Research of fuzzy implications via fuzzy linear regression in data analysis for a fuzzy model

Abstract

The effect of eutrophication is characterized by dense algal and plant growth due to the enrichment of nutrients for photosynthesis. As a result, it often plays an important role to the formation of plants that float in the surface of a water body. When nutrients are increasing in aquatic ecosystems, the photosynthetic plants grow rapidly. As a result, the algae limit the amount of dissolved oxygen required for respiration by other species in the water. Multi-criteria analysis has helped us towards the understanding and estimation of all physical, chemical and biological functions. In this paper, the examined water body, as a rich and variable system, is an ideal case for our study. Our purpose is to investigate some of the factors responsible for eutrophication (water temperature, nitrates, total phosphorus, Secchi depth, chlorophyll-a) using fuzzy logic. In this method, there are infinite numbers of fuzzy implications which can be used, since the proposition can take any value in the close interval [0,1]; hence, the investigation of the most appropriate implication is required. In this paper, we propose a method of evaluating fuzzy implications constructing triangular fuzzy numbers for all of the studied factors coming from statistical data. The deviation of the true value is the key for the selection of the most appropriate fuzzy implication describing the functions and the mechanisms in this ecosystem.

Keywords

Fuzzy logic fuzzy implications fuzzy linear regression lake eutrophication

1. Introduction

Among the major problems of environmental degradation, the debate on promoting green growth is more intense and timely than ever. The role of researchers is an important reference in such an effort, especially when it is related to an environmental asset of the highest value, such as biodiversity. In addition to contributing biodiversity to the smooth functioning of ecosystems, the biological richness of species also involves significant economic and social components. It concerns the whole of the biological wealth of our planet, which everyone has the duty to preserve and protect both for his own survival and for moral reasons.

Eutrophication, as a common phenomenon in ecosystems is a part of the normal aging process of many lakes and ponds. The water body, fed on rich nutrients from a natural source, is described as eutrophic which means that it is rich in nutrients and therefore abundant in plant and animal, although eutrophication is an environmental problem that occurs in lakes or closed shallow bays under certain circumstances. In fact, there is an excessive increase in the nutrient concentration caused by the enrichment of water with nutrient effluents (nitrates and phosphates from fertilizers and detergents). Bacteria and algae grow in number, forming a coating on the water surfaces, causing shading in the water below the surface. Without light, photosynthetic organisms are killed at the bottom, providing even greater amount of food to other bacteria, which continue to grow. As the number of bacteria increases, the consumption of dissolved oxygen in water is dramatically increased, while production is reduced, with the result that oxygen is present for non-photosynthetic organisms, such as fishes. Fishes are the first organisms to die while still surviving bacteria thus giving life to the ecosystem. Result of eutrophication is the change in the flora and fauna of the wetlands but as well as the limited possibilities for recreation. Additional “solutions” have been proposed to effect the most prevalent being the privatization of wetlands, and the recruitment of individual initiative and individual motivation for the appropriate configuration of ecosystems in a manner so as to increase the value of the environment for humans.

Mediterranean lakes present complicated mechanisms compared to traditional eutrophic condition development methods that are dependent on concentrations of phosphorus and nitrates [15]. Multi-criteria analysis seems to be the most suitable tool in order to estimate the trophic levels of such complex water bodies.

Phosphorus predicting empirical equations have been developed by Vollenweider [17, 18] in order to estimate better the phosphorus loading in lakes. The relationship between phosphorus concentration and algal biomass, measured as chlorophyll-a concentration, is described by Sakamoto [14] and Dillon and Rigler [4]. Secchi disk transparency was used by Lasenby [9] to predict areal hypolimnetic oxygen deficit (AHOD). The research becomes easier if the trophic criteria could be described by predictive equations. In this way, the researchers would avoid the examination of all of the possible parameters for the determination of the trophic status of a water body.

