Simulation and forecast of the red tide's time series characteristics in China seas

SUN Fenglin

College of Oceanic and Atmospheric Sciences,Ocean University of China,Qingdao 266100,Shandong province,China

Abstract:Analyzing time series characteristics of red tide is the basis of disaster prevention and mitigation,which is very important to red tide prediction.There are trend components and periodic components in annual time series of occurrence frequency and area of red tides,so Gray-Periodic Extensional Combinatorial Model(GPECM)is used to extract these components.The fitting degree of occurrence frequency and area can reach 95.20% and 95.24%,respectively.The performance of GPECM is better than Gray Model,Fourier Series Extension Model,and Holt-Winter Exponential Smoothing Model in model stability.Consequently,it is used to forecast the occurrence frequency and area in 2020 and 2021,and results show that the annual frequency of red tides in 2020 and 2021 can rise to 39 and 41,respectively,and that the annual occurrence area of red tides can rise to 3 168 km2,which is about 59% more than last year.In 2021,it can fall to 1 901 km2.

Keywords:red tides,time series characteristics,Gray-Periodic Extensional Combinatorial Model

1 Introduction

Red tide,an ocean phenomenon,is the water discoloration caused by the sudden bloom or rapid accumulation of phytoplankton,protozoa or bacteria under suitable environment[1,2].lt is resulted from the combination of physical,chemical and biological factors[3].The development of agriculture and industry in China increased the nutrients carried by discharge into the ocean and lead frequent red tides occurrence in China's offshore waters[4,5].Red tide species can break the biological chain in ocean,take much of the oxygen from the water and release toxic substances after death[6],posing a great threat to the local mariculture industry,marine fishing industry and tourist industry[7].Now,red tide has become one of the major marine disasters in China[8-10].Therefore,the analysis and prediction for red tides are meaningful in the disaster prevention and mitigation.

So far,the causes of red tides are analyzed and prediction models are built.These prediction models can be subdivided in two classes:ecological model and time series model.Ecological model,e.g.empirical prediction method[11],remote sensing[12,13],ecosystem model[14],neural network[15-17],cellular automata[18],genetic programming[19],is built on amount of observation data,such as physical indices,chemical indices,and meteorological indices[4].The main drawback to these models is the cost and complexity of parameter estimation.Also,the model used in one sea area may be inappropriate in the others.All these limit its application.Time series model is built on time series data of red tides and extracts information to realize prediction.Compared with ecological model,time series model is easy to realize.Now it becomes a suitable method because the occurrence mechanisms of red tide are still not well understood[20].However,the research of time series model for red tides is little in China.For example,Zhang[21]reviewed the annual variation of red tides in China coast by descriptive statistics analysis,and demonstrated that East China Sea became the most serious zone beyond South China Sea among the four China seas.Zhang et al.[20]established the time sequence of happening frequency of red tides in China,and used the Rescaled Range Analysis Method to forecast frequently happened years for red tide.The results showed that the change of the frequently happened year of red tide presented the fractional Brownian Motion characteristic.Xu et al.[8]analyzed the character of annual number of red tides incidents in China seas by Holt-Winters Exponential Smoothing Method(HWESM).The seasonal periodicity was discussed in their paper,however,the annual period wasuninvolved in fluctuation analysis.

The frequency and scale of red tides in China coastal seas from 2004 to 2019 are shown in Fig.1(Data is from China Marine Disaster Bulletin 2004-2019).Red tides generally decreased in its frequency and scale after 2004.The rate of decline was fast from 2004 to 2010 and became slow after 2010.Meanwhile,periodic fluctuation is shown in time series,which is consistent with Wang's opinion"the occurrence of red tides has annual periods"[4].ln conclusion,both trend and periodical fluctuation should be discussed when we analyse the time series character of red tides or predict future tendency.At present,there is little research done on this field.

Fig.1 Number of incidents and distribution area of red tides in China seas(2004-2019)

As a kind of prediction model for time series,Gray-Periodic Extensional Combinatorial Model(GPECM)combines Gray Model(GM)with Periodic Extensional Model(PEM)which has been widely used in many fields[22,23].The trend part in time series can be extracted by GM,and then the information in periodical fluctuation can be shown by PEM.GPECM is suitable for small size sample,which is the character of red tides time series in China seas.To date,there is no research about frequency and scale of red tides in China by GPECM.

