Donghui Li,Cong Shen ,Xiaopeng Dai,Xinghui Zhu,Jian Luo,Xueting Li,Haiwen Chen and Zhiyao Liang

Abstract: Data fusion can effectively process multi-sensor information to obtain more accurate and reliable results than a single sensor.The data of water quality in the environment comes from different sensors,thus the data must be fused.In our research,selfadaptive weighted data fusion method is used to respectively integrate the data from the PH value,temperature,oxygen dissolved and NH3 concentration of water quality environment.Based on the fusion,the Grubbs method is used to detect the abnormal data so as to provide data support for estimation,prediction and early warning of the water quality.

Keywords: Adaptive weighting,multi-source sensor,data fusion,loss of data processing,grubbs elimination.

1 Introduction

Multi-sensor information fusion is an emerging interdisciplinary discipline that has been formed since the 1970s.It has been widely used in high-tech fields such as military,national defense,and aerospace [Wu (2016);Guo (2017)],and has become a hot area attracting much attention.The primary goal of a multi-sensor system is to unite,correlate,and assemble the data from sensors with related database information to obtain more comprehensive,higher,and more reliable data information.However,many problems should be encountered in areas of data correlation,sensor uncertainty,and data management.While the fundamental problem is the inherent uncertainty in the sensor measurement system.This uncertainty arises not only from inaccuracies and noise from measurements,but also from inconsistencies of different sensors.In the past,strategies were used to model this uncertainty and fuse different types of data to get consistent decisions.In the early 1980s,Bar-shalom [Bar-shalom (1981)] studied the correlation between two sensor subsystems and gave the formula for calculating the mutual covariance matrix.In the 1990s,Carlson proposed the well-known federal Kalman filter algorithm [Carlson (1990)],which uses the upper bound of variance to eliminate correlation and uniform information distribution principles.The algorithm avoids the calculation of cross-covariance matrix,but it is conservative in some respects.

In 1994,Kim [Kim (2002)] proposed a maximum likelihood fusion estimation algorithm that requires random variables to be normally distributed so as to construct a likelihood function.Later,considering the significance of linear minimum variance,Sun et al.[Sun and Deng (2004)] proposed three fusion algorithms,that are matrix weighting,diagonal weighting and scalar weighting.In recent years,Olfati-Saber [Olfati-Saber (2008)] proposed a distributed algorithm based on consistency strategy-Kalman Consensus filter.And it has attracted extensive attention due to the simple distributed architecture as well as the scalability and robustness of the algorithm.Zhangpin and other people,after analyzing how to deal with the uncertainty and inconsistency of sensor nodes' data in sensor network data fusion,have proposed an optimized Bayesian estimation of multi- sensor data fusion method-a Bayesian fusion algorithm based on Kalman filter.According to the ways that filter is applied to sensor data,fusion data,or to both,three different techniques were proposed:forward filtering,backward filtering,and forward-backward filtering.

In the late 1970s,concepts and nouns concerning the information synthetic began to appear in some publicly-published documents,and in the later long period,the term “data fusion” was commonly used [Zuo (2005)].Until the 1990s,the term “information fusion”,after consideration of the diversity of sensor information,was then widely adopted [Ling,Chen,Gu et al.(2000)].The concept of information fusion has been described in many ways.The literature Bar-shalom [Bar-shalom (1981)] gives many concepts of fusion based on specific behaviors and related fields of application.The literature Liu et al.[Liu,Zhao,Liu et al.(2005)] proposes a general description of information fusion.Based on which above,the literature Mao et al.[Mao,Xiao and Yang (2015)] gives the following description:The form of information fusion is a framework,which organizes,correlates,and synthesizes multi-source information through specific mathematical methods and technical tools in order to obtain high-quality and useful information.And a precise definition of “high quality” depends on the application object.Nagesware [Rao (2000)] of Tunisia proposed a method of physical system fusion based on physics rules,and achieved satisfactory results in the detection of methane hydroxide;Serge [Moigne (2000)] from France proposed the way of fusing wind speed and direction,which is better for solving the wind field problem;Reboul et al.[Reboul and Bruge (2000)] from the United States developed a fusion system (Landsat),which can be used to monitor the changes in the vegetation and so on.

