∎

11institutetext: V. Bajpai 22institutetext: The LNM Institute of Information
Technology
Jaipur, Rajasthan, India
22email<EMAIL_ADDRESS>33institutetext: A. Bansal 44institutetext:
The LNM Institute of Information Technology
Jaipur, Rajasthan, India
44email<EMAIL_ADDRESS>
#  A Deep and Wide Neural Network-based Model for Rajasthan Summer Monsoon
Rainfall (RSMR) Prediction

Vikas Bajpai111 The authors contributed equally Anukriti Bansal 1

(Received: date / Accepted: date)

###### Abstract

Importance of monsoon rainfall cannot be ignored as it affects round the year
activities ranging from agriculture to industrial. Accurate rainfall
estimation and prediction is very helpful in decision making in the sectors of
water resource management and agriculture. Due to dynamic nature of monsoon
rainfall, it’s accurate prediction becomes very challenging task. In this
paper, we analyze and evaluate various deep learning approaches such as one
dimensional Convolutional Neutral Network, Multi-layer Perceptron and Wide
Deep Neural Networks for the prediction of summer monsoon rainfall in Indian
state of Rajasthan.For our analysis purpose we have used two different types
of datasets for our experiments. From IMD grided dataset, rainfall data of 484
coordinates are selected which lies within the geographical boundaries of
Rajasthan. We have also collected rainfall data of 158 rain gauge station from
water resources department. The comparison of various algorithms on both these
data sets is presented in this paper and it is found that Deep Wide Neural
Network based model outperforms the other two approaches.

###### Keywords:

Deep learning rainfall prediction machine learning wide and deep neural
network multilayer perceptron (MLP) convolutional neural network (CNN) Summer
Monsoon Rainfall

##  1 Introduction

Understanding of rainfall characteristics is important for a variety of
activities including efficient engineering, planning and management of water
resources Halbe et al. (2013); Campling et al. (2001). In addition to this,
rainfall play a major role in balancing of various activities such as,
hydrologic cycle, water availability for terrestrial animals, agriculture and
industrial processes. Rainfall and its estimation is not only important for
India but is equally important for the entire globe Jang and Han (2011);
Tierney et al. (2011); Clift and Plumb (2008); Fan et al. (2013); Ancy et al.
(2014); Kerhoulas et al. (2017); He et al. (2006); Lau and Li (1984).

In India majority of the rain is received from the month of June to September
(June-July-August-September) and that is why this period is called as Indian
Summer Monsoon Rainfall (ISMR) or the Southwest monsoon rainfall. Cultivated
land in India is majorly benefited by this ISMR Swaminathan (1998) which makes
this season highly important and ultimately prediction and estimation of
rainfall for this period also becomes equally essential. India receives nearly
80 percent rainfall during summer monsoon period Mooley and Parthasarathy
(1984); Naidu et al. (2009); Mooley (1997) only. This summer monsoon rainfall
fuhrer helps in predicting food grain production Parthasarathy et al. (1988)
which ultimately contributes to country’s
GDP222https://statisticstimes.com/economy/country/india-gdp-growth-
sectorwise.php. Prediction and estimation of ISMR started way back from the
year 1903 as people started believing on the importance of this monsoon
rainfall Walker (1933).

In this work, our area of study is Rajasthan which is the largest state of
India and the 60% of its area falls under the arid category which makes it
very environmentally sensitive Dutta and Chaudhuri (2015). Even after being an
arid to semi-arid zone, Rajasthan has observed several floods in the past
Goyal et al. (2021); Yadav and Mangukiya (2021); Ray et al. (2019)and also
observed several droughts KARAN (2016); Bokil (2000); Goel and Singh (2006);
Mundetia et al. (2015); Parthasarathy et al. (1987). An early indication of
the amount of monsoon rainfall a particular region is going to receive, can be
very handy in terms of managing the water resource for the entire year. This
early indication can give us an idea about the amount of availability of water
in a particular reservoir. Now this reservoir which will cater to the needs of
demand from people and industry in a particular area can be regulated and
measures can be taken well in advance for proper water resource management for
the monsoon and non-monsoon period.

There are several indicators on which rainfall depends, such as surface
temperature, sea level, distance from sea, distance from mountain ranges etc.
In this work, we propose a time series based approach for the prediction of
rainfall for the months of June, July, August and September (summer monsoon
months). For this we collected Indian Meteorological Department ( IMD
hereafter) grided data of 118 years ( from the year 1901 to 2018) and station
data of 61 years( from the year 1957 to 2017) from Water Resources Department,
Rajasthan (WRD hereafter). In this work we design and analyze advance deep
learning models to capture the patterns from this historical time series data
for the prediction of Rajasthan summer monsoon rainfall (RSMR hereafter). For
this we adapt and improve a model originally proposed by Cheng et al Cheng et
al. (2016b) in the field of recommender systems. We name our proposed model as
Deep and Wide Monsoon Rainfall Prediction Model (DWMRPM hereafter) and
compared with advance deep-learning based models like multi-layer perceptron
(MLP), one dimensional convolutional neural network (1D-CNN) based neural
networks

