# An ensemble neural network approach to forecast Dengue outbreak based on
climatic condition

Madhurima Panja1, 000Equal Contributions, Tanujit Chakraborty1, 0,
222Corresponding author: Mail<EMAIL_ADDRESS>Sk Shahid
Nadim3, Indrajit Ghosh4, Uttam Kumar1, Nan Liu5
1 Spatial Computing Laboratory, Center for Data Sciences, International
Institute of Information Technology Bangalore, India.
2 Department of Science and Engineering, Sorbonne University Abu Dhabi and
Sorbonne Center for Artificial Intelligence.
3 Odum School of Ecology, University of Georgia, Athens, Georgia, USA.
4 Department of Epidemiology and Biostatistics, University of Georgia, Athens,
Georgia, USA.
5 Duke-NUS Medical School, National University of Singapore, Singapore,
Singapore.

###### Abstract

Dengue fever is a virulent disease spreading over 100 tropical and subtropical
countries in Africa, the Americas, and Asia. This arboviral disease affects
around 400 million people globally, severely distressing the healthcare
systems. The unavailability of a specific drug and ready-to-use vaccine makes
the situation worse. Hence, policymakers must rely on early warning systems to
control intervention-related decisions. Forecasts routinely provide critical
information for dangerous epidemic events. However, the available forecasting
models (e.g., weather-driven mechanistic, statistical time series, and machine
learning models) lack a clear understanding of different components to improve
prediction accuracy and often provide unstable and unreliable forecasts. This
study proposes an ensemble wavelet neural network with exogenous factor(s)
(XEWNet) model that can produce reliable estimates for dengue outbreak
prediction for three geographical regions, namely San Juan, Iquitos, and
Ahmedabad. The proposed XEWNet model is flexible and can easily incorporate
exogenous climate variable(s) confirmed by statistical causality tests in its
scalable framework. The proposed model is an integrated approach that uses
wavelet transformation into an ensemble neural network framework that helps in
generating more reliable long-term forecasts. The proposed XEWNet allows
complex non-linear relationships between the dengue incidence cases and
rainfall; however, mathematically interpretable, fast in execution, and easily
comprehensible. The proposal’s competitiveness is measured using computational
experiments based on various statistical metrics and several statistical
comparison tests. In comparison with statistical, machine learning, and deep
learning methods, our proposed XEWNet performs better in 75% of the cases for
short-term and long-term forecasting of dengue incidence.

###### keywords:

Dengue; wavelet transform; forecasting; MODWT; neural networks; ensemble.

##  1 Introduction

Dengue, a fatal viral disease, is a major public health concern in many
tropical and subtropical regions of the globe [8]. According to a recent
article [82], the global number of dengue fever cases has increased from 23
million in 1990 to 104 million in 2017. While this infectious disease mainly
affects under-developed and developing countries, ongoing climate change and
globalization contribute significantly to increasing dengue transmission in
previously dengue-free nations. The spectrum of dengue infection ranges from
febrile disease, including high fever, retro-ocular pain, muscle, and joint
pain, skin rashes, and headaches, to severe manifestations of hemorrhagic
fever and shock syndrome [37]. Due to the unavailability of specific antiviral
treatment or ready-to-use licensed vaccines, public health officials primarily
rely on supportive treatments and insecticide-based vector control
interventions to limit the spread of dengue infection [50]. While the disease
is endemic in several regions of Africa, the western Pacific, and Southeast
Asia, these areas recurrently experience a high risk of infection when the
mosquito population surges near the human habitat. Climatic factors such as
rainfall, temperature, and relative humidity have been found to influence the
rapid spread of the dengue virus [63]. In such cases, the Aedes mosquitoes
undergo shorter times of breeding and maturity period, resulting in higher
population growth [77]. Moreover, owing to the shrinkage of the incubation
period in the Aedes vectors, their ability to infect humans with the dengue
virus increases rapidly [31]. Recent studies [79, 81] have shown that
meteorological conditions significantly influence the size of dengue
dispersion through their impacts on the life cycle developments, biting rates,
infective, and survivability rates of vectors. Similar results evaluating the
effect of heavy rainfall on rapid growth in vector populations have been
discussed in [39].

With the increasing spread of dengue infection due to climatic changes and
unplanned urbanization, there has been an unprecedented burden on the
healthcare infrastructure. Studies have identified that early detection and
treatment of severe cases can significantly reduce the risk of health
complications and mortality [24]. Moreover, to plan and allocate resources,
accurate predictions of infected individuals, or incidence, are crucial. For
instance, knowing the expected number of cases and when they will occur
enables planning through education and awareness campaigns, resource
reallocation to high-risk regions, or medical retraining to detect symptoms
and treat them correctly before peak transmission [72]. Due to this, public
health authorities have to rely on early warning systems to disseminate
available resources in the dengue-endemic areas optimally. The effectiveness
of accurate dengue forecast in designing pre-epidemic preparedness strategies
has attracted many researchers to attempt dengue incidence and outbreak
prediction with different degrees of success [33, 11, 46]. These efforts were
put into establishing an accurate early warning system to inform healthcare
officials about an upcoming outbreak. The type of models used to predict
dengue cases varied diversely. Previous works have used mechanistic models
[65], some others rely on statistical models [44], data-driven machine
learning techniques [36], and few others have used ensemble forecasting
systems [80, 23]. The use of mechanistic models based on deterministic
differential equations with detailed biology of dengue transmission has often
failed to generate reliable forecasts because of the data needed to
parameterize the models. Another study by [66] incorporated data from social-
media activity and internet searches to forecast dengue incidence. To this
end, forecasters have primarily relied on the dengue cases and environmental
factors and used statistical and machine learning models with a higher degree
of success than the mechanistic models [13, 80, 23].

Numerous statistical time series and machine learning models have been
developed for solving the dengue forecasting problem. Time series approaches,
either using covariates or historical incidence data alone, have been widely
used in several studies [48, 64]. Even among the statistical methodologies,
the auto-regressive integrated moving average (ARIMA) model is the most
suitable technique for lagged historical incidence data observations to
generate one-step and multi-step ahead forecasts. While the statistical
frameworks primarily focus on parametric methods to model the complexities of
dengue incidence cases, recent studies manifest the use of advanced machine
learning and deep learning approaches to generate reliable epidemic forecasts
[36, 9]. Another emerging technique in this regard is the ensemble framework
that enhances the forecasting efficiency of individual models. In [23], the
authors have proposed an ensemble approach combining negative binomial
regression, ARIMA, and generalized linear auto-regressive moving average model
through a vector auto-regressive structure to generate early forecasts for the
dengue outbreak. On the other hand, multiple studies have manifested that
climate variation is one of the critical drivers of dengue outbreaks. In [43],
the researchers have utilized the seasonal ARIMA model to study the
interrelationship of weather data and dengue incidence series in Mexico. In
addition, recent studies have recommended using a generalized additive model
with lags in the number of cases and meteorological variables to capture the
seasonal patterns of dengue outbreak [6]. Moreover, the use of a superensemble
framework in [18] has increased accuracy in the dengue forecast by combining
Earth observational data, seasonal climate forecasts, and state-of-the-art
models. Moreover, recent studies have utilized artificial neural networks
[84], auto-regressive neural networks (ARNN) [6], and hybrid ARIMA-ARNN model
[13] for univariate forecasting of dengue surveillance data for severely
affected regions. Convolutional neural network (CNN), Transformer, long short-
term memory (LSTM), and attention-enhanced LSTM (LSTM-ATT) models were
compared with traditional machine learning models on weather-based dengue
fever forecasting for Vietnam in [54].

