CN106529818A - Water quality evaluation prediction method based on fuzzy wavelet neural network - Google Patents

Abstract

The invention provides a water quality evaluation prediction method based on a fuzzy wavelet neural network and aims to solve problems of slow convergence speed during water quality prediction, poor approximation effect and inaccurate prediction result existing in a BP neural network in the prior art. According to the method, a fuzzy wavelet neural network prediction model is constructed through utilizing the known water quality analysis index quantity, prediction index quantity and fuzzy rule quantity, and the fuzzy wavelet neural network prediction model comprises an input layer, a subordinate layer, a fuzzy rule layer, a wavelet layer, an output layer and a defuzzication layer; a subordinate function parameter and a wavelet parameter of the wavelet layer are adjusted, a cost function is further defined, a BP algorithm based on a gradient descent method is utilized to carry out parameter adjustment, problems of low convergence speed, easy-to-generate concussion effects and local optimum can be avoided, model stability is improved, an initial parameter is optimized through employing an artificial bee colony algorithm, and the method is mainly applicable to water quality index prediction.

Description

Water quality assessment Forecasting Methodology based on Fuzzy Wavelet Network

Technical field

The present invention relates to hydrology evaluation and foreca field, the more particularly to water quality assessment based on Fuzzy Wavelet Network are pre- Survey method.

Background technology

Water quality prediction is that waters functional areas are set up in water pollution control unit, is polluted using water quality index is corresponding to land-based area Corresponding relation between source, obtains the technology of target water quality information.Both at home and abroad in water environment and water pollution control, to water quality mould The research and application of type is achieved and is developed on a large scale very much.Water quality prediction method mainly has simulation of water quality model, mathematical statistical model and people Artificial neural networks model, applied research of the traditional BP cerebellar model arithmetic computer in terms of water quality prediction with evaluation is achieved with very big Development, but exist that convergence rate is relatively slow, generalization ability is poor, the not high enough shortcoming of precision of prediction, it is impossible to reach satisfied prediction As a result.

The content of the invention

For above-mentioned situation, to overcome the defect of prior art, the present invention is provided and a kind of is based on fuzzy wavelet nerve net The water quality assessment Forecasting Methodology of network, it is therefore intended that solution BP neural network convergence rate when water quality prediction is carried out is slower, approaches Effect is poor, and predict the outcome not accurately problem.

Its technical scheme is：

A, with known water quality analysis indexes number as m, prediction index number as o, number of fuzzy rules be n build fuzzy wavelet Neural network prediction model, the Fuzzy Wavelet Network forecast model include input layer, are subordinate to layer, fuzzy rule layer, little Ripple layer, output layer conciliate obscuring layer；

The input layer is used to be input into known water quality analysis indexes, namely input variable：x₁, x₂..., x_m；

Described to be subordinate to layer for calculating the angle value that is subordinate to of each input variable, membership function is：

Wherein m is input variable number, and n is number of fuzzy rules, i.e. the hidden nodes of third layer, c_ij、d_ijGauss is subordinate to The center of function and width, η_j(x_i) it is membership function of i-th linguistic variable relative to j-th strip rule；

Corresponding number of fuzzy rules n of its nodes of the fuzzy rule layer, each node represent a fuzzy rule, each node The output of fuzzy rule layer is expressed as follows：

μ_j(x)=η_j(x₁)*η_j(x₂)*…η_j(x_m), j=1,2 ..., n；

The wavelet layer introduces wavelet function, improves calculating and the approximation capability of network model, small echo using wavelet function It is defined as follows：

ψ_jX () is formed by mother wavelet function ψ (x) translation and extension, wherein a_j={ a_1j,a_2j,…a_mj, b_j={ b_1j, b_2j,…b_mjFlexible and shift factor is represented respectively, it is as follows that morther wavelet is taken as Mexico's hat wavelet：

J-th wavelet network of wavelet layer is output as：

Wherein,a_ij、b_ijFor wavelet parameter；

The output layer is the product of the output of fuzzy rule layer and the output of small echo layer network,

K_j=μ_j(x)*y_j=η_j(x₁)*η_j(x₂)*…η_j(x_m)*ω_jψ_j(z),

The ambiguity solution layer is used for the output for calculating whole neutral net, and which is expressed as：

B, to membership function parameter c_ij、d_ij, wavelet layer wavelet parameter ω_j、a_ij、b_ijIt is adjusted, defines cost function For：

