Maximizing daily rainfall prediction accuracy with maximum overlap discrete wavelet transform‐based machine learning models

Abstract

Rainfall is an important phenomenon for various aspects of human life and the environment. Accurate prediction of rainfall is crucial for a wide range of sectors, including agriculture, water resources management, energy production, disaster management and many more. The ability to predict rainfall in an accurate fashion enables stakeholders to make informed decisions and take necessary actions to mitigate the impacts of natural disasters, water scarcity and other issues related to rainfall. In addition, advances in rainfall prediction technologies have the potential to contribute to sustainable water management and the preservation of water resources by providing the necessary information for decision‐makers to plan and implement effective water management strategies. Hence, it is important to continuously improve the accuracy of rainfall prediction. In this paper, the integration of the maximum overlap discrete wavelet transform (MODWT) and machine learning algorithms for daily rainfall prediction is proposed. The main objective of this study is to investigate the potential of combining MODWT with various machine‐learning algorithms to increase the accuracy of rainfall prediction and extend the forecast time horizon to 3 days. In addition, the performances of the proposed hybrid models are contrasted with the models hybridized with commonly used discrete wavelet transform (DWT) algorithms in the literature. For this, daily rainfall raw data from three rainfall observation stations located in Turkey are used. The results show that the proposed hybrid MODWT models can effectively improve the accuracy of precipitation forecasting, based on model evaluation measures such as mean square error (MSE) and Nash‐Sutcliffe coefficient of efficiency (CE). Accordingly, it can be concluded that the integration of MODWT and machine learning algorithms have the potential to revolutionize the field of daily rainfall prediction.

Full text

RESEARCH ARTICLE
Maximizing daily rainfall prediction accuracy with
maximum overlap discrete wavelet transform-based
machine learning models
Kübra Küllahcı| Abdüsselam Altunkaynak
Department of Civil Engineering Hydraulics and Water Resources Division, Istanbul Technical University, Maslak, Turkey
Correspondence
Kübra Küllahcı, Department of Civil
Engineering Hydraulics and Water
Resources Division, Istanbul Technical
University, Maslak 34469, Istanbul,
Turkey.
Email: onerk@itu.edu.tr
Abstract
Rainfall is an important phenomenon for various aspects of human life and
the environment. Accurate prediction of rainfall is crucial for a wide range of
sectors, including agriculture, water resources management, energy produc-
tion, disaster management and many more. The ability to predict rainfall in an
accurate fashion enables stakeholders to make informed decisions and take
necessary actions to mitigate the impacts of natural disasters, water scarcity
and other issues related to rainfall. In addition, advances in rainfall prediction
technologies have the potential to contribute to sustainable water management
and the preservation of water resources by providing the necessary information
for decision-makers to plan and implement effective water management strate-
gies. Hence, it is important to continuously improve the accuracy of rainfall
prediction. In this paper, the integration of the maximum overlap discrete
wavelet transform (MODWT) and machine learning algorithms for daily rain-
fall prediction is proposed. The main objective of this study is to investigate the
potential of combining MODWT with various machine-learning algorithms to
increase the accuracy of rainfall prediction and extend the forecast time hori-
zon to 3 days. In addition, the performances of the proposed hybrid models are
contrasted with the models hybridized with commonly used discrete wavelet
transform (DWT) algorithms in the literature. For this, daily rainfall raw data
from three rainfall observation stations located in Turkey are used. The results
show that the proposed hybrid MODWT models can effectively improve the
accuracy of precipitation forecasting, based on model evaluation measures
such as mean square error (MSE) and Nash-Sutcliffe coefficient of efficiency
(CE). Accordingly, it can be concluded that the integration of MODWT and
machine learning algorithms have the potential to revolutionize the field of
daily rainfall prediction.
Received: 27 October 2023 Revised: 29 March 2024 Accepted: 28 May 2024
DOI: 10.1002/joc.8530
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any
medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
© 2024 The Authors. International Journal of Climatology published by John Wiley & Sons Ltd on behalf of Royal Meteorological Society.
Int J Climatol. 2024;1–22. wileyonlinelibrary.com/journal/joc 1

KEYWORDS
hybrid, machine learning, maximum overlap discrete wavelet transform, prediction,
preprocessing, rainfall, wavelet
1|INTRODUCTION
Rainfall prediction is important for several reasons, par-
ticularly in terms of its impact on hydrology and water
resources. Accurate rainfall prediction can help in flood
forecasting and management; by predicting heavy rainfall
events, appropriate measures can be taken to minimize
flood damage (Bezak et al., 2016; Bui et al., 2019). It is
also important in water resources planning and manage-
ment during drought conditions (Deo et al., 2018;
Mouatadid et al., 2018) in maximizing crop yields and
reducing water usage (Hartmann et al., 2016). Accurate
rainfall prediction is essential for managing water
resources such as lakes, reservoirs and rivers. It can help
authorities make decisions about water storage, release
and distribution (Serinaldi & Kilsby, 2012), and, in effi-
cient water supply management by predicting the avail-
ability of water for various uses (Ali et al., 2018; Bagirov
et al., 2017; Zeynoddin et al., 2018). Rainfall prediction is
also critical for the management of hydropower genera-
tion. Accurate prediction of rainfall patterns can facilitate
the optimization of the generation of hydropower and
ensure the stability of the electrical grid (Haddad, 2011).
In precipitation forecasting, numerical models based
on physical mechanisms of natural events, statistical
models and their combinations have been used for years
to examine the relationship between precipitation and
geographic coordinates such as latitude and longitude
(Chegaar & Chibani, 2001) and other atmospheric
parameters such as temperature, humidity, pressure and
wind speed (Ali et al., 2018; Giebel & Kariniotakis, 2017;
Shahrban et al., 2016; Yu et al., 2016). Numerical models
for precipitation forecasting require substantial amounts
of data, leading to high computational costs (Gouda
et al., 2019; Mousavi et al., 2017). The statistical models
offer a methodology of extracting the characteristic fea-
tures of historical rainfall time series and using these
characteristics to forecast future trends in rainfall (Ashby
et al., 2005). On the other hand, importantly, Chong et al.
(2020) highlighted that certain statistical techniques may
be inadequate for predicting rainfall due to the tendency
of historical data to undergo significant transformations
in a relatively brief period of time.
The utilization of artificial intelligence and machine
learning algorithms in rainfall forecasting research has
emerged as a significant approach for modelling complex,
nonlinear phenomena, over the past few decades
(Altunkaynak & Küllahcı,2022; Chadalawada
et al., 2017; Küllahcı& Altunkaynak, 2023a,2023b;
Mandal & Jothiprakash, 2012; Wang & Altunkaynak,
2012) These cutting-edge technologies have proven to be
highly effective in accurately predicting rainfall patterns,
and overcoming the limitations of traditional statistical
methods (Altunkaynak & Nigussie, 2017; Jaiswal &
Malhotra, 2018). By combining multiple individual
models or techniques, these hybrid methods have the
potential to produce more robust and accurate predic-
tions, leading to improved outcomes and advancements
in the prediction field (Heidary & Abad, 2021; Küllahcı&
Altunkaynak, 2023a,2023b; Li et al., 2018; Ouyang
et al., 2016; Pandey et al., 2019; Partal & Ki¸si, 2007; Solgi
et al., 2014; Song et al., 2021; Yin et al., 2023; Zhao
et al., 2021). A selection of studies on the utilization of
both machine learning and signal processing techniques
in the prediction of rainfall time series can be found in
Table 1.
In the present contribution, we introduce an original
approach to rainfall analysis. We apply the maximum
overlap discrete wavelet transform (MODWT) signal
decomposition algorithm to enhance the accuracy of
daily rainfall predictions and extend the prediction time
horizon. Additionally, hybrid MODWT models are con-
trasted with hybrid discrete wavelet transform (DWT)
models. To the best of authors' knowledge, the MODWT
algorithm, as a signal decomposition method has not
been used in conjunction with different prediction
modelling methods. This study represents a pioneering
effort by integrating the MODWT (maximal overlap dis-
crete wavelet transform) with six distinct prediction tech-
niques. These methods include Artificial Neural
Networks (ANN), K-Nearest Neighbours (K-NN),
Extreme Learning Machine (ELM), Fuzzy Logic,
XGBoost (eXtreme Gradient Boosting) and a Deep Learn-
ing approach known as Long-Short Term Mem-
ory (LSTM).
The motivation of this study is to research and find
answers to the following three questions:
1. Can the MODWT improve rainfall predicting perfor-
mance when integrated with ML algorithms?
2. Can MODWT integrated with machine learning algo-
rithms achieve higher prediction accuracy compared
to models integrated with DWT in daily rainfall
prediction?
2KÜLLAHCIand ALTUNKAYNAK

