Particle swarm optimization method for fractional order prediction of water quality parameters

Abstract

The application is a particle swarm optimization water quality parameter fractional order prediction method, which comprises the following steps: 1, solving the fractional order r value through an algorithm; 2, performing non-equidistant transformation on a random oscillation sequence to convert the non-equidistant sequence into an equidistant sequence; 3, establishing a random oscillation sequence grey prediction model; 4, performing inverse transformation on the solved equidistant sequence to obtain a grey prediction model of the non-equidistant random oscillation sequence; and 5, obtaining each parameter of the water quality predicted by the model. The application considers that the fractional order model can better utilize new information, meet the new information priority principle, better fit and predict the equidistant random oscillation sequence, introduces a smoothing and oscillation average relative error weighted objective function, optimizes the fractional order r value through a particle swarm algorithm, establishes a non-equidistant random oscillation sequence fractional order grey prediction model, and improves the effectiveness and practicability of fitting and predicting the water quality parameters.

Description

Technical Field

[0001] This invention belongs to the field of water quality prediction and assessment technology, specifically a fractional-order prediction method for water quality parameters using particle swarm optimization. Background Technology

[0002] With economic development and accelerated urbanization, water pollution has become increasingly serious. Understanding water quality trends is crucial for the control and prevention of water pollution. Water quality monitoring and forecasting are the most important technologies in water environment management. In the absence of online measurement equipment, it is necessary to develop and apply water quality forecasting technologies. To indirectly obtain water quality parameter information, researchers have proposed various models, including mathematical statistics (Non-patent literature 1: Prediction of the alga growth in areservoir based on the stepwise cluster analysis. China Environmental Science, authors Sun Dongmei, Zhang Kai), grey system theory (Non-patent literature 2: Application of GreySystem Dynamic Model Group GM(1,1) in Water Quality Prediction of Qinhuai River. SICHUAN ENVIRONMENT, authors Hu Zhibing, Pang Yong, Song Weiwei, and Shao Yongxu), neural network models (Non-patent literature 3: A Back Propagation Neural Network Model based on kalman filter for water quality prediction, 2015 11th International Conference on Natural Computation, authors Yanfei Zhao, Zhihong Zou and Shenglong Wang), and water quality simulation models (Non-patent literature 4: Sensitivity Analysis and Water Quality Modeling of a Tidal River Using a Modified Streeter-Phelps Equation with HEC-RAS-Calculated Hydraulic Characteristics. Environmental Modeling). Various water quality parameter prediction methods exist, including those developed by Fan C, Wang WS, and Liu FR. Among these methods, the Grey Model (GM), established using grey system theory, is widely used for water quality prediction.However, extensive research has shown that integer-order grey models (GM(1,1) models) cannot fully utilize the advantages of new information and do not satisfy the principle of prioritizing new information. Fractional-order calculus, on the other hand, has good memory properties and better reveals the inherent laws of the system object, thus gaining widespread attention from scholars. Wu Lifeng believes that fractional-order calculus embodies an "In between" idea and proposed a fractional-order cumulative grey model (Non-patent literature 5: Discussion on Fractional-order Cumulative Grey Prediction Model. Proceedings of the 25th National Grey System Conference. Ed. China Higher Science and Technology Center, author Wu Lifeng). The selection of the fractional-order order r value is generally optimized using algorithms. Commonly used algorithms include particle swarm optimization, genetic algorithms, neural networks, and firefly algorithms. The objective function commonly used in algorithm optimization is mostly an error evaluation method, such as root mean square error (RMSE), mean absolute error (MAE), mean bias error (MBE), and mean absolute percentage error (MAPE).

[0003] In summary, existing methods for predicting water quality parameters have the following drawbacks:

[0004] 1) The trend of water quality parameters generally follows a non-exponential function, i.e., a random oscillation sequence; while the traditional NEROGM(1,1) model for water quality prediction is only applicable to conventional random oscillation sequences, and its prediction effect is poor when the water quality parameters show a random oscillation trend with fractional order.

[0005] 2) Although the fractional gray model can make better use of new information and satisfy the principle of prioritizing new information, it still has problems such as the accuracy of the fractional gray model being affected by the random error of the data itself and the cumbersome solution of the model itself.

