A novel dissolved oxygen prediction model based on enhanced semi-naive Bayes for ocean ranches in northeast China

Study area and data sources

The study area includes a total of 19 marine ranches in the Bohai and Yellow Seas in northeastern China, as shown in Fig. 1, all of which are within the northern temperate zone. Marine ranches focus on aquatic product processing and logistics, mariculture, pelagic fisheries, and the farming of sea bass, salmon, kelp, scallops, abalone, sea cucumbers, and other marine crops. These marine ranches are equipped with various water quality monitoring sensors, which can collect water quality parameters, including DO, water depth, chlorophyll content, and temperature in real time; we choose one of them, DO, as the research object of this paper.

Bayesian decision and an extremely large a posteriori hypothesis

Bayesian decision making is essentially a method for achieving classification by calculating probabilities. By counting the set of possible values Y of all samples, the set of attributes X corresponding to a sample y and the nearest set of observed attribute values x, the probability of occurrence of each possible category y can be calculated, and by selecting the maximum value of the probability, it is then possible to predict the most likely category (Jiang et al., 2019b). This principle provides a theoretical basis for the application of Bayesian decision making to DO prediction.

The goal of the Bayesian classifier is to find the hypothesis y ∈ Y that has the highest likelihood given the dataset X. The most likely hypothesis is called the maximum a posteriori hypothesis and is denoted as h_MAP. Given a training set and a set of test instances {x₁, x₂, …, x_m}, h_MAP is required to predict which class the test instance belongs to. According to the idea of the largest posterior hypothesis, the goal of the Bayesian optimal classifier is to select the event with the largest posterior probability P(y|X) as the predicted class in the attribute set X, as in Eq. (1).

(1) $$\begin{array}{cc}{h}_{MAP}\hfill & =\underset{y\in Y}{argmax}P(y|X)\hfill \\ \hfill & =\underset{y\in Y}{argmax}{\displaystyle \frac{P(y)P(X|y)}{P(X)}}\hfill \end{array}$$where P(y) refers to the probability of occurrence of sample y in sample set Y and is called the prior probability of y. P(X) refers to the prior probability of the training dataset X. P(y|X) represents the probability of observing sample y under the condition that attribute X appears. P(X|y) refers to the probability that attribute X holds given sample y.

Since the denominator is a constant that does not depend on y, it can be simplified, as in Eq. (2).

(2) $$\begin{array}{cc}{h}_{MAP}\hfill & =\underset{y\in Y}{argmax}P(X|y)P(y)\hfill \\ \hfill & =\underset{x\in X}{argmax}P(x_1,x_2,\cdots ,x_m|y)P(y)\hfill \end{array}$$

However, the complete estimation of P(x₁, x₂, …, x_m|y) requires computing the joint probabilities on all attributes, which is an “NP-hard” problem that cannot be solved in the range of polynomial time complexity.

Semi-naive Bayesian algorithm

To avoid the combinatorial explosion problem caused by the direct calculation of joint probabilities and to enable the model to be solved efficiently within the range of polynomial time complexity, it is necessary to introduce the “conditional independence assumption”, which is based on Bayesian theory, assume “conditional independence among attributes” and thus obtain the naive Bayesian formulation in (3).

(3) $$P(x_1,x_2,\cdots ,x_m|y)=\prod _{j=1}^{m}P(x_j|y)$$

The objective function h_MAP of the naive Bayesian formulation is described in (4).

(4) $$h_{MAP}=\underset{y\in Y}{argmax}P(y)\prod _{j=1}^{m}P(x_j|y)$$where x_j is the jth attribute value, and P(y) and P(x_j|y) in the above formula can be found with Eq. (5).

(5) $$\begin{array}{cc}P(y)\hfill & ={\displaystyle \frac{{\sum }_{i=1}^n\delta (y_i,y)}{n}}\hfill \\ P(x_j|y)\hfill & ={\displaystyle \frac{{\sum }_{i=1}^n\delta (x_{ij},x_j)\delta (y_i,y)}{{\sum }_{i=1}^n\delta (y_i,y)}}\hfill \end{array}$$where y_i denotes the class label of the ith training instance. x_ij denotes the jth attribute value of the ith training instance, and δ(y_i,y) is a binary function that is 1 when y_i = y and 0 otherwise.

The naive Bayesian model does not consider the relationships among attributes, which is often difficult to maintain when forecasting in practice. Therefore, by relaxing the assumption of conditional independence among attributes and considering the interactions among attributes, an enhanced semi-naive Bayesian classification model (NSB) is formed by assuming that all attributes are dependent on one attribute, as in Eq. (6). The dependent attributes are called “super-parent” attributes.