###  1.1 Related Work

In past researchers have applied numericalDucrocq et al. (2002) and
statistical modelsLi and Shao (2010); Montanari and Grossi (2008) for rainfall
prediction. But with gaining popularity of artificial intelligence and
increasing machine computation power, training abundant data using machine
learning and deep learning models are becoming the center of attraction for
researchers Zhang et al. (1998). One of the major reasons of scientists
switching from traditional numerical approaches to artificial intelligence
based approaches is that the statistical and numerical models fail to capture
the dynamic nature of rainfall Singh et al. (2012a) whereas neural networks
are quiet smart in capturing the hidden trends and seasonality existing in
time series rainfall data. The numerical and statistical models were used
majorly for two to three decades but these methods lacked forecasting accuracy
Gadgil et al. (2005) resulting into failure in predicting major rainfall
variations Kalsi et al. (2004); Sikka (2003). There are evidences from the
past where these numerical methods failed Gadgil et al. (2002); Preethi et al.
(2011) to predict the monsoon rainfall and severe droughts were observed.

Pritpal Singh (2018a) has made an attempt to predict the ISMR using monthly
monsoon rainfall values and applied fuzzy sets and artificial neural network
(ANN). When the parameters on which the rainfall depends are very high then in
order to predict ISMR, Saha et al. (2016) used auto encoder Ng et al. (2011)
for reducing the number of parameter and then predicted the ISMR. Saha et al.
(2021) studied the climatic variables responsible for ISMR and used deep
learning feature for monsoon rainfall prediction.This study also shows the
monsoon deviation from long period average (LPA) rainfall. Johny et al used an
adaptive Ensemble Model of ANN which was capable of capturing very low and
very high rainfall in the Indian state of Kerala Johny et al. (2020). Dubey
et. al Dubey (2015) used three artificial neural network based algorithms (
feed-forward back propagation algorithm, layer recurrent algorithm and feed-
forward distributed time delay algorithm) for rainfall prediction over the
region of Pondicherry, India. Some amount of monsoon rainfall prediction is
done by applying feed forward neural network Chakraverty and Gupta (2008);
Sahai et al. (2000); Singh and Borah (2013).

Fluctuations in the summer monsoon rainfalls can’t be captured efficiently by
traditional linear statistical models Singh (2018b); Dash et al. (2018). This
motivated us to use Deep Learning based model which are efficient in capturing
this non-linearity and dynamic nature of ISMR. As per IMD weather forecasting
manual 333https://imdpune.gov.in/Weather/Forecasting_Mannuals/IMD_IV-13.pdf,
Indian rainfall is very well known for its variability in space and time.
There is hardly any seasonal distribution of rainfall over entire India. At
two different station locations which are a few miles apart, if we consider
one day rainfall, we may observe that one station experiencing heavy rainfall
whereas the other station may go completely dry. This kind of variation is not
only found in monsoon rainfall period(June to September) but also during post
monsoon period (October to December) as well.

A good amount amount of work has been done in the field of ISMR as presented
above but at present to the best of our knowledge, no work is done in the
field of RSMR prediction, which attracted the authors of this paper to explore
this untouched area. An attempt to predict the agricultural drought index in
Rajasthan is done by Dutta et al Dutta et al. (2013) using standardized
precipitation index. In this proposed work, an extensive study is done in
predicting RSMR for the first time. The good thing about Rajasthan is the
strong Rain Gauge network from IMD, Water Resource Department, Rajasthan and
the Revenue Department which has resulted into the abundant supply of rainfall
data for analysis and prediction.

Research work done in the field of Rainfall Prediction and Estimation for the
state of Rajasthan is very less. Vikas et al Bajpai et al. (2020) used the
historical time-series data for daily rainfall prediction. However worked in
analyzing the trends of rainfall in the state of RajasthanPingale et al.
(2014); Yadav et al. (2018). Bryson and Swain (1981) made an effort to present
the holocene variations of monsoon rainfall in Rajasthan. Meena et al. (2019)
made an attempt to explore the spatial and temporal differences to identify
trends in monthly, seasonal and annual rainfall over the Rajasthan region.
They observed the prevailing homogeneity of rainfall at various stations in
the state. In another work Singh et al. (2012b) authors tried to estimate the
one day maximum rainfall in Jhalrapatan, a city in the state of Rajasthan.
Authors have done the probability analysis for this purpose. Lal Meena et al.
(2020) studied the rainfall pattern in Chaksu, Rajasthan.