[0006] 3) When fractional gray models use algorithms for optimization, the objective function is mostly an error evaluation method, which cannot fully utilize the information contained in the sequence. Summary of the Invention

[0007] This invention provides a particle swarm optimization method for fractional-order prediction of water quality parameters. While considering that fractional-order models can better utilize new information, satisfy the principle of prioritizing new information, and better fit and predict equally spaced random oscillating sequences, it introduces a smoothing and oscillation average relative error weighted objective function. The particle swarm optimization algorithm is used to optimize the fractional-order value r, establishing a fractional-order grey prediction model NEROFGM(r,1) for non-equally spaced random oscillating sequences. This invention proposes a water quality parameter prediction method suitable for water quality parameters exhibiting non-equally spaced random oscillation trends. The model established in this method is more consistent with actual conditions, improving the effectiveness and practicality of fitting and predicting water quality parameters, and overcoming the aforementioned shortcomings of existing water quality parameter prediction methods.

[0008] The technical solution of this invention is described below in conjunction with the accompanying drawings:

[0009] A fractional-order prediction method for water quality parameters using particle swarm optimization includes the following steps:

[0010] Step 1: Solve for the fractional order r using an algorithm;

[0011] Step 2: Perform a non-equal spacing transformation on the random oscillating sequence to convert the non-equal spacing sequence into an equal spacing sequence;

[0012] Step 3: Establish the ROFAGM(r,1) model, a fractional-order grey prediction model for random oscillating sequences;

[0013] Step 4: Perform an inverse transformation on the solved equally spaced sequence to obtain the fractional gray prediction model NEROFAGM(r,1) model for the non-equally spaced random oscillating sequence;

[0014] Step 5: Input the water quality data measured by the sensor into the model to obtain the parameters for predicting water quality.

[0015] Furthermore, the specific method for step one is as follows:

[0016] 11) Set the running parameters for the particle swarm optimization algorithm;

[0017] 12) Construct the objective function for particle swarm optimization; select the optimal fractional-order r value for the fractional-order model under the weighted average relative percentage error of minimum smoothing and oscillation average relative error, i.e.

[0018]

[0019] In the formula, q is the weight coefficient between sequences. To obtain the fitted values ​​of the oscillating sequence for the model, x (0) (k) is the fitted value of the actual oscillation sequence. To obtain the smooth sequence fitting value for the model, y (0)(k) represents the simulated value of the actual smoothed sequence;

[0020] Based on the objective function, the algorithm is used to optimize the fractional order r and find the optimal value.

[0021] Furthermore, the specific method for step 11) is as follows:

[0022] Learning factors c1 = 2, c2 = 2; dynamic inertia weight factor w = 0.8; maximum number of algorithm iterations is 3000; particle range r ∈ [0, 1]; population size is 50; precision is set to 0.000001.

[0023] Furthermore, the specific method for step two is as follows:

[0024] Let X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n If ] is a non-equally spaced sequence, then the corresponding equally spaced sequence is:

[0025]

[0026] in:

[0027]

[0028] In the formula, △x (0) (t i )=μ(t i )[x (0) (t i )-x (0) (t i-1 The total difference for each time period, i = 2, 3, ..., n; Let i be the unit time difference coefficient between each time period and the average time period, i = 2, 3, ..., n; Let i be the average time interval, i = 2, 3, ..., n; n be the value of X. (0) The number of elements.

[0029] Furthermore, the specific method for step three is as follows:

[0030] 31) The equally spaced random oscillation sequence is transformed into an equally spaced smooth sequence by amplitude compression;

[0031] 32) Accumulate and generate a new data sequence by performing calculations on the equally spaced smoothed sequence;

[0032] 33) The model parameters are obtained through least squares estimation, thereby establishing a discrete fractional grey model;

[0033] 34) Obtain a prediction model for equally spaced random oscillation sequences.