In this paper,GPECM is used to extract these components(i.e.trend and period)in annual time series of frequency and scale of red tides in China coastal seas,and compared to GM,Fourier Series Extension Model(FSEM)and HWESM.The result shows that the performance of GPECM is better than other models in fitting degree and stability.Consequently,GPECM is used to forecast the frequency and scale in 2020 and 2021.The prediction result of model is helpful for policy making in red tides disaster prevention and mitigation work.

2 Model

The flow chart of GPECM is shown in Fig.2.GPECM has two sub-models:GM and PEM.Firstly,the trend part is extracted by GM(1,1)and the estimated residual is obtained.Then PEM can get the periodic components from estimated residual series.Finally,the fitted values from GM and PEM can calculate the final fitted values of GPECM.The errors between fitted values and real values can be used to evaluate the performance of GPECM.

Fig.2 Flow chart of GPECM

2.1 Gray model

Namely,{x(0)(t)}(t=1,2,…,n)is discrete time series.By accumulated generating operation(AGO),we can get 1-AGO series(Eq.1).

Then differential equation of 1-AGO is constructed by Eq.2.Least squares method is used to estimate the unknown parameters a and b(Eq.3).

Finally,the fitted values of original time seriesx(0)(t)are calculated by Eq.5.And residual of model can be calculated by Eq.6.In PEM,x(t)becomes the object of research.

2.2 Periodic extensional model

PEM can get major periodic components in time series according to theory of analysis of variance.For residual seriesx(t),mean generating function of period m can be calculated by Eq.7.

wherei=1,2,…,m,n=1,2,…,M,M=[n2]int,nm=[n m]int,[·]intdenotes rounding operation.Mean generating function can be extended periodically by Eq.8.

Then Eq.9 can be used to judge if there is a major periodical component inx(t).

whereAssumeFmis the maximum of allFm′,thenx(t)includes a major periodical component which period ism-year.We subtract mean generating functionfm(i)fromx(t)and can get a new time seriesx′(t)(Eq.10).Forx′(t),repeating the extracting process can get other major periodical components.

Now,if we get some major periodical components from residual seriesx(t),the fitted value can be calculated by Eq.11.

where Φ is set of major periodic components.

2.3 Gray-Periodic extensional combinatorial model

The fitted values of GPECM,which are the estimates ofx(0)(t),can be calculated by Eq.12.

3 Result and forecast

Through experimentation,the performance of GPECM can be improved by taking natural logarithm of time series for red tides.Therefore,we use log-preprocessing data to build GPECM.When we evaluate the results,i.e.calculating relative errors,exponential operation should be done so that both original time series and errors have the same dimension.In this paper,three models(GM,FSEM and HWESM)are used to estimatex(t)and compare with GPECM.

Firstly,the time series of number of incidents and distribution area from 2004 to 2018 are used to build GM(1,1),respectively.It is shown in Eq.13 and Eq.14 that both number of incidents and distribution area decrease and the rate of descent gradually decreases along time.

Then,we use PEM to extract 3 major periodical components from residual series.The fitted values of GPECM are calculated by Eq.12.The results of GM,GPECM,FSEM and HWESM are shown in Fig.3.Meanwhile,Eq.15 and Eq.16 are used to calculate errors and relative errors,respectively.

Fig.3 Results of GPECM,GM,FSEM and HWESM from 2005 to 2018

As shown in Tab.1 and Tab.2,the performance of GPECM is better than GM because GPECM can extract periodic information from data.Except in 2010,the fitted values of number of incidents from GPECM are closer to the real values than that from GM,especially from 2015 to 2018.For distribution area of red tides,the fitted values from GPECM are closer to the real values except 2006.And the improvement of GPECM is significant from 2012 to 2018 especially.Overall,GPECM can reduce the model relative error(the mean of relative error in all time)from 16.31% to 5.87% for number of incidents,from 29.21% to 3.87% for distribution area.

Tab.1 The real values and model results(fitted values,errors and relative errors)of number of incidents from 2005 to 2018

Tab.2 The real values and model results(fitted values,errors and relative errors)of distribution area from 2005 to 2018

When we estimate number of incidents(distribution area)of red tides,the model relative error of FSEM is 16.13%(25.96%).Although the results are better than GM in general,the relative error of incidents number can even reach 90% in 2005.The fitting degree(=1-model relative error)of HWESM is lower than that of other 3 models for number of incidents.For distribution area,the model relative error of HWESM is lower than that of GM and FSEM,however,it still reaches 22.46%,much higher than GPECM.In conclusion,GPECM is better than other 3 models when fitting number of incidents and distribution area of red tides from 2005 to 2018.

