A COMPARATIVE STUDY OF USING
ADAPTIVE NEURAL FUZZY INFERENCE
SYSTEM (ANFIS), GAUSSIAN PROCESS
REGRESSION (GPR), AND SMRGT MODELS
IN FLOW COEFFICIENT ESTIMATION.
Ruya mehdi*
Gaziantep University, Civil Engineering Department, Yeditepe st., no 85088,
sahinbey dist., Gaziantep, Turkey.
ruya.mehdi1991@gmail.com
Ayse Yeter GUNAL
2Gaziantep University, Civil Engineering Department, Osmangazi district,
University Street, 27410 Sehitkamil / Gaziantep, Turkey .
agunal@gantep.edu.tr
Reception: 04/03/2023 Acceptance: 25/04/2023 Publication: 23/05/2023
Suggested citation:
Ruya M. And Ayse Yeter G. (2023). A Comparative Study of Using Adaptive
Neural Fuzzy Inference System (ANFIS), Gaussian Process Regression
(GPR), and SMRGT Models in Flow Coefficient Estimation. 3C Tecnología.
Glosas de innovación aplicada a la pyme, 12(2), 125-146. https://doi.org/
10.17993/3ctecno.2023.v12n2e44.125-146
https://doi.org/10.17993/3ctecno.2023.v12n2e44.125-146
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.44 | Iss.12 | N.2 April - June 2023
125
ABSTRACT
Estimating the flow coefficient is a crucial hydrologic process that plays a significant
role in flood forecasting, water resource planning, and flood control. Accurate
prediction of the flow coefficient is essential to prevent flood-related losses, manage
flood warning systems, and control water flow. This study aimed to predict the flow
coefficient for a period of 19 years (2000-2019) in the Aksu River Sub-Basin in Turkey,
using historical climatic data, including precipitation, temperature, and humidity,
provided by The Turkish State of Meteorological Service (TSMS). The study utilized
three different approaches, namely, the Adaptive Neural Fuzzy Inference System
(ANFIS), Simple Membership function and fuzzy Rules Generation Technique
(SMRGT), and Gaussian Process Regression (GPR), to predict the flow coefficient.
The models were evaluated using several statistical tests, such as Root Mean Square
Error (RMSE), Coefficient of Determination (R2), Mean Absolute Error (MAE), and
Mean Square Error (MSE), to determine their accuracy. Based on the evaluation
criteria, it is concluded that the Simple Membership Functions and Fuzzy Rules
Generation Technique (SMRGT) model has superior flow coefficient estimation
performance than the other models.
KEYWORDS
ANFIS, SMRGT, Flow coefficient, Prediction, Gaussian process regression.
INDEX
ABSTRACT
KEYWORDS
1. INTRODUCTION
2. MATERIALS AND METHODS
2.1. Area of Study and Dataset
2.2. Climate Properties of the Study Area
2.2.1. Precipitation
2.2.2. Temperature
2.2.3. Relative Humidity
2.3. Methods
2.3.1. ANFIS Model Development
2.3.2. Simple Membership Functions and Fuzzy Rules Generation
Technique (SMRGT)
2.3.3. Gaussian Process Regression (GPR)
2.4. Models Evaluation
3. RESULTS AND DISCUSSION
4. CONCLUSION
REFERENCES
https://doi.org/10.17993/3ctecno.2023.v12n2e44.125-146
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.44 | Iss.12 | N.2 April - June 2023
126
ABSTRACT
Estimating the flow coefficient is a crucial hydrologic process that plays a significant
role in flood forecasting, water resource planning, and flood control. Accurate
prediction of the flow coefficient is essential to prevent flood-related losses, manage
flood warning systems, and control water flow. This study aimed to predict the flow
coefficient for a period of 19 years (2000-2019) in the Aksu River Sub-Basin in Turkey,
using historical climatic data, including precipitation, temperature, and humidity,
provided by The Turkish State of Meteorological Service (TSMS). The study utilized
three different approaches, namely, the Adaptive Neural Fuzzy Inference System
(ANFIS), Simple Membership function and fuzzy Rules Generation Technique
(SMRGT), and Gaussian Process Regression (GPR), to predict the flow coefficient.
The models were evaluated using several statistical tests, such as Root Mean Square
Error (RMSE), Coefficient of Determination (R2), Mean Absolute Error (MAE), and
Mean Square Error (MSE), to determine their accuracy. Based on the evaluation
criteria, it is concluded that the Simple Membership Functions and Fuzzy Rules
Generation Technique (SMRGT) model has superior flow coefficient estimation
performance than the other models.
KEYWORDS
ANFIS, SMRGT, Flow coefficient, Prediction, Gaussian process regression.