3. In case of the success of the integrated models, which
MODWT hybridized machine learning method pro-
vides the best performance for daily rainfall
prediction?
These questions are investigated to assess the poten-
tial of the MODWT method through a daily rainfall-
predicting case study in Turkey. In exploring answers to
these questions, it is also expected that in addition
to evaluating the potential of MODWT for precipitation
forecasting, MODWT can be leveraged towards other
important hydrological forecasting applications
(e.g., groundwater level, evaporation, stream flow, water
quality).
The remainder of this study is structured as follows:
section 2provides a brief overview of the study area, data
and the methods employed in the analysis; section 3pre-
sents the key findings and a discussion of their implica-
tions, and section 4offers concluding remarks and
suggestions for future research directions.
2|MATERIALS AND METHODS
2.1 |Study area and data
The present study utilizes daily precipitation data from
three distinct precipitation observation stations, namely
Diyarbakır, Şanlıurfa and Adıyaman, located in the
Southeastern Anatolia region of Turkey. The Diyarbakir,
Şanlıurfa and Adıyaman stations, designated with the
numbers 17280, 17270 and 17265 are located at coordi-
nates 3753050.300N–4012009.700 E, 3709038.900N–
3847010.700E and 3745019.100 N–3816039.000E, respec-
tively. The data used in this study was obtained from the
Turkey State Meteorological Service (MGM, 2020) and
consisted of daily precipitation measurements. Figure 1
depicts the geographic positions of the meteorological
stations employed in the investigation. The observational
data from three stations cover a time period of 51 years,
from January 1970 to June 2021. The time series of daily
rainfall data for each of the stations are depicted in
Figure 2a–c, respectively. Table 2provides descriptive sta-
tistics for both model calibration and test data sets. The
division of data into training and test sets is a critical step
in machine learning algorithms, as it facilitates accurate
model performance evaluation. During the training
phase, the model learns the most suitable parameters that
capture patterns and relationships within the training
data. These parameters enable the model to make predic-
tions based on input features derived from the test data
by establishing internal representations and decision
boundaries. Subsequently, the model's predictions are
compared with the actual target values in the test data to
assess its performance. In this study, the k-fold cross-
validation method was employed to ensure model
TABLE 1 A few of the rainfall time series studies related to the hybrid usage of machine learning and signal processing technique.
References
Study
area
Temporal
scale
Decomposition
method Prediction method
Performance
evaluation
Feng et al. (2015) China Monthly DWT SVM R, RMSE, MAE, NSE
Altunkaynak and Nigussie
(2015)
Turkey Daily DWT, SA MLP RMSE, CE, SS
Amiri et al. (2016) Iran Monthly DWT ANN MAE, RMSE, SDR, IA
Tao et al. (2017) China Monthly EMD LSSVM NSE, RAE
Ghamariadyan et al. (2019) Australian Monthly DWT ANN RMSE, MAE, d
r
,R
Bojang et al. (2020) Taiwan Monthly SSA LSVR, RF RMSE, NSE
Wu et al. (2021) China Monthly,
annual
DWT ARIMA, LSTM RMSE, MAE, R
2
Wang et al. (2021) China Monthly WPD BPNN, GMDH,
ARIMA
RMSE, MAE, R, NSE
Singh et al. (2024) India Monthly DWT ANN RMSE, CE, R
2
Narimani et al. (2022) South
Korea
Daily SSA, EMD LightGBM, XGBoost RMSE, NSE, MAE, R
2
Abbreviations: ARIMA, autoregressive integrated moving average; BPNN, back-propagation neural network; CE, coefficient of efficiency; d
r
, refined index of
agreement; EMD, empirical modal decomposition; GMDH, group method of data handing; IA, index of agreement; LSSVM, least squares support vector
machine; LS-SVR, least-squares support vector regression; MAE, mean absolute error; MLP, multilayer perceptron; NSE, Nash–Sutcliffe; R, correlation
coefficient; RAE, relative absolute error; RF, random forest; RMSE, root-mean-square error; SA, season algorithm; SDR, standard deviation of residuals; SS,
skill score; SSA, singular spectrum analysis; WPD, wavelet packet decomposition.
KÜLLAHCIand ALTUNKAYNAK 3

accuracy and optimize hyperparameters. Cross-
validation, particularly k-fold cross-validation, assesses
model accuracy and determines optimal hyperpara-
meters. This technique requires splitting the data into
multiple subsets (folds) for both training and validating
the machine-learning model. In this study, daily rainfall
data is partitioned into two parts: training and testing.
The first 28 years of observed daily rainfall data, consti-
tuting 55% of the total data were allocated for model cali-
bration (from January 1970 to May 1997). The remaining
23 years of observed daily rainfall data, representing 45%
of the dataset (from May 1997 to June 2021), were
reserved for evaluating model performance. To ensure a
robust assessment of model accuracy, k-fold cross-
validation with k=5 is employed. This approach
enhances the reliability of our model evaluation by sys-
tematically rotating through different subsets of data for
training and validation, thereby reducing the risk of over-
fitting and providing a more comprehensive assessment
of model generalization performance.
2.2 |Discrete wavelet transform
vs. maximal overlap discrete wavelet
transform
This section provides an introduction to the traditional
discrete wavelet transform (DWT) and the maximal over-
lap discrete wavelet transform (MODWT). Additionally,
the proposed MODWT decomposition technique is pre-
sented in this section.
Wavelet analysis is a commonly utilized technique for
preprocessing signals. It was introduced, in contrast to
Fourier analysis (Daubechies, 1990), to extract both tem-
poral and frequency information and to overcome certain
limitations of stationary and nonstationary time series
modelling. The fundamental concept behind the discrete
wavelet transform (DWT) algorithm involves applying
both high-pass and low-pass filters to the original signal
simultaneously, followed by a downsampling operation.
This approach allows for the separation of the signal into
different frequency bands, with the low-pass filter
capturing the low-frequency components (called approxi-
mation) and the high-pass filter extracting the high-
frequency components (called detail).
The maximal overlap discrete wavelet transform
(MODWT) is a modified version of the discrete wavelet
transform (DWT) that is specifically engineered to
achieve a more comprehensive signal decomposition,
particularly for nonstationary signals (Percival &
Wladen, 2000). Unlike the DWT, the MODWT uses a fil-
ter bank that has a longer impulse response and an over-
lap between adjacent sub-bands, which allows for a more
accurate and complete decomposition of a signal into dif-
ferent frequency bands. In the MODWT, the filters are
designed to be maximally decimated, which means that
the subsampling step is delayed until the end of the
decomposition process, thus ensuring that all of the data
is used in the decomposition. The resulting sub-bands
have the same length as the original signal, and the over-
lap between adjacent sub-bands allows for a more accu-
rate reconstruction of the original signal. The details and
FIGURE 1 Study area.
[Colour figure can be viewed at
wileyonlinelibrary.com]
4KÜLLAHCIand ALTUNKAYNAK