The evaluation of the trophic state in a water body contains both randomness and fuzziness. For this reason, the research approach in such complex systems would have a lot of interest, with the methods of fuzzy logic. Fuzzy systems can operate in ambiguity and uncertainty environment and give results that are meaningful to humans. They approach human logic, and are ideal tools for decision making. A characteristic advantage of fuzzy logic is that it can analyze systems that are quite complex. Fuzzy logic gives a satisfactory solution to the so-called principle of incompatibility: ‘As the complexity of a system increases, our ability to make accurate and meaningful statements about its behavior decreases until we reach a threshold beyond which accuracy and significance become almost mutually exclusive features’. It is obvious that this principle is the result of the quantum principle of Heisenberg’s indeterminacy.

In classic logic, the implication depends only on whether the premise is true or false. All propositions in classic logic have the values 0 or 1, holds or does not hold. On the other hand, in fuzzy logic, the true or false of a fuzzy proposition takes values in the set {0, 1}. Similarly, fuzzy implications generalize those of classical logic [2].

From physical point of view, fuzzy implications depict the rules governing the ecosystem. The aim of this study is to investigate which of the studied ‘rules’ represents more accurately the studied water body. The key to the uniqueness of this method is the ability to select the most appropriate implication among others for each study case, after a detailed analysis and calibrate the fuzzy inference systems in order to construct an accurate predicting model. In this way, the researchers can understand in a better way the mechanisms which affect the biological and chemical functions in the ecosystem. Generally, the main goal of the paper is to gain further knowledge, so the new ecosystem can be managed and preserved with great efficiency. In a previous paper [6], we categorized the studied parameters in linguistic variables and formed trapezoid fuzzy numbers. From these constructed fuzzy numbers, we calculated the deviations and then calculated the best implication that represents the studied parameters. In this paper, we avoid making assumptions by using linguistic variables, so the approach is more realistic. We are going to calculate the fuzzy numbers via fuzzy linear regression for all of the studied parameters. Using this mathematical model, we construct a fuzzy inference system using the most appropriate implication.

2. Methodology

2.1 Study area

Located in south-eastern Thessaly in central Greece, Lake Karla is an important natural ecosystem undergoing reconstruction, also operating as a reservoir. The trophic state of the lake is characterized as hypertrophic with all the resulting negative impacts on biodiversity. In 1962, the need of flood protection was the cause of lake drainage projects and the creation of a smaller reservoir in part [16]. The restoration of the water body started in 2000. The largest environmental project in the Balkans has stimulated the interest of many researchers. Listed in the network of Natura 2000 and characterized as a Permanent Wildlife Refuge by Greek Law, this newly re-established water body is considered as a vital aquatic ecosystem. Karla’s Lake restoration project faces multiple challenges in the hydro-ecological management [1]. It is a project of local development and national importance with many positive impacts on the lake, in Thessaly and the wider environmental development. The general characteristics of this artificial lake and its drainage area are given by Fig. 1.

Figure 1.

Location and general characteristics of the study area.

2.2 Database and model application in Lake Karla

The three sampling stations in the reservoir were selected taking into account their representativeness in the lake ecosystem, the degree of pollution by point and non-point sources and the availability of accessibility to them. Sampling at all three stations was performed monthly between March 2012 and July 2013 (17 months). Water samples taken on a 20-day basis were based on the fact that during the hot season (Summer–Autumn) phytoplankton blooming phenomena took place [3, 5]. In this paper, we will process the average of the stations’ data in seventeen data sets.

2.3 Description of the fuzzy logic model

The general form of the studied model is as follows [13]:

$\displaystyle Y=A_{0}+A_{1}x_{1}+\ldots+A_{n}x_{n}$ (1)

where $Y$ is the dependent variable, $x_{i}$ the independent variables and $A_{i}$ , $i=1,2,\ldots,n$ , are the symmetrical fuzzy numbers. The equations for the studied model are considered:

$\displaystyle Y_{1}=A_{0}+A_{1}x_{11}+A_{2}x_{21}+\ldots+A_{n}x_{n1}$ $\displaystyle Y_{2}=A_{0}+A_{1}x_{12}+A_{2}x_{22}+\ldots+A_{n}x_{n2}$ (2) $\displaystyle\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ $\displaystyle Y_{m}=A_{0}+A_{1}x_{1m}+A_{2}x_{2m}+A_{n}x_{nm}$

where $A_{0},A_{1},\ldots,A_{n},i=1,2,\ldots,n$ are symmetrical triangular fuzzy numbers.