WhereinWith u_iThe respectively desired output and reality output of network, o are output variable number, using being declined with gradient BP algorithm based on method carries out parameter adjustment, and to avoid, convergence rate is slow, easily sink into concussion effect and local optimum, increases Model stability, optimizes initial parameter using artificial bee colony algorithm, comprises the steps：

Step 1：Initialization honeybee populations, Apiss sum SN, gathering honey honeybee respectively account for SN/2 with honeybee is followed, maximum search number of times Limit, iterationses iter=0, maximum iteration time maxCycle；All Apiss are investigation honeybee pattern, randomly generate SN Individual feasible solution；

Step 2：Each several part parameter c of initialization network model_ij、d_ij、ω_j、a_ij、b_ij；

Step 3：By each parameter assignment to network model；

Step 4：Using training sample training network model；

Step 5：Fitness value is calculated, bee colony is divided into into gathering honey honeybee and two kinds of honeybee, initialization flag vector trial is followed I ()=0, records continuous dwell times of the gathering honey honeybee in same nectar source；

Step 6：The new nectar source of gathering honey honeybee Local Search, calculates fitness value, if being better than current nectar source, updates current gathering honey The nectar source position that honeybee is located, makes trial (i)=0, otherwise updates trial (i)=trial (i)+1；

Step 7：Calculating follows honeybee select probability, finds new nectar source with this probability per honeybee is only followed, and is converted into gathering honey honeybee Neighborhood search is carried out, fitness value is calculated, is judged whether to retain nectar source, is updated trial (i)；

Step 8：If trial (i)>Limit, then execution step 9, otherwise execution step 10；

Step 9：I-th gathering honey honeybee is abandoned current nectar source and referred to as investigates honeybee, randomly generates new nectar source in solution space；

Step 10：The globally optimal solution that the current all Apiss of record are found, iter=iter+1；

Step 11：If iter>MaxCycle, then obtain network model's parameter optimization initial value, otherwise return to step 4；

In algorithm, each nectar source represents one of search space solution, for the problem containing D variable, then i-th nectar source Position is Xⁱ=[x_i1,x_i2,…,x_iD]^T, the feasible solution for randomly generating is as follows：

Wherein, i ∈ { 1,2 ..., SN }, j ∈ { 1,2 ..., D }；

C, the initial parameter value that optimizing is obtained is assigned to into network model, by water analysis index, namely input variable： x₁, x₂..., x_m, the input layer of network model is input to, obtains predicting output valve.

The present invention replaces the linear function of tradition T-S type fuzzy neural network conclusion parts with wavelet function, and small echo is become Change and organically combined with fuzzy neural network so that prediction network has fast convergence rate, and approximation capability is strong and can avoid falling into Enter the advantages such as local optimum, and optimize the initial value of parameter to be determined using artificial bee colony algorithm, it is to avoid because treating in its network really Determine that parameter is more, gradient calculation workload big, by the defect such as initial value affecting is larger so that the stability of water quality assessment forecast model Improve.

Description of the drawings

The topological diagram of the Fuzzy Wavelet Network that Fig. 1 is used by water quality prediction method of the present invention.

Fig. 2 is this method and tradition T-S types using absolute average error and relative average error as in the case of evaluation index Fuzzy neural network and BP neural network correction data.

Fig. 3 is the inventive method water quality prediction result coordinate diagram.

Fig. 4 is T-S type fuzzy neural network water quality prediction result coordinate diagrams.

Fig. 5 is BP neural network water quality prediction result coordinate diagram.

Network evolution procedure charts of the Fig. 6 for this patent, tradition T-S types fuzzy neural network and BP neural network.

Specific embodiment

The specific embodiment of the present invention is described in further detail below in conjunction with accompanying drawing.

In one embodiment：As shown in figure 1, herein using the fuzzy neural network based on T-S models, fuzzy logic has one Type and two type two types, traditional fuzzy systems can not process the uncertainty of fuzzy rule, therefore in the face of complicated be System, it is impossible to set up effective and reasonable fuzzy rule.Two fuzzy systems mainly include Mamdani types and T-S types, T-S patterns paste Model uses IF-THEN fuzzy rules, the premise part of each rule to include premise variable and fuzzy set, and its effect is fixed An adopted Fuzzy subspaee, conclusion part are typically a linear function.Research shows that T-S types network is in terms of study accurately It is better than Mamdani networks.Traditional Wavelet neutral net is that the non-linear Sigmoid functions in BP neural network are adopted non-thread Property wavelet basiss replace, the fitting of nonlinear function by carrying out linear superposition with the nonlinear wavelet base for being taken realizing, i.e., It is fitted with the finite term of wavelet series.For the advantage with reference to fuzzy neural network and wavelet neural network, present invention small echo Function replaces the linear function of tradition T-S pattern type conclusion parts, forms new fuzzy wavelet-neural network model, and this is fuzzy little The fuzzy rule of ripple neural network model can be described as R_n：If x₁is A_n1and x₂is A_n2and…and x_m is A_nm,