comparison of DWT and MODWT can be found in Cor-
nish et al. (2006).
For a daily rainfall signal R=Rt,t=0,1,…,k−1g
f, ini-
tially, the primary-stage approximations and details
should be calculated,
Dj,nX
k−1
t=0
~
h

j,tWj,n+tmod k,ð1Þ
Aj,n=X
k−1
t=0
~
g
j,t
~
Vj,n+tmod k:ð2Þ
The elements of jth level MODWT scaling and wave-
let coefficients Vjand Wjcan be written as, respectively,
Vj,n=X
k−1
t=0
~
g
j,tRn−tmod kj=1,2:3,…,L,ð3Þ
Wj,n=X
k−1
t=0
~
h

j,tRn−tmod k,ð4Þ
where kis the length of rainfall signal, ~
g
j,tand
~
h

j,tare jth
level low- and high-pass filters yielded by periodizing ~
gj,t
FIGURE 2 Time series of
the daily rainfall data obtained
from (a) Station 17280
(Diyarbakır), (b) Station 17270
(Şanlıurfa) and (c) Station 17265
(Adıyaman). [Colour figure can
be viewed at
wileyonlinelibrary.com]
KÜLLAHCIand ALTUNKAYNAK 5

and
~
hj,tto length k, respectively, and ~
gj,tand
~
hj,tare jth
level MODWT low- and high-pass filter.
Ultimately, the original rainfall time series signal can
be expressed in relation to the approximations and details
as follows:
RtðÞ=X
L
j=1
Dj+Aj:ð5Þ
Figure 3shows a flowchart of three-level MODWT for
rainfall time series.
2.3 |Artificial neural network
Artificial neural networks (ANNs) are simplified mathe-
matical models that capture various aspects of the func-
tions and structure of the human brain. Although ANNs
are originated from preliminary studies focused on devel-
oping mathematical models inspired by biological sys-
tems, their development has been influenced by various
mathematical and computational principles beyond
direct emulation of biological systems (Anderson
et al., 2001; Argatov, 2019; Avramidis & Wu, 2007;
Niarakis, 2022). ANNs are composed of several key com-
ponents, including
•The input layer often termed the input vector com-
prises input elements representing independent vari-
ables. Ideally, the number of neurons in the input
layer aligns with the number of inputs, but this is not
always necessary or optimal. Neural networks possess
the ability to process inputs of varying dimensions
thanks to techniques like feature engineering or
dimensionality reduction. These methods enhance the
network's flexibility and accuracy by effectively trans-
forming and extracting meaningful information from
the input data, enabling more robust and efficient
learning.
•Hidden layers in artificial neural networks play a piv-
otal role in processing input data to create more mean-
ingful representations, allowing the network to
capture intricate relationships. While there is no fixed
rule governing the optimal configuration of hidden
layers, their design significantly impacts the network's
TABLE 2 Related statistical
properties for rainfall stations.
Station Data Min Max Mean Standard deviation Skewness
Diyarbakır Total 0.00 71.60 3.35 5.33 3.29
Training 0.00 71.60 2.27 5.24 3.48
Testing 0.00 58.60 4.02 5.43 3.11
Şanlıurfa Total 0.00 90.50 3.47 5.74 3.64
Training 0.00 64.70 3.10 5.64 3.80
Testing 0.00 90.50 4.61 5.83 3.50
Adıyaman Total 0.00 105.90 4.44 7.11 3.47
Training 0.00 80.10 3.53 6.93 3.14
Testing 0.00 105.90 4.31 7.30 3.80
FIGURE 3 A flowchart of
three-level MODWT for rainfall
time series. [Colour figure can
be viewed at
wileyonlinelibrary.com]
6KÜLLAHCIand ALTUNKAYNAK

performance. Traditionally, trial-and-error methods
have been employed to determine the ideal number of
hidden layers and neurons per layer, as mentioned by
(Altunkaynak, 2007). However, it is crucial to note that
alongside these empirical techniques, more systematic
approaches like grid search, random search, Bayesian
optimization, gradient-based optimization and evolu-
tionary optimization have gained prominence for
hyperparameter tuning (Bergstra & Bengio, 2012; Joy
et al., 2016; Sergeyev et al., 2017). These methods offer
a more structured and efficient means of exploring the
vast architectural space, leading to improved model
performance and convergence.
•Weighted connections between nodes in adjacent
layers, which allow information to flow between the
layers.
•An output layer comprising one or more elements that
represent the dependent variable(s) being predicted by
the network (Walczak, 2014).
For a comprehensive understanding of artificial neu-
ral networks, it is essential to have knowledge about acti-
vation functions. Activation functions are mathematical
operations applied to the output of each neuron in a neu-
ral network to determine its output behaviour. In an arti-
ficial neural network, the sum of the products of inputs
and their corresponding weights is calculated, and
finally, an activation function is applied to obtain the out-
put of that layer and provide it as input to the next layer.
Therefore, the selection of a suitable activation function
can significantly impact the effectiveness of a neural net-
work in solving a specific problem. While numerous
types of activation functions can be used in artificial neu-
ral networks, the most preferred functions are linear,
hyperbolic tangent, sigmoid and step functions (Bueno &
Salmeron, 2009). In recent years, activation functions
such as ReLU, Leaky ReLU, Parameterized ReLU and
SoftMax have gained popularity in the literature. The
ReLU activation function, in particular, is favoured for its
mathematical simplicity, ease of derivative computation
leading to accelerated training processes, mitigation of
the vanishing gradient problem, and ability to induce
sparsity by zeroing out negative inputs. These character-
istics contribute to its widespread adoption in neural net-
work architectures across various domains (Hayou
et al., 2019; Sharma et al., 2020).
2.4 |Fuzzy Logic
Fuzzy Logic, developed by Lotfi A. Zadeh in the 1960s,
allows for graded evaluation rather than strict true or
false determination (Zadeh, 1965,1968,1978) It finds
wide applications in engineering problems such as con-
trol systems, pattern recognition, image processing and
decision-making (Altunkaynak, 2010; Özger & Şen, 2007;
Şen & Altunkaynak, 2004; Xiong et al., 2001) The algo-
rithm involves fuzzification, inference and defuzzifica-
tion processes. The algorithm involves fuzzification,
inference and defuzzification processes. Two main fuzzy
inference methods are Mamdani and Takagi-Sugeno. The
selection of appropriate parameters in TS fuzzy logic
modelling can be a challenging task (Mamdani, 1974;
Takagi & Sugeno, 1985). Therefore, ANFIS (Adaptive
Network-based Fuzzy Inference System) methodology,
originally introduced by Jang and Roger (1993), is com-
monly utilized to estimate the parameters of the member-
ship and consequent functions.
2.5 |K-Nearest Neighbour
K-Nearest Neighbour (KNN) is a widely used machine
learning algorithm for both classification and regression
tasks, known for its ease of applicability. KNN assigns
the label or value of the majority class among its
knearest neighbours for classification tasks or computes
the average of the values of its knearest neighbours for
regression tasks. KNN is classified as a nonparametric
algorithm since it does not presuppose any hypotheses
regarding the underlying distribution of the data (Fix &
Hodges, 1951). It is also a lazy learning algorithm, which
means that KNN does not have a training stage in the tra-
ditional sense. KNN stores the entire training dataset,
and at the time of inference, it computes predictions
directly (Hellman, 1970). KNN identifies the knearest
neighbours to the new observation based on a distance
metric.
The K-NN algorithm requires the computation of the
distance metric between a predicted data point and
the known data points in the training set. While there
exists a plethora of distance metrics, this study will focus
on Euclidean distance that is commonly employed in the
K-NN algorithm. The Euclidean distance dbetween two
point x,yin a multidimensional space can be calculated
using Equation (6) as follows:
dx,yðÞ=Xn
i=1xi−yi
jj
2

