Exploring non-linear modes of the subtropical Indian Ocean Dipole using autoencoder neural networks

The subtropical Indian Ocean Dipole (SIOD) significantly influences climate variability, predominantly within parts of the Southern Hemisphere. This study applies an autoencoder—a type of artificial neural network (ANN)—known for its ability to capture intricate non-linear relationships in data through the process of encoding and decoding—to analyze the spatiotemporal characteristics of the SIOD. The encoded SIOD pattern(s) is compared to the conventional definition of the SIOD, calculated as the sea surface temperature (SST) anomaly difference between the western and eastern subtropical Indian Ocean. The analysis reveals two encoded patterns consistent with the conventional SIOD structure, predominantly represented by the SST dipole pattern south of Madagascar and off Australia’s west coast. During different analysis periods, distinct variability in the global SST patterns associated with the SIOD was observed. This variability underscores the SIOD’s dynamic nature and the challenges of accurately defining modes of variability with limited records. One of the ANN patterns has a substantial congruence match of 0.92 with the conventional SIOD pattern, while the other represents an alternate non-linear pattern within the SIOD. This implies the potential existence of additional non-linear SIOD patterns in the subtropical Indian Ocean, complementing the traditional model. When global temperature and precipitation are regressed onto the ANN temporal patterns and the conventional SIOD index, both appear to be associated with anomalous climate conditions over parts of Australia, with several other consistent global impacts. Nevertheless, due to the non-linear nature of the ANN patterns, their effects on local temperature and precipitation vary across different regions as compared to the conventional SIOD index. This study highlights that while the conventional SIOD pattern is consistent with the ANN-derived SIOD pattern, the climate system’s complexity and non-linearity might require ANN modeling to advance our comprehension of climatic modes.


Introduction
The subtropical Indian Ocean dipole (SIOD) is one of the climatic modes in the subtropical Indian Ocean that impact climate variability, predominantly in the Southern Hemisphere [1].It is defined conventionally as the difference in sea surface temperature (SST) anomaly between the western (55 • E-65 • E, 37 • S-27 • S) and eastern (90 • E-100 • E, 28 • S-18 • S) regions of the southern Indian Ocean [1].The SIOD typically emerges between November and December, reaches its maximum intensity during January-February, and gradually dissipates around March-April [1,2].The adjustment of the Mascarene high during austral summer contributes to the dynamics that trigger the SIOD [1,2].The SST pattern of the SIOD has been noted to influence weather patterns over parts of Australia, southern Africa [3], and other regions around the globe [4].Shifts in the Mascarene High during SIOD events alter the local Hadley cell, leading to changes in the strength of the monsoon circulation system during the Indian summer monsoon event [5].Hence, the improved representation of the spatiotemporal characteristics of the SIOD is crucial for more accurate climate predictions and the development of climate models.
Behera and Yamagata [1] proposed the SIOD index-defined as the SST anomaly difference between the western and eastern subtropical Indian Ocean.This was indeed a significant milestone in our understanding of the climate variability in the subtropical Indian Ocean.Principal component analysis (PCA) was instrumental in identifying the leading mode of SST variability, which exhibited a dipole structure, thus leading to the definition of the SIOD Index [1].However, PCA might not optimally define the SIOD due to its inherent limitations.The application of PCA assumes linearity and orthogonality of the patterns, which may not always hold true for complex and non-linear climate phenomena such as SIOD [6].Consequently, the definition may fail to capture all relevant aspects of SIOD variability, such as shifts in its center of action or the potential for multiple, distinct SIOD patterns.These observations raise questions about the current linear model's ability to accurately and comprehensively define the SIOD, thus limiting our understanding and prediction of its climatic impacts.
As a solution to this, the application of artificial neural networks (ANN) in climatic studies offers a promising avenue [7].ANNs are computational models that can effectively handle complex non-linear systems [8].ANNs have the potential to provide a more holistic and dynamic representation of SIOD, addressing the limitations of conventional linear models [8].In particular, autoencoder ANNs have been used successfully in capturing the underlying structure and features of complex datasets [9].In this study, an autoencoder ANN is utilized to examine the spatiotemporal characteristics of the SIOD.The goal of this study is not only to understand the potential improvements in the SIOD's definition but also to ascertain why ANN might be a superior tool for representing such climatic modes.
By expanding our analytical approach to include ANN, this study anticipates contributing a more nuanced understanding of the SIOD and its impacts, with potential ramifications for climate science.Therefore, this study aims to offer insights into how advanced modeling techniques can better equip us to navigate the inherent complexities of climate systems.