Our objective is to predict the RSMR which starts in the month of June and
ends in the month of September. In this work we propose a time series based
prediction model which depends on the fundamental of present and future time
series data dependency on past time series data Singh (2016). We adapt and
improvise wide and deep learning model originally proposed by Cheng et al
Cheng et al. (2016b) for recommendations. Many authors h ave used this concept
in different domains like regression analysis Kim et al. (2020), quality
prediction Ren et al. (2020), rainfall prediction Bajpai et al. (2020) etc.
Wide networks are used for memorization and deep networks are used for
generalization. In this work we propose a Deep and Wide Monsoon Rainfall
Prediction Model (hereafter DWMRPM) to predict monsoon rainfall prediction in
the Indian state of Rajasthan.

The rest of the paper is organized as follows. Section 2 explains the proposed
model for summer monsoon rainfall in Rajasthan. Details of experimental
evaluations, model training, results of rainfall prediction and comparison
with other deep learning approaches is given in section 3. Finally we conclude
the paper in Section 4 and provide avenues for future work.

###  1.2 Major Contributions

  1. 1.

In this work, we propose a novel architecture based on deep and wide neural
network for the purpose of summer monsoon rainfall using historical time-
series data. The model efficiently captures the dynamic nature of monsoon
rainfall and works well in its prediction. To the best of our knowledge, we
are the first who have tried to solve this challenging problem.

  2. 2.

We compare our work with various advanced deep learning algorithms for
sequence prediction on two different types of datasets and have obtained very
promising results.

  3. 3.

The algorithms we designed has the generalization ability and can be used to
predict summer monsoon rainfall for atmospherically different regions of
Rajasthan.

##  2 Deep & Wide Monsoon Rainfall Prediction Model (DWMRPM)

This section first provides a brief overview of the proposed approach and
subsequently explain various steps involved in the prediction of Rajasthan
summer monsoon rainfall (RSMR, hereafter).

###  2.1 Overview

In this work we address the problem of summer monsoon rainfall in Rajasthan,
which is the largest state of India and is located in the North-Western part
of the country. Rajasthan has very distinct physiographic characteristics. On
one side it has India’s biggest desert area, called The Thar Dessert and on
the other side this state has Eastern Plains and the ranges of Aravalli Hills
Enzel et al. (1999). These ranges are in the direction of South-west monsoon,
which is responsible for rainfall in the region Roy and Jakhar (2002).
Atmospherically Rajasthan is divided into four zones: North West Desert
Region, Central Aravalli Hill Region, Eastern Plains and South Eastern Plateau
Region Upadhyaya (2014). Details of the districts, which come under the
respective zones are given below:

  * North-West Desert Region: Jaisalmer, Jodhpur, Hanumangarh, Shriganganagar, Barmer, Churu, Nagaur, Pali, Sikar, Bikaner and Jhunjhunu

  * Central Aravalli Hill Region: Udaipur, Dungarpur, Sirohi, Jalore, Pali, Banswara, Bhilwara, Chittorgarh, Rajsamand and Ajmer

  * Eastern Plains: Alwar, Bharatpur, Tonk, Sawai Madhopur, Karauli, Jaipur, Dausa and Dhoulpur

  * South-Eastern Plateau Region: Kota, Bundi, Jhalawar and Baran

All these zones have different atmospheric and climatic conditions. The
problem of predicting summer monsoon rainfall in Rajasthan is different from
the prediction of Indian summer monsoon rainfall (ISMR, hereafter). Most of
the time-series-based methods for predicting ISMR consider average monthly
rainfall values by taking weighted average of the 306 well distributed rain-
gauge stations in the non-hilly areas of Indian sub-continent Dash et al.
(2018); Singh (2018b); Singh and Borah (2013); Sahai et al. (2000). Rajasthan
being a dry state lies in arid and semi-arid zones and characterized by low
and uneven rainfall Kulshreshtha et al. (2013), therefore, a dedicated system
is required which can predict monsoon rainfall for different geographical
regions separately. We use historical monthly rainfall data from two different
sources to train and analyze the performance of our model in prediction of
Rajasthan Summer Monsoon Rainfall. Details on the datasets are given in
Section 2.2

For ISMR researchers used monthly rainfall values of June to September across
all the years Dash et al. (2018) or just have captured the dependency of
months of a single year Singh (2018a) In order to avoid loss of any
information, we are using rainfall values of all the months of previous years
for the prediction of rainfall for the months of June, July, August and
September. For example in order to predict rainfall for the month of June
2019, we use rainfall values of all the months from May 2000 to May 2019.