1
2,ð6Þ
where (xi,yi) are the variables of vectors xand y, respec-
tively, in the two-dimensional vector space, nis the num-
ber of variables and dis the Euclidean distance. The
common use of Euclidean distance in K-NN is due to its
simplicity and effectiveness, as it considers differences in
all data dimensions. Selecting the optimal number of
KÜLLAHCIand ALTUNKAYNAK 7

neighbours (k) is crucial for K-NN's performance, which
is achieved through cross-validation. k-fold cross-
validation partitions the data into training and test sets
across multiple folds, allowing for thorough model evalu-
ation. By testing various kvalues during each fold, the
impact on performance metrics such as accuracy and pre-
cision can be analysed, aiding in the selection of the opti-
mal kvalue.
2.6 |Extreme Learning Machine
Extreme Learning Machine (ELM) is a type of
machine learning algorithm that belongs to the family of
artificial neural networks (ANNs). ELMs are designed to
address some of the limitations of traditional ANNs, such
as long training times, the need for fine-tuning, and the
risk of overfitting. The Extreme Learning Machine (ELM)
proposed by Huang et al. (2006) is characterized by a
feedforward neural network architecture with a single
hidden layer consisting of one neuron. For more detailed
information, refer Gumaei et al. (2019) and Huang
et al. (2015,2011).
2.7 |Long-Short Term Memory Neural
Network
Long Short-Term Memory (LSTM) is a specialized variant
of recurrent neural networks (RNN) that is specifically
engineered to address the issue of vanishing gradients, a
persistent challenge encountered in conventional RNNs.
By Hochreiter and Schmidhuber (1997), like traditional
RNNs, LSTMs are designed to model sequential data by
maintaining a hidden state that captures the current
“context”of the input sequence. However, LSTMs differ
from traditional RNNs in that they incorporate memory
cells, which allow the network to selectively store and
retrieve information over long periods of time. The mem-
ory cell is controlled by various gates, which regulate the
flow of information into and out of the cell, allowing
the LSTM to learn and model long-term dependencies in
the input sequence (Lecun et al., 2015).
2.8 |Extreme Gradient Boosting
(XGBoost)
Ensemble modelling involves the generation of models
through tree-based methods such as random forests, extra
trees, adaptive and gradient boosting techniques
(Friedman, 2001). This study uses XGBoost to benchmark
a multi-stage ensemble model, enhancing performance
compared to gradient-boosted decision trees (GBDT).
XGBoost creates decision trees faster due to its paralleli-
zation feature, optimizing loop steps for efficient execu-
tion and handling missing values efficiently (Chen &
Guestrin, 2016; Li et al., 2019; Wang et al., 2020).
2.9 |Performance evaluation criteria
The evaluation of machine learning methods often relies
on multiple performance metrics, which allow for a com-
prehensive and reliable assessment of their predictive
capabilities. In the present study, we employed four dis-
tinct diagnostic metrics to compare the performance of
the methods under consideration, namely the mean
square error (MSE), coefficient of efficiency (CE),
mean absolute error (MAE) and correlation coefficient
(R). Each of these metrics captures different aspects of
the model's accuracy and ability to fit the data, and their
combined use enables a more robust evaluation of the
machine learning algorithms. Table 3presents the equa-
tions and intervals for the performance metrics employed
in this study.
The expressions presented here are commonly used to
evaluate the performance of prediction models. In these
expressions, Pprepresents the predicted values, Porepre-
sents the observed values, Ppand Porepresent the mean
of the predicted and observed values, respectively, and
nrepresents the number of samples.
3|RESULTS AND DISCUSSION
3.1 |Model development
This study proposes a combined model of machine learn-
ing algorithms with data processing tools MODWT to
improve daily rainfall prediction accuracy. Hybrid
MODWT models are developed to improve the prediction
accuracy of daily rainfall for three stations in Turkey over
an extended time horizon of up to 3 days. In addition to
the hybrid MODWT models, independent models and
hybrid DWT models are also created for comparison pur-
poses. The present study aims to develop an enhanced
prediction framework and obtain predictive results for
daily rainfall at three stations in Turkey, with an
extended time horizon of up to 3 days. To achieve this
objective, a series of methodological steps are performed,
which can be summarized as follows:
1. Determining the number of previous inputs for the
predictive model is a fundamental step in timeseries
research. In order to identify the lags that are
8KÜLLAHCIand ALTUNKAYNAK

correlated with the original series, autocorrelation
functions (ACFs) are generated. Subsequently, an arti-
ficial neural network (ANN) model is applied, taking
into account the lag times obtained from the ACFs.
The purpose of this step was to determine the optimal
number of lags to be used in the model, which is criti-
cal for achieving an accurate and reliable prediction
of the target variable. By considering different lag
times acquired from the ACFs, the ANN model is able
to effectively identify the lag times that are most
strongly correlated with the original series, thereby
optimizing the predictive performance of the model.
2. In the modelling stage, the stand-alone ML algo-
rithms, ANN, Fuzzy, KNN, ELM, LSTM and XGBoost
are developed to predict daily rainfall. Then, the
MODWT decomposition technique is incorporated
into the stand-alone models to improve the prediction
accuracies. In the last step of the modelling stage, dis-
crete wavelet transform (DWT) is included in the
independent models in order to make a comparison
with the hybridized MODWT model results.
3. Finally the accuracy of all models is evaluated with
respect to the performance metrics. Moreover, the
scatter and Taylor diagrams are presented to provide
the intercomparison of the generated models.
The flowchart of the current research is shown in
Figure 4. The dashed lines in the figure correspond to the
primary phases and preprocessing steps involved in
developing a machine-learning model. On the other
hand, the solid lines depict the operational stages of the
model.
3.1.1 | Stage 1: Lag time determination
Optimizing the lag time is an utmost important compo-
nent of effective time series forecasting. The lag time
denotes the delay between the occurrence of an event and
the manifestation of its impact in the time series data. The
appropriate lag time is determined by identifying the opti-
mal number of past observations to be incorporated into
the model to achieve accurate predictions. The literature
describes two common methods for input selection: a trial
and error approach to determine the optimal number of
lagged time series data for the best forecast performance,
and using the autocorrelation function (ACF) method to
identify the most correlated lagged variables.
In this study, the ACF method was used to select
input variable combinations, which is a frequently used
approach in hydrological forecasting studies (Kothe
et al., 2019; Mislan et al., 2015). The ACF method identi-
fied that lag time-1 has the highest autocorrelation, as
illustrated in Figure 5. However, during model develop-
ment, the Artificial Neural Network (ANN) algorithm
was employed to determine whether lags larger than
1 have a positive impact on the model. The general archi-
tecture of the employed Artificial Neural Network (ANN)
algorithm is as follows: it comprises three layers, each
consisting of 50 neurons. The activation function utilized
throughout the network is the tanh. The optimization
algorithm employed is Adam, with a learning rate set at
0.001. Additionally, the training process iterates
250 epochs. This configuration is selected to ensure a
structured and robust framework for neural network
modelling, allowing for effective learning and adaptation
to complex patterns within the data.
Representing other stations, the results of the model
for station 17280 are presented in Table 4.
Initially, a lag time of daily (t) is deemed as a single
input for the analysis, after which additional lag times
are sequentially integrated up to a maximum of six con-
secutive lag times. Table 4shows that the inclusion of
more lag times leads to an increase in the model's perfor-
mance from one to two lag times. (CE
(t)
:0.54,
CE
(t−1,t)
:0.60). Furthermore, as evidenced by the
TABLE 3 Performance evaluation
criteria. Metrics Maximum Minimum Equation
MSE ∞0MSE=1
nP
n
i=1
Ppi −Poi