Albeit the vast application of statistical, machine learning, deep learning,
and ensemble framework in dengue forecasting, these conventional models suffer
from some significant drawbacks. For instance, the statistical methods,
especially the linear ARIMA and ETS methods, are insufficient to model the
non-linear complexities in real-world dengue datasets. Although advanced
machine learning and deep learning frameworks can handle the irregular
behavior of the epidemic series using canonical architectures, their
forecasting performance severely diminishes for long-term prediction problems.
Moreover, these data-driven methods suffer from the problem of overfitting,
i.e., it learns the patterns and the noise in the training series to such an
extent that it negatively impacts the forecasting performance for an unseen
dataset. In the case of ensemble frameworks, although the forecast accuracy
enhances w.r.t the constituent models, the appropriate selection of models and
their weights, often termed ‘forecast combination puzzle’, pose a significant
challenge to its universal success. To overcome these drawbacks, wavelet
transformation has been considered a proper mathematical technique for decades
[75, 60]. The efficacy of the wavelet decomposition to utilize high and low-
frequency filters for appropriate signal extraction from noisy time series
datasets has led to its popularity in forecasting literature [2, 83, 30].
Several studies have utilized the maximal overlapping version of discrete
wavelet transformation (MODWT) to model the non-stationary behavior of time
series datasets from various domains, namely energy [3], meteorology [55], and
natural calamities [51] to name a few. The wavelet-based forecasting technique
has also gained popularity in epidemic forecasting. In [14], the researchers
have proposed a hybrid ARIMA and wavelet forecasting approach to generate
real-time forecasts for the novel Covid-19 pandemic. Recently, an ensemble
wavelet neural network (EWNet) architecture for modeling univariate epidemic
datasets has been proposed [58]. This study utilizes a MODWT decomposition
technique on the historical incidence data and generates epidemic forecasts by
modeling multiple transformed series using auto-regressive neural networks in
an ensemble framework. The EWNet model has been shown to improve the forecast
accuracy for various epidemics compared to state-of-the-art forecasters.
However, this framework is incompetent to handle the causal relationship
between epidemic incidence and other climatic factors. Our study establishes
that rainfall is a crucial driver of dengue ecology by utilizing robust
causality tests; hence, it should be considered in the dengue forecasting
framework. Our proposed model extends the EWNet framework to handle the effect
of the causal covariate in its architecture. The proposed XEWNet utilizes an
ensemble framework to model the wavelet-transformed dengue incidence series
and rainfall data using auto-regressive neural networks. Experimental results
suggest significant improvement of our proposed model over the EWNet
architecture and other benchmark forecasters for short-term and long-term
dengue forecasting in the San Juan, Iquitos, and Ahmedabad regions. We also
provide the appropriate choices of hyperparameters of the proposed
architecture for different horizons in the geographical regions. The
statistical tests also validate the robustness of our proposed model.
Furthermore, we establish that the current formulation of the XEWNet framework
is more suitable as an early warning system for dengue-prone regions than the
state-of-the-art forecasters. Moreover, the proposed model generates the most
reliable dengue forecasts with one year lead time; hence this can be utilized
by public health officials in designing effective dengue combat policies.

The remaining sections of the paper are organized as follows. First, in
Section 2, we discuss the datasets considered in this study, understand the
causal relationship between rainfall and dengue incidence datasets, and
elaborate on the development of the ensemble model. Next, a detailed
experimental evaluation of the proposed XEWNet framework for the short-term
and long-term forecast is discussed in Section 3. Then, in Section 4, we
manifest the robustness of our proposed model using statistical tests.
Eventually, in Section 5, we conclude the paper with discussions about the
results, policy recommendations, and future scope of work.

##  2 Materials and Methods

This study is focused on the weekly laboratory-confirmed dengue incidence
cases in three tropical and sub-tropical regions: San Juan, Iquitos, and
Ahmedabad. To assess the impact of climatic conditions on dengue outbreaks, we
consider the data on the weekly distribution of rainfall for these cities. We
retrieved the dengue incidence data for San Juan and Iquitos from the National
Oceanic and Atmospheric Administration (NOAA)
Repository111https://dengueforecasting.noaa.gov/, which was initially provided
by the collaboration of the U.S. government, public-sector agencies of Peru,
and the U.S. universities. On the other hand, the precipitation data for these
regions are accumulated by the U.S. climatology agency and the NOAA. The data
on dengue incidence (per 10000 population) and rainfall in the Ahmedabad
region has been collected from [27]. In the case of San Juan, the weekly data
is considered from the beginning of 1991 till the end of 2012. For Iquitos and
Ahmedabad, the period ranging from 2002 to 2012 and 2005 to 2012,
respectively, have been contemplated in this study. In the adjoining section,
we describe the global characteristics of these datasets in detail. The data
and codes used in this study are made publicly accessible at
https://github.com/mad-stat/XEWNet for the reproducibility of this work.

Figure 1: Dengue cases (red) and rainfall (blue) data of San Juan, Iquitos,
and Ahmedabad (incidence per 10000 population).

###  2.1 Dengue Surveillance Data

This study uses three multivariate time series datasets (weekly) to generate a
short-term (26 weeks) and long-term (52 weeks) forecast of dengue incidence in
the San Juan, Iquitos, and Ahmedabad regions. Previous studies have used
several traditional time series models and modern deep learning architectures
to forecast the dengue outbreak in San Juan and Iquitos cities; for details,
see [11, 44, 42, 23]. In this study, we first intend to test the hypothesis
that rainfall plays a crucial role in dengue transmission. For doing so, we
mainly focus on regions within the subtropical belt and receiving moderate to
heavy rainfall. For instance, San Juan situated along the southern side of the
Atlantic Ocean across the northeastern coast of Puerto Rico experiences a
tropical monsoon climate with frequent rainfall throughout the year. The
available data shows that the average rainfall in this region is 4.78 mm,
ranging from 0 to 53.39 mm from 1991 to 2012. Iquitos, the northeastern
Peruvian city, is situated south of the Amazon River. This region experiences
a hot and humid equatorial climate with wet summers. Nevertheless, the town
encountered constant precipitation, with average weekly rainfall of 9.09 mm,
ranging from 0 to 26.07 mm from 2002 to 2012. On the other hand, the north-
western metropolitan city of India, Ahmedabad, is located along the banks of
the Sabarmati River. A tropical monsoon climate prevails in this city
throughout the year, with moderate precipitation during the monsoon season.
This region encountered rainfall ranging from 0 to 422.62 mm from 2005 to
2012. A thematic plot representing rainfall distribution and dengue incidence
cases for the studied areas is presented in Fig. 1. In the subsequent section,
we provide a detailed description of the global characteristics of the dengue
datasets considered in this study that will be useful for building a dengue
forecasting system for its probable usage in national public health systems.

Table 1: Dengue incidence datasets and their corresponding ACF, PACF plots for three regions Region |  Data |  ACF plot |  PACF plot
---|---|---|---
San Juan |  |  |
Iquitos |  |  |
Ahmedabad |  |  |

###  2.2 Preliminary Data Analysis

Dengue surveillance datasets considered in this study are free from any
missing observations and are of different lengths. Although it comprises some
outliers; however, the values are plausible and thus kept as it is. The
autocorrelation function (ACF) and partial autocorrelation function (PACF)
plots of the dengue caseloads presented in Table 1 indicate the presence of
serial correlation between the observations and its lagged values. Thus, these
plots represent the retrospective nature of dengue cases. Furthermore, we
analyze specific statistical features of the dengue datasets to unfold the
general structure of the disease outbreak. Based on the recommendations of
[21], we examine the stationarity, seasonality, linearity, normality, and
long-term dependency of three dengue datasets. We perform Kwiatkowski-
Phillips-Schmidt-Shin (KPSS) test to study the stationarity of the dengue
cases in the three regions [69]. This non-parametric unit root test checks the
stationarity of the observed time series against a deterministic trend.
Another essential property of any time process is seasonality, which
determines the replicating patterns of a time series within a fixed period. We
utilize a combined Ollech and Webel’s test to verify the seasonality of the
series [56]. Subsequently, to examine the linearity of the time series
datasets, we used Terasvirta’s neural network test [71]. We also verify
whether the distribution of dengue incidence cases follows a normal
distribution using the Anderson-Darling test [53]. Additionally, we check the
long-term dependency or self-similarity using the Hurst exponent [40].
Furthermore, we employ the non-parametric Granger causality (GC) test [35] to
verify the causal relationship between rainfall and dengue outbreak. This
statistical hypothesis test evaluates whether rainfall helps to forecast
dengue cases by measuring the ability of rainfall data to predict the
disease’s future dynamics. Alongside the GC test, we also consider the wavelet
coherence approach to examine the time-localized interaction between the
observed series [34]. The wavelet coherence model decomposes the time-indexed
data into time-frequency components and provides useful visualization to
identify the oscillatory behavior for the corresponding time series. The
results of the above-stated statistical tests presented in Table 2 reveal
specific exciting properties of the dengue datasets. To begin with, we can
infer from the KPSS test results that the dengue series are non-stationary
except for Iquitos. Secondly, all the dengue incidence series exhibit non-
linear and non-gaussian characteristics as observable from the p-values of the
Terasvirta test and Anderson-Darling test, respectively. The results from
Ollech and Webel test for seasonality narrates that the dengue incidence of
Iquitos exhibits seasonality; however, the data for the other two regions is
non-seasonal. We also observe that the Hurst exponent value for all the
datasets is $>$ 0.5; hence the dengue incidence cases have similarities to
their historical observations. Moreover, for performing the GC test, we use
single differencing on the non-stationary incidence datasets to meet the
stationarity assumption of the GC test and set the order of the test to two,
which suggests up to $2^{nd}$ order of lags to include in the auxiliary
variable, i.e., rainfall. The computed p-value from Table 2 for the GC test of
order two indicates a statistically significant causal relationship between
rainfall and dengue cases in San Juan and Iquitos region. However, for the
Ahmedabad region, the p-value does not indicate that rainfall Granger cause
dengue incidence. In the case of non-linear and non-stationary data, the GC
test sometimes fails to quantify the flow of information between time series.
A possible remedy to improve the GC test for non-stationary time series is to
increase the differencing order, and lag order in the test [70]. In this case,
we increase the order of lags to four to verify the causality structure in the
oscillatory time series data. From the p-values of the GC test with order
four, we can infer a significant causal relationship exists between rainfall
and dengue incidence cases for all the datasets. However, a stronger causality
check can be produced using a wavelet coherence plot for the non-stationary
and non-linear time series of dengue cases vs rainfall [1]. The wavelet
coherence plot in Fig. 2 can be interpreted as follows: (a) Time is plotted
along the horizontal axis, and the vertical axis represents scale; (b) Areas
on the time-frequency space where movements of two-time series co-occur are
located by the wavelet coherence; (c) Significant relationship between the two
series are marked with warmer colors (red), whereas, the colder colors (blue)
represents low dependency between the series; (d) Area on the time-frequency
space lying beyond the boundary marked with colder shades signifies no
relationship between the observed time series; (e) The arrows on the plot
indicates the direction of the relationship between the variables such that
right-down (right-up) or left-up (left-down) direction symbolizes that the
first (second) series is the leading cause for the second (first). Thus,
similar to the results of the GC test (with order four), the wavelet coherence
plot also depicts that the rainfall and dengue incidence series exhibit
similar oscillatory behavior for all the regions. Motivated by these results,
we use rainfall data as an exogenous causal variable to forecast dengue cases
for these three locations.