[0034] Furthermore, the specific method for step 31) is as follows:

[0035] Let the data sequence right when say It is a monotonically increasing sequence; when say It is a monotonically decaying sequence; when have and say Let be an oscillating sequence; Then T = Mm is called an oscillating sequence. The amplitude;

[0036] Suppose a random oscillating sequence If the amplitude is T, then Y (0) =[y (0) (1),y (0) (2),…,y (0) [(n-1)] is called A first-order smooth sequence; where, k = 1, 2, ..., n-1, is called... The first-order smoothing operator;

[0037] The specific method for step 32) is as follows:

[0038] Let the data sequence Y (r) =[y (r) (1),y (r) (2),…,y (r) [(n-1)] is the sequence Y (0) =[y (0) (1),y (0) (2),…,y (0) The cumulative generation sequence of [n-1] is then k = 1, 2, ..., n-1 or described as:

[0039] Y (r) =A1Y (0) (4)

[0040] In the formula, This is called a fractional-order cumulative generating matrix;

[0041] The Gamma function is defined as follows:

[0042]

[0043] When x≥0, formula (5) has Γ(x+1)=xΓ(x)=x! ;

[0044] The specific method for step 33) is as follows:

[0045] Combined with Y (0) and Y (1) The discrete fractional gray model FAGM(r,1) is established as follows:

[0046] y (r) (k+1)=β1y (r) (k)+β2 (6)

[0047] In the formula, β1 and β2 are model parameters, whose values ​​can be estimated using the least squares method; the specific formula is as follows:

[0048]

[0049] In the formula,

[0050] When y (r) (1)=y (0) When (1), the reduction formula of the discrete grey model FAGM(r,1) is as follows:

[0051]

[0052] The specific method for step 34) is as follows:

[0053] Let the data sequence For data sequences If the sequence is generated by cumulative subtraction, then... k = 2, ..., n-1 or can be described as:

[0054]

[0055] In the formula, To generate a cumulative decreasing matrix;

[0056] The restoration function expression for the discrete fractional grey model FAGM(r,1) is as follows:

[0057]

[0058] For equally spaced random oscillating sequences, the ROFAGM(r,1) model is obtained:

[0059]

[0060] In the formula, C = x (0) (2) = const; for The model fit value.

[0061] Furthermore, the specific method for step four is as follows:

[0062] The ROFAGM(r,1) model (11) is transformed into NEROFAGM(r,1):

[0063]

[0064] In the formula, For X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n The model fit value of )]; These are the model's predicted values; The total difference for each time period, j = 2, 3, ..., n + m; Let j = 2, 3, ..., n+m, where j is the unit time difference coefficient between each time period and the average time period. Let i be the average time interval, i = 2, 3, ..., n; m be the number of predicted values; rearrange formula (12) to obtain the NEROFAGM(r,1) model:

[0065]

[0066] The beneficial effects of this invention are as follows:

[0067] 1) The NEROFAGM(r,1) model described in this invention effectively expands the application range of solving random oscillating sequences from integer order to fractional order. The NEROGM(1,1) model can only simulate objects without fractional order characteristics well, while the NEROFAGM(r,1) model, through a particle swarm optimization method based on smoothing and oscillation average relative error weighting, makes it applicable not only to objects with fractional order characteristics, but also further explores the hidden patterns in the oscillation sequence, greatly expanding the application range of the model;

[0068] 2) The particle swarm optimization method based on the weighted average relative error of smoothing and oscillation proposed in this invention also has certain scalability. The conventional error evaluation methods used in existing literature are all for evaluating the error of the fitted and predicted random oscillating sequences; while the method proposed in this invention weights the random oscillating sequences and the smoothed sequences after data processing, making better use of the information contained in the sequences.

[0069] 3) The water quality prediction method proposed in this invention is applicable to water quality parameter monitoring and prediction, especially for systems where water quality parameters cannot be monitored in real time due to technical, physical and economic reasons, and has certain guiding significance for water quality environment management. Attached Figure Description

[0070] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0071] Figure 1 A flowchart of the modeling process for the NEROFAGM(r,1) model in water quality parameter prediction methods;

[0072] Figure 2a The fitted curves are for two different objective functions.

[0073] Figure 2b The average relative error curves of the fitted values ​​for two different objective functions are shown.

[0074] Figure 3a The graph shows the predicted values ​​for two different objective functions.