2
CE 1 −∞
CE=1−P
n
i=1
Ppi −Poi
ðÞ
2
P
n
i=1
Ppi −Poi
ðÞ
2
MAE ∞0MAE=1
nP
n
i=1
Ppi −Poi

R1−1
R=P
n
i=1
Ppi −Ppi
ðÞ
×Poi −Poi
ðÞ

2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
n
i=1
Ppi −Ppi
ðÞ
2

s×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
n
i=1
Poi −Poi
ðÞ
2
r
2
6
6
6
6
4
3
7
7
7
7
5
KÜLLAHCIand ALTUNKAYNAK 9

presented values in Table 4, it was observed that the
inclusion of more than two lag times did not contribute
to the accuracy of the models. Based on these observa-
tions and in accordance with the ACF results, the num-
ber of time lags was determined to be two.
3.1.2 | Stage 2: Time series decomposition
Wavelet decomposition is a signal processing technique
that decomposes a signal into a set of wavelet basis
functions. The basis functions are generated by dilating
and translating a single mother wavelet function, which
is usually chosen to have compact support in both the
time and frequency domains. Despite numerous efforts in
the literature, identifying a universal wavelet type that is
applicable to all time series remains a challenging task
due to the lack of established guidelines for mother wave-
let selection. In this respect, this study employed four dis-
tinct wavelet types, namely Symlets wavelets, Daubechies
wavelets, Fejer-Korovkin and Coiflet wavelets, to deter-
mine the optimal wavelet type that best matches the
FIGURE 4 Flowchart of the
model development processes.
[Colour figure can be viewed at
wileyonlinelibrary.com]
10 KÜLLAHCIand ALTUNKAYNAK

observations. To identify the optimal level of decomposi-
tion, three to six levels were tested for each mother wave-
let. The ELM model was selected for this process due to
its fast processing capability. Hyperparameters used in
the ELM model are number of neurons in input layer,
35, number of neurons in hidden layer, 35, activation
function, tanh, and learning rate, 0.001.
Table 5presents the bold values in the results that
indicate the optimal combination of wavelet function and
decomposition level for daily rainfall prediction. The
results demonstrate that the Symlets wavelets produce the
highest accuracy, while the Fejer-Korovkin wavelets yield
the lowest accuracy. Furthermore, among the various
decomposition levels tested for Symlets wavelets, the third
level achieves the best results, with an MSE of 0.84 and an
CE of 0.97166. These results were used in subsequent pre-
dictions for daily rainfall by combining Symlets wavelet
decomposition with various machine learning methods.
Figure 7illustrates the obtained subseries (called Level
1, Level 2 and Level 3) and the approximation through
Symlets wavelet decomposition, respectively.
In order to be able to compare the MODWT model
results obtained during the modelling phase, the time
series is further decomposed into subseries with DWT in
stand-alone models. As in MODWT, different wavelet
types and levels have been tested and the best perfor-
mance has been obtained for the Haar wavelet and three
subseries.
FIGURE 5 Autocorrelation function graph of three stations. [Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 4 The performance of
different lag-time for Station 17280.
Model Input combination Output
Training Test
MSE CE MSE CE
ANN tt+1 16.94 0.46 15.74 0.54
ANN t−1,tt+1 15.69 0.51 13.76 0.60
ANN t−2, t−1,tt+1 15.49 0.50 13.94 0.59
ANN t−3, t−2, t−1,tt+1 15.76 0.50 14.24 0.58
ANN t−4, t−3, t−2, t−1,tt+1 15.25 0.51 14.01 0.59
ANN t−5, t−4, t−3, t−2, t−1,tt+1 15.50 0.51 14.36 0.58
ANN t−6, t−5, t−4, t−3, t−2, t−1,tt+1 15.40 0.51 14.31 0.58
Note: Bold indicates the prediction results obtained using models with different inputs and the data of the
model that gives the highest prediction.
KÜLLAHCIand ALTUNKAYNAK 11

3.1.3 | Stage 3: Prediction models
In this stage, six different machine learning
(ML) techniques were applied for daily rainfall prediction
in Turkey, including ANN, Fuzzy logic, K-NN, ELM,
XGBoost and LSTM hybridized with the MODWT signal
decomposition technique. The training process in
machine learning algorithms engages learning and
adjusting their parameters based on input data. The tun-
ing of hyperparameters is a crucial step in evolving reli-
able ML models. Tuning hyperparameters reduces
overfitting and enhances the model's generalizability to
new data (Bardenet et al., 2013). Selecting the best hyper-
parameters is also a significant component of improving
the accuracy of the model. There are various methods for
hyperparameter selection and finding the optimal solu-
tion. Grid search and random search are among the
many methods developed for hyperparameter optimiza-
tion. Random search methods randomly select different
hyperparameter values for a specified number of itera-
tions, while grid search explores all potential values
within a predefined range of hyperparameters through
trial and error to find the best solution. In this study, a
grid search technique with five-fold cross-validation was
used to explore the possible values of hyperparameters
(Kuhn & Johnson, 2013). This approach is particularly
advantageous when dealing with a large number of
hyperparameters or a high-dimensional search space,
offering improved efficiency compared to an exhaustive
random search (Yu & Zhu, 2020). Table 6contains the
necessary hyperparameters and trial intervals for each
algorithm. During grid search, each combination of
hyperparameters on the grid is evaluated by training and
validating the model using the cross-validation set. By
thoroughly exploring all possible combinations of
TABLE 5 The performance of different mother wavelets and
their corresponding different decomposition levels for daily rainfall
prediction.
MODWT function Level
Test
MSE CE
Symlets wavelets 3 0.84 0.97166
4 0.88 0.97033
5 0.86 0.97079
6 0.88 0.97031
Daubechies wavelets 3 1.00 0.96626
4 1.10 0.96282
5 1.09 0.96308
6 1.00 0.96601
Fejer-Korovkin wavelets 3 1.64 0.94427
4 1.62 0.94502
5 1.61 0.94551
6 1.68 0.94308
Coiflets wavelets 3 0.98 0.96691
4 1.04 0.96470
5 0.86 0.97070
6 0.89 0.96699
Note: Bold indicates the wave type and properties that give the best results
by examining the models obtained in different wave types and categories in
order to decide on the wave type to be used in the proposed MODWT
separation method.
TABLE 6 Hyperparameters of models for all algorithms.
Method Parameter Value
ANN Network type Feed Forward Neural
Network with Back
Propagation
Neurons used for
hidden layer
[3, 15, 35]
Activation function [sigmoid, tanh, linear]
Optimization
algorithm
[adam, sgd, rmsprop]
Learning rate [0.001, 0.01, 0.1]
Number of epochs [100, 500, 1000]
Batch size [16, 64, 128]
KNN Max_k [50]
Distance metric [euclidean, manhattan,
minkowski]
Weights [Uniform, distance]
ANFIS Number of fuzzy sets [2, 3, 5]
Membership function [Triangular, Trapezoidal,
Gaussian]
Training algorithm [“gradient descent,”“least
squares,”“hybrid”]
XGBoost Number of trees [1, 100]
Maximum depth [1, 30]
Learning rate [0.1, 0.95]
ELM Number of neurons in
input layer and
hidden
[20, 35, 100]
Activation functions [sigmoid, tanh, sine,
radial]
Learning rate [0.001, 0.01, 0.1]
LSTM Number of Hidden
units
[100, 250, 500]
Dropout rate [0.1, 0.2, 0.3]
Initial learning rate [0.001, 0.005, 0.01]
Activation functions [sigmoid, tanh, relu]
Backpropagation
algorithm
[adam, sgd, rmsprop]
12 KÜLLAHCIand ALTUNKAYNAK