Wherein, x₁, x₂..., x_mFor input variable, y₁, y₂..., y_nFor the output of wavelet function, A_ijIt is subordinate to letter for Gauss Number, represents the i-th rule of j-th input variable, and n is number of fuzzy rules；The small echo of above-mentioned Fuzzy Wavelet Network model It is defined as follows：

ψ_jX () is formed by mother wavelet function ψ (x) translation and extension, wherein a_j={ a_1j,a_2j,…a_mj, b_j={ b_1j, b_2j,…b_mjFlexible and shift factor is represented respectively, it is as follows that morther wavelet is taken as Mexico's hat wavelet：

The output of wavelet neural network is represented by：

Wherein,ψ_jThe j-th unit wavelet function of (x) for hidden layer, ω_jFor input layer and the weight of hidden layer Coefficient, wavelet neural network (WNN) with preferable approximation capability, relative to other kinds of multilayer perceptron and radial direction base net Network etc. has the characteristics of easily training, while the initial value of wavelet neural network parameter has considerable influence to its convergence rate, it is excellent Initial parameter after change can increase the stability and convergence rate of network.

The impact that each wavelet function is exported to Fuzzy Wavelet Network model (FWNN), IF- are indicated by formula (1) The fuzzy model of THEN rule formats can pass through constantly the membership function parameter of study adjustment premise part, conclusion part and stretch Contracting and shift factor come perfect, therefore wavelet function can improve calculating and the approximation capability of FWNN.

This example selects 6 kinds of indexs of water quality analysis, respectively ammonia nitrogen amount, dissolved oxygen, COD, permanganic acid Potassium index, total phosphorus, total nitrogen, input variable of above-mentioned six indexs as FWNN, and water grade output then as FWNN, Namely input layer number m=6, the output node number o=1 of ambiguity solution layer, while number of fuzzy rules n=4 of fuzzy rule layer, Fuzzy Wavelet Network model is set up under MatlabR2010b environment, and using equivalent interpolation water quality index normal data life Into training sample, training data 350 is built, separately initial parameter, ABC parameters is optimized using artificial bee colony (ABC) algorithm It is defined as：SN=40, Limit=8, maxCycle=50, then it is (4m+1) × n=100 that FWNN needs the number of parameters of optimization Individual, then each solution is 100 dimensional vectors.

The initial value of Optimization of Fuzzy wavelet neural network (FWNN) is as follows：

Step 1：Initialization honeybee populations, Apiss sum 40, gathering honey honeybee respectively account for 20 with honeybee is followed, maximum search number of times Limit=8, iterationses iter=0, maximum iteration time maxCycle=50；All Apiss are investigation honeybee pattern, at random Produce 40 feasible solutions；

Step 2：Each several part parameter c of initialization network model_ij、d_ij、ω_j、a_ij、b_ij；

Step 3：Fuzzy Wavelet Network (FWNN) is given by each parameter assignment；

Step 4：Using training sample training Fuzzy Wavelet Network (FWNN)；

Step 5：Fitness value is calculated, bee colony is divided into into gathering honey honeybee and two kinds of honeybee, initialization flag vector trial is followed I ()=0, records continuous dwell times of the gathering honey honeybee in same nectar source；

Step 6：The new nectar source of gathering honey honeybee Local Search, calculates fitness value, if being better than current nectar source, updates current gathering honey The nectar source position that honeybee is located, makes trial (i)=0, otherwise updates trial (i)=trial (i)+1；

Step 7：Calculating follows honeybee select probability, finds new nectar source with this probability per honeybee is only followed, and is converted into gathering honey honeybee Neighborhood search is carried out, fitness value is calculated, is judged whether to retain nectar source, is updated trial (i)；

Step 8：If trial (i)>8, then execution step 9, otherwise execution step 10；

Step 9：I-th gathering honey honeybee is abandoned current nectar source and referred to as investigates honeybee, randomly generates new nectar source in solution space；