[0075] Figure 3b The graph shows the average relative error of the predicted values ​​for two different objective functions. Detailed Implementation

[0076] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0077] Example 1

[0078] See Figure 1 This embodiment provides a fractional-order prediction method for water quality parameters using particle swarm optimization, comprising the following steps:

[0079] Step 1: Solve for the fractional order r using an algorithm;

[0080] The specific method is as follows:

[0081] 11) Set the running parameters for the particle swarm optimization algorithm;

[0082] Learning factors c1 = 2, c2 = 2; dynamic inertia weight factor w = 0.8; maximum number of algorithm iterations is 3000; particle range r ∈ [0, 1]; population size is 50; precision is set to 0.000001.

[0083] 12) Construct the objective function for particle swarm optimization; select the optimal fractional-order r value for the fractional-order model under the weighted average relative percentage error of minimum smoothing and oscillation average relative error, i.e.

[0084]

[0085] In the formula, q is the weight coefficient between sequences. To obtain the fitted values ​​of the oscillating sequence for the model, x (0) (k) is the fitted value of the actual oscillation sequence. To obtain the smooth sequence fitting value for the model, y (0) (k) represents the simulated value of the actual smoothed sequence;

[0086] Based on the objective function, the algorithm is used to optimize the fractional order r and find the optimal value.

[0087] Step 2: Perform a non-equal spacing transformation on the random oscillating sequence to convert the non-equal spacing sequence into an equal spacing sequence;

[0088] The specific method is as follows:

[0089] Let X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n If ] is a non-equally spaced sequence, then the corresponding equally spaced sequence is:

[0090]

[0091] in:

[0092]

[0093] In the formula, △x (0) (t i )=μ(t i )[x (0) (t i )-x (0) (t i-1 The total difference for each time period, i = 2, 3, ..., n; Let i be the unit time difference coefficient between each time period and the average time period, i = 2, 3, ..., n; Let i be the average time interval, i = 2, 3, ..., n; n be the value of X. (0) The number of elements.

[0094] Step 3: Establish the ROFAGM(r,1) grey prediction model for random oscillating sequences;

[0095] The specific method is as follows:

[0096] 31) The equally spaced random oscillation sequence is transformed into an equally spaced smooth sequence by amplitude compression;

[0097] Let the data sequence right when say It is a monotonically increasing sequence; when say It is a monotonically decaying sequence; when have and say Let be an oscillating sequence; Then T = Mm is called an oscillating sequence. The amplitude;

[0098] Suppose a random oscillating sequence If the amplitude is T, then Y (0) =[y (0) (1),y (0) (2),…,y (0) [(n-1)] is called A first-order smooth sequence; where, k = 1, 2, ..., n-1, is called... The first-order smoothing operator;

[0099] 32) Accumulate and generate a new data sequence by performing calculations on the equally spaced smoothed sequence;

[0100] Let the data sequence Y (r) =[y (r) (1),y (r) (2),…,y (r) [(n-1)] is the sequence Y (0) =[y (0) (1),y (0) (2),…,y (0) The cumulative generation sequence of [n-1] is then k = 1, 2, ..., n-1 or described as:

[0101] Y (r) =A1Y (0) (4)

[0102] In the formula, This is called a fractional-order cumulative generating matrix;

[0103] The Gamma function is defined as follows:

[0104]

[0105] When x≥0, formula (5) has Γ(x+1)=xΓ(x)=x! ;

[0106] 33) Model parameters are obtained through least squares estimation, thereby establishing a discrete fractional gray model; enhancing the regularity of the sequence and improving its noise resistance.

[0107] Combined with Y (0) and Y (1) The discrete fractional gray model FAGM(r,1) is established as follows:

[0108] y (r) (k+1)=β1y (r) (k)+β2 (6)

[0109] In the formula, β1 and β2 are model parameters, whose values ​​can be estimated using the least squares method; the specific formula is as follows:

[0110]

[0111] In the formula,

[0112] When y (r) (1)=y (0) When (1), the reduction formula of the discrete grey model FAGM(r,1) is as follows:

[0113]

[0114] 34) Obtain a prediction model for equally spaced random oscillation sequences.

[0115] Let the data sequence For data sequences If the sequence is generated by cumulative subtraction, then... k = 2, ..., n-1 or can be described as:

[0116]

[0117] In the formula, To generate a cumulative decreasing matrix;

[0118] The restoration function expression for the discrete fractional grey model FAGM(r,1) is as follows:

[0119]

[0120] For equally spaced random oscillating sequences, the ROFAGM(r,1) model is obtained:

[0121]

[0122] In the formula, C = x(0) (2) = const; for The model fit value.

