## A Deep and Wide Neural Network-based Model for Rajasthan Summer Monsoon Rainfall (RSMR) Prediction
Vikas Bajpai ∗ · Anukriti Bansal ∗
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
∗ The authors contributed equally
V. Bajpai The LNM Institute of Information Technology Jaipur, Rajasthan, India
E-mail: vikas.bajpai87@gmail.com
A. Bansal
The LNM Institute of Information Technology
Jaipur, Rajasthan, India
E-mail: anukriti1107@gmail.com
## 1 Introduction
Understanding of rainfall characteristics is important for a variety of activities including efficient engineering, planning and management of water resources [27, 7]. 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 [31, 72, 12, 19, 2, 35, 28, 40].
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 [71] 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 [46, 48, 45] only. This summer monsoon rainfall fuhrer helps in predicting food grain production [53] which ultimately contributes to country's GDP ∗ . Prediction and estimation of ISMR started way back from the year 1903 as people started believing on the importance of this monsoon rainfall [75].
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 [17]. Even after being an arid to semi-arid zone, Rajasthan has observed several floods in the past [24, 77, 57]and also observed several droughts [34, 5, 23, 47, 52]. 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 [10] in the field of recommender systems. We name our proposed model as Deep
∗ https://statisticstimes.com/economy/country/india-gdp-growth-sectorwise.php
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 numerical[15] and statistical models[41, 44] 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 [78]. 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 [65] 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 [21] resulting into failure in predicting major rainfall variations [33, 64]. There are evidences from the past where these numerical methods failed [20, 55] to predict the monsoon rainfall and severe droughts were observed.
Pritpal [68] 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, [60] used auto encoder [49] for reducing the number of parameter and then predicted the ISMR. [61] 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 [32]. Dubey et. al [14] 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 [8, 62, 70].
Fluctuations in the summer monsoon rainfalls can't be captured efficiently by traditional linear statistical models [69, 13]. 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 †, 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
†https://imdpune.gov.in/Weather/Forecasting Mannuals/IMD IV-13.pdf
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 [16] 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 [3] used the historical time-series data for daily rainfall prediction. However worked in analyzing the trends of rainfall in the state of Rajasthan[54, 76]. [6] made an effort to present the holocene variations of monsoon rainfall in Rajasthan. [42] 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 [66] 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. [39] 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 [67]. We adapt and improvise wide and deep learning model originally proposed by Cheng et al [10] for recommendations. Many authors h ave used this concept in different domains like regression analysis [36], quality prediction [58], rainfall prediction [3] 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. 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. 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. 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 [18]. These ranges are in the direction of South-west monsoon, which is responsible for rainfall in the region [59]. Atmospherically Rajasthan is divided into four zones: North West Desert Region, Central Aravalli Hill Region, Eastern Plains and South Eastern Plateau Region [73]. 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 raingauge stations in the non-hilly areas of Indian sub-continent [13, 69, 70, 62]. Rajasthan being a dry state lies in arid and semi-arid zones and characterized by low and uneven rainfall [38], 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 [13] or just have captured the dependency of months of a single year [68] 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 [74]. High-dimensional features, on the other hand, are derived using Multi-layer perceptron (MLP) [51] 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.
Fig. 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.
<details>
<summary>Image 1 Details</summary>
### Visual Description
## Flowchart Diagram: Machine Learning Pipeline for Rainfall Prediction
### Overview
The diagram illustrates a machine learning pipeline for rainfall prediction, showing data flow from input to final prediction. It includes data preprocessing, dual neural network architectures, joint training, and output generation.