Table 2: Global characteristics of the dengue datasets. In the case of
statistical tests, the p-value ($<$ 0.1) within the bracket with ∗ indicates
statistical significance.

Features and Tests San Juan Iquitos Ahmedabad Length 1196 (1991-2012) 598
(2002-2012) 424 (2005-2012) Train : Test (short-term) 1170 : 26 572 : 26 398 :
26 Train : Test (long-term) 1144 : 52 546 : 52 372 : 52 KPSS test Non-
stationary ($<$ 0.01∗) Stationary ($>$ 0.1) Non-stationary (0.02∗) Terasvirta
test Non-linear (0.032∗) Non-linear ($<$ 0.01∗) Non-linear (0.002∗) Ollech and
Webel test Non-seasonal Seasonal Non-seasonal Anderson-Darling test Non-normal
($<$ 0.01∗) Non-normal ($<$ 0.01∗) Non-normal ($<$ 0.01∗) Hurst Exponent Long-
term dependent (0.739) Long-term dependent (0.603) Long-term dependent (0.673)
Granger Causality test Causality found Causality found Causality not found
(order = 2, Cases Vs. Rainfall) (0.065∗) (0.012∗) (0.573) Granger Causality
test Causality found Causality found Causality found (order = 4, Cases Vs.
Rainfall) (0.032∗) (0.030∗) (0.049∗)

Figure 2: Wavelet coherence plot of San Juan, Iquitos, and Ahmedabad. Colder
(blue) to warmer (red) colors indicate an increasingly significant
interrelationship between dengue and rainfall. Arrows represent the direction
of the relationship, and the cold areas beyond the boundary have no
significant relationship.

###  2.3 Development of Proposed XEWNet Model

This work introduces an ensemble wavelet neural network with an exogenous
factor(s) (XEWNet) framework for forecasting dengue incidence based on past
disease dynamics and rainfall patterns. The proposed model overcomes the
deficiencies of individual neural networks and improves the forecasting
performance by including precipitation data. This section summarizes the
notion of the maximal overlap discrete wavelet transformation (MODWT) used in
the network, followed by a detailed description of the proposed XEWNet
architecture.

####  2.3.1 MODWT Algorithm

Real-world datasets comprise signals and noises whose effective analysis
yields valuable information about the generating process. For quantifying
these structures present in the signal, a precise mathematical operation is
required to look at the data through the noise. Wavelet transform is a
powerful signal decomposition technique that uses specialized functions called
wavelets to analyze the signal [61]. The key feature of a wavelet is its
short-lived wave-like oscillations which are localized in time. This
characteristic enables the wavelet transform to represent a signal in both
time and scale (frequency) domains, unlike the Fourier transform. Broadly,
there are two types of wavelet transformations: (i) Continuous wavelet
transform (CWT) and (ii) Discrete wavelet transform (DWT). While the former
variant allows the wavelet to vary continuously, resulting in an infinite
number of wavelets, the latter samples the wavelet discretely w.r.t time and
frequency. Hence, our study focuses on the DWT approach since the dengue
incidence data is sampled from a finite interval at discrete time points. Use
of DWT as a localized approximator can be found in several domains, namely in
energy [52, 7], climatology [55], material science [47], epidemics [58], and
economics [45]. A detailed description of DWT is given in Appendix 6.1.
Despite having vast applicability, the DWT approach has certain drawbacks,
such as restricting the size of the signal to be strictly a power of 2.
Moreover, the DWT also downsamples, i.e., a simple shift in the signal can
significantly impact the signal energy of the DWT by scale and shift. These
hinder the application of DWT in specific real-world situations, and to
overcome these drawbacks of the DWT approach, a maximal overlapping version of
the DWT (MODWT), also termed as the non-decimated wavelet transform has been
proposed in the literature [61, 75, 60]. The MODWT algorithm is a linear
filtering approach that transforms a signal into a time-dependent wavelet and
scaling coefficients related to variations over a set of scales. Unlike the
DWT, its maximal overlapping version is well-defined for signals of varying
lengths and retains the downsampled values at each stage of decomposition.
These appealing properties of MODWT make it ideal for analyzing time series
from different domains [47, 7, 76, 55]. Our study utilizes the multi-
resolution analysis (MRA) of the shift-invariant MODWT approach for
transforming non-stationary real-world dengue incidence series. The
mathematical formulation of the MODWT algorithm is presented in Appendix 6.2.
Since the real-world dengue incidence series exhibits high-frequency
components for short time intervals and low-frequency components for a long
period, this decomposition technique is more relevant. Thus in our study, the
MODWT-based MRA technique has been applied to decompose the non-stationary,
long-term dependent series into simpler components and extract the relevant
information. The following subsection describes the proposed XEWNet model that
integrates the wavelet transformation with an ensemble of neural networks to
enhance the dengue outbreak’s predictability.

Mathematically, given a dengue incidence time series
$Y=\\{Y_{1},Y_{2},\ldots,Y_{N}\\}$ with $N$ historical data points, the MODWT
algorithm decomposes the original series into details and smooth coefficients:

$Y_{t}=\sum_{l=1}^{L}D_{l,t}+S_{L,t},$ (1)

where $D_{l,t}$ denotes details coefficients (irregular fluctuations or high-
frequency components) at scale $l$, $(l=1,2,\ldots,L,\;\text{ where
}L=\log_{e}(N)-1)$ and $S_{L}$ is the smooth coefficient (overall trend or
low-frequency components) of the original series [60].

Figure 3: The proposed XEWNet workflow: (a) To predict dengue incidence cases,
we provide a weekly time series of dengue cases ($Y_{t}$) and rainfall
($X_{t}$) in the training period; (b) We perform a MODWT based MRA
transformation on Y and generate multiple series of details and smooth
coefficients; (c) We begin to train numerous auto-regressive neural networks
to individually model the transformed series along with rainfall dataset in
the input stack; (d) Each of the neural networks is trained with a single
hidden layer having a pre-specified number of nodes inside the hidden stack;
(e) The output stack comprises of one-step ahead forecast generated by
individual neural networks. These predictions are combined to generate the
final out-of-sample forecast.

####  2.3.2 Proposed Model

In our proposed XEWNet framework, we simultaneously utilize the MODWT
algorithm as a pre-processing step to extract the local spectral and temporal
information from the input signal. A pivotal characteristic of the multi-
resolution analysis of the MODWT approach is that it helps decompose the input
signal into non-identical frequency bands exploited for forecasting in the
proposed model.