INDEX
ABSTRACT
KEYWORDS
1. INTRODUCTION
2. MATERIALS AND METHODS
2.1. Area of Study and Dataset
2.2. Climate Properties of the Study Area
2.2.1. Precipitation
2.2.2. Temperature
2.2.3. Relative Humidity
2.3. Methods
2.3.1. ANFIS Model Development
2.3.2. Simple Membership Functions and Fuzzy Rules Generation
Technique (SMRGT)
2.3.3. Gaussian Process Regression (GPR)
2.4. Models Evaluation
3. RESULTS AND DISCUSSION
4. CONCLUSION
REFERENCES
https://doi.org/10.17993/3ctecno.2023.v12n2e44.125-146
1. INTRODUCTION
Hydrology is the study of the entire water cycle, with the most critical aspect being
the section where rainfall causes water to flow. The flow is crucial in designing flood
protection measures for urban and agricultural areas, as well as determining the
amount of water that can be extracted from a river for irrigation or water supply.
Turkey is located in a region that is prone to natural disasters, such as floods and
earthquakes, and the amount of rainfall, particularly during the rainy season, is a
significant factor in climate change. The rainy season is when floods and landslides
are most likely to occur, and several factors, such as the condition of the catchment
area [1], rain duration and intensity [2], land cover [3], topographic conditions [4], and
drainage network capacity [5], can contribute to floods. However, climate change is
the fundamental cause of these disasters. Urban floods, including flash floods, are
considered the most distressing types of floods. Flood forecasts for Turkey indicate
that 51% of flood events occur in late spring and early summer, with a significant
portion observed during winter and a small portion in autumn. The Black Sea,
Mediterranean, and Marmara Regions have the highest frequency of flood occurrence
in that order.
In flood forecasting, the flow coefficient is the most important factor to consider.
Flow coefficient is the ratio of the volume of water that drains superficially throughout
rainfall to the total volume of precipitation over a specified period [6,7]. The flow
coefficient, an essential tool in hydrologic processes of countless urban and rural
engineering projects, etc. [8], can indicate the quantity of water flowing from specific
precipitation and reflect the influence of natural geomorphological elements on the
flow. Flow coefficients are also useful when contrasting watersheds to determine how
various landscapes convert precipitation into rainfall events. [9,10] Precipitation is one
of the most crucial variables when assessing and determining the flow coefficient [7,
11]. Precipitation may refer to a single rainfall event or an interval in which multiple
rainfall events occur. Initial losses and infiltration capacity are attained when
precipitation intensifies—consequently, flow increases, leading to a greater flow
coefficient. In addition to precipitation properties such as intensity, duration, and
distribution, specific physical aspects of watersheds, such as soil type, vegetation,
slope, and climate, influence the occurrence and volume of the flow. The flow
coefficient can be estimated by employing tables in which the flow is related to the
surface type. According to [12], the effective study of the coefficient is a highly
complex operation due to many influencing variables. This implies that the flow
coefficients reported in the literature transmit less information than is required [9] and
that their values, when tabulated as if they were constant, may not reflect reality.
Since the accurate estimation of the flow coefficient is crucial to our existence,
improving models incorporating meteorological, hydrologic, and geological variables is
necessary. Thus, effective water management and operation of water structures will
be possible. Several models are used to model such a process. These models are
separated into experimental models, conceptual black box models, or grey box
models, and physically-based distributional models, or white box models.
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Experimental models (black box model) do not explicitly account for the physical laws
of the processes and only connect the input and output via the conversion function.
The second group consists of conceptual models, which are based on limited studies
of the existing processes in the basin hydrology system, as opposed to the
distributional physically-based models; their development has not been based on the
total number of physical processes but rather on the designer's comprehension of the
system's behavior. The third group consists of distributional physically-based models;
these models attempt to account for all the processes within the desired hydrological
system by applying physical definitions. In contrast, physically based models provide
a more realistic approach by mathematically representing the real phenomenon. Even
though physically-based models appear to be more suitable for modeling purposes,
they lack acceptability because of their fundamental uncertainty and high
computational cost.
Reports indicate that machine learning techniques such as ANN and FIS effectively
model such complexities (flow coefficient). Their simplicity and capacity for dealing
with nonlinearity without understanding the entire system distinguish them from
others. Numerous examples in the literature demonstrate that fuzzy logic (FL)-based
systems excelled at modeling different hydrological events such as precipitation,
runoff, streamflow, etc. Due to the presence of uncertainty and vagueness in these
domains, FL-based systems are well-suited for modeling.
This study proposes one of the pertinent machine learning algorithms, the Adaptive
Neuro-Fuzzy Inference System (ANFIS), for estimating the flow coefficient. The
ANFIS model employs Tagaki-Sugeno-Kang (TSK) first order [13,14]. As a flow
coefficient prediction, the hybrid learning algorithm is selected from various algorithms
for supervised learning. The widespread use of hybrid learning algorithms justifies
their selection. An advantage of ANFIS is that it is a combination of ANN and fuzzy
systems employing ANN learning capabilities to acquire fuzzy if-then rules with
suitable membership functions, which can learn something from the inaccurate data
that has been input and leads to the inference. Another benefit is that it can effectively
utilize neural networks' self-learning and memory capabilities, resulting in a more
sustainable training process [15].