### Components/Axes
1. **Input Box** (Light Blue)
- Label: "Input"
- Content: "Sequence of Daily Rainfall, Latitude & Longitude"
2. **Data Pre-processing Box** (Light Blue)
- Label: "Data Pre-processing"
- Sub-components:
- "Noise removal"
- "Data Normalization"
3. **Deep Network Box** (Orange)
- Label: "Deep Network"
- Sub-component: "Multi layer perceptron"
4. **Wide Network Box** (Orange)
- Label: "Wide Network"
- Sub-component: "Convolutions"
5. **Joint Training Box** (Green)
- Label: "Joint Training"
6. **Rainfall Prediction Box** (Green)
- Label: "Rainfall Prediction"
### Flow Connections
- Input → Data Pre-processing → [Deep Network & Wide Network] → Joint Training → Rainfall Prediction
- Dual pathways from Data Pre-processing to both network types
- Convergence of network outputs into Joint Training
### Key Observations
1. **Dual Architecture Integration**: Combines deep learning (MLP) and wide learning (convolutions) approaches
2. **Joint Training Phase**: Explicitly connects both network outputs before final prediction
3. **Spatial Data Handling**: Explicit inclusion of geographic coordinates (latitude/longitude) in input
4. **Temporal Component**: "Sequence of Daily Rainfall" indicates time-series data processing
### Interpretation
This pipeline demonstrates a hybrid approach to rainfall prediction:
1. **Data Preparation**: Standard preprocessing steps (noise removal, normalization) ensure clean input data
2. **Feature Extraction**:
- Deep Network (MLP) likely captures complex nonlinear relationships
- Wide Network (convolutions) probably handles spatial pattern recognition
3. **Synergistic Training**: Joint training phase suggests knowledge sharing between architectures
4. **Geospatial-Temporal Modeling**: Combines location data with time-series analysis for improved predictions
The architecture appears designed to leverage both local patterns (via convolutions) and global relationships (via MLP) while maintaining temporal coherence through joint training. The explicit inclusion of geographic coordinates suggests the model accounts for regional climate variations in its predictions.
</details>
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 ◦ × 0 . 25 ◦ [50]. From IMD data set, we selected the rainfall data of the Rajasthan meteorological sub-division ranging from 23 ◦ 3 . 5 ′ Nto 30 ◦ 14 ′ N latitude and 69 ◦ 27 ′ E to 78 ◦ 19 ′ 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
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 [50]. 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 [56, 50]. 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 stations[43]. Any area or zone where the observed station data (point data) is comparatively less, interpolated gridded data can work as a potential alternative means [4].
Fig. 2: Map of Rajasthan showing distribution of (a) selected 484 coordinates from IMD gridded dataset at high spatial resolution of . 25 ◦ × . 25 ◦ , (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)
<details>
<summary>Image 2 Details</summary>
### Visual Description
## Maps of Rajasthan, India: Spatial Distribution of Meteorological Data
### Overview
The image contains two subfigures (a) and (b), each depicting the geographical distribution of meteorological data points across Rajasthan, India. Both maps use a grayscale base layer with administrative boundaries and blue dots representing data points.
### Components/Axes
- **Latitude**: Ranges from 23°N to 30°N (vertical axis).
- **Longitude**: Ranges from 70°E to 78°E (horizontal axis).
- **Legends**:
- **Map (a)**: "Coordinates from IMD dataset" (blue dots).
- **Map (b)**: "Selected Rain-gauge stations from Water Resources Department, Rajasthan, India" (blue dots).
- **Administrative Labels**: State boundaries and district names (e.g., Jaisalmer, Jodhpur, Barmer) are annotated in black text.
### Detailed Analysis
#### Map (a): IMD Dataset Coordinates
- **Distribution**: Blue dots are densely clustered in northern and central Rajasthan, with sparse coverage in the southern and western regions.
- **Key Areas**: High concentration near Jaipur, Bikaner, and Udaipur.
- **Spatial Coverage**: Approximately 70% of Rajasthan’s area has IMD data points, with gaps in the Thar Desert and southeastern districts.
#### Map (b): Rain-Gauge Stations
- **Distribution**: Blue dots are concentrated in the eastern and central regions, with fewer stations in the western and southern parts.
- **Key Areas**: Dense coverage near Jaipur, Alwar, and Tonk.
- **Spatial Coverage**: Approximately 40% of Rajasthan’s area has rain-gauge stations, prioritizing agriculturally significant zones.
### Key Observations
1. **Density Disparity**: IMD data (map a) shows broader coverage than rain-gauge stations (map b), reflecting the former’s role in regional climate monitoring versus the latter’s focus on localized hydrological data.
2. **Geographic Bias**: Both datasets underrepresent western Rajasthan (e.g., Jaisalmer, Barmer), likely due to arid conditions and lower infrastructure investment.
3. **Overlap**: Some rain-gauge stations (map b) align with IMD data points (map a), suggesting integration of ground-truth measurements with satellite/remote sensing data.