Step 10：The globally optimal solution that the current all Apiss of record are found, iter=iter+1；

Step 11：If iter>50, then obtain the network model's initial parameter value after optimizing, otherwise return to step 4；

In algorithm, each nectar source represents one of search space solution, for the problem containing D variable, then i-th nectar source Position is Xⁱ=[x_i1,x_i2,…,x_iD]^T, the feasible solution for randomly generating is as follows：

Wherein, i ∈ { 1,2 ..., SN }, j ∈ { 1,2 ..., D }；

Then the network parameter initial value obtained by artificial bee colony algorithm is assigned to into Fuzzy Wavelet Network (FWNN), by water analysis index namely input variable：x₁, x₂..., x_m, the input layer of network model is input to, is predicted Output valve.

In order to verify the beneficial effect of this patent method, herein using absolute average error (MAE) and relative average error (MAPE) as evaluation index：

Fuzzy Wavelet Network is contrasted with tradition T-S types fuzznet and BP neural network, is tested Sample group sequence number 1-50, every group of packet contain 6 input variables, and concrete data and group number are as follows：

Build with identical input number of nodes, traditional T-S types fuzzy neural network of identical output node number, BP nerve net Network, above-mentioned two network model and Fuzzy Wavelet Network (FWNN), will with identical input number of nodes and output node number Above-mentioned 1-50 groups data are separately input in each network model for training, and, as abscissa, network is defeated for the group number with test sample Go out index water grade Fig. 3 to Fig. 5 is obtained as vertical coordinate, and it is the mean error (MAE) according to the output of each network, relatively average Error (MAPE) makes form such as Fig. 2, can be apparent from Fuzzy Wavelet Network and possess more accurately predicting, its error amount It is less.

Traditional T-S types fuzzy neural network, BP neural network, Fuzzy Wavelet Network (FWNN) model are all needed after setting up The adjustment entered by line parameter, the parameter value for needing continuous iteration more to be optimized during the adjustment of parameter, therefore with In above-mentioned three kinds of network modeies are built, during parameter adjustment, its training error and iterationses build such as Fig. 6 coordinates, wherein training Error is vertical coordinate, and during parameter optimization, its iterations is abscissa, it can be seen that Fuzzy Wavelet Network (FWNN) possesses Convergence rate, namely the process of its acquisition optimized parameter faster is more efficient and convenient, is better than traditional T-S types fuzznet Network, BP neural network.

Claims (1)

1. the water quality assessment Forecasting Methodology based on Fuzzy Wavelet Network, it is characterised in that, comprise the steps:

A, with known water quality analysis indexes number as m, prediction index number as o, number of fuzzy rules be n build fuzzy wavelet nerve Network Prediction Model, the Fuzzy Wavelet Network forecast model include input layer, be subordinate to layer, fuzzy rule layer, wavelet layer, Output layer conciliates obscuring layer；

The input layer is used to be input into known water quality analysis indexes, namely input variable：x₁, x₂..., x_m；

Described to be subordinate to layer for calculating the angle value that is subordinate to of each input variable, membership function is：

$\begin{array}{ccc}{\mathrm{\η}}_j\left(x_i\right)=e^{-\frac{{(x_i-c_{ij})}^2}{2d_{ij}^2}}& i=1,2,\mathrm{...},m,& j=1,2,\mathrm{...},n\end{array}$

Wherein m is input variable number, and n is number of fuzzy rules, i.e. the hidden nodes of third layer, c_ij、d_ijGauss member function Center and width, η_j(x_i) it is membership function of i-th linguistic variable relative to j-th strip rule；

Corresponding number of fuzzy rules n of its nodes of the fuzzy rule layer, each node represent a fuzzy rule, and each node is obscured Rules layer output is expressed as follows：

μ_j(x)=η_j(x₁)*η_j(x₂)*…η_j(x_m), j=1,2 ..., n；

The wavelet layer introduces wavelet function, improves calculating and the approximation capability of network model, small echo definition using wavelet function It is as follows：

${\mathrm{\ψ}}_j\left(x\right)=\frac{1}{\sqrt{\left|a_j\right|}}\mathrm{\ψ}\left(\frac{x-b_j}{a_j}\right),a_j\≠0,j=1,2,...,n$