[0123] Step 4: Perform an inverse transformation on the solved equally spaced sequence to obtain a grey prediction model for the non-equally spaced random oscillating sequence;

[0124] The ROFAGM(r,1) model (11) is transformed into NEROFAGM(r,1):

[0125]

[0126] In the formula, For X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n The model fit value of )]; These are the model's predicted values; The total difference for each time period, j = 2, 3, ..., n + m; Let j = 2, 3, ..., n+m, where j is the unit time difference coefficient between each time period and the average time period. Let i be the average time interval, i = 2, 3, ..., n; m be the number of predicted values; rearrange formula (12) to obtain the NEROFAGM(r,1) model:

[0127]

[0128] Step 5: Input the water quality data measured by the sensor into the model to obtain the various parameters for predicting water quality.

[0129] Example 2

[0130] This embodiment uses the method of Embodiment 1 to predict the fractional order of water quality parameters, as detailed below:

[0131] The study used pH parameters measured offline in 2018, as shown in Table 1. Table 1 shows the data for 9 days from May 16th to October 14th. A new isometric pH sequence was obtained through non-equidistant sequence transformation, as shown in Table 1. The optimal fractional order r values ​​obtained from the two different objective functions are also different. The r value obtained using MAPE as the objective function is 0.892422; the r value obtained using a weighted average relative error of smoothing and oscillation with a weight of 0.9 as the objective function is 0.97646. For ease of calculation, r values ​​of 0.892 and 0.976 were taken respectively. Figure 2a , Figure 2bAs shown, the fitted values ​​of both models can approximate the original sequence very well. The absolute and relative errors between the fitted values ​​obtained by constructing the new objective function and the actual values ​​are relatively small. These data indicate that the proposed method of solving for the fractional-order r value based on smoothing and oscillatory average relative error weighting as the objective function is effective and feasible in water quality fitting. Figure 3a , Figure 3b As shown, the new objective function significantly outperforms the MAPE objective function in prediction. Fitting the original sequence to the original sequence yields good results for known original sequences. However, some errors occur when predicting data using the original sequence. Although some errors exist in the prediction process, the average relative error of NEROFAGM(r,1) is less than 20%. These data indicate that the proposed NEROFAGM(r,1) is effective and feasible in water quality prediction.

[0132] Table 1: Original non-equally spaced sequences and transformed equally spaced sequences

[0133]

[0134] Table 2: Fitting values ​​for different objective functions

[0135]

[0136] The objective function is a weighted average of the smoothed and oscillating relative errors (r = 0.976):

[0137] Fitted sequence:

[0138] Residual:

[0139] Relative error:

[0140] Average relative simulation error:

[0141] Table 3: Predicted values ​​for different objective functions

[0142]

[0143]

[0144] The preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the scope of protection of the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, any person skilled in the art can make equivalent substitutions or changes based on the technical solution and inventive concept of the present invention within the scope of the technology disclosed in the present invention. These simple modifications are all within the scope of protection of the present invention.

[0145] It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable way without contradiction. In order to avoid unnecessary repetition, the present invention will not describe the various possible combinations separately.

[0146] Furthermore, various different embodiments of the present invention can be combined in any way, as long as they do not violate the spirit of the present invention, they should also be regarded as the content disclosed by the present invention.