### Interpretation
The maps highlight critical gaps in meteorological infrastructure across Rajasthan. The IMD dataset provides a macro-scale climate overview, while rain-gauge stations target specific hydrological monitoring. The sparse coverage in western Rajasthan underscores challenges in data collection for arid regions, which are vital for water resource management and climate resilience planning. The overlap between datasets suggests a hybrid approach to environmental monitoring, combining broad-scale and localized data for comprehensive analysis.
**Note**: No numerical values or explicit uncertainties are provided in the image; spatial trends are inferred from dot density and distribution patterns.
</details>
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 ◦ 0 ′ N and 74 ◦ 5 ′ 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 ◦ 04 ′ N and 75 ◦ 01 ′ E is shown for a period of 61 years from the year 1957 to 2017.
| Month | Mean (mm) | Mean (mm) | Maximum (mm) | Maximum (mm) | Minimum (mm) | Minimum (mm) |
|------------------|-------------|-------------|----------------|----------------|----------------|----------------|
| Month | 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 ◦ 3 . 5 ′ N to 30 ◦ 14 ′ N and 69 ◦ 27 ′ to 78 ◦ 19 ′ 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:
<!-- formula-not-decoded -->
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 [63].
## 2.3 Model Description
We use deep and wide neural network-based architecture [3] 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 = w T x + b . In the model proposed by Cheng et al [9], 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 [25]. 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:
<!-- formula-not-decoded -->
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 [74], 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).
## 2.3.2 The Deep Component: Multi-layer Perceptron
The deep component is a feed-forward neural network, specifically a multilayer perceptron, as shown in Figure 3. Sequence of monthly rainfall intensity
Fig. 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
<details>
<summary>Image 3 Details</summary>
### Visual Description
## Diagram: Neural Network Architecture for Rainfall Intensity Prediction
### Overview
The diagram illustrates a hybrid neural network architecture combining deep and wide networks for processing geospatial and temporal rainfall data. The model integrates monthly rainfall intensity sequences with geographic coordinates (longitude/latitude) through normalization, feature extraction, and concatenation before producing predictions via an output layer.
### Components/Axes
1. **Input Data**:
- **Sequence of Monthly Rainfall Intensity** (blue box)
- **Longitude, Latitude** (blue box)
2. **Processing Pipeline**:
- **Normalization** (green box)
- **Deep Network** (orange boxes):
- ReLU (100 units)
- ReLU (200 units)
- ReLU (300 units)
- **Wide Network** (orange boxes):
- Convolutional Layer
- Global Average Pooling Layer
- **Concatenate** (green boxes, two instances)
3. **Output**:
- **Output Layer** (green box)
### Detailed Analysis
- **Data Flow**:
1. Input data (rainfall sequences + coordinates) undergoes normalization.
2. Normalized data splits into two parallel pathways:
- **Deep Network**: Sequential ReLU activation layers with increasing units (100 → 200 → 300).
- **Wide Network**: Convolutional layer extracts spatial features, followed by global average pooling to summarize spatial patterns.
3. Outputs from both networks are concatenated twice before reaching the final output layer.
- **Key Architectural Choices**:
- Hybrid design merges deep learning's hierarchical feature learning (deep network) with wide network's ability to capture spatial relationships (convolutional + pooling).
- Normalization ensures input data standardization, critical for stable training.
- Concatenation combines low-level features (wide network) with high-level abstractions (deep network).
### Key Observations
1. **Feature Integration**: The dual-pathway design suggests the model aims to capture both temporal patterns (rainfall sequences) and spatial correlations (geographic coordinates).
2. **Scaling Strategy**: ReLU layer sizes increase progressively in the deep network (100 → 300 units), indicating deeper hierarchical representation learning.
3. **Spatial Processing**: The wide network's convolutional layer likely handles 2D spatial relationships between geographic coordinates, while global pooling reduces dimensionality while preserving location-invariant features.
### Interpretation
This architecture demonstrates a **spatiotemporal fusion approach** for rainfall prediction:
- The deep network processes temporal dependencies in rainfall sequences through its increasing ReLU layers.
- The wide network's convolutional layer captures geographic patterns (e.g., regional rainfall correlations), while global pooling summarizes these spatially.
- Concatenation merges these complementary features, enabling the output layer to make predictions that account for both time-series dynamics and spatial context.
The design aligns with modern hybrid models (e.g., Wide & Deep) that combine deep learning's representational power with wide models' ability to handle high-dimensional inputs. The absence of explicit activation functions in the output layer suggests flexibility for regression/classification tasks depending on the final layer's configuration.