(6) $$P(y|\mathbf{x})\propto \sum _{i=1}^{d}P(y,x_i)\prod _{j=1}^{d}P(x_j|y,x_i)$$

To avoid the case where P(y,x_i) and P(x_j|y,x_i) are equal to 0, it is necessary to estimate them with the Laplace equation as $\hat{P}(y,x_i)$ and $\hat{P}(x_j|y,x_i)$:

$$\begin{array}{cc}\hat{P}(y,x_i)\hfill & ={\displaystyle \frac{|D_{y,x_i}|+1}{|D|+N\times N_i}}\hfill \\ \hat{P}(x_j|y,x_i)\hfill & ={\displaystyle \frac{|D_{y,x_i,x_j}|+1}{|D_{y,x_i}|+N_j}}\hfill \end{array}$$where | $D_{y,x_i}$| refers to the number of predicted values y and the number i of attribute values x_i. Z represents the size of the training set Diff_train of the difference series, D represents the total number of samples, and L represents the length of the sliding window (Bravo et al., 2015; Kim et al., 2014). The size of D is equal to Z − L. N refers to the total number of possible values of the prediction y, and N_i refers to the total number of possible values of the ith attribute. | $D_{y,x_i,x_j}$| refers to the number of predicted values y, where the ith attribute has a value of x_i and the jth attribute has a value of x_j, and N_j refers to the total number of possible values of the jth attribute.

Difference series

A time series with smoothness means that its time plots are approximately horizontal over a long period of time and maintain a stable variance. For example, white noise series and other time series with smoothness do not have trends or seasonality, do not change with time and have value ranges that are relatively easy to determine, and their DO and other water quality parameter data have the characteristics of nonsmoothness. To reduce interference from the irregular fluctuation of DO data with the model prediction, this paper uses the first-order difference to preprocess the DO series, as in Eq. (7).

(7) $$\begin{array}{cc}Diff\hfill & =\{dif{f}_1,dif{f}_2,\dots ,dif{f}_n\}\hfill \\ \hfill & =\{d{o}_2-d{o}_1,d{o}_3-d{o}_2,\dots ,d{o}_m-d{o}_{m-1}\}\hfill \end{array}$$

The semi-naive Bayesian model is a classification model, and water quality data, such as DO, are a continuous type of data, so the semi-naive Bayesian model cannot be used directly to predict DO data. In this paper, we analyze the distribution of the data after the first-order difference, as shown in Fig. 2, and find that the DO data after the first-order difference have an approximately normal distribution. DO data with the characteristics of a normal distribution can be filtered by setting the frequency threshold γ for the lower frequency attribute values to make more accurate predictions. In this paper, the possible values of the difference series are regarded as a finite class, and the DO time series can be predicted effectively by combining a semi-naive Bayesian classification model with a finite number of values from the difference series as the prediction target.

Single pasture prediction evaluation

In this paper, we first selected DO data from a marine pasture and implemented an enhanced semi-naive Bayesian prediction model with GoLang programming. In this dataset, there are 12,589 records in the testing data and 113,297 records in the training data. Additionally, the same water quality dataset is predicted with the long short-term memory (LSTM) (Bi, Liu & Li, 2020) and the RBFNN models (Moradi et al., 2020), and the prediction results are compared with those of the enhanced semi-naive Bayesian prediction model. To quantitatively represent the prediction effects of the different algorithms, the root mean square error (RMSE) (Hyndman & Koehler, 2006), mean absolute percentage error (MAPE) (de Myttenaere et al., 2016) and mean absolute error (MAE) (Willmott & Matsuura (2005)) are used as error functions; they are described in Eqs. (9)–(11). The prediction error of each model is calculated and shown in Table 1. The NSB model proposed in this paper has improved in prediction accuracy over similar algorithms.

(9) $$\begin{array}{cc}RMSE\hfill & =\sqrt{{\displaystyle \frac{{\sum }_{t=1}^m{({\hat{y}}_t-y_t)}^2}{m}}}\hfill \end{array}$$

(10) $$\begin{array}{cc}MAPE\hfill & ={\displaystyle \frac{1}{m}{\sum }_{t=1}^m\left|{\displaystyle \frac{{\hat{y}}_t-y_t}{y_t}}\right|}\hfill \end{array}$$

(11) $$\begin{array}{cc}MAE\hfill & ={\displaystyle \frac{{\sum }_{i=1}^m\left|{\hat{y}}_t-y_t\right|}{m}}\hfill \end{array}$$

To more intuitively represent the difference in the prediction accuracies of the different prediction algorithms, a comparison graph of the algorithms is drawn and shown in Fig. 4. The figure shows that the prediction values obtained by the enhanced semi-naive Bayesian prediction model proposed in this paper can generally fit the actual DO in the experimentally selected marine pasture well. The graph shows that the DO content in the marine pasture water environment dropped suddenly at approximately 2:00 pm; the model proposed in this paper was able to adjust the prediction results in time, and the prediction results fluctuated around the true value. The enhanced semi-naive Bayesian model can predict the DO more smoothly and accurately when the real value of the DO is stable.