We determined the degree $h$ to which we expect the data $((x_{1j},x_{2j},\ldots,x_{nj}),y_{j})$ to be included in the referred number $Y_{i}$ , that is:

$\displaystyle\mu_{Y_{i}}(y_{i})\geqslant h,\,\,\,i=1,2,\ldots,m$ (3)

We also want the spread of chlorophyll-a fuzzy numbers $Y_{i},i=1,2,\ldots,m,$ be as low as possible. The symmetric fuzzy numbers, $A_{i},i=0,1,2,\ldots,n$ , have the following form:

$\displaystyle\mu_{A_{i}}(x)=L\left({\frac{x-r_{i}}{c_{i}}}\right),\,\,c_{i}>0$ (4)

where $r_{i}$ is the centre and $c_{i}$ the spread, $i=0,1,2,\ldots,n$ , and the function $L(x)$ is defined:

$\displaystyle L:R\to R,\,\,\,L(x)=\max\{0,\,\,1-\left|x\right|\}$ (5)

The calculation of numbers $A_{i}$ was performed by computing the numbers $r_{i}$ and $c_{i}$ , where $i=0,1,2,\ldots,n$ . This gives the following linear regression:

$\displaystyle\min\left\{{mc_{0}+\sum_{i=1}^{m}{\sum_{j=1}^{n}{c_{j}\left|{x_{% ji}}\right|}}}\right\}$ $\displaystyle y_{i}\geqslant-(1-h)\left({c_{0}+\sum_{j=1}^{n}{c_{j}\left|{x_{% ji}}\right|}}\right)+r_{0}+\sum_{j=1}^{n}{r_{j}x_{ji}},i=1,2,\ldots,m$ (6) $\displaystyle y_{i}\leqslant(1-h)\left({c_{0}+\sum_{j=1}^{n}{c_{j}\left|{x_{ji% }}\right|}}\right)+r_{0}+\sum_{j=1}^{n}{r_{j}x_{ji}},i=1,2,\ldots,m$

For this model, we are going to use the following form of the above method:

$\displaystyle Y=A_{0}+A_{1}x$ (7)

where $Y$ is the dependent parameter and $x$ the independent. In our first step, in this study, we consider as the independent parameters the following: water temperature, nitrates, total phosphorus and Secchi depth. For the dependent parameter we suggest the value of chlorophyll-a. So the examined pairs are: temperature-chl-a, nitrates-chl-a, total phosphorus-chl-a and Secchi depth-chl-a. Having these pairs, we now construct all the fuzzy numbers that represent the chl-a levels, applying the Eq. (2.3), where $m=$ 17, as we mentioned in Section 2.2, and $n=$ 2.

Since everything depends on everything, our next step is to consider the chl-a as an independent variable and the others as dependent. In other words, we apply the fuzzy linear regression firstly when $X\to Y$ and secondly when $Y\to X$ and then we take a pair of $({x,y})$ where $x, y$ are true values. In this way, we are going to construct the fuzzy numbers of the rest of our studied parameters.

In order to understand which of the studied couple has the strongest relation in the eutrophication prediction, we are going to calculate the measure of fuzziness in each and every constructed fuzzy number by the following relation:

$\displaystyle M=\sqrt{M_{1}^{2}+M_{2}^{2}}$ (8)

where $M_{1}$ the measure of fuzziness of the numbers coming from $X\to Y$ and $M_{2}$ the measure of fuzziness of the numbers coming from $Y\to X$ .

The indicator of measure of fuzziness is calculated as following:

$\displaystyle M_{1}=\sqrt{[(c_{0})^{2}+(c_{1})^{2}]}$ (9) $\displaystyle M_{2}=\sqrt{[(d_{0})^{2}+(d_{1})^{2}]}$

where $c_{0}$ and $c_{1}$ (resp. $d_{0}$ and $d_{1}$ ) are the spreads of fuzzy numbers that have been previously calculated of coming from $X\to Y$ (resp. $Y\to X$ ).