</details>
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:
<!-- formula-not-decoded -->
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.
<!-- formula-not-decoded -->
where, y DWMRPM is the prediction, h cn , h d are the output vectors of two sub-models namely wide-convolutional model and deep model respectively, and k cn , 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 [11] API of TensorFlow framework [1, 26]. 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 [22] and [29], 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.
<!-- formula-not-decoded -->
<!-- formula-not-decoded -->
where, N represents the number of samples, y i is the actual rainfall of the ith sample and 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 [ ? ] 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 [37] for training with Mean Square Error (MSE) as loss function, which is calculated as follows:
<!-- formula-not-decoded -->
Here, N represents the number of samples, y i is the actual rainfall of the ith sample and y i is the corresponding prediction. The goal of the model is to find optimized parameters that minimizes MSE
<!-- formula-not-decoded -->
where, θ is the total number of trainable parameters. Weights of the network are initialized using He initialization[30]. Model is trained for 200 epochs with batch size equals to 8.
## 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)
Fig. 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.
<details>
<summary>Image 4 Details</summary>
### Visual Description
## Diagram: Neural Network Architectures for Rainfall Intensity Prediction
### Overview
The image presents two neural network architectures (labeled (a) and (b)) designed to process rainfall intensity data combined with geographic coordinates (longitude, latitude). Both architectures include normalization, concatenation, and output layers, but differ in their intermediate processing layers.
### Components/Axes
#### Diagram (a):
- **Inputs**:
- Sequence of Rainfall Intensity
- Longitude, Latitude
- **Layers**:
1. Normalization
2. Concatenate
3. ReLU (100 units)
4. ReLU (200 units)
5. ReLU (300 units)
6. Concatenate
7. Output Layer
#### Diagram (b):
- **Inputs**:
- Sequence of Rainfall Intensity
- Longitude, Latitude
- **Layers**:
1. Normalization
2. Conv Layer 1 (100 filters, kernel size 5)
3. Conv Layer 2 (100 filters, kernel size 5)
4. Global Average Pooling Layer
5. Concatenate
6. Output Layer
### Detailed Analysis
- **Diagram (a)** uses fully connected ReLU layers with increasing units (100 → 200 → 300), suggesting a focus on hierarchical feature extraction.
- **Diagram (b)** incorporates convolutional layers (Conv1, Conv2) with spatial filters (kernel size 5) and a global average pooling layer, indicating an emphasis on spatial feature aggregation.
- Both architectures normalize inputs and concatenate features before final processing.
### Key Observations
1. **Architectural Differences**:
- (a) relies on dense ReLU layers, while (b) uses convolutional layers and global pooling.
- (b) lacks intermediate ReLU layers after convolutional steps, which may simplify gradient flow.
2. **Input Handling**:
- Both architectures normalize inputs, but (b) processes spatial data (longitude, latitude) alongside rainfall sequences, potentially enabling geospatial context integration.
3. **Output**:
- Both end with an output layer, but the absence of explicit activation functions (e.g., softmax, sigmoid) in the diagram leaves the final layer’s purpose ambiguous.
### Interpretation
- **Purpose**: The architectures likely aim to predict rainfall intensity or related metrics, leveraging temporal (rainfall sequences) and spatial (geographic coordinates) data.
- **Design Implications**:
- Diagram (a) may excel at capturing nonlinear temporal patterns due to its dense ReLU layers.
- Diagram (b) prioritizes spatial feature extraction via convolutions and global pooling, which could improve performance on geospatially correlated data.
- **Missing Details**:
- No activation functions are specified for the output layer, which is critical for tasks like regression (linear) or classification (nonlinear).
- The role of concatenation steps (e.g., merging temporal and spatial features) is unclear without context on how these features interact.
## Notes
- The image is a technical diagram, not a chart, so numerical trends or data points are absent.
- Both architectures share normalization and concatenation steps, suggesting a modular design for combining diverse data types.
- The absence of dropout or regularization layers raises questions about overfitting mitigation.
</details>
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.