Prediction accuracy evaluation on data from multiple pastures

The accuracy of the prediction model can only indicate the fitting effect of the prediction model to known data, while the prediction of DO data by intelligent marine ranching is mainly concerned with the actual forecasting effect on future data. Hence, the generalized performance is the main index for measuring the actual forecasting ability of the model.

Willmott’s index of agreement (WIA) is an index proposed by Willmott (1981) and is a standardized measure of the degree of the model prediction error. A WIA result of 1 indicates that the estimated value matches the actual value perfectly, while a result of 0 indicates that the estimated value does not match the actual value at all. The model is generally considered to have predictive significance when the WIA is greater than 0.6. In this paper, the WIA is used to evaluate the generalized performance of the established DO prediction model; the WIA is described in Eq. (12).

(12) $$\begin{array}{c}WIA=1-{\displaystyle \frac{{\sum }_{i=1}^n{(O_i-P_i)}^2}{{\sum }_{i=1}^n{(\left|P_i-\overline{O}\right|+\left|O_i-\overline{O}\right|)}^2},0\le WIA\le 1}\hfill \end{array}$$where O denotes the observed value, P denotes the predicted value, and ō denotes the average observed value. The average observation is calculated in Eq. (13).

(13) $$\begin{array}{c}\overline{O}=\sum _{i=1}^n{\displaystyle \frac{O_i}{n}}\hfill \end{array}$$

To further illustrate that the model proposed in this paper has a good ability to predict using data from multiple marine ranches, the DO data of sixteen marine ranches are predicted in this paper, as shown in Table 2. The WIA results in the table are all greater than 0.9 for each pasture, indicating that the enhanced semi-naive Bayesian algorithm can be used to predict the DO data for each marine pasture. Moreover, the error values remained low for each marine range.

To observe the prediction from the proposed model on singular and missing values, the prediction on the DO data from each farm on any day is plotted in Fig. 5. Ranches No. 3, No. 9 and Nos. 13 to 15 show that the enhanced semi-naive Bayesian classification algorithm proposed in this paper has a good prediction effect on smoother data and mutant data. This is because when the data are smooth, the DO difference series varies less, the transformation pattern has previously appeared, and thus, a smooth prediction can be made. For mutated data, a typical feature is a sudden increase or decrease in the values, while the data before and after the mutation are in stable ranges; this feature is reflected in the difference series as a number of values with large absolute values. Since the sliding window method is used, the model is able to predict a mutation with a higher probability for the next moment after the mutation occurs, as the end point of the sliding window has a larger absolute value, which is expressed in the prediction image as a lag of one moment for the predicted mutation. When the mutated data return to normal, most of the data in the sliding window are not mutated, and so the model continues to make normal predictions. For the other pastures, the predictions fluctuate around the true values when the DO data are cyclical. This feature is partly influenced by the frequency threshold γ. Since the frequency threshold restricts the values with a low probability of occurrence in the difference series and there are more possible values for difference changes in the cyclically varying data, resulting in a lower frequency in each difference series, only the more common, large and stable changes are retained after the frequency threshold restriction; thus, the model predicts large fluctuations around the true value. The second reason for the prediction fluctuations is the influence of the variation in the “super-parent” property. The “super-parent” property of the enhanced semi-naive Bayesian model is not fixed, and the values of each difference series can become the “super-parent” property within a sliding window. This feature leads to the “super-parent” attribute changing over time in the sliding window during the cyclical change process, and the “super-parent” attribute directly determines the prediction results of the model; it also leads to large fluctuations in the prediction results.

DM method for statistical tests

The DM test can be used to statistically quantify the degree of difference between any two models (Harvey, Leybourne & Newbold, 1997; Diebold & Mariano, 1995; Chen, Wan & Wang, 2014). By combining different error functions, the DM test results can be classified as DM-MAE, DM-MAPE, etc. The larger the absolute value of the test result is, the more significant the variability of the two models. In this paper, we compare the DM test results of the enhanced semi-naive classification model with those from similar models (as shown in Tables 3 and 4). The results show that the model proposed in this paper differs significantly from similar algorithms, and since the error in the algorithm in this paper is smaller than that of similar algorithms, it can be concluded that the model proposed in this paper is statistically superior to similar models.

Effects of the model parameters on the prediction results

Two parameters exist in the enhanced semi-naive Bayesian model proposed in this paper: one is the length L of the extracted attributes when building the model from the first-order difference series, and the other is the threshold γ specified in the calculation of the semi-naive Bayesian model. In this subsection, different parameter values are set independently, and the variability in the prediction effect with different parameter settings is represented in the form of a three-dimensional scatter plot, as shown in Fig. 6. The experimental results show that the prediction accuracy improves as γ and L increase. The accuracy is weakly correlated with the parameter settings when the values are greater than a certain value. However, in the actual experiments, an increase in the L value is accompanied by an increase in the model prediction time. Therefore, to improve the efficiency of the algorithm, it is more effective to set a relatively large γ than to increase the value of parameter L.