Having all these representing fuzzy numbers, now we are going to apply the following symmetric fuzzy implications. The existence of observations allows us to assume that in the ideal fuzzy inference system, the true value of the implication $x\Rightarrow y$ has to be equal to 1, since the values of $x$ and also of $y$ are related to observations, that is, they have to do with verified relations of assumed cause and implied effect. Starting from this finding, the evaluation of the fuzzy implications was based on the deviation of the true values of each implication from 1. The fuzzy implication [2, 8] assigns a true value $J({x,y})$ to the fuzzy proposition “if $p$ then $q$ ” for every true value $({x,y})$ of the fuzzy propositions $p, q$ . It is a function $J:[{0,1}]\times[{0,1}]\to[{0,1}]$ having the form $J(x,y)=n(x)\vee y$ where $n$ is a fuzzy negation and $\vee$ a $t$ -norm (see [8]).

Let us consider the following example:

We have two linguistic variables as input the temperature and as output the levels of chlorophyll. We consider the following implication considering this linguistic parameter:

$\displaystyle\textit{high temperature}\Rightarrow\textit{high chlorophyll}$

This fuzzy implication gives us infinitively many other implications. For readers convenience we state a numerical example:

$\displaystyle 26^{\circ}\text{C}\Rightarrow 78.5\,\text{mg/m}^{3}$

Each implication of the above has a true value which can be found. Firstly, we find the true values of these linguistic variables so these above can be transformed in the following (Fig. 2).

So, this implication between the input and output parameter has a true value and can be found selecting the most appropriate implication.

In this paper, we used symmetric and asymmetric implications [2, 8]:

$\displaystyle J_{\textit{Mamdani}}\left({x,y}\right)=\text{min}\left\{{x,y}\right\}$ $\displaystyle J_{\textit{Larsen}}\left({x,y}\right)=x\,y$ $\displaystyle J_{\textit{Zadeh}}\left({x,y}\right)=\text{max}\left(\text{min}% \left\{{x,y}\right\},1-x\right)$ (10) $\displaystyle J_{\textit{Reichenbach}}\left({x,y}\right)=1--x+x\,y$ $\displaystyle J_{\textit{Lukasiewicz}}\left({x,y}\right)=\text{min}\left({1-x+% y,1}\right)$

To illustrate the last implication, we use the t-norm (Probor) $x\nabla y=x+y-xy$ , and the class of fuzzy complements $n_{\lambda}(x)=\frac{1-x}{1+\lambda x},\lambda>-1$ . So, we deduce the implication [5]:

$\displaystyle J(x,y)=n_{\lambda}(x)\nabla y=n_{\lambda}(x)+y-n_{\lambda}(x)y=% \,\frac{1-x+y(1+\lambda x)-y(1-x)}{1+\lambda x},\lambda>-1.$ (11)

In a previous paper [5] the linguistic parameters were used in order to categorize the fuzzy numbers of all the parameters and then find the true values. However, in this paper, instead of the linguistics, the linear regression was the used method for the estimation of the fuzzy numbers and the true values for this ecosystem. Moreover, in the paper of Ellina et al. [6], using linguistic parameters (low, medium and high) we found the most appropriate implications that describe the studied ecosystem, although, in that paper, we used trapezoid fuzzy numbers constructed by literature and legistation but not a method such as fuzzy linear regression to represent our parameters.

Figure 2.

Numerical example with linguistic parameters for the temperature and chlorophyll.

3. Results and discussion

In Tables 1 and 2, the fuzzy numbers calculated by the use of fuzzy linear regression are presented. Particularly, the fuzzy numbers of Table 1 are the results of the paper of Ellina and Kagalou [5]. In this case, the parameter of chlorophyll-a represents the dependent variable in the examined couples. The fuzzy numbers of Table 2 represent all the fuzzy numbers calculated via fuzzy linear regression, although the relation between the examined parameters in these couples is different from the one of Table 1. In this case, the variable of chlorophyll-a is the independent one and the four others are the dependent variables. This comes up from the consideration that in such an extremely changeable ecosystem there are no clear relations among the biotic and abiotic parameters that influence its trophic status.