hyperparameters within the defined grid, grid search aids
in identifying the best set of hyperparameters that yield
the highest performance metric.
3.2 |Model results
3.2.1 | Stand-alone models results
Table 7displays the results of the stand-alone models in
terms of different performance indicators. Evident from
Table 7is, as the prediction time increases, it can be
observed that the MSE and MAE values increase, while
the CE and R values decrease. As the correlation value
decreases with increasing lead times, it is reasonable to
expect that the predictive performance of the models
will be lower for a wider future time horizon. Based on
themodelresults,thehighestperformanceforT+1
lead time at station 17280 is observed in the LSTM
model with CE
LSTM
=0.51, while the lowest perfor-
manceisfoundintheFuzzymodelwithCE
Fuzzy
=0.41.
Fortheothermodels,theobtainedresultsarebetween
these two values close to each other. When the results
for station 17270 are examined, the highest perfor-
mance is observed in the ANN model with
CE
ANN
=0.61, while the lowest performance is seen in
the Fuzzy model with CE
Fuzzy
=0.53. For the remaining
models, obtained results are close to each other between
these two values. Finally, when the results for station
17265 for T+1 lead time are considered, the highest
performanceisobservedintheLSTMmodelwith
CE
LSTM
=0.43, while the Fuzzy model provided lowest
performance with CE
Fuzzy
=0.35. Overall, it is observed
that for T+1 lead time, only a few models achieve the
acceptable success criterion of CE =0.5 at all three sta-
tions, while the other models perform below this value.
Altunkaynak (2010)notedthatthereisnowidely
accepted standard for evaluating model performance
based on the CE value. However, according to Donigian
and Love (2012) and Altunkaynak (2010) CE values fall-
ing within the ranges of 0.65–0.75 and 0.75–0.85 can be
deemed fair and good, respectively. Furthermore, a CE
value exceeding 0.85 is regarded as indicative of very
good model prediction performance. For the subsequent
lead times (T+2) and (T+3), the prediction perfor-
mances of the models are below the acceptable level,
indicating that stand-alone models need to be used in
conjunction with preprocessing techniques to achieve
high-accuracy prediction results for daily rainfall at the
three stations.
TABLE 7 The comparative performance evaluation of stand-alone machine learning models using selected indicators.
Station 17280 (test) 17270 (test) 17265 (test)
Models MSE CE MAE RMSE CE MAE RMSE CE MAE R
t+1 ANN 14.76 0.50 1.59 0.71 13.76 0.60 1.44 0.77 30.81 0.42 2.11 0.65
Fuzzy 17.26 0.41 1.97 0.64 16.08 0.53 1.79 0.73 34.51 0.35 2.67 0.60
K-NN 15.09 0.49 1.56 0.70 14.59 0.57 1.40 0.76 30.97 0.42 2.11 0.65
ELM 15.07 0.49 1.62 0.70 13.85 0.59 1.44 0.77 31.74 0.41 2.18 0.64
XGBoost 15.95 0.46 1.67 0.68 14.93 0.56 1.51 0.75 31.90 0.40 2.22 0.63
LSTM 14.53 0.51 1.43 0.72 14.18 0.58 1.29 0.77 30.35 0.43 1.95 0.66
t+2 ANN 18.38 0.38 2.08 0.61 18.40 0.46 1.95 0.68 38.06 0.29 2.78 0.54
Fuzzy 21.12 0.28 2.50 0.53 21.56 0.37 2.35 0.61 41.72 0.22 3.24 0.47
K-NN 18.38 0.38 2.08 0.61 19.50 0.43 1.88 0.65 38.84 0.27 2.71 0.52
ELM 18.85 0.36 2.19 0.60 18.88 0.45 2.03 0.67 38.89 0.27 2.89 0.52
XGBoost 19.55 0.34 2.18 0.58 20.56 0.40 2.07 0.63 39.75 0.26 2.92 0.51
LSTM 17.95 0.39 1.97 0.63 17.96 0.47 1.78 0.69 37.96 0.29 3.00 0.55
t+3 ANN 20.59 0.30 2.36 0.55 21.13 0.38 2.26 0.62 43.02 0.19 3.41 0.44
Fuzzy 23.20 0.21 2.78 0.46 24.71 0.27 2.65 0.53 45.73 0.14 3.70 0.38
K-NN 20.79 0.30 2.31 0.54 22.60 0.34 2.19 0.58 42.38 0.21 3.06 0.46
ELM 20.98 0.29 2.52 0.54 24.91 0.27 2.63 0.52 45.12 0.15 3.54 0.40
XGBoost 21.72 0.26 2.50 0.51 23.60 0.31 2.39 0.55 43.16 0.19 3.30 0.44
LSTM 20.76 0.30 2.29 0.55 20.98 0.38 2.27 0.62 41.26 0.23 3.10 0.48
KÜLLAHCIand ALTUNKAYNAK 13