These methods (ANFIS and other fuzzy systems) lack a definitive method for
determining the number of fuzzy rules and membership functions (MF) required for
each rule [13]. In addition, they have no learning algorithm for refining MF that can
minimize output error. Therefore, Toprak in 2009 [16] proposed a new method known
as the Simple Membership functions and fuzzy Rules Generation Technique
(SMRGT). This new technique takes into account the physical cause-and-effect
relationship and is designed to assist those who struggle to select the number, form,
and logic of membership functions (MFs) and fuzzy rules (FRs) in any fuzzy set.
Gaussian Process Regression (GPR) is a statistical learning theory and Bayesian
theory-based machine learning technique. It is well-suited for handling complicated
regression tasks, such as high dimensions, a small number of samples, and
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Experimental models (black box model) do not explicitly account for the physical laws
of the processes and only connect the input and output via the conversion function.
The second group consists of conceptual models, which are based on limited studies
of the existing processes in the basin hydrology system, as opposed to the
distributional physically-based models; their development has not been based on the
total number of physical processes but rather on the designer's comprehension of the
system's behavior. The third group consists of distributional physically-based models;
these models attempt to account for all the processes within the desired hydrological
system by applying physical definitions. In contrast, physically based models provide
a more realistic approach by mathematically representing the real phenomenon. Even
though physically-based models appear to be more suitable for modeling purposes,
they lack acceptability because of their fundamental uncertainty and high
computational cost.
Reports indicate that machine learning techniques such as ANN and FIS effectively
model such complexities (flow coefficient). Their simplicity and capacity for dealing
with nonlinearity without understanding the entire system distinguish them from
others. Numerous examples in the literature demonstrate that fuzzy logic (FL)-based
systems excelled at modeling different hydrological events such as precipitation,
runoff, streamflow, etc. Due to the presence of uncertainty and vagueness in these
domains, FL-based systems are well-suited for modeling.
This study proposes one of the pertinent machine learning algorithms, the Adaptive
Neuro-Fuzzy Inference System (ANFIS), for estimating the flow coefficient. The
ANFIS model employs Tagaki-Sugeno-Kang (TSK) first order [13,14]. As a flow
coefficient prediction, the hybrid learning algorithm is selected from various algorithms
for supervised learning. The widespread use of hybrid learning algorithms justifies
their selection. An advantage of ANFIS is that it is a combination of ANN and fuzzy
systems employing ANN learning capabilities to acquire fuzzy if-then rules with
suitable membership functions, which can learn something from the inaccurate data
that has been input and leads to the inference. Another benefit is that it can effectively
utilize neural networks' self-learning and memory capabilities, resulting in a more
sustainable training process [15].
These methods (ANFIS and other fuzzy systems) lack a definitive method for
determining the number of fuzzy rules and membership functions (MF) required for
each rule [13]. In addition, they have no learning algorithm for refining MF that can
minimize output error. Therefore, Toprak in 2009 [16] proposed a new method known
as the Simple Membership functions and fuzzy Rules Generation Technique
(SMRGT). This new technique takes into account the physical cause-and-effect
relationship and is designed to assist those who struggle to select the number, form,
and logic of membership functions (MFs) and fuzzy rules (FRs) in any fuzzy set.
Gaussian Process Regression (GPR) is a statistical learning theory and Bayesian
theory-based machine learning technique. It is well-suited for handling complicated
regression tasks, such as high dimensions, a small number of samples, and
https://doi.org/10.17993/3ctecno.2023.v12n2e44.125-146
nonlinearity, and it has a substantial potential for generalization. Gaussian process
regression has many favorable circumstances over neural networks, including simple
implementation, self-adaptive acquisition of hyper-parameters, flexible inference of
non-parameters, and probabilistic significance of its outcome. Results are less
affected by bias and easier to read thanks to the GPR's seamless integration of
hyperparameter estimates, model training, and security assessments. Processes with
a Gaussian (GP) distribution take it for granted that the overall distribution of the
model's probabilities is Gaussian.
The objectives of this study are to (1) compare the predictive power of the ANFIS,
SMRGT, and GPR models and (2) select the model and algorithm with the highest
degree of accuracy and the lowest error rate. This is the first attempt to compare the
abovementioned models to determine the flow coefficient.
2. MATERIALS AND METHODS
2.1. AREA OF STUDY AND DATASET
The Aksu River basin is located in the Antalya Basin, southwest of Turkey. The total
length of the Aksu River is approximately 145 km, with headwaters Akdag situated
within Isparta Province and discharges to the Mediterranean from the Antalya-Aksu
border. The southern part of the basin is narrower than the north. Two different
climatic types, Mediterranean and continental climates, are observed in the Aksu
River basin. The north part has low precipitation throughout the year, and the
northwest and northeast mountain areas are the highest areas and have lower
temperatures, intense precipitation, and snow, whereas the south plain areas are
generally warmer with intense rainfall and evaporation. Several measurement data
are collected to support the study. The primary data are obtained from TSMS (Turkish
State of Meteorological Service). The data processed for this study are precipitation,
temperature, and humidity.