Real-world dengue incidence series exhibit distinctive characteristics such as
non-stationary, non-linearity, long-term dependency, and non-normality in the
majority of cases (as reported in Table 2). Thus, to model the underlying
mechanism that generates these real-world irregular time series, we transform
them using MODWT decomposition with Daubechies orthogonal wavelets of length
2, i.e., the Haar filter. This linear MODWT transformation has time-frequency
localization capacity and it can efficiently separate the long-range
dependency, non-stationarity, and trends in signals [85]. Thus the MODWT-based
MRA decomposition of the series $Y_{t}$ generates a set of $L+1$ new
uncorrelated random variables, namely the wavelet coefficients. In our
proposed methodology, instead of using the actual signal $Y_{t}$, we model
these $L+1$ random variables with distinct neural networks in an ensemble
framework. These neural networks’ architecture and learning mechanisms enable
them to model and control the complexities of the $L+1$ non-linear wavelet
coefficients. Therefore, the application of neural networks in the proposed
framework can aptly handle the non-linearity present in the three dengue
incidence datasets. Moreover, to capture the causal relationship of input
series and exogenous rainfall variable ($X_{t}$), we model $X_{t}$ with each
of the $L+1$ decomposed series of $Y_{t}$ inside their corresponding network.
The forecasts generated from these $L+1$ neural networks are then aggregated
to generate the future dynamics of $Y_{t}$. Thus $h$-step ahead forecasts of
$Y_{t}$ denoted by $\hat{Y}_{N}^{(h)}$, based on $N$ historical observations,
can be mathematically expressed as:

$\hat{Y}_{N}^{(h)}=\sum_{l=1}^{L}\hat{D}_{l,N}^{(h)}+\hat{S}_{L,N}^{(h)},$ (2)

where $\hat{D}_{l,N}^{(h)}\;(l=1,2,\ldots,L)$ and $\hat{S}_{L,N}^{(h)}$ are
the forecasts generated by $L+1$ neural networks. We call this model combining
$L+1$ auto-regressive neural networks in an ensemble setting with an exogenous
variable as XEWNet$(p,k)$ model. Thus, our task now reduces to designing an
optimal neural network for each transformed series in XEWNet($p,k$) model. In
each of the base models in the ensemble, we utilize a feed-forward auto-
regressive neural network (ARNN) architecture comprising three layers - one
input layer with $(p+1)$ nodes, one hidden layer having $k$ nodes, and an
output layer. Each of these neural networks is employed for modeling $p$
lagged observations of the wavelet coefficients and an exogenous variable to
generate a one-step forecast of the corresponding series. Hence, the output
generated by $L+1$ neural networks with the above-stated architecture can be
mathematically expressed as

$\displaystyle\hat{D}_{l,t}^{(1)}$
$\displaystyle=\beta_{o,l}+\sum_{\tilde{h}=1}^{k}\gamma_{\tilde{h},l}\phi\left(\tilde{\beta_{l}}+\sum_{i=0}^{p-1}\alpha_{i,l}D_{l,t-i}+\alpha_{x,l}x_{t}\right);\;(l=1,2,\ldots,L),$
$\displaystyle\hat{S}_{L,t}^{(1)}$
$\displaystyle=\eta_{o}+\sum_{\tilde{h}=1}^{k}\delta_{\tilde{h},L}\phi\left(\tilde{\eta_{L}}+\sum_{i=0}^{p-1}\lambda_{i,L}S_{L,t-i}+\lambda_{x,L}x_{t}\right);$

where $\\{\beta_{o,l}\\},\\{\tilde{\beta}_{l}\\}$ denotes the bias terms, the
weights $\\{\alpha_{i,l},\alpha_{x,l}\\}$ are the connections of the lagged
inputs and exogenous variable with the hidden neurons, respectively,
$\\{\gamma^{{}^{\prime}}_{l}\\}$ denotes the $k$-dimensional weight vector
between the hidden layer and the output layer for the neural networks of the
details coefficients. The corresponding parameters of the neural network for
modeling the smooth series of $Y_{t}$ are given by $\eta_{o},\tilde{\eta_{L}}$
as the bias, $\\{\lambda_{i,L},\lambda_{x,L}\\}$ denote the input node to
hidden node connections, and $\delta^{{}^{\prime}}_{L}$ denotes the hidden
layer to output layer weights. The logistic sigmoidal activation function is
denoted by $\phi$ [67]. We initialize the connection weights at random
starting values and train them using gradient descent back-propagation
approach [67]. The above-stated procedure generates a one-step-ahead forecast,
whereas we feed back the latest forecast and remove the last lagged
observation from the model iteratively for computing a multi-step-ahead
forecast. Eventually, we forecast the original series by utilizing the outputs
from different models, as stated in Eqn. 2.

The proposed XEWNet model has three hyperparameters, namely, $L+1$ denoting
the number of levels of wavelet decomposition, $p$ indicating the number of
lagged observations of the target variable, and $k$ quantifying the total
number of hidden neurons in the hidden layer. Previous studies have identified
that the essence of the choice of wavelet decomposition level is to reveal the
complex variability of the transformed signal [60, 5]. MODWT eliminates
alignment artifacts, and the decomposition level $L+1$ consists of $(L+1)N$
coefficients rather than the original series of size $N$ (whereas simple DWT
has $N$ coefficients for any given $L+1$). The MODWT pyramid algorithm
requires $N\log_{2}(N)$ multiplications, which is the same as needed by the
Fast Fourier Transform (FFT) algorithm [61]. However, during the practical
implementation, to reduce the complexity of computing MODWT coefficients and
reduction of redundant wavelet coefficients, Percival and Walden [61]
suggested using $N\log_{e}(N)$ multiplications since the MODWT algorithm
requires $\mathcal{O}(N\log N)$ operations similar to FFT algorithm [59].
Therefore, an increase in the decomposition level may increase the possibility
of creating redundant wavelet coefficients and also increase the computational
complexity of the model. Also, a reduction in the decomposition levels may
reduce the effectiveness of the MODWT approach as opposed to its algorithmic
complexity. Hence following this conventional approach, we set
$L+1=\lfloor\log_{e}\text{(length of training set})\rfloor$ in our model,
where $\lfloor\cdot\rfloor$ denotes the floor function. We utilize the Akaike
information criterion (AIC) [25] for determining the best value of $p$, and
for stabilizing the neural network we specify $k=\left[\frac{p}{2}+1\right]$
this restricts the over-fitting and under-fitting of the network [58, 41]. A
schematic workflow of the proposed XEWNet model is presented in Fig. 3. As the
figure shows, the target time series (indicated in blue) is first transformed
using a MODWT-based MRA into several levels. The lagged values of these levels
and the exogenous variable (marked in red) are passed through the input stack.
The connection weights connect each input set to the hidden nodes (dashed line
with arrows). The outputs from the hidden stack are processed in the output
layer, which eventually generates the one-step ahead forecast from the
interconnected network. These predictions from several networks are finally
aggregated to generate the XEWNet output. Applying wavelet decomposition to
diagnose the main signal from the given series and using an auto-regressive
neural network with auxiliary information in an ensemble framework for
enhancing the forecast accuracy justifies the nomenclature of the proposed
model as XEWNet. The implementation technique of the proposed XEWNet is
summarized in Algorithm 1.

Input :

Two in-sample time series ($Y_{t},X_{t}$) of length $N$ denoting the target
and exogenous variables.

Output :

Forecast of target series for $h$ time steps.

  1. 1.

Transform the target series, $Y_{t}$, into corresponding wavelet coefficients
by applying multi-resolution analysis of maximal overlap discrete wavelet
transformation (MODWT) using a haar filter.

  2. 2.

Generate $L+1$ uncorrelated time series objects by extracting $L$ detail
series and one smooth series from the above-stated decomposition.

  3. 3.

Select $L+1$ as $\log_{e}(N)$, where $N$ denotes the size of training dataset.

for  _$l\leftarrow 1$ to $L$_ do

Model $l^{th}$ detail series with $X_{t}$ using an auto-regressive neural
network.
Design the network with one input layer having $p+1$ nodes, one hidden layer
with $k$ neurons, and an output layer.
Input $X_{t}$ and $p$ lagged values of the $l^{th}$ detail series in the feed-
forward architecture, which subsequently passes through $k$ hidden neurons and
generates a one-step-ahead forecast.
Select $p$ by minimizing the Akaike information criterion (AIC) and
$k=\left[\frac{\displaystyle{p}}{\displaystyle{2}}+1\right]$, for stable
learning architecture.
return _One-step ahead forecast for the corresponding details series._

end for

  1. 4.

Model the smooth series of $Y_{t}$ and $X_{t}$ with a similar auto-regressive
neural network as defined above and generate the subsequent forecast.

  2. 5.

Form an ensemble framework to combine the forecasts generated from different
series and formulate the final one-step-ahead prediction.

  3. 6.