The research of information fusion theory involves many basic theories.From the perspective of the algorithm,it can be roughly divided into two categories:probability statistics method and artificial intelligence method.Among them,probability statistics method is represented by Bayes [Guan (2013)] and its deformation methods.In artificial intelligence methods,Bayes estimation [Guan (2013)],evidence reasoning [Zhang (2016)] fuzzy theory [Guo (2013)] as well as neural network [Jia (2017)] in total account for 85 percent of the entire information fusion algorithm.And machine learning methods such as vector-supporting machines [Lu (2016);Ding,Zhang,Zhang et al.(2018);Xie (2014)],genetic algorithms [Li (2017)],rough sets [Guo (2017)] and some other artificial intelligence methods[Liu,Dong,Liu et al.(2018);Zhang,Cai,Liu et al.(2018);Sun,Cai,Li et al.(2018);Cui,Zhang,Cai et al.(2018)] have also been applied in information fusion.In our research,we also draw some methods [Cai,Wang,Zheng et al.(2013);Liu,Cai,Xu et al.(2015);Huang,Liu,Zhang et al.(2018)] in other articles.In this paper,on the basis of the research method of Li et al.[Li,Zhang,Xu et al.(2018);Liu,Tang,Li et al.(2019)],adaptive weighted data fusion method is adopted to eliminate the fused data by using Grubbs,so as to obtain data that can be actually used.

2 Adaptive weighted data fusion

2.1 General weighted data fusion

At present,the data fusion theory has been widely used in the fields of state estimation.Among the related algorithms,the weighted fusion algorithm is mature,and many research results have proved that the algorithm is optimal,unbiased,and has the smallest mean square error.The key to apply the weighted fusion algorithm lies in the determination of the weight coefficient,which in turn is inversely proportional to the variance measurement of sensors.Assume that there are N sensors of different accuracy,of which the variances areσ12,σ22,…,σn2,and the measured values of the sensors areX1,X2,…,Xn,thus the result of the weighted fusion is:

From the above equation,it can be seen that the measurement data with large variance gives a smaller weight,while the data with small variance gives a larger weight,so this data fusion method can obtain a more reliable measurement result than the method of the arithmetic average.

2.2 Adaptive weighted data fusion ideas

Assume that the mean square errors of the sensed data of the N sensors areσ12,σ22,...,σn2,respectively,the measured values of the sensor nodes are X1,X2,...,Xn,correspondently,and the weight factors of each sensor is W1,W2,...,W3,respectively.Since the data are independent of each other and belongs to unbiased estimation of X,the truth factor and weight factor of X after fusion would respectively satisfy the following relationships:

The total mean square error is:

X1,X2,...,Xnare independent from each other and are unbiased estimates of X.Thus conclusion can be drawn:

p ≠ q,p=1,2,...,n;q=1,2,...,n.And σ2can be written as:

According to the above formulas,the mean square error2σ is a multivariate quadratic function,so2σ must have a minimum value.According to the extreme value theory of the multivariate function,the minimum weight factor is:

So,the correspondent minimum mean square error is:

After adaptive weighted data processing,each sensor data from test points are fused into one data and sent to the monitoring and management center for applying water quality prediction and warning service.

The weighted data fusion algorithm does not require prior knowledge of the data from sensor measurement,and the data fusion results can be obtained merely by the data values through local decision after the fusion of a single sensor.