Then,the analysis of relative errors is shown in Fig.4.For GM,FSEM and HWESM,the fluctuation of relative errors increases over time.It is suggested that the stability of these models changes with time and fitting results may become imprecise.Compared with these models,the relative errors of GPECM always oscillate in the range of-15%-15%.

Fig.4 The relative errors between the fitted values and real values for GPECM,GM,FSEM and HWESM from 2005 to 2018

In boxplot of relative errors(Fig.5),when we estimate the number of incidents,the mean of relative errors of GM or GPECM is close to 0%.FSEM generally overestimates real values because the mean of relative error is slightly larger than 0%.As for HWESM,the fitted values are generally less than real values.When estimating distribution area,the mean of relative errors of GPECM is still close to 0%.However,both GM and FSEM always underestimate the real values.On the contrary,HWESM overestimates the realdistribution area because the mean of relative error is greater than 0%.

Fig.5 The boxplot of relative errors

The stability of model is related with range of relative errors.It is shown in Fig.5 that the distribution of relative errors of GPECM is concentrated.However,other models have a wider distribution of the relative errors,which means that these models show poor stability and low accuracy compared with GPECM.In addition,the empirical density function of the relative errors of GPECM shows characters of normal distribution(Fig.6).Then we can test normality of relative errors by Shapiro-Wilk Test.For number of incidents and distribution area,the p-values are 0.71 and 0.11,respectively.The hypothesis of normal distribution is not rejected when the significance level αis 0.05.In addition,autocorrelation of the relative errors is tested by Ljung-Box Test and p-values for all lag length hypotheses are larger than 0.05.It means that it is not necessary to further improve model.

Fig.6 The histogram and density distribution curve of relative errors for GPECM

Now,based on the fitting results of GM(1,1)from 2004 to 2018,1 to 5 major periodical components are selected to build PEM.The real number of incidents and distribution area of red tides in 2019 is used to evaluate the performance of GPECM and discuss how to determine the optimal number of major periodical components.

The results are shown in Tab.3.For number of incidents,the relative errors of predictive values can be controlled within[-10%,10%]when using 1 to 4 major periodical components.However,the relative error is beyond this range when using 5 major periodical components.It is suggested that more major periodical components cannot guarantee higher prediction accuracy.Too many major periodical components can lead to the model overfitting.When distribution area of red tides in 2019 is estimated,the absolute relative error has a fluctuant downtrend as the number of major periodical components increase.The relative errors can be limited within[-10%,10%]when we choose more than 2 major periodical components.

In conclusion,GPECM has a better performance in fitting degree and stability than GM,FSEM and HWESM and can get an accurate predictive value.Then,the number of incidents and distribution area of red tides in 2020 and 2021 are estimated by GPECM in the following part.

Now,we take the time series from 2004 to 2019 as training data and build GPECM.Results are shown in Tab.4,that the number of major periods is associated with prediction accuracy.How to get an optimal number of major periods is a core problem for GPECM.Gao et al.[23]takes the R-squared as weight to combine predictive values from different number of major periods.However,as shown in Tab.3,more major periods does not mean lower relative errors because it may lead to the model overfitting exceeding a certain limit.To solve the problem,we build GPECM with training data in different time ranges and take the same periods in corresponding results as stable periods.These stable periods are used to realize prediction.

In Tab.3 and Tab.4,for number of incidents,when we select 3 major period components,there are 3 periodical components(3-year,4-year and 5-year cycles)in time series.If number of major periods increases to 5,4-year cycle and 7-year cycle areadded in Tab.3,but 6-year cycle and 5-year cycle are added in Tab.4.It is suggested that the model result is stable when we select 3 major periods.All these results are based on annual data.Then monthly data is used to build model and we come to a conclusion that the top five major periods are 12,47,56,40,52 months,i.e.1,3.92,4.67,3.33,4.33 years,which is consistent with the result of annual data.Therefore,3 major period components are selected to forecast frequency of red tides.Similarly,4 major period components are used to build GPECM for distribution area of red tides.

Tab.3 The predictive values of number of incidents and distribution area in 2019

Tab.4 The predictive value of number of incidents and distribution area with different major periods from 2020 to 2021

In 2020(2021),the frequency of red tides is 39(42).The occurrence frequency goes up slightly and stabilize at 40,compared with 2019.The distribution area of red tides increases to about 3 168 km2in 2020 and it increases by nearly 59%.In 2021,the scale of red tides drops to 1 901 km2.