Generate multi-step ahead forecast iteratively by replacing the distant
historical observation with the latest forecast.

Algorithm 1 Proposed Ensemble Wavelet Neural Network with exogenous factor(s)
(XEWNet($p,k$))

##  3 Numerical Experiments and Results

In this section, we verify the practical utility of the proposed XEWNet
framework by conducting extensive experimentation on three dengue datasets. In
the subsequent subsections, we discuss several accuracy measures used in this
study (refer to Section 3.1), benchmark models from different paradigms (refer
to Section 3.2), implementation of the proposed architecture (refer to Section
3.3), and performance evaluation of the proposal in comparison with the state-
of-the-art forecasters (refer to Section 3.4).

###  3.1 Performance Metrics

This study adopts four popularly used key performance indicators, namely Root
Mean Square Error (RMSE), Mean Absolute Error (MAE), Symmetric Mean Absolute
Percent Error (SMAPE), and Mean Absolute Scaled Error (MASE), to evaluate the
deviation between the forecasts and the ground truth. The formulae for these
metrics are as follows:

$\displaystyle RMSE$
$\displaystyle=\sqrt{\frac{1}{h}\sum_{i=1}^{h}(y_{i}-\hat{y}_{i})^{2}};\;\;MAE=\frac{\displaystyle
1}{\displaystyle h}\sum_{i=1}^{h}|y_{i}-\hat{y}_{i}|;$ (3) $\displaystyle
SMAPE$
$\displaystyle=\frac{1}{h}\sum_{i=1}^{h}\frac{|\hat{y}_{i}-y_{i}|}{(|\hat{y}_{i}|+|y_{i}|)/2}\times
100;\;\;MASE=\frac{1}{h}\frac{\sum_{i=N+1}^{N+h}|\hat{y}_{i}-y_{i}|}{\frac{1}{N-\mathcal{s}}|y_{i}-y_{i-\mathcal{s}}|};$

where $y_{i}$ and $\hat{y}_{i}$ are the ground-truth and its corresponding
forecast, respectively, $N$ represents the training size, $h$ is the forecast
horizon, and $\mathcal{s}$ denotes the seasonality of the series. By general
convention, the model having the least values of these metrics is the “best”
performing model [49].

###  3.2 Benchmark Forecasters

We utilize several state-of-the-art methods from different paradigms to
compare the proposed model’s effectiveness with other baseline forecasters.
This analysis has two primary objectives: firstly, to assess the importance of
rainfall data in predicting dengue transmission, and secondly, to check
whether an ensemble wavelet framework has the potential to generate more
reliable forecasts compared to individual forecasters. To verify these two
hypotheses, we compare our proposed XEWNet approach with two types of models.
One class of models is trained exclusively with the historical dengue
incidence dataset, and the other class of models receives both dengue and
rainfall datasets as input; however, their formulation does not possess any
hybridization or ensemble framework. In this regard we consider statistical
models specifically exponential smoothing (ETS) [78], auto-regressive
integrated moving average (ARIMA) [10], self-exciting threshold auto-
regressive (SETAR) [73], Theta [4], Wavelet ARIMA (WARIMA) [2], TBATS (T:
Trigonometric. B: Box-Cox transformation. A: ARIMA errors. T: Trend. S:
Seasonal components) [22], and Bayesian structural time series (BSTS) [68];
machine learning methods, such as artificial neural network (ANN) [62] and
auto-regressive neural network (ARNN) [28]; hybrid models, namely ARIMA-ARNN
[12], and ARIMA-WARIMA [14]; and ensemble models viz, ARIMA-ETS-THETA [15],
ARNN-THETA-ETS [15], and ensemble wavelet neural network (EWNet) [58] to
forecast the dengue cases based on its historical observations. We further
extend the experimentation by providing rainfall data as a covariate to some
of the traditional models like exponential smoothing with exogenous variables
(ETSX) [41] and auto-regressive integrated moving average with exogenous
variables (ARIMAX) [41]; auto-regressive neural network with exogenous
variables (ARNNX) [41]; and deep learning frameworks including Neural basis
expansion analysis with exogenous variables (NBeatsX) [57], Temporal
convolutional networks with exogenous variables (TCNX) [17], Transformers with
exogenous variables (TransformersX) [74], and Block recurrent neural network
with exogenous variables (BlockRNNX) [38]. We summarize the implementation
technique for the above-stated models in the subsequent section.

###  3.3 Model Implementation

The implementation of the proposed XEWNet framework is done using R
statistical software. To begin with, we partition the dengue and rainfall
datasets into training (in-sample) and test (out-of-sample) segments. Two test
sets of fixed lengths (26 weeks and 52 weeks) are considered in this study to
extrapolate the forecast efficiency of the proposal for different horizons.
Given the in-sample data we employ MODWT based MRA approach with ‘haar’ filter
on the dengue dataset using modwt function of “wavelets” package in R. We
specify the number of levels of decomposition as $L+1=\left[\log_{e}(\text{
length of training set })\right]$ and boundary condition as periodic. This
transformation generates $L$ wavelet coefficients (details) series and one
scale coefficients (smooth) series of the original dengue dataset. In the
followed-up step, we deploy several auto-regressive neural networks locally to
model the transformed series and rainfall data as an auxiliary variable using
the nnetar function of the “forecast” package in R. To restrict the redundancy
of neural networks; we specify the architecture of the network as having $p+1$
neurons in the input layer and $k$ hidden nodes distributed in a single hidden
layer. The size of the input layer ($p+1$) is determined using the minimum AIC
and $k$ is specified as
$\left[\frac{\displaystyle{p}}{\displaystyle{2}}+1\right]$. Thus, each of
these $L+1$ neural nets models $p$ lagged values of the transformed series and
rainfall as a covariate to generate a one-step forecast. Eventually, we
combine the local predictions from different models and generate the final
(ensemble) output utilizing an ensemble approach. This method is then used
iteratively to generate a multi-step ahead forecast. These out-of-sample
forecasts of the desired length are subsequently compared with the ground
truth using the accuracy metrics defined in Eqn. 3 and the corresponding
values pertaining to specific regions are reported in Tables 3 \- 5. For the
implementation of the baseline forecasters like ETS, ETSX, ARIMA, ARIMAX,
TBATS, Theta, ARNN, and ARNNX, we have used the “forecast” package of R
software. Other models such as SETAR, WARIMA, and ANN are trained using the
“tsdyn”, “WaveletARIMA”, and “nnfor” packages in R. In case of hybrid ARIMA-
ARNN and ARIMA-WARIMA we have utilized the implementation provided in [13] and
[14], respectively. The deep learning prototypes were executed using the
“darts” library in Python [38]. Lastly, ensemble methods are trained using the
standard implementation details available in [15, 58].

Table 3: Performance metrics with 1 year and 6 months test data set for San
Juan. Parameters of XEWNet ($p,k$) are reported for 52 weeks-26 weeks.

Paradigm Model 52 weeks forecast 26 weeks forecast RMSE MAE SMAPE MASE RMSE
MAE SMAPE MASE Statistical ETS [78] 86.28 57.17 85.78 04.71 12.78 11.48 54.64
01.95 Models ETSX [41] 87.17 57.53 86.17 04.74 12.14 10.63 51.64 01.80 ARIMA
[10] 80.18 53.94 80.43 04.45 17.29 16.01 67.35 02.72 ARIMAX [41] 80.09 53.90
80.36 04.44 17.25 15.96 67.19 02.71 SETAR [73] 85.37 52.23 83.31 04.64 12.14
10.80 52.21 01.83 Theta [4] 87.13 57.58 86.39 04.75 12.32 10.88 52.42 01.84
WARIMA [2] 84.46 55.77 83.10 04.60 19.06 17.52 70.80 02.98 TBATS [22] 86.27
57.17 85.78 04.71 12.79 11.49 54.65 01.95 BSTS [68] 82.26 54.33 80.38 04.48
14.73 13.52 60.56 02.30 Machine Learning ANN [62] 74.20 45.59 72.73 04.09
17.86 16.43 68.14 02.79 Models ARNN [29] 86.65 57.76 86.21 04.70 12.42 08.26
42.90 01.40 ARNNX 87.59 59.40 97.85 04.90 14.74 09.55 53.11 01.62 Deep
Learning NBeatsX [57] 104.6 82.58 117.9 05.14 81.54 69.33 65.19 03.52 Models
TCNX [17] 119.7 97.72 172.4 06.09 137.4 125.4 157.1 06.37 TransformersX [74]
73.99 49.61 52.68 03.47 90.06 74.05 60.91 03.76 BlockRNNX [38] 78.27 55.39
57.99 03.47 111.3 93.92 84.14 04.77 Hybrid ARIMA-ARNN [13] 80.04 53.91 80.41
04.43 17.49 16.18 67.77 02.75 Models ARIMA-WARIMA [14] 80.39 54.10 80.58 04.46
17.45 16.02 67.07 02.75 Ensemble ARIMA-ETS-Theta [15] 84.43 56.05 83.65 04.62
13.67 12.59 58.18 02.14 Models ARNN-Theta-ETS [15] 88.44 58.38 87.98 04.81
12.02 10.35 50.63 01.76 EWNet [58] 71.52 45.04 58.70 03.71 08.62 06.94 36.06
01.18 Proposed Model XEWNet (10,6)-(10,6) 68.49 42.14 51.67 03.47 07.69 05.66
29.08 0.960

Table 4: Performance metrics with 1 year and 6 months test data set for
Iquitos. Parameters of XEWNet ($p,k$) are reported for 52 weeks - 26 weeks.

