean Squared Error and the Nash–Sutcliffe model Efficiency coefficient for the neural network at a membership level equal to 1.

MSE (mg/L)^{2} | NSE | |
---|---|---|

Train | 1.04 | 0.59 |

Validation | 1.22 | 0.61 |

Test | 1.42 | 0.51 |

**Table 2.**Percentage of data (%) captured within each fuzzy interval for the Fuzzy Neural Network model.

μ | Train | Validation | Test |
---|---|---|---|

1.00 | – | – | – |

0.80 | 20.02 | 16.10 | 15.85 |

0.60 | 40.05 | 34.63 | 33.41 |

0.40 | 60.07 | 52.44 | 52.44 |

0.20 | 80.10 | 80.00 | 80.24 |

0.00 | 99.51 | 98.78 | 99.02 |

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## Share and Cite

**MDPI and ACS Style**

Khan, U.T.; Valeo, C.
Optimising Fuzzy Neural Network Architecture for Dissolved Oxygen Prediction and Risk Analysis. *Water* **2017**, *9*, 381.
https://doi.org/10.3390/w9060381

**AMA Style**

Khan UT, Valeo C.
Optimising Fuzzy Neural Network Architecture for Dissolved Oxygen Prediction and Risk Analysis. *Water*. 2017; 9(6):381.
https://doi.org/10.3390/w9060381

**Chicago/Turabian Style**

Khan, Usman T., and Caterina Valeo.
2017. "Optimising Fuzzy Neural Network Architecture for Dissolved Oxygen Prediction and Risk Analysis" *Water* 9, no. 6: 381.
https://doi.org/10.3390/w9060381