2.3 Adaptive weighted data fusion algorithm

Assume that two different sensors are used to measure a constant quantity.The observed values are:z1=x+v1,z2=x+v2.Among them,vi(i= 1,2) is the random error during the obser vation.Assumingthe two sensor observations are assumed to be independent of each other.The assumed estimate of x and the observed value of zi(i= 1,2) are linearly related,andis the unbiased estimation value of x,thus:

?= (ω1,ω2)is the weight of the measured value for each sensor.Assume that the estimation error isWith the mean squared error of which the cost function is,it can be inferred that:

Since E (v1)=E (v2)=0and E (x)=E (),it can be inferred that:ω2=1- ω1

Then the cost function can be written as:

thus:

In order to ensure the smallest J,? is derivatived and thus:

The optimal weight is:

The optimal estimate is:

The above formula shows that when the values of the two sensors are given suitable,the optimal estimation can be obtained by integrating the observed values through the observer.To extend this conclusion to multiple sensors,the variances of measured values from the multiple sensor groups should be setted as σi(i=1,2...n) respectively,and they are independent from each other.The estimated value of the true value is,and is the unbiased estimate of x .The weighting factors of the sensors are ωi(i=1,2...n) respectively.According to the extremum value-seeking theory of multivariate function,the weighting factor corresponding to the minimum mean-square error can be obtained as:

3 Examples of adaptive weighted data fusion algorithm

This study mainly focused on the detection of PH value,dissolved oxygen degree,temperature,and ammonia nitrogen concentrations in aquaculture environment.Five sets of sensors were placed at the No.2 pond of Quyuan aquaculture base in Yueyang City,Hunan Province.Taking into account the instability of the quality of the sensor equipment itself,each set of sensors was divided into experimental Group 1 and experimental Group 2.Both experimental Group 1 and experimental Group 2 contain 4 sensors,a PH value sensor,a dissolved oxygen sensor,a temperature sensor,and an ammonia nitrogen concentration sensor.

In order to verify the fusion effect of the adaptive weighted data fusion,experiments were conducted in fish ponds.Fish pond data of October 2017 were recorded every 10 minutes.5 points of time,that are 8:05,8:15,8:25,8:35,and 8:45 on October 1,2017,were chosen to fuse.The extracted data are shown in Tabs.1-5.Sensor Groups 1-4 denote sensors in the four corners of the fish pond.Sensor group 0 denotes the sensor group in the middle of the fish pond.Sensor 0-4 (A) denotes the Group 1 sensor in the experiment.Sensor 0-4 (B) indicates the Group 2 experimental sensors.

Table 1:Data in 8:05 at 1th October 2017

Table 2:Data in 8:15 at 1th October 2017

Table 3:Data in 8:25 at 1th October 2017

Table 4:Data in 8:35 at 1th October 2017

Table 5:Data in 8:45 at 1th October 2017

3.1 Weighted data fusion of temperature data

Taking the temperature data out from Tab.1 separately for weighted fusion analysis,the data are shown in the following Tab.6.

Table 6:Data in 8:05 at 1th October 2017

The arithmetic average value of the measurement data from Group A is:

The corresponding standard error is:

The arithmetic average value of the measurement data from Group B is

The corresponding standard error is:

Since T(1)and T(2)are the measurement data from the same batch,and there has been no statistical measurement of temperature,that is,the variance of the previous measurement σ1=∞,thus it can be concluded that (σ1)-1= 0.

According to the batch estimation theory,the variance of the temperature fusion value can be obtained as:

H is the coefficient matrix of the measurement equation and H=[1,1];D is the covariance of the measurement noise and

The fusion value of the measurement data obtained by combining the above formula is:

Substituting the data from Tab.6,result can be obtained as:The fusion value after weighted information fusion at 8:05 is:T1+=18.6989,and variance σ1+=0.0636.

Similarly,the temperature data in Tabs.2-5 are fused through the same weighted information fusion method,and the following temperature and temperature data fusion results are obtained:

Table 7:Fusion data of temperature data in different time points at 1th October 2017

3.2 Weighted data fusion of dissolved oxygen data

The weighted data fusion analysis of the dissolved oxygen data in Tables from 1 to 5 is performed.The weighted information fusion method is the same as the temperature data fusion calculation method in 3.1.Thus,the following results can be obtained:

Table 8:Fusion data of dissolved oxygen in different time points at 1th October 2017

3.3 Weighted data fusion of PH data

The weighted fusion analysis is performed on the PH data from Tabs.1 to 5,and the weighted information fusion method is the same as the temperature data fusion calculation method in 3.1.The following results can be obtained:

Table 9:Fusion data of PH value in different time points at 1th October 2017

3.4 Weighted data fusion of NH3 concentration data

The weighted fusion analysis of the NH3concentration data from Tabs.1 to 5 is performed.The weighted information fusion method is the same as the temperature data fusion calculation method in 3.1.Thus,the following results can be obtained:

Table 10:Fusion data of NH3 value in different time points at 1th October 2017

4 Treatment of loss data in aquaculture environment

Loss error,also known as fault error,is an error that is clearly inconsistent with the facts.It is usually caused by issues of operator error,damage to the internal components of the system,loose wiring,detachment,and sudden external shocks.The presence of negligence error is a serious distortion of the measurement result and must be removed.This paper mainly uses Grubbs judgment criteria to process data.

4.1 The Grubbs guidelines

Assume that the measurement data sequence of a sensor from multiple independent detections toward a certain object can be defined as X1,X2,???,Xn,and the measurement data Xi(i=1,2,...,n)obey the normal distribution.Assume that this measurement column is setted according to the ascending order,that is:

and the measured value obeys the normal distribution,then

According to the principle of sequential statistics,the exact distribution of the Grubbs statistic is found:

Therefore,after giving the significant level of a (usually taking a=0.05 or a=0.01),we can use the table lookup method (Grubbs Threshold Table https://wenku.baidu.com/view/0f 3c083a172ded630a1cb6c8.html) to find out the critical value of the Grubbs statistic g0(n,a)。p [ gi≥ g0(n,a) ]= a as a small probability event,and it should not appear when Xi(i=1,2,???,n) obeys the state distribution.

Measure gi,the corresponding Grubbs statistic of the top value Xi(i=1o r n),if it satisfies gi≥ g0(n,a),it is considered that there is a significant difference in the distribution of the statistic gi,and the corresponding Xishould contain a delinquent error (negligence error,or gross error,which means an error that is obviously inconsistent with the facts).Xiis a suspicious value and should be eliminated.If gi＜ g0(n,a),it is considered that the corresponding Xihas no error negligence value Xi,and it cannot be rejected as a suspicious value.

4.2 Treatment of errors in aquaculture environment

In the aquaculture environment,the data collected by sensors will inevitably result in sparse errors due to various reasons.In this paper,we use Grubbs criterion to deal with the negligence errors in the environment.This section performed the Grubbs processing of negligence data on the combined data in Section 3,and the following table shows the fused data of temperature,PH,dissolved oxygen,and NH3concentration.

Table 11:Fusion data in different time points at 1th October 2017

4.2.1 Processing of temperature data gross errors

According to the known conditions,the Grubbs elimination is practiced to the temperature data of the first group,and the average value of five numbers in the temperature data is:

The residual error Vi=Xi-T1is calculated due to the average value,and the residual error of the first data group is shown in Tab.12:

Table 12:Residual errors of temperature data

The approximate error can be calculated by using the residual error of the temperature data:

Using the table look-up method (Grubbs Threshold Table https://wenku.baidu.com/view/0f3c083a172ded630a1cb6c8.html) to find out the critical value of Grubbs g0(n,a),and it is known that P[g≥g0(n,a)]=a (significant level a generally takes 0.05 or 0.01) is a small Probability event that should not occur when Xiobeys a normal distribution.Check the value table and it is known that g0(5,0.05)=1.764,g0(5,0.05)×1σ =0.829.At this time |Vi|max=0.378＜g0(5,0.05)×0.216=0.829,so the temperature data need not be eliminated and the subsequent calculation can be performed directly.