2.2. CLIMATE PROPERTIES OF THE STUDY AREA
2.2.1. PRECIPITATION
The most severe effect of climate change is a rise in the frequency and intensity of
extreme weather events in some parts of the world; the most obvious manifestation of
this is the recent rise in the frequency and intensity of extreme precipitation in various
parts of the world, which is causing infrastructure systems to become completely
inadequate. Precipitation ranks among the most crucial elements of climatic
parameters and atmospheric circulation, as well as the element that provides water to
the land and is the primary flow source. In this work, the precipitation stations' data
and locations are obtained from TSMS (Turkish State of Meteorological Service). 57
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precipitation observation stations (POSs) with (1793) records of monthly precipitation
data for 20 years are used. The precipitation increased in (Oct., Nov., Dec., Jan., and
Feb.) and the minimum precipitation recordings showed in (June, July, August. and
Sep.). The maximum monthly precipitation (907.2 mm) was recorded in (Nov. 2001).
In comparison, the minimum record for most of the years was (0.1 mm), especially in
August. The annual average rainfall was 963.60 mm based on 19 years of Aksu
meteorological station measurements (see Fig.1). The maximum annual rainfall was
1891.8mm in 2001.
Figure 1. Average annual precipitation values for 19 years.
2.2.2. TEMPERATURE
The region is influenced by both moist tropical (MT) and warm and dry tropical air
(CT) from the African and Arabian regions during the summer. (6996) Monthly
temperature data have been studied in Aksu meteorological stations; the temperature
showed an increase in (July, and Aug.), while the minimum temperature recordings
showed in (Dec., and Jan.). The maximum monthly temperature was (31.4 °C)
recorded in Aug. 2012, and the minimum record (- 5 °C) was shown in (Dec., and
Jan.) 2016 and 2017. The annual average temperature was 16.03 °C based on 19
years of Aksu meteorological station measurements (see Fig.2). The maximum annual
temperature was 16.92 °C in 2010.
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
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precipitation observation stations (POSs) with (1793) records of monthly precipitation
data for 20 years are used. The precipitation increased in (Oct., Nov., Dec., Jan., and
Feb.) and the minimum precipitation recordings showed in (June, July, August. and
Sep.). The maximum monthly precipitation (907.2 mm) was recorded in (Nov. 2001).
In comparison, the minimum record for most of the years was (0.1 mm), especially in
August. The annual average rainfall was 963.60 mm based on 19 years of Aksu
meteorological station measurements (see Fig.1). The maximum annual rainfall was
1891.8mm in 2001.
Figure 1. Average annual precipitation values for 19 years.
2.2.2. TEMPERATURE
The region is influenced by both moist tropical (MT) and warm and dry tropical air
(CT) from the African and Arabian regions during the summer. (6996) Monthly
temperature data have been studied in Aksu meteorological stations; the temperature
showed an increase in (July, and Aug.), while the minimum temperature recordings
showed in (Dec., and Jan.). The maximum monthly temperature was (31.4 °C)
recorded in Aug. 2012, and the minimum record (- 5 °C) was shown in (Dec., and
Jan.) 2016 and 2017. The annual average temperature was 16.03 °C based on 19
years of Aksu meteorological station measurements (see Fig.2). The maximum annual
temperature was 16.92 °C in 2010.
891,975
702,7
597,22
680,22
703,9
802,26
817,7
841
1005,5
975,23
1346,26
353,16
791,15
1203,4
1021,8
1268,2
1773,6
971,6
1891,8
633,35
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
P (MM)
Years
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Figure 2. Average annual temperature values.
2.2.3. RELATIVE HUMIDITY
(6680) of monthly relative humidity data have been considered from the Aksu
meteorological stations; it is recognized that the humidity increased in (Jan., and
Dec.), while the minimum humidity recordings showed in (July and Aug.). The
maximum monthly humidity was (97.7%) recorded in Jan. 2017, and the minimum
record was (2.4%) in Dec. 2017. the annual average humidity was 63.3% (see Fig.3).
The maximum annual humidity was 67.4 % in 2002, and the minimum was 58.85 % in
2013.
Figure 3. average annual relative humidity rates.