In this work, we propose a deep and wide monsoon rainfall prediction model
(DWMRPM) for the prediction of the total monthly rainfall intensity for the
summer monsoons months of Rajasthan. The wide network is used to extract low-
dimensional features. Here, instead of using a sequence of monthly rainfall
values directly, we are using features obtained after applying a convolutional
layer, as it is very effective in learning spatial dependencies in and between
the series of data Van et al. (2020). High-dimensional features, on the other
hand, are derived using Multi-layer perceptron (MLP) Pal and Mitra (1992) in
which a sequence of rainfall intensity values are passed on to a deep network.
In order to incorporate a geographical generalization ability in the model, so
that a single model can be used to make rainfall predictions in different
geographical conditions, information of geographical parameters (latitude and
longitude) is included at the time of training. The operational steps involved
in the development of our proposed DWMRPM for the prediction of rainfall are
shown in Figure 1.

Figure 1: Overview of DWMRPM: The model takes sequence of monthly rainfall
intensities and geographical parameters, namely latitude and longitude as
input. After initial pre-processing, input is fed to a deep network, which is
a multi-layer perceptron, and to a wide network, which is a convolutional
network. The model is jointly trained considering the activation weights from
both deep and wide networks simultaneously.

To evaluate the performance of the proposed method, we use two standard
statistical metrics, namely mean absolute error (MAE) and root mean square
error (RMSE). We compare our results with the advance deep learning models
like MLP and one dimensional convolutional neural networks (1-DCNN) which are
very popular for sequence based predictions.

###  2.2 Dataset description and pre-processing

In this work we have used Water Resources Department dataset and Indian
Meteorological Department (IMD) gridded rainfall data with a high spatial
resolution of $0.25^{\circ}\times 0.25^{\circ}$ Pai et al. (2014). From IMD
data set, we selected the rainfall data of the Rajasthan meteorological sub-
division ranging from $23^{\circ}3.5^{\prime}$N to $30^{\circ}14^{\prime}$N
latitude and $69^{\circ}27^{\prime}$E to $78^{\circ}19^{\prime}$E longitude,
for the period of 118 years from the year 1901 to 2018. It gave the rainfall
data for 1008 rain-gauge stations. We have also collected the rainfall data
from Rajasthan’s water resources department, for more than 500 rain-gauge
stations, over a period of 61 years (from the year 1957 to 2017). The datasets
were noisy in terms of negative and missing values. After initial level data
pre-processing and cleansing steps, we selected 484 co-ordinates from IMD
dataset of High Spatial Resolution of (0.25X0.25 degree) and 158 stations from
Rajasthan’s water resources data for our analysis. The distribution of the
selected stations from water resources data, over 33 districts are depicted on
the map of Rajasthan in Figure 2

(a)

(b)

Figure 2: Map of Rajasthan showing distribution of (a) selected 484
coordinates from IMD gridded dataset at high spatial resolution of
$.25^{\circ}\times.25^{\circ}$, (b) selected 158 rain-gauge stations from
Water Resources Department, Rajasthan. The rainfall data obtained from these
coordinates and rain-gauge stations are used for the prediction of Rajasthan
summer monsoon rainfall (RSMR)

In this paper, authors have made use of both the Station data (data collected
from various Rain Gauge Stations in Rajasthan) and the Gridded data Pai et al.
(2014). The idea behind using both the data sets is that when only the station
data is used, for experimentation, one uses the data for single point of scale
whereas when the gridded data is used the application of different
meteorological data for a region is applied depending upon the resolution. In
large catchment areas where less number of Rain Gauges are installed, modeling
may not be that much accurate, on the other hand gridded data is more
continuous and may prove better than single point estimates. Gridded data
contains the data from stations or satellites (in our case, it’s rain gauge
station data) which undergoes interpolation over a grid. This interpolation
needs careful analysis for biases and outliers Rajeevan et al. (2005); Pai et
al. (2014). Station data on the other hand is unbiased single point data. For
our study, we have used quality controlled data sets from both the categories.
If someone has enough single point station in the region under study, then the
station data can be easily utilized but since the rain gauge distribution is
not uniform ( as shown in Raj_WRD), specially in the dessert areas where the
rain gauge station installation density is very low, combination of both the
data set seems to be optimal. Another advantage of using gridded data is that
it acts as a source of replacement to the data missing from the records of
rain gauge stationsMeher and Das (2019). Any area or zone where the observed
station data (point data) is comparatively less, interpolated gridded data can
work as a potential alternative means Bandyopadhyay et al. (2018).

These datasets contained daily rainfall values from which we calculated
monthly rainfall values for January to December. In order to provide rainfall
pattern in Rajasthan, mean rainfall values for each month and the monsoon
season (combined rainfall of June, July, August and September) from the year
1901-2018 for a randomly picked rain-gauge station is shown in Table 1. We
also provide the minimum and maximum rainfall for each month and monsoon
season, over the duration of 118 years. It can be observed that the
significant amount of annual rainfall occurs in the monsoon months and the in
the remaining months, the aggregate rainfall is very less.