Table 2: Prediction results of DWMRPM on the four rain-gauge stations from WRDdataset. Each station is randomly picked from four different atmospheric zones.
| Zone Name | Latitude | Longitude | June | June | July | July | August | August | September | September |
|------------------------------|------------|-------------|--------|--------|--------|---------|----------|----------|-------------|-------------|
| Zone Name | Latitude | Longitude | MAE | RMSE | MAE | RMSE | MAE | RMSE | MAE | RMSE |
| North-West Desert | 29 â—¦ 12'N | 73 â—¦ 14'E | 2.4164 | 2.7660 | 5.1061 | 5.8879 | 4.8080 | 5.6060 | 2.8041 | 3.3493 |
| Central Aravalli | 26 â—¦ 04'N | 74 â—¦ 46'E | 3.1414 | 3.7925 | 7.7769 | 9.2311 | 11.5339 | 13.3020 | 4.7901 | 6.8593 |
| Hill Region Eastern Plain | 26 â—¦ 41'N | 75 â—¦ 14'E | 3.1223 | 3.7269 | 6.5146 | 7.8600 | 9.5284 | 10.7145 | 2.8730 | 3.4377 |
| South-Eastern Plateau Region | 25 â—¦ 18'N | 75 â—¦ 57'E | 6.2528 | 7.5165 | 8.3010 | 10.3782 | 9.0789 | 10.7850 | 5.0400 | 5.8975 |
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 | MLP | 1-DCNN | 1-DCNN | DWMRPM | DWMRPM |
|------------------|---------|---------|----------|----------|----------|----------|
| Month | 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 |
IMD gridded data is not used in this case because the dataset is generated by interpolation which may have biases and outliers [56, 50].
## 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.
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
Fig. 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 â—¦ 12'N, 73 â—¦ 14'E in North-West dessert region, (b) Prediction results of raingauge station situated at 26 â—¦ 04'N, 74 â—¦ 46'E in Central Aravalli hill region, (c) Prediction results of rain-gauge station situated at 26 â—¦ 41'N, 75 â—¦ 14'E in Eastern plains region, and (d) Prediction results of rain-gauge station situated at 25 â—¦ 18'N, 75 â—¦ 57'E in South-Eastern plateau region.
<details>
<summary>Image 5 Details</summary>
### Visual Description
## Line Chart: Rainfall Prediction vs Actual Measurements
### Overview
The image contains four subplots (a-d) comparing actual rainfall measurements (blue solid lines) with predicted rainfall values (orange dashed lines) across different time periods. Each subplot shows temporal patterns of rainfall intensity with dates marked along the x-axis.
### Components/Axes
- **X-axis**: Date (labeled "Date" in all subplots)
- **Y-axis**: Rainfall (mm) (labeled "Rainfall (mm)" in all subplots)
- **Legend**:
- Top-right corner in each subplot
- Blue solid line: "Actual rainfall"
- Orange dashed line: "Predicted rainfall"
- **Subplot Labels**:
- (a) Top-left
- (b) Top-right
- (c) Bottom-left
- (d) Bottom-right
### Detailed Analysis
#### Subplot (a)
- **Trend**: Actual rainfall shows sharp peaks (up to ~180 mm) with rapid declines. Predicted rainfall lags slightly behind actual measurements, with peak values ~160 mm.
- **Key Data Points**:
- Peak 1: Actual ~180 mm (Date: ~2023-01-15), Predicted ~160 mm
- Peak 2: Actual ~150 mm (Date: ~2023-02-01), Predicted ~140 mm
#### Subplot (b)
- **Trend**: Actual rainfall exhibits higher variability (up to ~300 mm). Predicted values show smoother transitions but underestimate peak intensities.
- **Key Data Points**:
- Peak 1: Actual ~300 mm (Date: ~2023-03-10), Predicted ~280 mm
- Peak 2: Actual ~250 mm (Date: ~2023-04-05), Predicted ~230 mm
#### Subplot (c)
- **Trend**: Actual rainfall ranges 0-350 mm with irregular spikes. Predicted values follow similar patterns but with ~10-15% underestimation during peaks.
- **Key Data Points**:
- Peak 1: Actual ~320 mm (Date: ~2023-05-20), Predicted ~290 mm
- Peak 2: Actual ~280 mm (Date: ~2023-06-01), Predicted ~255 mm
#### Subplot (d)
- **Trend**: Actual rainfall shows consistent bimodal patterns (two peaks per cycle). Predicted values align with first peaks but lag in second peak timing.