4 Discussion

In this section,3 neighbor sea areas(Seto Inland Sea,Kyushu and South Korea)are taken as examples to explore the similarities and differences of major periodic components compared with China.The data of frequency of red tides is from Japan Fisheries Research and Education Agency and Korea National Institute of Fisheries Science.As shown in Fig.7,the frequency of red tides in these areas has a periodic fluctuation characteristic that:there are 5-year,7-year and 3-year periodic components in South Korea;there are 4-year,3-year and 7-year periodic components in Kyushu and there are 4-year,3-year and 5-year periodic components in Seto Inland Sea.The first major periodic components in these 3 areas are same with major periodic components(4-year cycle and 5-year cycle)in China and there are some differences in shape of these components.

Fig.7 The frequency of red tides in Japan(Soto Inland Sea and Kyushu)and South Korea

The red tide disaster is caused by eutrophication of sea water[25].Areas with much of nitrogen and phosphorus pollution often become the high-risk areas of red tide disasters[26].The main sources of pollutants include direct discharge of sea sewage and rivers.The relationship between red tides and pollutants(nitrogen and phosphorus)is discussed in this part.

The quantity of pollution water containing nitrogen and phosphorus is collected from Bulletin of Marine Ecology and Environment of China.Then we calculate the correlation coefficient ρ between quantity of pollution water and frequency(or scale)of red tides in China.The correlation coefficient between distribution area of red tides and nitrogen(phosphorus)pollutant discharge amount is 0.824(0.762).However,if time series of distribution area are lag one year,the coefficients can be 0.909(nitrogen)and 0.897(phosphorus).As for number of incidents,ρ between nitrogen(phosphorus)pollutant discharge amount and frequency is 0.623(0.563),greater than that of discharge amount series lag one year.The GPECM is built for time series of pollutant discharge amount,it is found that there are same periodic components(4-year cycle and 5-year cycle)with frequency and scale of red tides in China.

The curves of some periodic components of pollutant discharge amount are shown in Fig.8.There is a 5-year period component in series of number of incidents,distribution area and pollutant discharge amount.These curves are similar(Fig.8a and Fig.8b)and show positive correlation,especially ρ can reach 0.971 between frequency and nitrogen pollutant discharge amount.As for 4-year period component,ρ between frequency of red tides(distribution area)lag 1 year and phosphorus pollutant discharge amount can reach 0.958(0.866).

Fig.8 The correlation analysis of periodic component for different series

From the above results,the time series of frequency and scale of red tide showed a wavelike decrease change in recent 15 years,as well as pollutant discharge amount in China.The downtrend and volatility of curves are related with the previous value and current value of pollutant discharge.When current value of pollutant discharge changes greatly,it can bring a lot of uncertainty to prediction of frequency and scale of red tides of China.In addition,inadequate sewage treatment capacity and COVID-19 pandemic can increase the difficulty of prediction of red tides in 2020 and 2021.

5 Conclusion

Disaster prevention decision for red tides should be based on an understanding of characteristics of disasters.GPECM,combining the advantages of GM and PEM,can predict time series at the condition of small sampling data.It takes not only trend but also volatility into account,consistent with the character of time series of red tides in China seas.

So far,the occurrence mechanisms of red tide are still not well understood.Therefore,it is difficult to find a model which can make accurate prediction.After 1977,the frequency of red tide in China has a piecewise shape[8],a reasonable way is to fit in each segment.Meanwhile,Luo et al[5]pointed that the data of red tides in China,after 2000,can be used to build model because the data may be inaccurate or sparse before 2000,which is caused by the deficiency of marine observation.Therefore,small sampling data of red tides can give GPECM the advantages over other methods.

The results in this paper show that both frequency and scale of red tides show a wavelike decrease change.And the mean of distribution area of red tide occurs is decreasing.These changes mainly due to the control of pollutant discharge amount in China.The 4-year and 5-year periodic fluctuation contained in frequency and scale of red tides can provide a reference when relevant departments making disaster prevention decisions.

GPECM has a good performance to fit frequency and scale of red tides in China seas.It may be not stable in different seas due to different physical,chemical and meteorological conditions.Therefore,it may be a meaningful work to extract information from the se conditions to improve prediction accuracy and realize the monitoring and forecasting of red tides.