15,55
16,5
15,6
15,85
15,7
15,55
16,53
15,5
16,4
16,35
16,92
15,6
16,2
16,275
16,325
15,68
15,92
15,52
16,62
16,1
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
Temperature °C
Years
64,6
64,4
63,6
60,96
62,88
65,9
58,85
61,35
60,97
66,22
64,6
62
60,53
64,25
62,1
62,6
64,85
67,4
65,95
62,5
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
Humidity %
Years
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2.3. METHODS
2.3.1. ANFIS MODEL DEVELOPMENT
The Adaptive Neuro-Fuzzy Inference System (ANFIS) is a hybrid approach
combining the advantages of two intelligent methods, neural networks, and fuzzy
logic, to ensure qualitative and quantitative rationality. This new network can be
effectively trained to interpret linguistic variables by utilizing neural networks and fuzzy
logic. ANFIS implements a Sugeno-style first-order fuzzy system; it applies TSK
Takagi Sugeno and Kang rules in its architecture [17] and effectively handles
nonlinear real-time problems. ANFIS has been utilized extensively in disaster risk
management, rock engineering [18,19], health services, finance, and other real-time
areas [20–22]. It addresses regression and classification issues.
In first-order Sugeno's system, a typical set of IF/THEN rules for three inputs and
one output can be expressed as follows:
Rule 1: If x is A1 and y is B1, then f1 = p1 x + q1 y + r1 (1)
Rule 2: If x is A2 and y is B2, then f2 = p2 x + q2 y + r2 (2)
Rule 3: If x is A3 and y is B3, then f3 = p3 x + q3 y + r3 (3)
Generally, ANFIS is composed of five layers:
Input Layer:
Nodes in the input layer stand in for the system's input variables. Each input node
is associated with a membership function connecting an input value to a fuzzy set. If
there are n input variables, the input layer can be denoted as:
(4)
Where Input variables are denoted by
and their corresponding nodes
in the input layer are denoted by .
Fuzzification Layer:
The multiplicators and transmitters of this layer are their nodes. This product
signifies the firing strength of a rule. Let be the membership function of node
for input with parameters
. The output of the fuzzification layer can be
denoted as:
(5)
Where is the number of membership functions per input variable and
is
the degree of membership of the input variable in the fuzzy set.
y1=x1,y2=x2⋯yn=xn
x1,x2,⋯,xn
y1,y2,⋯,yn
Aij(x)
(i)
(j)
(pij)
uij =Aij(xj), for i = 1 to m and j = 1 to n
(m)
(uij)
jth
ith
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2.3. METHODS
2.3.1. ANFIS MODEL DEVELOPMENT
The Adaptive Neuro-Fuzzy Inference System (ANFIS) is a hybrid approach
combining the advantages of two intelligent methods, neural networks, and fuzzy
logic, to ensure qualitative and quantitative rationality. This new network can be
effectively trained to interpret linguistic variables by utilizing neural networks and fuzzy
logic. ANFIS implements a Sugeno-style first-order fuzzy system; it applies TSK
Takagi Sugeno and Kang rules in its architecture [17] and effectively handles
nonlinear real-time problems. ANFIS has been utilized extensively in disaster risk
management, rock engineering [18,19], health services, finance, and other real-time
areas [20–22]. It addresses regression and classification issues.
In first-order Sugeno's system, a typical set of IF/THEN rules for three inputs and
one output can be expressed as follows:
Rule 1: If x is A1 and y is B1, then f1 = p1 x + q1 y + r1 (1)
Rule 2: If x is A2 and y is B2, then f2 = p2 x + q2 y + r2 (2)
Rule 3: If x is A3 and y is B3, then f3 = p3 x + q3 y + r3 (3)
Generally, ANFIS is composed of five layers:
Input Layer:
Nodes in the input layer stand in for the system's input variables. Each input node
is associated with a membership function connecting an input value to a fuzzy set. If
there are n input variables, the input layer can be denoted as:
(4)
Where Input variables are denoted by and their corresponding nodes
in the input layer are denoted by .
Fuzzification Layer:
The multiplicators and transmitters of this layer are their nodes. This product
signifies the firing strength of a rule. Let be the membership function of node
for input with parameters . The output of the fuzzification layer can be
denoted as:
(5)
Where is the number of membership functions per input variable and is
the degree of membership of the input variable in the fuzzy set.
y1=x1,y2=x2⋯yn=xn
x1,x2,⋯,xn
y1,y2,⋯,yn
Aij(x)
(i)
(j)
(pij)
uij =Aij(xj), for i = 1 to m and j = 1 to n
(m)
(uij)
jth
ith
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Rule Layer:
The nodes in this layer calculate the rule's firing strength relative to the total
firing strength of all rules.
(6)
Defuzzification Layer:
This layer's nodes are adaptive with node functions.
(7)
Where is the output of Layer 3 and are the parameter set? Parameters
of this layer are referred to as consequent parameters.
Output Layer:
All inputs are combined at a single fixed node to produce the final output. We can
model the output layer as:
(8)
In this model, 7-year data is used, where the training data were Precipitation,
temperature, and humidity (input variables) data from January 2013 to December
2017 (5 years). On the other hand, the testing data from January 2018 to December
2019 (2 years), in this case, by trial and error, is 70%:30%. Training is conducted
using a membership function such as the Gaussian membership function (gaussmf).
In this step, a fuzzy inference system (FIS) is generated and evaluated, which can
produce MSE and MARE.