ψ_jX () is formed by mother wavelet function ψ (x) translation and extension, wherein a_j={ a_1j,a_2j,…a_mj, b_j={ b_1j,b_2j,… b_mjFlexible and shift factor is represented respectively, it is as follows that morther wavelet is taken as Mexico's hat wavelet：

$\mathrm{\ψ}\left(x\right)=\frac{1}{\sqrt{\left|a\right|}}(1-x^2)e^{\left(\frac{-x^2}{2}\right)},$

J-th wavelet network of wavelet layer is output as：

$y_j={\mathrm{\ω}}_j{\mathrm{\ψ}}_j\left(z\right),{\mathrm{\ψ}}_j\left(z\right)={\Σ}_{i=1}^m\frac{1}{\sqrt{\left|a_{ij}\right|}}(1-z_{ij}^2)e^{-\frac{z_{ij}^2}{2}},$

Wherein,a_ij、b_ijFor wavelet parameter；

The output layer is the product of the output of fuzzy rule layer and the output of small echo layer network,

K_j=μ_j(x)*y_j=η_j(x₁)*η_j(x₂)*…η_j(x_m)*ω_jψ_j(z),

${\mathrm{\ψ}}_j\left(z\right)={\Σ}_{i=1}^m\frac{1}{\sqrt{\left|a_{ij}\right|}}(1-z_{ij}^2)e^{-\frac{z_{ij}^2}{2}},$

The ambiguity solution layer is used for the output for calculating whole neutral net, and which is expressed as：

$u={\mathrm{\Σ}}_{j=1}^n{\mathrm{\μ}}_j\left(x\right)y_j/{\mathrm{\Σ}}_{j=1}^n{\mathrm{\μ}}_j\left(x\right);$

B, to membership function parameter c_ij、d_ij, wavelet layer wavelet parameter ω_j、a_ij、b_ijIt is adjusted, defining cost function is：

$E=\frac{1}{2}{\mathrm{\Σ}}_{i=1}^o{(u_i^d-u_i)}^2$

WhereinWith u_iThe respectively desired output and reality output of network, o are output variable number, using with gradient descent method are The BP algorithm on basis carries out parameter adjustment, and to avoid, convergence rate is slow, easily sink into concussion effect and local optimum, increases model Stability, optimizes initial parameter using artificial bee colony algorithm, comprises the steps：

Step 1：Initialization honeybee populations, Apiss sum SN, gathering honey honeybee respectively account for SN/2 with honeybee is followed, maximum search number of times Limit, iterationses iter=0, maximum iteration time maxCycle；All Apiss are investigation honeybee pattern, randomly generate SN Individual feasible solution；

Step 2：Each several part parameter c of initialization network model_ij、d_ij、ω_j、a_ij、b_ij；

Step 3：By each parameter assignment to network model；

Step 4：Using training sample training network model；

Step 5：Fitness value is calculated, bee colony is divided into into gathering honey honeybee and two kinds of honeybee is followed, initialization flag vector trial (i)= 0, record continuous dwell times of the gathering honey honeybee in same nectar source；

Step 6：The new nectar source of gathering honey honeybee Local Search, calculates fitness value, if being better than current nectar source, updates current gathering honey honeybee institute Nectar source position, make trial (i)=0, otherwise update trial (i)=trial (i)+1；

Step 7：Calculating follows honeybee select probability, and finding new nectar source with this probability per honeybee is only followed, and be converted into gathering honey honeybee is carried out Neighborhood search, calculates fitness value, judges whether to retain nectar source, updates trial (i)；

Step 8：If trial (i)>Limit, then execution step 9, otherwise execution step 10；

Step 9：I-th gathering honey honeybee is abandoned current nectar source and referred to as investigates honeybee, randomly generates new nectar source in solution space；

Step 10：The globally optimal solution that the current all Apiss of record are found, iter=iter+1；

Step 11：If iter>MaxCycle, then obtain network model's parameter optimization initial value, otherwise return to step 4；

In algorithm, each nectar source represents one of search space solution, for the problem containing D variable, then i-th nectar source position For X^I=[x_i1,x_i2,…,x_iD]^T, the feasible solution for randomly generating is as follows：

$X_{ij}=X_j^{\min }+rand(0,1)(X_j^{\max }-X_j^{\min })$

Wherein, i ∈ { 1,2 ..., SN }, j ∈ { 1,2 ..., D }；

C, the initial parameter value that optimizing is obtained is assigned to into network model, by water analysis index, namely input variable：x₁, x₂..., x_m, the input layer of network model is input to, obtains predicting output valve.