Paradigm Model 52 weeks forecast 26 weeks forecast RMSE MAE SMAPE MASE RMSE
MAE SMAPE MASE Statistical ETS [78] 04.86 02.68 97.25 01.32 01.86 01.64 85.85
01.14 Models ETSX [41] 04.86 02.68 97.23 01.32 01.86 01.64 85.86 01.14 ARIMA
[10] 07.19 06.52 126.4 03.23 06.34 05.98 126.8 04.15 ARIMAX [41] 07.17 06.50
126.5 03.22 06.34 05.97 127.0 04.14 SETAR [73] 05.82 04.66 112.4 02.31 04.31
03.74 107.1 02.60 Theta [4] 04.87 02.67 97.35 01.32 01.85 01.63 85.81 01.13
WARIMA [2] 08.69 07.71 126.5 03.82 07.20 06.07 118.8 04.21 TBATS [22] 07.76
07.19 129.7 03.56 06.97 06.56 129.7 04.55 BSTS [68] 08.06 06.37 122.7 03.15
02.37 02.08 91.19 01.44 Machine Learning ANN [62] 05.58 04.33 109.8 02.22
05.15 04.42 110.8 03.06 Models ARNN [29] 05.08 02.52 103.7 01.25 01.63 01.36
86.52 0.940 ARNNX [41] 12.21 08.32 121.8 01.91 01.62 01.36 86.52 0.940 Deep
Learning NBeatsX [57] 12.62 8.44 110.2 01.93 13.94 11.01 114.4 01.77 Models
TCNX [17] 44.48 29.93 145.5 06.87 32.39 22.11 91.99 03.56 TransformersX [74]
11.95 08.05 119.9 01.85 08.52 06.71 62.68 01.08 BlockRNNX [38] 12.55 08.82
129.4 02.02 13.08 10.34 86.81 01.66 Hybrid ARIMA-ARNN [13] 06.70 05.89 122.3
02.91 05.70 05.29 121.7 03.67 Models ARIMA-WARIMA [14] 06.94 06.17 123.3 03.05
05.82 05.29 120.3 03.67 Ensemble ARIMA-ETS-Theta [15] 05.11 03.54 104.3 01.75
03.09 02.67 97.75 01.85 Models ARNN-Theta-ETS [15] 04.87 02.90 99.07 01.43
02.21 01.94 89.54 01.35 EWNet [58] 04.78 03.01 99.30 01.49 02.25 01.95 89.10
01.35 Proposed Model XEWNet (7,5)-(1,2) 04.73 02.50 96.41 01.25 01.98 01.57
79.95 01.09

Table 5: Performance metrics with 1 year and 6 months test data set for
Ahmedabad. Parameters of XEWNet ($p,k$) are reported for 52 weeks-26 weeks.

Paradigm Model 52 weeks forecast 26 weeks forecast RMSE MAE SMAPE MASE RMSE
MAE SMAPE MASE Statistical ETS [78] 14.26 10.18 101.4 01.98 05.48 05.22 113.3
03.96 Models ETSX [41] 14.26 10.18 101.5 01.98 05.48 05.22 113.3 03.95 ARIMA
[10] 15.01 09.91 99.18 01.93 03.54 03.26 94.05 02.47 ARIMAX [41] 15.03 09.91
99.14 01.93 03.52 03.23 93.71 02.45 SETAR [73] 11.79 09.64 94.55 01.88 08.18
07.62 125.9 05.77 Theta [4] 14.05 10.09 100.4 01.97 05.66 05.40 114.6 04.09
WARIMA [2] 10.51 09.36 99.32 01.83 08.95 08.20 129.1 06.21 TBATS [22] 14.26
10.18 101.5 01.99 05.48 05.22 113.2 03.95 BSTS [68] 24.90 17.21 165.9 03.36
67.06 57.98 198.4 43.92 Machine Learning ANN [62] 14.95 10.02 100.5 01.84
08.70 08.15 129.1 06.17 Models ARNN [29] 17.86 11.40 156.8 02.22 02.05 02.69
119.5 01.28 ARNNX [41] 17.29 10.96 143.5 02.14 02.07 02.71 122.7 01.29 Deep
Learning NBeatsX [57] 10.64 06.72 92.91 01.92 09.32 06.84 38.44 0.760 Models
TCNX [17] 11.37 07.02 78.70 01.37 14.56 11.87 64.47 01.31 TransformersX [74]
09.99 06.91 78.99 01.35 11.91 08.91 48.24 0.990 BlockRNNX [38] 11.37 07.03
78.70 01.37 14.56 11.87 64.47 01.31 Hybrid ARIMA-ARNN [13] 15.00 09.91 99.20
01.93 03.55 03.27 94.22 02.48 Models ARIMA-WARIMA [14] 15.21 09.91 100.8 01.93
03.60 03.20 97.88 02.47 Ensemble ARIMA-ETS-Theta [15] 14.40 10.06 100.8 01.96
04.88 04.63 108.4 03.50 Models ARNN-Theta-ETS [15] 14.62 10.03 100.8 01.95
04.30 04.07 103.2 03.08 EWNet [58] 10.45 06.71 86.83 01.31 11.74 09.25 54.16
01.02 Proposed Model XEWNet (10,6)-(16,9) 09.98 06.55 78.34 01.28 02.04 02.36
128.5 01.79

###  3.4 Model Performance Comparisons

This section elaborates on the efficiency of the proposed XEWNet model and
other benchmark forecasters to serve as an early warning system for dengue
outbreaks. The experimental results obtained by implementing the above-stated
models for San Juan, Iquitos, and Ahmedabad regions are reported in Tables 3
\- 5, respectively. From the accuracy measures for the San Juan region (Table
3), it is evident that our proposed architecture outperforms all other
baseline models for both the forecast horizons (short and long-term
trajectories). Particularly, for short-term prediction, the XEWNet framework
is 18% more accurate than the second-best performing forecaster, EWNet. This
improvement in forecast accuracy is predominantly due to the use of rainfall
data as a covariate. We observe this phenomenon for ARNNX, which generates the
most reliable dengue forecast among the baselines with covariates, but its
performance drops significantly w.r.t. our proposed model. The performance of
the deep learning frameworks is not satisfactory for 26 weeks of prediction;
their long-term forecasts are competitive with the proposed model.

In the case of dengue incidence of Iquitos, the short-term predictions
generated by the attention-mechanism-based TransformersX and ARNNX frameworks
surpass all other baselines. However, these models generate less accurate
outcomes for long-term forecasting compared to the proposed XEWNet approach
which generates consistently better long-term out-of-sample predictions. The
efficiency of the wavelet transformation is found in extracting signal from
noise, resulting in reliable long-term predictions for Iquitos data. For the
dengue forecasting task of the Ahmedabad region, we observe that the deep
learning-based models provide competitive forecasts with 26 weeks lead time.
The short-term dengue predictions for this region produced by the XEWNet
framework transcends all the models from different paradigms w.r.t the RMSE
and MAE scores. However, the other two accuracy measures of the NBeatsX model
are minimum. Based on the accuracy metrics for long-term forecasts of this
region (Table 5), we can detect that TransformersX and EWNet perform
significantly better than the benchmark methods. However, compared to the
proposed XEWNet framework, their forecasting ability diminishes by 9%. We
diagrammatically present the long-term forecasts generated by the proposed
XEWNet model for the three geographical regions in Fig. 4. The short-term (26
weeks) prediction of the proposed forecaster is provided in Fig. 5. Although
the experimental results suggest that the proposed XEWNet method generates
more accurate dengue forecast for the three regions in comparison to the
baseline forecasters as expressed by the evaluation metrics, the plots in Fig.
4 and 5 demonstrates that in some instances the out-of-sample point estimates
generated by the XEWNet model cannot be well fitted with real incidence cases.
However, our task is to predict the number of dengue cases each week (in each
location) based on the climatic condition and an understanding of the
relationship between climate and dengue dynamics has been established in this
study. This can improve research initiatives and resource allocation to help
fight life-threatening epidemics with our proposed early warning system.