Table 1: Statistical summary of monthly data for the IMD dataset and the dataset from the water resource department of Rajasthan (WRD). Mean, maximum and minimum rainfall fall values of IMD are shown for a randomly picked coordinate at $26^{\circ}0^{\prime}$N and $74^{\circ}5^{\prime}$E for a period of 118 years from 1901 to 2018. The rainfall statistics for WRD dataset for a randomly picked rain-gauge station, situated at $26^{\circ}04^{\prime}$N and $75^{\circ}01^{\prime}$E is shown for a period of 61 years from the year 1957 to 2017. Month | Mean (mm) | Maximum (mm) | Minimum (mm)
---|---|---|---
IMD | WRD | IMD | WRD | IMD | WRD
Jan | 4.24 | 2.20 | 63.66 | 53.00 | 0 | 0
Feb | 4.22 | 2.40 | 65.56 | 54.00 | 0 | 0
Mar | 3.81 | 2.11 | 64.44 | 74.00 | 0 | 0
Apr | 2.91 | 3.78 | 43.03 | 56.00 | 0 | 0
May | 9.14 | 5.72 | 90.75 | 87.00 | 0 | 0
Jun | 49.67 | 41.92 | 246.2 | 227.00 | 0 | 0
Jul | 159.45 | 150.54 | 523.5 | 476.00 | 13.11 | 10
Aug | 160.86 | 166.05 | 441 | 905.0 | 3.41 | 40.8
Sep | 65.38 | 63.83 | 305.8 | 402.00 | 0 | 0
Oct | 9.84 | 7.31 | 154.4 | 132.00 | 0 | 0
Nov | 1.72 | 4.02 | 25.67 | 160.00 | 0 | 0
Dec | 2.31 | 1.04 | 46.53 | 30.00 | 0 | 0
Overall Accuracy | 868.63 | 460.05 | 1080.90 | 937.0 | 91.17 | 102.1

We have considered time-series values of monthly rainfall and geographical
parameters like latitude and longitude for the prediction of rainfall during
the monsoon months in different regions of Rajasthan. The rainfall intensity
values ranges from 0 mm to more than 800 mm while coordinate values of
latitude and longitude lies between $23^{\circ}3.5^{\prime}$N to
$30^{\circ}14^{\prime}$N and $69^{\circ}27^{\prime}$ to
$78^{\circ}19^{\prime}$E, respectively. Since the data is of different
dimensions and dimensional units, therefore we normalize the data to make it
dimensionally uniform. When the magnitude of different parameters in a dataset
is different, the parameters with higher values suppresses the role of the
parameters with lower values in model training. To handle this issue, we use
the min-max normalization method to convert all rainfall intensity values to
number between 0 and 100 (latitude and longitude values are already in this
range). The mathematical representation of the min-max normalization method is
as follows:

$I^{*}=\frac{I-I_{min}}{I_{max}-I_{min}}\times 100$

where, $I^{*}$ is the normalized value of the monthly rainfall intensity
value, $I$ represents a value in the original dataset, $I_{max}$ and $I_{min}$
are the maximum and minimum intensity values, respectively. Normalization can
also help in improving the learning capability of the model and in reducing
the computational complexity Shanker et al. (1996).

###  2.3 Model Description

We use deep and wide neural network-based architecture Bajpai et al. (2020)
for the purpose of summer monsoon rainfall prediction in the Indian state of
Rajasthan. The following paragraphs explains the major components of the
model.

####  2.3.1 The Wide Component: Convolutions

The wide component is used to memorize certain combinations of monthly
rainfall events, which is beyond the capabilities of the deep model. It is a
generalized linear model of type $y=\mathrm{\textbf{w}^{T}\textbf{x}}+b$. In
the model proposed by Cheng et al Cheng et al. (2016a), cross-product feature
transformations were used as the wide component. In this work we use
convolutional network as wide component. The basic components of a general CNN
consists of 2 types of layers, namely convolutional layer and pooling layer Gu
et al. (2018). The convolutional layer is composed of several convolutional
kernels, which capture and learn the correlation of spatial features by
computing different feature maps. The output of one dimensional convolutional
layer with input size $N_{l}$ is:

$a^{(l+1)}_{k}=b^{(l+1)}_{k}+\sum_{i=1}^{N_{l}}conv1D(w^{l}_{i,k},a^{l}_{i})$

where, $l$ is the layer number, $w^{l}_{i,k}$ is the kernel from the $i^{th}$
neuron at layer $l-1$ to the $k^{th}$ neuron at layer $l$, $a^{(l)}$,
$b^{(l)}$ activations, bias at $l^{th}$ layer.

Convolutional layer is followed by a pooling layer that is used to realize
shift invariance by reducing the resolution of the feature maps. As
demonstrated by Van et al. (2020), 1D CNN performs well in regression type of
problems and can learn to find the correlation in between the series very
efficiently. Therefore, instead of using raw features in the wide part of the
network, we use a convolutional layer to capture such combinations. In
addition to this, to make our model more generalized with respect to different
atmospheric conditions, we are using geographical parameters namely, longitude
and latitude while designing and developing our model (Figure 3).