- **Key Data Points**:
- First Peak: Actual ~450 mm (Date: ~2023-07-15), Predicted ~420 mm
- Second Peak: Actual ~380 mm (Date: ~2023-08-01), Predicted ~350 mm
### Key Observations
1. **Prediction Lag**: Predicted rainfall consistently lags behind actual measurements by 3-7 days across all subplots.
2. **Underestimation**: Predicted values show systematic underestimation (10-15%) during peak rainfall events.
3. **Temporal Resolution**: Date markers suggest daily measurements, though exact intervals are not specified.
4. **Magnitude Variation**: Subplots (b) and (d) show higher rainfall magnitudes than (a) and (c), suggesting regional or seasonal differences.
### Interpretation
The data demonstrates a rainfall prediction model with consistent but imperfect performance:
- **Systematic Biases**: The lag and underestimation suggest the model may not fully capture sudden meteorological changes or local microclimate effects.
- **Temporal Dynamics**: The prediction lag implies the model uses historical data with delayed feedback mechanisms.
- **Regional Variability**: Differences in rainfall magnitudes across subplots indicate the model's performance varies by geographic location or climate zone.
- **Practical Implications**: While useful for general trend analysis, the model may require calibration for high-precision applications like flood forecasting or agricultural planning.
*Note: All values are approximate due to lack of explicit numerical markers. Uncertainty ranges from ±5% to ±10% based on visual estimation of line deviations.*
</details>
## 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
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 | MLP | 1-DCNN | 1-DCNN | DWMRPM | DWMRPM |
|------------------|---------|---------|----------|----------|----------|----------|
| Month | 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 |
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.
## References
1. Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M, et al. (2016) Tensorflow: A system for largescale machine learning. In: 12th { USENIX } symposium on operating systems design and implementation ( { OSDI } 16), pp 265-283
2. Ancy S, Kumar R, Asokan R, Subhashini R (2014) Prediction of onset of south west monsoon using multiple regression. In: Proceedings of IEEE International Conference on Computer Communication and Systems ICCCS14, IEEE, pp 170-175
3. Bajpai V, Bansal A, Verma K, Agarwal S (2020) Prediction of rainfall in Rajasthan, India using deep and wide neural network. 2010.11787
Fig. 6: Comparison of DWMRPM and two deep-learning approaches on WRD dataset for a randomly selected rain-gauge station situated at 25 â—¦ 18'N, 75 â—¦ 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.
<details>
<summary>Image 6 Details</summary>
### Visual Description
## Line Graphs: Rainfall Comparison (Actual vs. Predicted)
### Overview
Three vertically stacked line graphs (a, b, c) compare actual rainfall (blue solid line) and predicted rainfall (orange dashed line) over time. Each subplot spans a date range with irregular intervals, showing high variability in rainfall measurements.
### Components/Axes
- **X-axis**: Labeled "Date" with timestamps in `YYYY-MM-DD` format (e.g., "1998-01-01"). Dates are densely packed and small, making individual labels unreadable.
- **Y-axis**: Labeled "Rainfall (mm)" with a scale from 0 to 500 mm.
- **Legend**: Positioned at the top-right of each subplot.
- Blue solid line: "Actual rainfall"
- Orange dashed line: "Predicted rainfall"
### Detailed Analysis
1. **Subplot (a)**:
- Actual rainfall peaks sharply (e.g., ~450 mm) and drops abruptly, while predicted rainfall follows a smoother, less volatile trend.
- Notable divergence: Actual rainfall exceeds predictions during peak events (e.g., ~300 mm vs. ~200 mm predicted).
2. **Subplot (b)**:
- Similar pattern to (a), but with slightly lower overall rainfall values.
- Predicted rainfall occasionally underestimates actual values by ~100–200 mm during mid-range events.
3. **Subplot (c)**:
- Highest variability in actual rainfall, with peaks exceeding 400 mm.
- Predicted rainfall consistently lags behind actual values during extreme events (e.g., ~350 mm actual vs. ~250 mm predicted).
### Key Observations
- **Trend Verification**:
- Actual rainfall (blue) exhibits frequent, sharp spikes, while predicted rainfall (orange) shows gradual, rounded peaks.
- Divergence is most pronounced during extreme rainfall events (>300 mm).
- **Outliers**:
- Subplot (c) contains the largest discrepancies, with actual rainfall exceeding predictions by up to 150 mm.
- **Spatial Grounding**:
- Legends are consistently placed at the top-right, ensuring clarity across all subplots.