To reduce computations that are too large in the pre-processing, the data is
normalized into the range (0-1) using the following equation:
(9)
Where the normalized data is the original data, and , are the maximum and
minimum values of the original data, respectively.
ith
¯
W
=
W1+W2+W3
W1
¯
Wi f =¯
W⋅(pi x +qiy +ri)
{pi,qi,ri}
f=
n
∑
i=1
¯
Wif
i
¯x=
x−m
n−m
¯x
x
n
m
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Figure 5. A framework of the ANFIS model.
2.3.2. SIMPLE MEMBERSHIP FUNCTIONS AND FUZZY
RULES GENERATION TECHNIQUE (SMRGT)
The fuzzy-Mamdani method is used to construct the SMRGT model. A combination
of expert judgment and data-driven experimentation determines both the fuzzy subset
and the variable ranges in this model. This method streamlines incorporating the
event's physics into a fuzzy model. The steps involved in the SMRGT procedure are
as follows:
1. Define input and output variables: The first step is to define the input and
output variables of the fuzzy logic system. This work used three inputs
(Precipitation, temperature, and humidity) with one output (flow coefficient).
2. Determine membership functions: Membership functions (MFs) map input
values to fuzzy sets. Five MFs were used and labeled as; Very low, Low,
Medium, High, and Very high. Also, this step involves selecting the shape of
the membership function, a triangular shape was selected.
3. Determine the key values: in this step, the unit width (UW), core value (Ci), the
number of right-angled triangles (nu), the expanded base width (EUW), and
key values (Ki) of the fuzzy sets were determined. Equations [10–18] were
used to calculate the key values. Table 1 shows the obtained key values.
These key values are the inputs of the model.
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Figure 5. A framework of the ANFIS model.
2.3.2. SIMPLE MEMBERSHIP FUNCTIONS AND FUZZY
RULES GENERATION TECHNIQUE (SMRGT)
The fuzzy-Mamdani method is used to construct the SMRGT model. A combination
of expert judgment and data-driven experimentation determines both the fuzzy subset
and the variable ranges in this model. This method streamlines incorporating the
event's physics into a fuzzy model. The steps involved in the SMRGT procedure are
as follows:
1. Define input and output variables: The first step is to define the input and
output variables of the fuzzy logic system. This work used three inputs
(Precipitation, temperature, and humidity) with one output (flow coefficient).
2. Determine membership functions: Membership functions (MFs) map input
values to fuzzy sets. Five MFs were used and labeled as; Very low, Low,
Medium, High, and Very high. Also, this step involves selecting the shape of
the membership function, a triangular shape was selected.
3. Determine the key values: in this step, the unit width (UW), core value (Ci), the
number of right-angled triangles (nu), the expanded base width (EUW), and
key values (Ki) of the fuzzy sets were determined. Equations [10–18] were
used to calculate the key values. Table 1 shows the obtained key values.
These key values are the inputs of the model.
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(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Figure 5. The parameters of the triangular MF.
Vr = (P,T,H) max −(P,T,H) min
C
i=K3 =
Vr
2
−(P,T,H)
min
U
W=
Vr
nu
O
=
UW
2
EU W =U W + 0
K
4=Ki =Ci+ 1 =
(Ci −(P,T,H) min
2)
+ (P,T,H)
min
K
2=Ci−1=(P,T,H) max −
(
(P,T,H) max −
Ki
2)
K
1=(P,T,H) min +
(EU W
3)
K
5= (P,T,H)max −
EU W
3
Ci
1
0
K1
UW
K5
EUW
Ci-1
Ci+1
MEMBERSHIP DEGREE
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Table 1. key values of the SMRGT model.
4. Generate fuzzy rules: Fuzzy rules map input values to output values. Each rule
consists of an antecedent (input) and a consequent (output). In this step, 125
rules were set in pertinent physical conditions such as "IF", "AND", and
"THEN."
5.
Run the model: MATLAB software was selected. As an operator, the Mamdani
algorithm is implemented. The centroid method was selected for the
defuzzification procedure.
Input and output files prepared and added to the
program with (.dat) extension. Then the program with the (.fis) extension is
loaded. The (.m) extension file is prepared for running the prepared program.
Model results can be obtained by running this file with the (.m) extension.
Preparing the program with this procedure will reduce the trial and error
process. Then the table of the fuzzy set was created.
2.3.3. GAUSSIAN PROCESS REGRESSION (GPR)
Numerous disciplines employ a potent instrument that can be considered a
generalized regression model. This paper uses a Gaussian Process Regression
(GPR) model to predict the flow coefficient values. Before explaining the Gaussian
Process Regression, it is required to describe a regression model. In regression,
observation's output is considered to be a function of the variables
input, plus
some noise .
(19)
The fundamental regression function is forecasted based on the input parameters
and the given outputs. Once the regression model has been developed, a new value
for the output variable can be determined for a given input variable. This is why
regression models are widely used [23-25]. For GPR, it is assumed that the
regression function
is derived from a Gaussian Process (GP) with a zero mean
function and the covariance/kernel function .