In Table 3 the measures of fuzziness of every examined couple are presented. Obviously, the measure of fuzziness in the relation between water temperature and chl-a is the smallest of all the other examined couples. The smaller this index is, the more confident we are for the prediction of the chl-a levels. Similar results are derived in the paper of Ellina and Kagalou [5]. However, the big number of fuzziness in the other studied parameters does not mean that they do not influence the levels of chlorophyll-a as well, but there is uncertainty in trophic state’s prediction.

Table 1
Fuzzy numbers from fuzzy linear regression with the output of chl-a [5]

Calculated fuzzy numbers $X\to Y$
Chl-a – Water temperature	${\rm A}_{0}$ (0, 0)	${\rm A}_{1}$ (8.45, 6.715)
Chl-a – Nitrates	${\rm A}_{0}$ (29.015, 0)	${\rm A}_{1}$ (526.45, 485.881)
Chl-a - Total phosphorus	${\rm A}_{0}$ ( $-$ 1.15, 25.11)	${\rm A}_{1}$ (3036.64, 1768.7)
Chl-a - Secchi depth	${\rm A}_{0}$ (218.4, 181)	${\rm A}_{1}$ (19.516, 0)

Table 2

Fuzzy numbers from fuzzy linear regression with the input of chl-a

Calculated fuzzy numbers $Y\to X$
Water temperature – Chl-a	B ${}_{0}$ (14.508, 9.909)	B ${}_{1}$ (0.471, 0)
Nitrates – Chl-a	B ${}_{0}$ (0.212, 0.1735)	B ${}_{1}$ (0.00084, 0.0000265)
Total phosphorus – Chl-a	B ${}_{0}$ (0.00786, 0.01285)	B ${}_{1}$ (0.000498, 0.00029)
Secchi depth – Chl-a	B ${}_{0}$ (0.2425, 0.2425)	B ${}_{1}$ (0, 0)

Table 3

Measure of fuzziness of the examined couples

Measure of fuzziness	M ${}_{1}$	M ${}_{2}$	M
Water temperature – chl-a	6.715	9.9090	11.969
Nitrates – chl-a	485.881	0.1735	485.881
Total phosphorus – chl-a	1768.878	0.0128	1768.87
Secchi depth – chl-a	181	0.2425	181

We calculate the deviations of every implication mentioned above from unity [6]:

$\displaystyle\sigma_{p,i}=\sqrt{(1-\mu_{p,i}^{(1)})^{2}+(1-\mu_{p,i}^{(2)})^{2% }+\ldots+(1-\mu_{p,i}^{(j_{p})})^{2}}$ (12)

where $\mu_{p,i}^{(s)}$ the true values, $p=1,2,3,4$ the examined independent parameters (water temperature, NO ${}_{3}$ , TP, Secchi depth), $i=1,2,3,4,5,6$ the implications (Mamdani, Larsen, Zadeh, Lukasiewicz, Reichenbach, Probor), $s=1,2,\ldots,j_{p}$ and $j_{p}$ is the number of true values. It should be mentioned that in the last implication, we computed $\lambda$ so as Eq. (12) is minimized for every studied parameter, and we took $\lambda=$ $-$ 0.99. In Table 4, all the calculated deviations of every examined parameter from the chl-a can be observe. For readers convenience, we state in Appendix a numerical example, concerning the couple of temperature and chlorophyll-a.

Table 4

Deviations of the fuzzy implications per parameter

Implications	Mamdani	Larsen	Zadeh	Lukasiewicz	Reichenbach	Probor
Water temperature – chl-a	3.0070	3.26	1.625	1.023	1.242	0.169
Nitrates – chl-a	3.1172	3.407	1.697	1.124	1.295	1.481
Total phosphorus – chl-a	3.1970	3.380	1.757	1.228	1.404	0.124
Secchi depth – chl-a	3.3112	3.552	1.977	1.491	1.655	0.064

In fuzzy inference systems, there is the possibility to choose the most appropriate implication. More precisely, we check the deviation of each implication and we choose the implication with the smallest deviation. In these systems, the smaller the deviation is, the most suitable implication we have for our examined study case. As it can be observed, the best implication that expresses our system is the implication deduced by Probor, since three out four examined couples have the smallest deviation in the method of Probor and only the couple of nitrates and chl-a is described best by the implication of Lukasiewicz.