4.2.2 Treatment of dissolved oxygen data gross errors

The Grubbs elimination on the dissolved oxygen data can be performed by using the same calculations as in Section Processing of Temperature Data Gross Errors,and in the dissolved oxygen data:

Table 13:Residual error of dissolved oxygen data

The approximate error can be calculated by using the residual error of the dissolved oxygen data:

Using the table look-up method (Grubbs Threshold Table https://wenku.baidu.com/view/0f3c083a172ded630a1cb6c8.html) to find out the critical value of Grubbs g0(n,a),and it is known that P[g≥g0(n,a)]=a (significant level a generally takes 0.05 or 0.01) is a small probability event that should not occur when Xiobeys a normal distribution.Check the value table and it is known that g0(5,0.05)=1.764,g0(5,0.05)×2σ =0.576.At this time |Vi|max=0.584＞g0(5,0.05)×0.327=0.576,thus for dissolved oxygen data at 8:45,it should be eliminated.The data after elimination is shown in the following tab:

Table 14:Dissolved oxygen data after Grubbs elimination

The eliminated data should be re-calculated using the Grubbs criteria,and=8.95,=0.018,and the values of g0(4,0.05)=1.496，g0(4,0.05)×=0.027 were obtained by checking numerical table.At this point |Vi|max=0.02＜g0(4,0.05)×0.018=0.027,thus,it is not necessary to eliminate,and the next calculation can be performed directly.

4.2.3 Handling gross errors of PH data

To perform Grubbs elimination on the PH value,the same calculation method as in section Processing of Temperature Data Gross Errors can be used in the PH value data:

Table 15:Residual error of PH data

The approximate error can be calculated by using the residual error of the dissolved oxygen data:

Using the table look-up method (Grubbs Threshold Table https://wenku.baidu.com/view/0f3c083a172ded630a1cb6c8.html) to find out the critical value of Grubbs g0(n,a),it is known that P[g≥g0(n,a)]=a (significant level a generally takes 0.05 or 0.01) is a small Probability event that should not occur when Xiobeys a normal distribution.Check the value table and it is known that g0(5,0.05)=1.764,g0(5,0.05)×2σ =0.1.At this time |Vi|max=0.066＜g0(5,0.05)×0.057=0.1,so the PH data need not be deleted.

4.2.4 Handling gross errors of NH3concentration data

To practice Grubbs elimination on the data of NH3concentration,the same calculation method as in section Processing of Temperature Data Gross Errors can be performed.In the NH3concentration data:

Table 16:Residual error of NH3 concentration data

The approximate error can be calculated by using the residual error of the dissolved oxygen data:

Using the table look-up method (Grubbs Threshold Table https://wenku.baidu.com/view/0f3c083a172ded630a1cb6c8.html) to find out the critical value of Globes g0(n,a),it is known that P[g≥g0(n,a)]=a (significant level a generally takes 0.05 or 0.01) is a small Probability event that should not occur when Xiobeys a normal distribution.Check the value table and it is known that g0(5,0.05)=1.764,g0(5,0.05)×4σ =0.01.In this case,|Vi|max=0.0076＜g0(5,0.05)×0.0057=0.01,thus,it is not necessary to eliminate the NH3concentration data.

5 Summary

This study has placed importance on the PH value,temperature,oxygen dissolved a value,and NH3concentration in the aquaculture environment.Self-adaptive weighted methods are used to perform respectively the first-level fusion of the factors affecting the aquaculture environment,obtaining fusion result of sensor data at a 10-minutes interval between 8:05 and 8:45.Limited by environment and network transmission,the method of Grubbs is used to detect abnormal data after fusion.Abnormal data of dissolved oxygen would be removed so as to provide a reliable data support for subsequent water quality judgment,prediction and early warning.

In this paper,achievements were made in aspects of data fusion and error data processing in the aquaculture environment.But the following work can also be conducted at levels of feature level fusion and decision level fusion so as to better consider the rationality of the integration at a more complete level.During the process of fusion,the combination of intelligent algorithms such as Bayesian,expert system and cluster analysis can bring more reliable results to the fusion.Meanwhile in processing the abnormal data,it would see comprehensive consideration in aspects of the eliminating the error data (using Wright criterion and histogram method,etc.) and the completion of missing data (using Newton interpolation and Lagrange interpolation method,etc.).Thus,we can constantly improve the fusion accuracy,and promote the development and application of data fusion.

Acknowledgement:This study was supported by National Key Research and Development Project (Project No.2017YFD0301506),National Social Science Foundation (Project No.71774052),Hunan Education Department Scientific Research Project (Project No.17K044;17A092).

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