(20)
Ci-1 (K2) Ci (K3) Ci+1 (K4) K1 K5
Precipitation
650
1100
1550
312.5
1887.5
Temperature
12.5
25
37.5
3.125
46.88
Humidity
25
50
75
6.25 93.75
ith
(yi)
(xi)
(εi)
yi=f(xi)+εi
(x)
(x,x′ )
f(x)∼GP (0,k(x,x′
))
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136
Table 1. key values of the SMRGT model.
4. Generate fuzzy rules: Fuzzy rules map input values to output values. Each rule
consists of an antecedent (input) and a consequent (output). In this step, 125
rules were set in pertinent physical conditions such as "IF", "AND", and
"THEN."
5. Run the model: MATLAB software was selected. As an operator, the Mamdani
algorithm is implemented. The centroid method was selected for the
defuzzification procedure. Input and output files prepared and added to the
program with (.dat) extension. Then the program with the (.fis) extension is
loaded. The (.m) extension file is prepared for running the prepared program.
Model results can be obtained by running this file with the (.m) extension.
Preparing the program with this procedure will reduce the trial and error
process. Then the table of the fuzzy set was created.
2.3.3. GAUSSIAN PROCESS REGRESSION (GPR)
Numerous disciplines employ a potent instrument that can be considered a
generalized regression model. This paper uses a Gaussian Process Regression
(GPR) model to predict the flow coefficient values. Before explaining the Gaussian
Process Regression, it is required to describe a regression model. In regression,
observation's output is considered to be a function of the variables input, plus
some noise .
(19)
The fundamental regression function is forecasted based on the input parameters
and the given outputs. Once the regression model has been developed, a new value
for the output variable can be determined for a given input variable. This is why
regression models are widely used [23-25]. For GPR, it is assumed that the
regression function is derived from a Gaussian Process (GP) with a zero mean
function and the covariance/kernel function .
(20)
Ci-1 (K2)
Ci (K3)
Ci+1 (K4)
K1
K5
Precipitation
650
1100
1550
312.5
1887.5
Temperature
12.5
25
37.5
3.125
46.88
Humidity
25
50
75
6.25 93.75
ith
(yi)
(xi)
(εi)
yi=f(xi)+εi
(x)
(x,x′ )
f(x)∼GP (0,k(x,x′
))
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It is also assumed that the noise has a Gaussian distribution. The function
is known as a kernel function. This function represents the covariance between the
and values in a regression model given and as inputs.
GPR offers numerous advantages over alternative regression models. For
instance, it offers an indication of the uncertainty of the predictions, which is crucial for
various practical applications. It can also model nonlinear relationships between input
and output variables and accommodate missing data.
The procedure of the Gaussian Process Regression (GPR) in MATLAB can be
summarised as follows:
i.
Load the data of three inputs and one output (The training data was average
monthly precipitation, temperature, and humidity records for 15 years) into
MATLAB. The input data should be a size N x D matrix, where N is the number
of data points and D is the number of input variables. The output data should
be a column vector of size N x 1.
ii.
Define the kernel function: the kernel function was defined using the 'make
kernel' function.
iii. Specify the prior distribution: the prior distribution over the Gaussian process is
specified using MATLAB's 'fitrgp' function. We can specify a mean function, a
kernel function, and hyperparameters for the kernel function.
iv.
Train the model: the GPR model is trained using the 'fitrgp' function. This
function estimates the hyperparameters of the kernel function from the training
data.
v.
Make predictions: the trained GPR model uses the' predict' function to predict
new input values. The 'predict' function returns a predicted mean and a
variance for each input value.
vi.
Load test data: 5 years of measurements of the abovementioned parameters
were selected and loaded. Then the prediction was made. Cross-validation
(v=5) was selected for GPR to protect the models against overfitting.
εi
(x,x′ )
x
x′
x
x′
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Figure 6. The generated code for the GPR model in MATLAB.
2.4. MODELS EVALUATION
Four parameters were used to evaluate the model's performance: Mean Absolute
Error (MAE), Root Mean Squared Error (RMSE), the coefficient of determination (R2),
and Mean Square Error (MSE). They were given in Eq. (5-7). MAE, MSE, and RMSE
are two measures of error. Thus ideal models would have MAE and RMSE values
equal to zero. The coefficient of determination is the proportion of variability the
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138
Figure 6. The generated code for the GPR model in MATLAB.
2.4. MODELS EVALUATION
Four parameters were used to evaluate the model's performance: Mean Absolute
Error (MAE), Root Mean Squared Error (RMSE), the coefficient of determination (R2),
and Mean Square Error (MSE). They were given in Eq. (5-7). MAE, MSE, and RMSE
are two measures of error. Thus ideal models would have MAE and RMSE values
equal to zero. The coefficient of determination is the proportion of variability the
https://doi.org/10.17993/3ctecno.2023.v12n2e44.125-146
regression line indicates to the variability of data for linear regression. A regression
line that is the mean value of data would have R2=0, while an ideal model would have
R2=1.