3.2.2 | Hybrid MODWT machine learning
models results
After obtaining the results from the stand-alone models
the daily rainfall time series data are decomposed using
the maximum overlap discrete wavelet transform
(MODWT), which is an alternative to the commonly used
discrete wavelet transform (DWT) preprocessing algo-
rithm. In this study, the daily time series data are divided
into four subseries (bands), with one being approxima-
tion and the remaining three being detail bands
(Figure 6).
FIGURE 6 Plots of three
bands and the approximation
decomposed by MODWT for
station 17280. [Colour figure can
be viewed at
wileyonlinelibrary.com]
TABLE 8 The comparative performance evaluation of hybrid MODWT machine learning models using selected indicators.
Station 17280 (test) 17270 (test) 17265 (test)
Models MSE CE MAE RMSE CE MAE RMSE CE MAE R
t+1 MODWT-ANN 0.81 0.97 0.39 0.99 0.78 0.98 0.36 0.99 1.71 0.97 0.53 0.98
MODWT-Fuzzy 1.95 0.93 0.82 0.97 1.27 0.96 0.39 0.98 1.96 0.96 0.56 0.98
MODWT-K-NN 1.26 0.96 0.48 0.98 1.45 0.96 0.47 0.98 2.96 0.94 0.66 0.97
MODWT-ELM 0.84 0.97 0.39 0.98 0.90 0.97 0.37 0.99 1.96 0.96 0.56 0.98
MODWT-XGBoost 1.04 0.96 0.47 0.98 1.31 0.96 0.47 0.98 2.32 0.96 0.65 0.98
MODWT-LSTM 0.87 0.97 0.47 0.99 1.09 0.97 0.45 0.99 1.92 0.96 0.58 0.98
t+2 MODWT-ANN 5.33 0.82 0.96 0.91 4.61 0.86 0.85 0.93 11.39 0.79 1.33 0.89
MODWT-Fuzzy 5.65 0.81 0.99 0.90 5.45 0.84 0.88 0.92 11.50 0.78 1.38 0.89
MODWT-K-NN 5.58 0.81 1.02 0.91 5.10 0.85 0.92 0.93 11.73 0.78 1.40 0.89
MODWT-ELM 5.09 0.83 0.95 0.91 4.75 0.86 0.87 0.93 11.13 0.79 1.33 0.90
MODWT-XGBoost 5.45 0.82 1.01 0.9 5.10 0.85 0.93 0.93 11.56 0.78 1.41 0.89
MODWT-LSTM 5.38 0.82 1.02 0.92 4.45 0.87 0.85 0.94 10.92 0.80 1.57 0.90
t+3 MODWT-ANN 5.98 0.80 1.00 0.90 5.75 0.83 0.99 0.92 13.28 0.75 1.52 0.87
MODWT-Fuzzy 6.72 0.77 1.13 0.88 6.67 0.80 1.04 0.90 13.77 0.74 1.53 0.87
MODWT-K-NN 6.71 0.77 1.20 0.88 6.42 0.81 1.10 0.91 13.71 0.74 1.59 0.87
MODWT-ELM 6.35 0.78 1.13 0.89 9.28 0.73 1.41 0.86 20.09 0.62 2.02 0.79
MODWT-XGBoost 6.77 0.77 1.22 0.88 6.43 0.81 1.12 0.91 13.55 0.75 1.60 0.87
MODWT-LSTM 6.14 0.79 1.15 0.89 5.78 0.83 1.08 0.92 12.19 0.77 1.46 0.88
14 KÜLLAHCIand ALTUNKAYNAK

ML models are applied to each subseries which are
divided into training and testing sets. The subseries
are predicted up to a lead time of 3 days. The results are
presented in Table 8.
Upon evaluation of the results of hybrid MODWT
models, it is found that CE values for t+1 lead time ran-
ged between 0.93 and 0.98 for all three stations. These
results clearly indicate that the hybrid MODWT models
provide perfect accuracy for t+1 lead time. The CE value
ranges between 0 and 1, and values close to 1 are indica-
tive of excellent performance. When evaluating hybrid
MODWT models for t+1 lead time, the MODWT-ANN
model achieved the lowest MSE and highest CE values
for the three stations. However, it is evident that other
models also performed well with CE values being rela-
tively close. When comparing the hybrid MODWT
models with stand-alone models, it is observed that the
CE values for t+1 lead time increased from approxi-
mately 0.50–0.97 with the hybrid models. The findings of
stand-alone models indicate that they are not sufficient
for even predicting t+1 lead time, whereas the hybrid
MODWT models exhibited “excellent”performance with
CE values above 0.97. According to the prediction results
for t+2 lead time in Table 8, the CE values range from
0.78 to 0.87 for all three stations. When the results for
t+2 lead time are examined based on the prediction
algorithm, the highest performance with a CE value of
0.83 is obtained with the MODWT-ELM model for station
17280, the highest performance with a CE value of 0.87 is
obtained with the MODWT-LSTM model for station
17270, and finally, the highest performance with a CE
value of 0.80 was obtained with the MODWT-LSTM
model for station 17265. Upon analysis of the results
obtained for the t+3 lead time, it is found that the
hybrid models exhibit “good”performance, with CE
values reaching up to 0.83 across all three stations. As the
time lag between the input data and the predicted values
increases, there is a slight decline in the models' predic-
tive performance owing to the diminishing correlation
with delays, or time steps. Nevertheless, the findings indi-
cate that the hybrid MODWT models constitute a viable
means of enhancing predictive accuracy, as well as
extending the temporal horizon of reliable precipitation
forecasting for up to 3 days.
3.2.3 | Hybrid DWT machine learning
models results
As part of the final stage of modelling, the results of the
hybrid DWT model are presented for comparison with
the hybrid MODWT model. First, the daily rainfall time
series is divided into four subseries (bands) using DWT,
with one approximation and three details. Similar to the
other models, the subseries are separated into training
and testing sets, and the hybrid DWT model was applied
to each subseries to generate individual predictions for
lead times ranging from 1 to 3 days. The MSE, CE, MAE
and Rvalues of the hybrid DWT models are calculated
for the testing (calibration) phase, for lead times ranging
from 1 to 3 days, at three stations. These results are pre-
sented in Table 9. When examining the results of the
hybrid DWT model (Table 9), the CE values for the t+1
lead time are obtained between 0.71 and 0.73 for station
17280, between 0.72 and 0.80 for station 17270, and
between 0.80 and 0.86 for station 17265. The results indi-
cate that the hybrid DWT models performed better than
stand-alone models, but lagged behind the hybrid
MODWT models. For the hybridized DWT model, the
t+2 lead time CE values ranged between 0.69 and 0.47
for the three stations, while for the t+3 lead time, all CE
values were below 0.5. The decomposition of daily rain-
fall time series into certain frequencies and scales using
the widely used DWT decomposition method in the liter-
ature has contributed positively to the prediction perfor-
mance and improved the prediction accuracy. However,
the results of the model show that the DWT hybrid
model could not achieve the performance success of the
hybrid MODWT model. It is recognized that
the MODWT hybrid model provides a reliable and suffi-
cient contribution to increasing and improving the pre-
diction accuracy, especially for the t+2 and t+3 lead
times.
Moreover, the assessment of the outcomes is facili-
tated by employing Taylor diagrams, which furnishes a
comprehensive evaluation of the models' performance
across multiple dimensions, thereby allowing for the pre-
cise determination of their level of accuracy. The findings
presented in Figure 7demonstrate that the stand-alone
models exhibited lower levels of accuracy compared to
the hybrid-DWT models. Remarkably, the hybrid-
MODWT models outperformed all other models across
all lead times. Although the Taylor diagrams provide
insights into the accuracy of the predicted time series,
they do not offer information about the distributions of
the predicted and observed time series. Hence, it is essen-
tial to consider the distributional characteristics of the
time series to gain a comprehensive understanding of
the model performance. Additionally, scatter plots were
employed to assess the statistical significance of the
results.
A scatter plot in statistical analysis constitutes a visual
depiction of the correlation between two distinct vari-
ables. It typically consists of a horizontal axis represent-
ing one variable and a vertical axis representing the
other, plotting data points as individual points on
KÜLLAHCIand ALTUNKAYNAK 15