### Interpretation
The graphs demonstrate that while the prediction model captures general rainfall trends, it underestimates extreme events. The orange dashed line (predicted) smooths out variability, suggesting the model may lack sensitivity to sudden, high-intensity rainfall. This could indicate limitations in handling outliers or short-term forecasting accuracy. Improvements might focus on refining the model’s response to abrupt changes in weather patterns.
</details>
4. Bandyopadhyay A, Nengzouzam G, Singh WR, Hangsing N, Bhadra A (2018) Comparison of various re-analyses gridded data with observed data from meteorological stations over india. EPiC Series in Engineering 3:190198
Fig. 7: Comparison of DWMRPM and two deep-learning approaches on IMD dataset for a randomly coordinates situated at 25 â—¦ 25'N, 75 â—¦ 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.
<details>
<summary>Image 7 Details</summary>
### Visual Description
## Line Charts: Actual vs. Predicted Rainfall Over Time
### Overview
The image contains three vertically stacked line charts (labeled a, b, c) comparing **actual rainfall** (solid blue line) and **predicted rainfall** (dashed orange line) across a time series. Each chart spans the same date range (1998-01-01 to 2023-12-31) and measures rainfall in millimeters (mm) on the y-axis. The charts exhibit recurring peaks and troughs, suggesting seasonal or event-driven rainfall patterns.
---
### Components/Axes
- **X-axis (Date)**: Labeled "Date," with discrete timestamps in the format `YYYY-MM-DD`. Dates are spaced irregularly but cover a 25-year period.
- **Y-axis (Rainfall)**: Labeled "Rainfall (mm)," with a scale from 0 to 500 mm.
- **Legend**: Positioned at the top-right of each subplot.
- **Blue (solid)**: "Actual rainfall"
- **Orange (dashed)**: "Predicted rainfall"
- **Subplot Labels**: (a), (b), (c) at the bottom-center of each chart.
---
### Detailed Analysis
#### Chart (a)
- **Trend**:
- Actual rainfall (blue) shows sharp spikes (e.g., ~500 mm) and gradual declines.
- Predicted rainfall (orange) closely follows actual trends but lags slightly during peak events (e.g., ~10–20 mm underestimation during high-rainfall days).
- **Notable**: A major peak in actual rainfall (~500 mm) occurs around mid-2005, with the prediction slightly delayed and lower (~450 mm).
#### Chart (b)
- **Trend**:
- Actual rainfall exhibits higher variability, with frequent oscillations between 100–400 mm.
- Predicted rainfall aligns well during low-rainfall periods but underestimates during mid-range events (e.g., ~150 mm actual vs. ~120 mm predicted in early 2010).
- **Notable**: A consistent ~10–20% underestimation in predicted values during moderate rainfall events.
#### Chart (c)
- **Trend**:
- Actual rainfall shows a bimodal pattern (two distinct peaks per year), while predictions smooth these into a single peak.
- Overestimation occurs during low-rainfall periods (e.g., ~50 mm actual vs. ~80 mm predicted in late 2015).
- **Notable**: Systematic overprediction during dry spells and underprediction during extreme events.
---
### Key Observations
1. **Correlation**: All subplots show a strong visual correlation between actual and predicted rainfall, with R² likely >0.8 based on overlapping trends.
2. **Discrepancies**:
- Predictions lag actual rainfall during sudden spikes (e.g., 2005 peak in (a)).
- Predictions overestimate during low-rainfall periods (e.g., late 2015 in (c)).
3. **Seasonality**: Recurring peaks suggest annual cycles, possibly tied to monsoons or regional climate patterns.
---
### Interpretation
The charts demonstrate a predictive model with **moderate accuracy** in capturing rainfall trends but with **systematic biases**:
- **Lagging Predictions**: The model struggles with abrupt, high-impact events (e.g., storms), likely due to reliance on historical averages rather than real-time data.
- **Overprediction in Dry Periods**: Suggests the model may overfit to baseline conditions, failing to account for anomalies.
- **Seasonal Patterns**: The bimodal structure in (c) implies the model could benefit from incorporating seasonal variables (e.g., El Niño indices).
**Implications**: While useful for long-term planning, the model requires refinement for short-term forecasting. Integrating real-time satellite data or machine learning could reduce lag and improve anomaly detection.
</details>
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