(21)
(22)
(23)
3. RESULTS AND DISCUSSION
In this study, the flow coefficient in the Aksu river basin was estimated by using
Adaptive Neural Fuzzy Inference System (ANFIS), Simple membership functions and
fuzzy Rules Generation Technique (SMRGT), and Gaussian Process Regression
(GPR) models. The results were compared with each other. The dataset belonging to
the years 2000–2019 was used in modeling the SMRGT and GPR. For ANFIS, seven
year's data from 2013-2019 were used; 70% was used for training and 30% for
testing. Monthly Precipitation (P), Temperature (T), and Relative Humidity (H) were
used as the input variables. To determine the success of the models used to estimate
the flow coefficient value, RMSE (root mean square error), MAE (mean absolute
error), MSE (mean square error), and R (correlation coefficient) were calculated, as
explained in the previous section. The performance of the model results is shown in
Table 2. When Table 2 was examined, all models gave similar results. According to
the RMSE, MAE, MSE, and R criteria, the best results were obtained in the SMRGT,
and the worst was in the GPR.
Table 2. The RMSE, MAE, MSE, and R2 statistics of all models.
In ANFIS analysis, Gaussian parabolic 5 × 5 × 5 Membership Functions (MFs) and
Grid partition section were analyzed with 100 iterations, assuming the output as linear.
M
AE =
1
n
n
∑
1
∣Ci, measured −Ci, estimated ∣
M
SE =
1
n
n
∑
1
(Ci, measured −Ci, estimated )
2
R MSE =
1
n
n
∑
1
(Ci, measured − Ci, estimated )∧2
Models Period RMSE MSE MAE R2
ANFIS
Training 1.92 37 1.01 993
Testing 15.67 2.45 12.15 561
All data 8.53 728 4.19 863
SMRGT All data 9.6 0.93 8.07 963
GPR
Training 26.9 7.24 20.26 0.61
Testing 20.1 4.05 15.79 0.55
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Variation and scatter graphs for the ANFIS method are shown in Fig.7. The correlation
coefficient of all data is seen as R: 0.863. As realized in the figure, ANFIS results were
close to the observed values.
Figure 7. ANFIS structure uses three inputs and 5 MFs for each input with a type of Gaussmf.
Figure 8. Scatter diagram of the trained data results.
y = 0,9958x + 0,0005
R² = 0,993
0
0,2
0,4
0,6
0,8
1
0 ,00 0 ,20 0 ,40 0 ,60 0 ,80 1 ,00
Actual
ANFIS
Training Data
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Variation and scatter graphs for the ANFIS method are shown in Fig.7. The correlation
coefficient of all data is seen as R: 0.863. As realized in the figure, ANFIS results were
close to the observed values.
Figure 7. ANFIS structure uses three inputs and 5 MFs for each input with a type of Gaussmf.
Figure 8. Scatter diagram of the trained data results.
y = 0,9958x + 0,0005
R² = 0,993
0
0,2
0,4
0,6
0,8
1
0 ,00 0 ,20 0 ,40 0 ,60 0 ,80 1 ,00
Actual
ANFIS
Training Data
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Figure 9. Scatter diagram of the tested data results.
Figure 10. Scatter plot for all data results.
y = 1,0389x + 0,0449
R² = 0,5614
0
0,2
0,4
0,6
0,8
1
0 ,00 0 ,20 0 ,40 0 ,60 0 ,80 1 ,00
Actual
ANFIS
Testing Data
y = 0,9902x + 0,017
R² = 0,863
0
0,2
0,4
0,6
0,8
1
0 ,00 0 ,20 0 ,40 0 ,60 0 ,80 1 ,00
Actual
ANFIS
All Data
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Figure 11. Variation graph of ANFIS model outcomes.
Simple Membership Functions and Fuzzy Rules Generation Technique (SMRGT)
model results are given in Figure 12. When the scatter graph was examined, it was
seen that the output values of the SMRGT model gave closer results to the actual
values; also, the correlation coefficient was 0.96. In Table 2, it was found that the
SMRGT model showed the best performance among all models. Compared to all
models, it can be seen from Table 2 that SMRGT model results had the low error rates
(RMSE: 9.6; MSE: 0.93; MAE: 8.07) and the highest correlation (R: 0.963). The
figures show that the SMRGT approach shows almost the same trend as the actual
values.
Figure 12. scatter diagram of SMRGT model.
0 ,00
0 ,20
0 ,40
0 ,60
0 ,80
1 ,00
1 ,20
010 20 30 40 50 60 70 80 90
Time
actual ANFIS
y = 1,0254x + 0,0634
R² = 0,9629
0,00 00
0,20 00
0,40 00
0,60 00
0,80 00
1,00 00
1,20 00
00,2 0,4 0,6 0,8 11,2
ACTUAL
SMRGT
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