the graph. The scatter diagram allows for visualizing pat-
terns or trends in the data, such as a positive or negative
correlation between the two variables. It can be helpful
for identifying outliers or clusters in the data, as well as
for identifying potential relationships or dependencies
between the variables. Scatter diagrams for the models
for station 17280 for lead time t+1 are shown in
Figure 8and provide a visualization of the relationship
between the observed data and the corresponding precip-
itation forecasts. Moreover, the 45diagonal line (also
known as the 1:1 line) is utilized as an effective explor-
atory tool to assess the degree of concordance between
the observed daily rainfall data and the corresponding
predictions generated by the proposed models. Based on
the analysis of Figure 8, it can be observed that all of the
stand-alone models yielded suboptimal performances in
t+1 prediction, and predictions indicated a poor correla-
tion between the predicted and observed rainfall values.
However, the hybrid-DWT model demonstrated an
improvement in the performance of the stand-alone
models due to wavelet decomposition. As expected, the
t+1 prediction had the best results. However, this
enhancement was not sufficient for accurate rainfall pre-
diction, as the scatter plots for t+1 showed inadequate
dispersion of data points. Figure 8revealed that the
hybrid MODWT model outperformed the hybrid DWT
models, exhibiting significant improvements in t+1 lead
time. Consequently, it can be inferred that the hybrid
MODWT model produced the best-scattered results
among the proposed models.
In summary, the proposed hybrid MODWT models
showed superior performance in forecasting future rain-
fall for all time spans (1, 2 and 3 days) compared to other
models evaluated.
4|CONCLUSION
This study has aimed to compare the performance of var-
ious hybrid models for daily rainfall prediction by incor-
porating preprocessing methods, MODWT and DWT,
with machine learning algorithms to improve accuracy
and extend the lead time of prediction. The accurate pre-
diction of daily rainfall is crucial for effective water
resource management, flood forecasting and agricultural
planning, among other applications. However, due to the
complex and nonlinear nature of meteorological systems,
accurate rainfall prediction remains a challenging task.
Hybrid models, which combine multiple techniques to
improve prediction accuracy, have shown promising
results in previous studies and are increasingly being uti-
lized in rainfall prediction research. Therefore, this study
TABLE 9 The comparative performance evaluation of hybrid DWT machine learning models using selected indicators.
Station 17280 (test) 17270 (test) 17265 (test)
Models MSE CE MAE RMSE CE MAE RMSE CE MAE R
t+1 DWT-ANN 7.88 0.73 1.16 0.86 6.80 0.80 0.98 0.89 8.14 0.85 1.19 0.92
DWT-Fuzzy 8.90 0.70 1.32 0.84 9.61 0.72 1.29 0.85 10.53 0.80 1.45 0.90
DWT-K-NN 8.57 0.71 1.22 0.84 8.34 0.76 1.16 0.87 8.20 0.85 1.16 0.92
DWT-ELM 8.30 0.72 1.21 0.85 8.03 0.76 1.10 0.88 7.66 0.86 1.21 0.93
DWT-XGBoost 8.68 0.71 1.22 0.84 8.90 0.74 1.17 0.86 8.64 0.84 1.22 0.91
DWT-LSTM 8.05 0.73 1.17 0.55 8.21 0.76 1.20 0.88 7.63 0.86 1.16 0.93
t+2 DWT-ANN 13.22 0.55 1.78 0.74 13.39 0.61 1.64 0.78 25.33 0.53 2.36 0.71
DWT-Fuzzy 15.71 0.47 1.94 0.68 14.43 0.58 1.73 0.76 20.64 0.61 2.09 0.78
DWT-K-NN 14.10 0.52 1.92 0.72 14.30 0.58 1.79 0.76 17.72 0.67 1.85 0.81
DWT-ELM 13.39 0.55 1.78 0.74 13.60 0.60 1.64 0.78 17.74 0.67 1.87 0.81
DWT-XGBoost 14.21 0.52 1.84 0.72 15.21 0.55 1.76 0.74 18.39 0.66 1.94 0.81
DWT-LSTM 13.45 0.54 1.83 0.74 13.37 0.61 1.59 0.78 16.76 0.69 1.88 0.82
t+3 DWT-ANN 16.49 0.44 2.06 0.66 15.20 0.56 1.88 0.75 22.51 0.58 2.21 0.76
DWT-Fuzzy 17.28 0.41 2.12 0.64 16.69 0.51 1.98 0.72 26.77 0.50 2.59 0.71
DWT-K-NN 17.62 0.40 2.19 0.64 14.30 0.58 1.79 0.76 22.25 0.58 2.22 0.76
DWT-ELM 16.56 0.44 2.03 0.66 19.25 0.43 2.19 0.66 28.27 0.47 2.55 0.68
DWT-XGBoost 17.76 0.40 2.11 0.63 17.68 0.48 2.04 0.69 22.92 0.57 2.32 0.76
DWT-LSTM 16.66 0.44 2.05 0.66 18.02 0.47 2.09 0.71 21.47 0.60 2.18 0.77
16 KÜLLAHCIand ALTUNKAYNAK

aimed to contribute to the ongoing efforts to improve
rainfall prediction accuracy and extend the lead time of
prediction by comparing the performance of the hybrid
MODWT model. Based on the findings of the study, it
can be inferred that several conclusions can be drawn:
•From the results of this study, it can be inferred that
the stand-alone machine learning methods had inade-
quate prediction performance as indicated by the per-
formance metrics. However, the implementation of the
discrete wavelet transform (DWT) showed some
improvement in the prediction accuracy. On the other
hand, the hybrid DWT approach did not meet the
expected accuracy levels, particularly when predicting
for time horizons t+2 and t+3, while the hybrid
MODWT models revealed a considerable increase in
model accuracies up to 3 days at all stations.
•The MODWT decomposition method is compatible
with various machine learning algorithms, as evi-
denced by the similar performance results obtained
from the six different algorithms used in the study.
This suggests that the MODWT method is a suitable
technique for use in different estimation algorithms,
thereby contributing to the development of more accu-
rate and effective predictive models.
•All the hybrid MODWT models are found as the best
model according to the Taylor diagrams and scatter
plots.
•The findings of the diagnostic assessment criteria indi-
cate that as the prediction lead times increase, there is
FIGURE 7 The Taylor diagrams of employed models' errors for station 17270, considering prediction horizons of (a) t+1, (b) t+2 and
(c) t+3. [Colour figure can be viewed at wileyonlinelibrary.com]
KÜLLAHCIand ALTUNKAYNAK 17

a decrease in the values of CE and R, while the values
of MSE and MAE increase for all of the stations. This
can be attributed to the decreasing autocorrelation,
which in turn affects the accuracy of the predictions.
These results suggest that longer prediction lead times
may require more sophisticated modelling techniques
that account for the decrease in autocorrelation over
time in order to achieve more accurate and reliable
predictions.
The success of the developed MODWT model high-
lights its potential for accurate long-term prediction and
suggests that it may be applied to various scientific fields
where such predictions are of importance. As such, it is
recommended that further investigations be carried out
to explore the potential of the MODWT model for accu-
rately predicting various hydrological variables in earth
sciences, and for improving predictions across various
time scales and regions. However, it is important to note
that this study is limited by the use of machine learning
applications, and the authors intend to address this limi-
tation by adopting multi-stage modelling approaches in
their future research on rainfall time series.
AUTHOR CONTRIBUTIONS
Kübra Küllahcı:Methodology; validation; visualization;
formal analysis; software; data curation; conceptualiza-
tion; writing –original draft; resources; investigation.
Abdüsselam Altunkaynak: Writing –review and edit-
ing; project administration; supervision; investigation.
ACKNOWLEDGEMENT
We sincerely thank the Turkey Meteorological Service for
providing precipitation data.
FIGURE 8 Scatter plots of models of lead time (t+1) for station 17280 (for the test set) (a) Stand-Alon models, (b) Wavelet models and
(c) MODWT models (the red line represents the 1:1 perfect line). [Colour figure can be viewed at wileyonlinelibrary.com]
18 KÜLLAHCIand ALTUNKAYNAK

CONFLICT OF INTEREST STATEMENT
The authors declare no conflicts of interest.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are avail-
able from the corresponding author upon reasonable
request.
ORCID
Kübra Küllahcıhttps://orcid.org/0000-0003-4699-5878
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How to cite this article: Küllahcı, K., &
Altunkaynak, A. (2024). Maximizing daily rainfall
prediction accuracy with maximum overlap
discrete wavelet transform-based machine learning
models. International Journal of Climatology,1–22.
https://doi.org/10.1002/joc.8530
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