Figure 4: Long-term (52 weeks) dengue cases forecast (green) generated by
proposed XEWNet and ground truth data (pink) for the three geographical
regions in the study. The blue dashed line marks the distinction between the
training and test set.

Figure 5: Short-term (26 weeks) dengue cases forecast (green) generated by
proposed XEWNet and ground truth data (pink) for the three geographical
regions in the study. The blue dashed line marks the distinction between the
training and test set.

The overall experimental evaluation of the proposed model and the benchmark
forecasters reveal some interesting observations. The accuracy metrics for
linear statistical models like ETS and ARIMA suggest their incapability to
handle the irregularities of real-world dengue datasets. Moreover, the
inclusion of rainfall data as an exogenous variable marginally improves the
forecast accuracy of these traditional models. This failure of ETSX and ARIMAX
is primarily attributed to approximating the complex relationship between
rainfall and dengue incidence cases by a linear function with a constant rate
of change. Although these problems are addressed more precisely by the deep
learning frameworks, their overall performance is unsatisfactory for long-term
forecasts. The limited availability of historical data is one of the potential
reasons for the collapse of these advanced models. Data set size creates a
barrier to the performance of the deep learners; however, our proposed XEWNet
approach can optimally model the complex non-linear relationship between the
observed and covariate series, thus resulting in improved predictions. Unlike
the deep learning approaches, the stable architecture of our proposal limits
the number of training parameters, thus restricting the model over-fitting.
Moreover, the use of MODWT-based MRA transformation generates the wavelet and
scale coefficients that can overlook the signal through noise resulting in
accurate long-term forecasts. Hence, experimentally we can infer that the
proposed XEWNet model can generate reliable point forecasts for dengue
incidence cases in the considered geographical regions with appropriate lead
time, more preferably for long-term forecasting.

##  4 Significance of Improvements and Validation of Results

This section focuses on determining the robustness of the proposed XEWNet
framework. We also describe the potential threats that impact the performance
of our prediction system. It is important to note that the validation of the
proposed model’s performance depends on these factors to a considerable
extent.

###  4.1 Statistical Significance of Performance

In this section, we determine the statistical significance of the competitive
forecasters with covariate (exogenous variable, i.e., rainfall) using several
robust statistical tests. We restrict our attention to RMSE and MAE
performance indicators for conducting these tests. Initially, we utilize the
non-parametric multiple comparisons with the best (MCB) approach to compare
the relative performance of different forecasting frameworks that include
exogenous variables [26]. This statistical methodology assigns a rank to the
competitive models based on their accuracy metric and computes their 95%
critical interval. The critical distance for the best performing forecaster
serves as the reference value for this test. The results of the MCB test
reported in Fig. 6 indicate that the XEWNet model is the ‘best’ among other
forecasters with an average rank of 1.33 (for RMSE) and 1.17 (for MAE). From
the RMSE plot (left), we notice that ETSX is the second best performing model
(rank 3.50) and the average rank of ARIMAX, ARNNX, and TransformersX are
precisely the same (rank 4.00). However, for the other deep learning models,
there is a considerable increase in the RMSE score. Therefore, the critical
interval of the proposed XEWNet model (shaded region) is considered the
reference value. Since the critical distance of deep learning frameworks,
namely NBeatsX, BlockRNNX, and TCNX, do not overlap with this reference value,
their performance is significantly worse than the proposed XEWNet model.
Moreover, for the MAE metric (right plot), we observe a considerable
difference between the average rank of the proposed XEWNet model and the other
forecasters. The non-overlapping nature of the reference value and the
critical interval of BlockRNNX and TCNX leads to the conclusion that their
performance is significantly worse than the best performing XEWNet model at a
95% level of significance.

Figure 6: Schematic visualization of the multiple comparisons with the best
(MCB) test. The left plot provides the result for the RMSE metric and the
right plot indicates the MAE metric-based output. For example, in the figure,
XEWNet-1.33 specifies the rank of the XEWNet model as 1.33 for the RMSE
metric. The blue lines indicate the critical distance of the model, the middle
point of this interval, which is denoted by black (significant) or red (not
significant), represents the mean rank, and the shaded region marks the
reference value.

Alongside the MCB testing approach, we also consider the non-parametric
Friedman test to validate our experimental results [32]. This distribution-
free test is more suitable for non-normal datasets, a principal characteristic
of real-world dengue incidence cases (refer to Table 2). Furthermore, the
Friedman test hypothesizes that the forecasters’ performance is equivalent
based on their average ranks. To conduct this statistical test, we compute the
mean rank of the models w.r.t. their accuracy measures. This distribution-free
test rejects the null hypothesis if the computed p-value is less than the
chosen significance level. From the Friedman test results reported in Fig. 7
w.r.t RMSE metric (left) and MAE metric (right), we obtain the computed
p-value $<$ 0.05. Thus following the Friedman test procedure, we reject the
null hypothesis of model equivalence and conclude that the performance of the
forecasters is significantly different. Furthermore, we perform posthoc
analysis using the Durbin-Conover test with a Holm correction for pairwise
comparison [19]. This test evaluates the null hypothesis that there is no
significant difference between the performance of each pair of forecasters
using multiple comparisons. The non-parametric Durbin-Conover testing
procedure rejects the null hypothesis if the computed p-value is less than the
chosen significance level (0.05) and concludes that the model’s performance is
significantly different. We report the significant p-values of this pairwise
test in Fig. 7 for the RMSE metric (left) and MAE metric (right). The median
rank of 6.21 (RMSE) and 4.39 (MAE) for the proposed XEWNet model are the least
among all the forecasters. There is a significant difference in performance
between XEWNet and BlockRNNX (p-value $<\;0.05$), XEWNet and NBeatsX (p-value
$<\;0.05$), and XEWNet and TCNX (p-value $<\;0.05$) in terms of RMSE metric.
For the MAE score, the computed p-values between proposed XEWNet and TCNX,
BlockRNNX, and NBeatsX are considerably lesser than $0.05$. Hence their
performance varies significantly w.r.t. the proposal. Thus combining the
results obtained from the non-parametric statistical tests, we can conclude
that the point estimates of future dengue incidence generated by the proposed
XEWNet framework are statistically validated and can be deployed by public-
health officials as an early warning system in dengue-prone regions.

Figure 7: Friedman test and Durbin-Conover test results for the forecasting
methods based on RMSE score (left) and MAE score (right). In the figure, the
value of the Friedman test statistic is reported in the top panel with the
confidence interval. The significant p-values for pairwise tests are provided
along the horizontal bars. The median ranks of the forecasters are marked in
red.

###  4.2 Threats to Validity

Finally, from the experimental evaluation (refer to Section 3.4), we observe
that the proposed XEWNet outperforms all the benchmark forecasters in 75% of
the cases. In particular, the proposal generates the best long-term forecasts
for all three geographical regions regarding the four accuracy measures. Our
study utilizes RMSE, MAE, SMAPE, and MASE metrics as the key performance
indicators based on their popularity in forecasting literature [41, 15].
Several other performance measures exist in the time series literature, and
the appropriate metric choice can influence the forecasting performance.
However, our experiments’ performance indicators are carefully selected by
including square, absolute, percentage, and scaled errors to ensure the
reliability of the experimental results. In future work, other accuracy
measures may need to be considered. Also, this study utilizes three publicly
available dengue incidence cases and rainfall datasets to demonstrate the
practical utility of our forecasting system. These time series datasets from
San Juan, Iquitos, and Ahmedabad have been used widely in previous studies for
dengue forecasting [23, 42, 27]. Our selected datasets are accumulated from
various dengue endemic regions across the world and they are diverse in size
and intensity of infection. For stronger generalizability of our findings, the
forecasting performance of the proposed XEWNet for other affected regions may
need to be further explored.

##  5 Discussion

Dengue fever is a viral mosquito-borne disease transmitted predominately by
the Aedes aegypti and Aedes albopictus mosquitoes. It infects an estimated 400
million people annually, with nearly half the world’s population at risk of
infection. Dengue cases are influenced by complex interactions of ecology,
environment, meteorological factors, and virus factors, among many others.
This data-driven study approaches the problem of forecasting dengue incidence
(weekly cases for three regions) with the help of past dengue case data and
rainfall data from the San Juan, Iquitos, and Ahmedabad regions. In addition,
with the use of causality tests, we confirmed that incorporating weather-based
dengue early warning into the dengue prediction system may significantly
enhance the system’s accuracy.