4. Conclusions

Around a quarter of country’s lakes present the phenomenon of eutrophication as a result of the concentration of nitrates and phosphates, mainly from agricultural and municipal wastewater in surface water. As complex systems, water bodies in a modelling process, cope with significant problems. For instance, the lack of data of the time series is one of the most recent but severe issues that a researcher confronts. Fuzzy logic can be used as a powerful tool in categorizing environmental status and describing multifaceted changes. This alternative method gives the opportunity to use the combination of many approaches such as traditional indices derived from crisp sets to constant parameters. The main advantage of this tool is the ability to unite many kinds of perceptions by offering stability between social, economic and biological impacts. This paper proposes a method in order to select the most appropriate fuzzy implication using real water quality observations in the restored Lake Karla (Thessaly, Greece). This approach presents progress when compared to the arbitrary choice of fuzzy implications. The importance of this method is great because the applications in the fuzzy inference systems will be held by an accurate predicting tool, choosing the best implication describing the studied ecosystem (i.e., Matlab software). In this project, we come to the conclusion that the most suitable implication describing this water body is the implication deduced by Probor.

Footnotes

Acknowledgments

An initial shorter version of the paper has been presented in the International Conference of Numerical Analysis and Applied Mathematics 2017 (ICNAAM 2017) in Thessaloniki, September 2017 and the extended abstract has been published in AIP Conference Proceedings . We would like to thank the referees for their helpful suggestions and remarks.

Appendix

For this numerical example we are going to use the following studied couple of temperature and chlorophyll:

•

Temperature: 14.4 ${}^{\circ}$ C

•

Chlorophyll-a: 76.28 mg/m ${}^{3}$

From Table 1 we take the fuzzy number of chlorophyll-a described by water temperature:

$\displaystyle y=(0,\,\,0)+(8.45,\,\,6.715)x=(8.45,\,\,6.715)x$

where $y$ is the fuzzy number of chlorophyll-a and $x$ the water temperature. Since we get $x=$ 14.4 from experimental data, replacing, we have the following triangular fuzzy number:

$\displaystyle y=(8.45,\,\,6.715)\times 14.4=(8.45\times 14.4,\,\,6.715\times 1% 4.4)=(121.68,\,\,96.696)$

At this point, we take the experimental value of chlorophyll-a (76.28 mg/m ${}^{3})$ and estimate the true value as follows.

Figure 3.

Numerical example with linguistic parameters for the temperature and chlorophyll.

Now, we take the fuzzy number of water temperature described by chlorophyll-a from Table 2:

$\displaystyle y=(14.508,\,\,9.909)+(0.471,\,\,0)x$

where $y$ is the fuzzy number of water temperature and $x$ the chlorophyll-a.

Since we get $x=$ 76.28 from experimental data, replacing, we have the following triangular fuzzy number:

$\displaystyle y=(14.508,\,\,9.909)+(0.471,\,\,0)\times 76.28=(50.4358,\,\,9.909)$

$t$ this point, we take the experimental value of temperature (14.4 ${}^{\circ}$ C) and estimate the true value as follows.

Figure 4.

Numerical example with linguistic parameters for the temperature and chlorophyll.

Applying the fuzzy implications in Chapter 2.3, we get the following results:

$\displaystyle 0.6263\Rightarrow 0.5015$

Working similarly, we take the following table, computing the deviation of the fuzzy implication of Mandami and continue this methodology for the rest data sets and for all of the studied couples in order to find all true values.

Table 5

Numerical example for the Mamdani deviations computation

Months	TEMPERATURE ( $x$ )	CHLOROPHYLL ( $y$ )	MAMDANI (min{ $x, y$ })
1	0.62630	0.5015	0.5015
2	0.62139	0.970415	0.62139
3	0.80364	0.37805	0.37805
4	0.76963	0.90890	0.76963
5	0.44321	0.695238	0.44321
6	0.30146	0	0
7	0.87211	0.8333057	0.8333057
8	0.51928	0.2306508	0.2306508
9	0.96397	0.2175059	0.2175059
10	0	0.9375315	0
11	0.1773034	0.4240494	0.1773034
12	0.0855451	0.9680556	0.0855451
13	0.4608318	0.3154622	0.3154622
14	0.8495972	0.5611587	0.5611587
15	0.4094777	0.2968893	0.2968893
16	0	0.3036862	0
17	0	0.0008110	0
			3.0070

References

Beklioglu

Romo

Kagalou

Quintana

and Becares

, State of the art in the functioning of shallow Mediterranean lakes: workshop conclusions, Hydrobiologia 584 (2007), 317–326.