Claims

1. A fractional-order prediction method for water quality parameters using particle swarm optimization, characterized in that, Includes the following steps: Step 1: Solve for the fractional order r using an algorithm; Step 2: Perform a non-equal spacing transformation on the random oscillating sequence to convert the non-equal spacing sequence into an equal spacing sequence; Step 3: Establish the ROFAGM(r,1) grey prediction model for random oscillating sequences; Step 4: Perform an inverse transformation on the solved equally spaced sequence to obtain a grey prediction model for the non-equally spaced random oscillating sequence; Step 5: Input the water quality data measured by the sensor into the model to obtain the various parameters for predicting water quality; The specific method for step one is as follows: 11) Set the running parameters for the particle swarm optimization algorithm; 12) Construct the objective function for particle swarm optimization; select the optimal fractional-order r value for the fractional-order model under the weighted average relative percentage error of minimum smoothing and oscillation average relative error, i.e. In the formula, q is the weight coefficient between sequences. To obtain the fitted values ​​of the oscillating sequence for the model, x (0) (k) is the fitted value of the actual oscillation sequence. To obtain the smooth sequence fitting value for the model, y (0) (k) represents the simulated value of the actual smoothed sequence; The optimal value is obtained by using an algorithm to optimize the fractional order r based on the objective function. The specific method for step three is as follows: 31) The equally spaced random oscillation sequence is transformed into an equally spaced smooth sequence by amplitude compression; 32) Accumulate and generate a new data sequence by performing calculations on the equally spaced smoothed sequence; 33) The model parameters are obtained through least squares estimation, thereby establishing a discrete fractional grey model; 34) Obtain a prediction model for equally spaced random oscillation sequences; The specific method for step 31) is as follows: Let the data sequence right when say It is a monotonically increasing sequence; when say It is a monotonically decaying sequence; when have and say Let be an oscillating sequence; Then T = Mm is called an oscillating sequence. The amplitude; Suppose a random oscillating sequence If the amplitude is T, then Y (0) =[y (0) (1),y (0) (2),…,y (0) [(n-1)] is called A first-order smooth sequence; where, Called The first-order smoothing operator; The specific method for step 32) is as follows: Let the data sequence Y (r) =[y (r) (1),y (r) (2),…,y (r) [(n-1)] is the sequence Y (0) =[y (0) (1),y (0) (2),…,y (0) The cumulative generation sequence of [n-1] is then Or it can be described as: AND (r) =A1Y (0) (4) In the formula, This is called a fractional-order cumulative generating matrix; The Gamma function is defined as follows: When x≥0, formula (5) has Γ(x+1)=xΓ(x)=x! ; The specific method for step 33) is as follows: Combined with Y (0) and Y (1) The discrete fractional gray model FAGM(r,1) is established as follows: y (r) (k+1)=β1y (r) (k)+β2 (6) In the formula, β1 and β2 are model parameters, whose values ​​can be estimated using the least squares method; the specific formula is as follows: In the formula, When y (r) (1)=y (0) When (1), the reduction formula of the discrete grey model FAGM(r,1) is as follows: The specific method for step 34) is as follows: Let the data sequence For data sequences If the sequence is generated by cumulative subtraction, then... Or it can be described as: In the formula, To generate a cumulative decreasing matrix; The restoration function expression for the discrete fractional grey model FAGM(r,1) is as follows: For equally spaced random oscillating sequences, the ROFAGM(r,1) model is obtained: In the formula, C = x (0) (2) = const; for The model fit value.

2. The fractional-order prediction method for water quality parameters using particle swarm optimization according to claim 1, characterized in that, The specific method for step 11) is as follows: Learning factors c1 = 2, c2 = 2; dynamic inertia weight factor w = 0.8; maximum number of algorithm iterations is 3000; particle range r ∈ [0, 1]; population size is 50; precision is set to 0.000001.

3. The fractional-order prediction method for water quality parameters using particle swarm optimization according to claim 1, characterized in that, The specific method for step two is as follows: Let X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n If ] is a non-equally spaced sequence, then the corresponding equally spaced sequence is: in: In the formula, △x (0) (t i )=μ(t i )[x (0) (t i )-x (0) (t i-1 The total difference for each time period, i = 2, 3, ..., n; Let i be the unit time difference coefficient between each time period and the average time period, i = 2, 3, ..., n; Let i be the average time interval, i = 2, 3, ..., n; n be the value of X. (0) The number of elements.

4. The fractional-order prediction method for water quality parameters using particle swarm optimization according to claim 1, characterized in that, The specific method for step four is as follows: The ROFAGM(r,1) model (11) is transformed into NEROFAGM(r,1): In the formula, For X (0) =[x (0) (t1),x (0) (t2),…,x (0) (t n The model fit value of )]; These are the model's predicted values; The total difference for each time period, j = 2, 3, ..., n + m; Let j = 2, 3, ..., n+m, where j is the unit time difference coefficient between each time period and the average time period. Let i be the average time interval, i = 2, 3, ..., n; m be the number of predicted values; rearrange formula (12) to obtain the NEROFAGM(r,1) model:

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