In this paper, we proposed a new variant of the wavelet-based forecasting
technique using the ARNN model, namely the XEWNet model. The proposed model is
first theoretically built using a MODWT transformation applied to dengue
incidence time series data and building locally auto-regressive neural
networks on the decomposed time series. In the final stage, an ensemble of
local forecasts is used to generate the out-of-sample forecasts of the XEWNet
model. The final output of the model gives accurate and simultaneous estimates
of the outbreak for the short-term (26 weeks) and long-term (52 weeks). This
forecasting system will effectively implement invaluable control measures to
control or even eliminate the cyclical dengue epidemic in the Latin American
and Asian subcontinents considered in this study. For benchmark comparison, we
have deployed several statistical, machine learning, and deep learning models
using various statistical accuracy metrics. The results presented here
indicate that XEWNet performs superior in most cases. Thus, the point
estimates generated by our proposed XEWNet are more reliable and accurate for
their practical usage in real-time dengue monitoring systems. However, since
the proposed XEWNet model builds an ensemble of several auto-regressive neural
nets, thus, there is a dependency between data sizes and model performances.
When a lot of training data (dengue surveillance) is available and one wants
to forecasts for the long-term (say, 52 weeks), our proposed framework seems
more accurate and effective in generating reliable forecasts. Further studies
to improve the long-term sustainability of forecast precision and improvement
of the XEWNet for generating probabilistic forecasts will help maintain a
forecasting model’s performance. Future research is also needed to address
critical public health needs, produce probabilistic forecasts, and understand
other major features that causally impact dengue incidence cases globally. An
attempt to reveal and study the dengue mortality data and propose to study
other key indicators such as peak intensity, epidemic duration, and rate of
emergence of dengue epidemic can be considered as the future scope of this
study.

### Conflict of interest

The authors declare that they have no conflict of interest.

### Data availability statement

All the codes and data are available in our GitHub repository:
https://github.com/mad-stat/XEWNet.

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##  6 Appendix

###  6.1 Discrete Wavelet Transform (DWT)

The Daubechies class of wavelets comprising orthogonal wavelets defines a
discrete wavelet transform [20]. To formulate a DWT, we consider the
Daubechies family of wavelets and denote the wavelet filters and the scaling
filters as $\\{a_{m}:m=0,1,\ldots,\mathcal{M}-1\\}$ and
$\\{b_{m}:m=0,1,\ldots,\mathcal{M}-1\\}$, respectively, where $\mathcal{M}-1$
is the length of the filter. The wavelet filters and scaling filters are
restricted to satisfy the orthonormality property:

$\sum_{m=0}^{\mathcal{M}-1}a_{m}^{2}=\sum_{m=0}^{\mathcal{M}-1}b_{m}^{2}=1,$
(4)
$\sum_{m=0}^{\mathcal{M}-1}a_{m}a_{m+2n}=\sum_{m=0}^{\mathcal{M}-1}b_{m}b_{m+2n}=0\;;\;\forall\;\;n\in\mathbf{Z}_{\neq
0}.$ (5)

In the Wavelet literature, Eqn. 4 is referred as unit energy assumptions and
Eqn. 5 as even-length scaling assumptions [61]. Furthermore, the wavelet
filters $\\{a_{m}\\}$ and the scaling filters $\\{b_{m}\\}$ are also linked by
the following constraint(s):

$b_{m}\equiv(-1)^{m+1}a_{\mathcal{M}-1-m}\;\;\;\;\textrm{ or
}\;\;\;\;a_{m}\equiv(-1)^{m}b_{\mathcal{M}-1-m};\;\;\text{for}\;\;m=0,1,\ldots,\mathcal{M}-1.$

Thus, scaling filters are referred to as the “quadrature mirror” filter
corresponding to wavelet counterparts. The above-stated formulation of the DWT
coefficients is popularly known as the “pyramid algorithm” [60]. Suppose we
represent the time series as $Y=\\{Y_{t}:t=1,2\ldots,N\\}$. Considering the
$l^{th}$ stage input to the pyramid algorithm as
$\\{W_{l-1,t}:t=0,\ldots,N_{l-1}-1\\}$, where $N_{l}={N}/{2^{l}}$ and
$W_{0,t}\equiv Y_{t}$, then following [75], the corresponding outputs of the
algorithm are the $l^{th}$ level scaling and wavelet coefficients of the
series $\\{Y_{t}\\}$ denoted by

$U_{l,t}=\sum_{m=0}^{\mathcal{M}_{l}-1}a_{l,m}Y_{(2^{l}(t+1)-1-m)\text{ mod
}N}\;\;\;\;\text{ and
}\;\;\;\;W_{l,t}=\sum_{m=0}^{\mathcal{M}_{l}-1}b_{l,m}Y_{(2^{l}(t+1)-1-m)\text{
mod }N},$

where $\mathcal{M}_{l}=(2^{l}-1)(\mathcal{M}-1)+1$. These $l^{th}$ level
scaling and wavelet filters satisfy the orthonormality properties as stated in
Eqn. (4) and Eqn. (5) alongside

$\sum_{m=0}^{\mathcal{M}_{l}-1}b_{l,m}=2^{{l}/{2}}\textrm{ and
}\sum_{m=0}^{\mathcal{M}_{l}-1}a_{l,m}=0.$

The pyramid algorithm formulation of the DWT has been applied in several
domains [52, 16] for the efficient extraction of signal from noise.

###  6.2 Multi-resolution analysis of MODWT algorithm

The maximal overlap discrete wavelet transform (MODWT) is a modified version
of DWT in that it is a non-orthogonal and redundant transformation [61]. The
wavelet filters $\\{\tilde{a}_{m}\\}$ and the scaling filters
$\\{\tilde{b}_{m}\\}$ of MODWT can be defined as normalized versions of DWT
filters as follows:

$\tilde{a}_{l,m}=\frac{a_{l,m}}{2^{l/2}}\;\;\;\;\textrm{ and
}\;\;\;\;\tilde{b}_{l,m}=\frac{b_{l,m}}{2^{l/2}}.$ (6)

Following [61], the MODWT wavelet and scaling coefficients of level $l$,
obtained as the $l^{th}$ stage output of the pyramid algorithm, are given by:

$\tilde{U}_{l,t}=\sum_{m=0}^{\mathcal{M}_{l}-1}\tilde{a}_{l,m}Y_{(t-m)\textrm{
mod }N}\;\;\;\;\textrm{ and
}\;\;\;\;\tilde{W}_{l,t}=\sum_{m=0}^{\mathcal{M}_{l}-1}\tilde{b}_{l,m}Y_{(t-m)\textrm{
mod }N},$ (7)

where $\mathcal{M}_{l}=(2^{l}-1)(\mathcal{M}-1)+1$. Thus, the MODWT
coefficients at level $l$ are defined as the convolutions of the original time
series $\\{Y\\}$, and it has the same length as that of $\\{Y\\}$. Meanwhile,
Eqn. 7 can be expressed in matrix notation as:

$\tilde{U}_{l}=\overline{\overline{{u}}}_{l}Y\;\;\;\;\textrm{ and
}\;\;\;\;\tilde{W}_{l}=\overline{\overline{{w}}}_{l}Y,$

where the $N\times N$ matrices $\overline{\overline{{u}}}_{l}$ and
$\overline{\overline{{w}}}_{l}$ comprises of the wavelet coefficients
$\\{\tilde{a}_{l,m}\\}$ and scaling coefficients $\\{\tilde{b}_{l,m}\\}$,
respectively. Thus, the original signal $\\{Y\\}$ can be represented as:

$Y=\sum_{l=1}^{L}\overline{\overline{{u}}}_{l}^{T}\tilde{U}_{l}+\overline{\overline{{w}}}_{L}^{T}\tilde{W}_{L}=\sum_{l=1}^{L}D_{l}+S_{L},$

where the details coefficient
$D_{l}=\overline{\overline{{u}}}_{l}^{T}\tilde{U}_{l}$ denotes the irregular
fluctuations of the series $\\{Y\\}$ at scale $l\;\forall\;(l=1,2,\ldots,L)$
and the smooth $S_{L}=\overline{\overline{{w}}}_{L}^{T}\tilde{W}_{L}$
indicates the overall trend of the original signal at scale $L$ [60]. The
above-stated multi-resolution analysis (MRA) of MODWT utilizes a zero-phase
filter to partition a signal across details and smooth coefficients. This
scale-based additive decomposition generates good time resolution and poor
frequency resolution for high-frequency signals and vice-versa for low-
frequency signals.