Botzoris

Papadopoulos

and Papadopoulos

B.K.

, A method for the evaluation and selection of an appropriate fuzzy implication by using statistical data, Fuzzy Economic Review XX(2) (2015), 19–29.

Chamoglou

Papadimitriou

and Kagalou

, Key-descriptors for the functioning of a Mediterranean reservoir: the case of the new Lake Karla-Greece, Environmental Processes 1 (2014), 127–135.

Dulon

P.J.

and Rigler

F.H.

, The phosphorus-chlorophyll relationship in lakes, Limnology Oceanography 19 (1974), 767–773.

Ellina

and Kagalou

, Selection of the most appropriate parameter for the chlorophyll-a estimation of an artificial lake via fuzzy linear regression, European Water 55 (2016), 105–114.

Ellina

Papaschinopoulos

and Papadopoulos

B.K.

, Fuzzy inference systems: selection of the most appropriate fuzzy implication in terms of statistical data, Environmental Processes 4 (2017), 923–935.

Ellina

Papaschinopoulos

and Papadopoulos

B.K.

, Research of fuzzy implications via fuzzy linear regression in a eutrophic waterbody, AIP Conference Proceedings 1978 290007 (2018), doi: 10.1063/1.5043914.

Klir

G.J.

and Yuan

, Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddler River, NJ.

Lasenby

D.C.

, Development of oxygen deficits in 14 southern Ontario lakes, Limnology Oceanography 20 (1975), 993–999.

10.

R.-S.

S.-L.

and Hu

J.-Y.

, Analysis of reservoir water quality using fuzzy synthetic evaluation, Stochastic Environmental Research & Risk Assessment 13(5) (1999), 327–336.

11.

OECD(1982)Eutrophication of Waters: Monitoring, Assessment and Control. Organisation of Economic Cooperation and Development, Paris, p. 154

12.

Papadopoulos

B.K.

and Sirpi

M.A.

, Similarities and distances in fuzzy regression models, Soft Computing 8(8) (2004), 556–551.

13.

Papadopoulos

B.K.

and Sirpi

M.A.

, Similarities in fuzzy regression models, Journal of Optimization Theory and Applications 102(2) (1999), 373–383.

14.

Samakoto

, Primary production by phytoplankton community in some Japanese lakes and its dependence on lake depth, Archiv für Hydrobiologie 62 (1966), l–28.

15.

Scheffer

Hosper

Meijer

Moss

and Jeppesen

, Alternative equilibria in shallow lakes, Trends in Ecology & Evolution 8 (1993), 275–279.

16.

Sidiropoulos

Papadimitriou

Stabouli

Loukas

Mylopoulos

and Kagalous

, Past, present and future concepts for conservation of the re-constructed Lake Karla (Thessaly-Greece), Fresenius Environmental Bulletin 21(10a) (2012), 3027–3034.

17.

Vollenweider

R.A.

, Possibilities and limits of elementary models concerning the budget of substances in lakes, Archiv für Hydrobiologie 66 (1969), 1–36.

18.

Vollenweider

R.A.

, Advances in defining critical loading levels for phosphorus in lake eutrophication, Memorie dell Istituto Italiano di Idrobiologia 33 (1976), 53–83.

Research of fuzzy implications via fuzzy linear regression in data analysis for a fuzzy model

Abstract

Keywords

1. Introduction

2. Methodology

2.1 Study area

2.3 Description of the fuzzy logic model

Table 1 Fuzzy numbers from fuzzy linear regression with the output of chl-a [5]

Footnotes

Acknowledgments

Appendix

References

Table 1
Fuzzy numbers from fuzzy linear regression with the output of chl-a [5]