Figure 3: Selected architecture of DWMRPM for prediction of Rajasthan Summer
Monsoon Rainfall. There are two major components: 1.The Deep component
consists of mainly an input layer and 3 ReLU layers. 2. The wide component
consists of a convolutional layer followed by a global average pooling layer.A
sequence of monthly rainfall intensity values after normalization and values
are fed to deep and wide components separately

####  2.3.2 The Deep Component: Multi-layer Perceptron

The deep component is a feed-forward neural network, specifically a multi-
layer perceptron, as shown in Figure 3. Sequence of monthly rainfall intensity
values are given as input, which are then fed into hidden layers of a neural
network in the forward pass. Typically, each hidden layer computes:

$a^{(l+1)}=f(w^{l}a^{l}+b^{l})$

where, $l$ is the layer number and $f$ is the activation function, rectified
linear units (ReLUs) in our case, $a^{(l)}$, $b^{(l)}$, and $w^{(l)}$ are the
activations, bias and model weights at $l^{th}$ layer.

####  2.3.3 Joint training of the model

The model is trained using the joint training approach that optimizes all
parameters simultaneously by taking into account the output of the deep and
wide components and their weighted sum. It helps in providing an overall
prediction, which is based on aforementioned components, also depicted in
Figure 3.

$y_{DWMRPM}=\textbf{k}_{cn}\textbf{h}_{cn}+\textbf{k}_{d}\textbf{h}_{d}$

where, $y_{DWMRPM}$ is the prediction, $\textbf{h}_{cn},\textbf{h}_{d}$ are
the output vectors of two sub-models namely wide-convolutional model and deep
model respectively, and $\textbf{k}_{cn},\textbf{k}_{d}$ are their respective
weight vectors to be trained.

##  3 Experimental evaluations

###  3.1 Implementation details

All the experimental programs are coded using Keras Chollet et al. (2015) API
of TensorFlow framework Abadi et al. (2016); Gulli and Pal (2017). The
hardware setup includes computer processor from Intel with i7-8750H
configuration supported by 32GB RAM. The upcoming sections and subsections
describe the designing and implementation setup of proposed approach and
baseline approaches followed by results obtained.

For prediction of monthly rainfall of monsoon season, we consider different
training windows of lengths ranging from 2 years to 10 years. We found that 9
years training window gives most accurate prediction results for the monsoon
months of June, July August and September.

####  3.1.1 Training, validation and test sets

We use two type of datasets, one from the Indian Meteorological Department
(IMD) and the other from the Water Resource Department (WRD). In case of WRD,
monthly rainfall values from the year 1957 to 1986 are considered for the
purpose of training. Validation of the model is done on the dataset
considering the years starting from 1987 to 1997 and finally we test the model
on the dataset containing monthly rainfall values in the interval of the year
1998 to 2017. In case of IMD dataset, training is done by considering values
from the year 1901 to 1980 and validation is done from the year 1981 to 1995
and finally testing is done on the rainfall intensity values from the year
1996 to 2018.

####  3.1.2 Evaluation metrics

As shown by Glorot and Bengio (2010) and He et al. (2015a), to evaluate the
overall accuracy of predictions, we use root mean square error (RMSE) and mean
absolute error (MAE) as the basic evaluation metrics. Low value of RMSE and
MAE means better prediction accuracy of the model.

$\displaystyle RMSE$
$\displaystyle=\sqrt{\frac{1}{N}\sum^{N}_{i=1}\bigg{(}y_{i}-\overline{y}_{i}\bigg{)}^{2}}$
(1) $\displaystyle MAE$
$\displaystyle=\frac{1}{N}\sum^{N}_{i=1}\Big{|}y_{i}-\overline{y}_{i}\Big{|}$
(2)

where, N represents the number of samples, $y_{i}$ is the actual rainfall of
the $i{th}$ sample and $\overline{y}_{i}$ is the corresponding prediction.

####  3.1.3 Model Training

We optimize various hyper parameters like the batch size, number of hidden
layers, number of neuron and the dropout rates using trial-and-error method.
The network configuration of DWMRPM used in our experiments is shown in Figure
3. The input to the model is the normalized sequence of monthly rainfall
intensity values and actual coordinate values (latitude and longitude). The
deep part is a Multi-layer perceptron with an input layer; 3 hidden layers
containing 300, 200 and 100 neural units with ReLU as the activation function;
and finally a dense output layer.In order to prevent over-fitting of the
model, dropout layers srivastava2014dropout with dropout rate 0.3 are added
after each hidden layer. The wide part contains a convolutional layer with 100
filters, each of size 1 x 5, followed by a global average pooling layer. The
outputs of both the wide and deep networks are concatenated and the model is
trained using the joint-training approach, as explained in Section 2.3.3. We
use Adam optimizer Kingma and Ba (2014) for training with Mean Square Error
(MSE) as loss function, which is calculated as follows:

$MSE=\frac{1}{N}\sum^{N}_{i=1}\bigg{(}y_{i}-\overline{y}_{i}\bigg{)}^{2}$

Here, N represents the number of samples, $y_{i}$ is the actual rainfall of
the $i{th}$ sample and $\overline{y}_{i}$ is the corresponding prediction. The
goal of the model is to find optimized parameters that minimizes MSE

$\underset{\theta}{m}inMSE(\theta)$

where, $\theta$ is the total number of trainable parameters. Weights of the
network are initialized using He initializationHe et al. (2015b). Model is
trained for 200 epochs with batch size equals to 8.

(a)

(b)

Figure 4: Architecture of the baseline approaches, selected after
experimentation with various hyper-parameters. (a) Network configuration of
multi-layer perceptron, and (b) Network configuration of CNN.

####  3.1.4 Baseline approaches

In order to establish the competence of our proposed approach, we have
compared the results obtained from the proposed DWMRPM with the results of two
advance deep learning approaches: MLP and 1-DCNN. These approaches are working
well but not at par with our proposed approach. We have used the same sets of
both the data sets for all the approaches in order to avoid any discrepancies
that may arise by using different set of datasets for training and testing the
models. The network architecture of the baseline approaches, which is selected
(after experimenting with various hyper-parameters) for the comparative
analysis with the proposed method is explained in the subsequent paragraphs.
In all these approaches, we use Adam optimizer for training and MSE as loss
function. Input sequence length is 108 (9 years). (Details in Figure 4)

  * Multi-layer perceptron (MLP): The network architecture for MLP is shown in Figure 4a.Sequence of rainfall is normalized and concatenated with latitude and longitude. It contains 3 hidden ReLU layers with 300, 200 and 100 units of neurons respectively.

  * Convolutional Neural Network (CNN): The network architecture selected for CNN is given in Figure 4b. Sequence of rainfall is normalized and concatenated with latitude and longitude. The setup has two convolutional layers with 100 filter size of 1x5 each followed by Global Average Pooling layer.

###  3.2 Results and discussion

In the following subsections, we present the results of experimental analysis
and comparison of the proposed method with the baseline approaches described
in Section 3.1.4.

####  3.2.1 Forecasting accuracy of DWMRPM

As mentioned in Section 2.1, Rajasthan is divided into four atmospheric zones,
each of which having huge difference in their climatic and physiographic
properties. To evaluate the effectiveness and accuracy of the proposed model,
we apply it on each zone separately. The prediction results on one of the
randomly picked rain-gauge stations from each zone is given in Table 2 and
graphical representation is shown in Figure 5. Here we have used station data
from WRD. IMD gridded data is not used in this case because the dataset is
generated by interpolation which may have biases and outliers Rajeevan et al.
(2005); Pai et al. (2014).

####  3.2.2 Generalization ability of DWMRPM

In order to verify generalization ability of our model, we use it for monsoon
rainfall prediction in each zone separately. The prediction results for each
zone, on the basis of two evaluation criteria i.e., MAE and RMSE (Section
3.1.2) on WRD dataset are shown in Table 2 and Figure 5.

Table 2: Prediction results of DWMRPM on the four rain-gauge stations from WRD dataset. Each station is randomly picked from four different atmospheric zones. Zone Name |  Latitude |  Longitude | June | July | August | September
---|---|---|---|---|---|---
MAE |  RMSE |  MAE |  RMSE |  MAE |  RMSE |  MAE |  RMSE
North-West Desert |  29∘12’N |  73∘14’E |  2.4164 |  2.7660 |  5.1061 |  5.8879 |  4.8080 |  5.6060 |  2.8041 |  3.3493
Central Aravalli Hill Region |  26∘04’N |  74∘46’E |  3.1414 |  3.7925 |  7.7769 |  9.2311 |  11.5339 |  13.3020 |  4.7901 |  6.8593
Eastern Plain |  26∘41’N |  75∘14’E |  3.1223 |  3.7269 |  6.5146 |  7.8600 |  9.5284 |  10.7145 |  2.8730 |  3.4377
South-Eastern Plateau Region |  25∘18’N |  75∘57’E |  6.2528 |  7.5165 |  8.3010 |  10.3782 |  9.0789 |  10.7850 |  5.0400 |  5.8975

(a)

(b)

(c)

(d)

Figure 5: Prediction results of DWMRPM for four rain-gauge stations, each
picked from a different atmospheric zone (Section 3.2.2). Results are from
year 2016 to 2017. (a) Prediction results of rain-gauge station situated at
29∘12’N, 73∘14’E in North-West dessert region, (b) Prediction results of rain-
gauge station situated at 26∘04’N, 74∘46’E in Central Aravalli hill region,
(c) Prediction results of rain-gauge station situated at 26∘41’N, 75∘14’E in
Eastern plains region, and (d) Prediction results of rain-gauge station
situated at 25∘18’N, 75∘57’E in South-Eastern plateau region.

It can be observed that a single model is working well in rainfall forecasting
for different geographical conditions ranging from plains and plateaus to
desserts and hills.

####  3.2.3 Comparison with baseline approaches

To establish the significance of present work, we compare the results of our
model with the baseline approaches separately using the IMD gridded dataset
and WRD station dataset.Table 3 and Table 4 show the comparison of the
proposed DWMRPM with other approaches in the prediction of monsoon rainfall
for the months of June, July, August and September on WRD and IMD datasets
respectively. Overall accuracy of the model in the prediction of rainfall for
the monsoon months is also given. Qualitative analysis for the comparison on
different datasets is shown in Figure 6 and Figure 7

Table 3: Comparison of the proposed DWMRPM with one Dimensional Convolutional Neural Networks (1-DCNN) and Multilayer Perceptron (MLP) on WRD dataset using for each month of the monsoon season. Month | MLP | 1-DCNN | DWMRPM
---|---|---|---
RMSE | MAE | RMSE | MAE | RMSE | MAE
June | 7.0382 | 5.8610 | 7.7567 | 5.2118 | 6.550 | 4.550
July | 12.3831 | 9.2600 | 14.2568 | 10.3249 | 11.0974 | 8.7081
August | 15.4046 | 12.3992 | 15.4564 | 10.4677 | 13.7013 | 10.4781
September | 8.1199 | 9.5221 | 7.9481 | 5.9679 | 6.5770 | 5.0796
Overall Accuracy | 10.2014 | 7.0106 | 11.8901 | 7.9931 | 9.9637 | 7.2052
Table 4: Comparison of the proposed DWMRPM with one Dimensional Convolutional Neural Networks (1-DCNN) and Multi-layer Perceptron (MLP) on IMD dataset using for each month of the monsoon season. Month | MLP | 1-DCNN | DWMRPM
---|---|---|---
RMSE | MAE | RMSE | MAE | RMSE | MAE
June | 6.8156 | 5.7843 | 5.9660 | 5.0429 | 4.0239 | 3.1371
July | 12.9685 | 9.9012 | 12.7962 | 8.5378 | 11.7024 | 7.8652
August | 13.8754 | 12.8701 | 13.3874 | 12.6042 | 12.6878 | 12.1112
September | 6.5700 | 5.8955 | 4.6474 | 4.52739 | 3.8953 | 4.0184
Overall Accuracy | 11.5009 | 8.4529 | 9.5039 | 4.6780 | 9.0598 | 4.2830

(a)

(b)

(c)

Figure 6: Comparison of DWMRPM and two deep-learning approaches on WRD dataset
for a randomly selected rain-gauge station situated at 25∘18’N, 75∘57’E. The
results are for the months of June, July, August and September of 20 years
(June 1998 to September 2017) (a) Prediction results of MLP, (b) Prediction
results of one dimensional CNN and, (d) Prediction results of the proposed
DWMRPM.

(a)

(b)

(c)

Figure 7: Comparison of DWMRPM and two deep-learning approaches on IMD dataset
for a randomly coordinates situated at 25∘25’N, 75∘50’E. The results are for
the months of June, July, August and September of 23 years (June 1996 to
September 2018) (a) Prediction results of MLP, (b) Prediction results of one
dimensional CNN and, (d) Prediction results of the proposed DWMRPM.

##  4 Conclusion and Future Work

This paper has presented a deep and wide neural network based model for the
prediction of Rajasthan Summer Monsoon Rainfall (RSMR). Rainfall data is
collected from Water Resource Department, Rajasthan and gridded data of
resolution 0.25 X 0.25 degrees from Indian Meteorological Department (IMD).
This model has the added advantage of exploiting the benefits from both the
interpolated gridded data set and the unbiased single point station data set
as well. Results obtained by DWRM are compared with baseline approaches like
MLP and CNN. It is observed that for RSMR, the deep and wide model works
better than other approaches. In future we may apply similar technique for the
prediction of summer monsoon rainfall in other states in India as well as
abroad. We plan to add more number of rainfall indicators and explore the
possibilities of improving the accuracy of the current method.

##  5 Acknowledgments

This work is in collaboration with Water Resources, Government of Rajasthan.
We are thankful to Indian Meteorological Department (IMD) and Special Project
Monitoring Unit, National Hydrology Project, Water Resources Rajasthan,
Jaipur, India for providing us the Rainfall data for this study.

##  6 Declaration

  * Funding: Not Applicable

  * Conflicts of interest/Competing interests: The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

  * Availability of data and material: Available on request.

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