Data and methods
SST data used in encoding the non-linear SIOD pattern and, for calculating the traditional SIOD index (i.e. the SDI) is obtained from the NOAA Extended Reconstructed SST, version 5 (ERSSTv5) [10] from 1900 to 2022 at a monthly resolution from January to March when the SIOD is active.The horizontal resolution of the data is 2.0 • longitude and latitude.The SST data was deseasonalized to better identify and analyze the SIOD without the influence of regular seasonal patterns.This was done by subtracting the long-term mean (from 1900 to 2022) of each month from the data values in that same month.This method removes the monthly variations attributable to regular seasonal effects.
Monthly gridded observational temperature and precipitation data sets are obtained from the Climate Research Unit Version 4.06 [11] from 1901 to 2021.The spatial resolution of the data sets is 0.5 • longitude and latitude.The datasets produced by the Climate Research Unit, which is part of the University of East Anglia are publicly available at https://crudata.uea.ac.uk/cru/data/hrg/cru_ts_4.06/cruts.2205201912.v4.06.
Monthly 2 m temperature over the ocean and land and 10 m wind speed were also obtained from ERA5-reanalysis [12] from 1940 to 2022.The horizontal resolution of the ERA5 data is 0.25 • longitude and latitude.
Figure 1 shows the flowchart for applying the autoencoders ANN in deriving non-linear SST patterns in the subtropical Indian Ocean.The focus is to investigate the encoded dipole patterns representative of the conventional SIOD.Therefore, only the ANN patterns with sufficient congruence match of at least 0.90 with the spatial pattern associated with the conventional SIOD index (i.e.SST anomaly difference between 55 • E-65 • E, 37 • S-27 • S, and 90 • E-100 • E, 28 • S-18 • S) were analyzed.The region used for encoding the spatial pattern of the SIOD covers the oceanic domains used to traditionally define the SIOD index including the adjacent oceanic domains.
From figure 1, the initial phase of applying the autoencoder involves the normalization of the input data to a range of [0,1], utilizing a MinMaxScaler, which is a data pre-processing tool in Python's scikit-learn library [13].This step is critical as it ensures the model's training process is not unduly influenced by the scale of the data, enhancing both the speed of convergence and the overall performance of the model [14].
After normalization, the autoencoder model's architecture is defined.The number of input and output neurons is dictated by the dimensionality of the SST data.The choice of the number of hidden layers was examined iteratively between 2 and 128 aiming to strike a balance between model complexity in detecting the SIOD pattern and the risk of overfitting.The size of the hidden layer determines the number of potential patterns that the model will learn from the SST data.Too many neurons might lead the model to overfitting, while too few could result in an oversimplified model that might miss crucial underlying patterns [15].The encoder, which compresses the input into a dense representation, and the decoder, which reconstructs the input data from this representation, are each represented as a dense layer.The encoder employs a rectified linear unit activation function, chosen for its ability to handle non-linearities effectively and alleviate the vanishing gradient problem [15].On the other hand, the decoder uses a sigmoid activation function to constrain its outputs to the [0, 1] range, matching the range of the normalized input data [16].
Once the autoencoder architecture is defined, the model is compiled using the Adam optimizer and mean squared error (MSE) as a loss function.Adam, an adaptive learning rate optimization algorithm, is particularly effective for large-scale, complex deep learning models, providing an efficient gradient descent optimization [17].The MSE measures the average squared differences between the predicted (i.e.reconstructed SST data) and actual values, providing a relevant metric for training the model, as the goal is to minimize the reconstruction error.The autoencoder is trained using a maximum of 50 epochs in a training and validation period.An epoch signifies one complete pass of the dataset through the algorithm.The early stopping, a form of regularization was used to avoid overfitting.This means training is halted as soon as the validation error increases.
The encoder part of the trained autoencoder is used to encode the normalized SST data into a lower-dimensional space.This step provides a dense representation of the input data that captures its most relevant spatial patterns, which is equivalent to the SST modes of variability in the subtropical Indian Ocean.
The decoder then uses this encoded data to reconstruct the input data, aiming to minimize the reconstruction error.The reconstruction is not perfect, but it ideally captures the most important patterns in the data.The resulting MSE serves as an indicator of how well the model has learned to recreate the original data.
To investigate both the temporal stability of the encoded SIOD patterns the analysis period is divided into two (1900-1960 and 1961-2022) and the same ANN algorithm is applied to the two periods, respectively.The SIOD patterns of the ANN algorithm are compared to the patterns from the convectional SIOD, and the impact of defining the SIOD pattern either way on global temperature and precipitation was investigated.

Results and discussion
From figure 2, at the 13th epoch number, the validation error increased while the gradient of the training loss became relatively flat, indicating a plateau, which necessitated stopping the model training.Thus, 13 epochs were used.Figure 3 shows the resulting encoded patterns with sufficient congruence matches with the conventional SIOD pattern.The black frames show the western and eastern subtropical Indian Ocean regions conventionally used to define the SDI.The congruence match between the spatial patterns of Node 1 and Node 2 is 0.5, which indicates that the two patterns are distinct.The spatial pattern of Node 2 has the highest congruence match of 0.92 with the conventional SIOD spatial pattern, while Node 1 matches the SIOD pattern with a 0.90 congruence coefficient.The closer match between the spatial pattern of Node 2 and the conventional SIOD is because its centers of action are relatively more aligned with the conventional regions (i.e. the black frames in figure 3) used to define the SIOD.
Figure 4 shows the regression of the time series of Nodes 1 and 2, and SDI onto global SST for different time periods.In the long-term (1900-2022), the SST in the eastern tropical and southern Pacific Ocean as well as in the South Atlantic Ocean are significantly associated with patterns of Node 2 and the SDI.However, considering the SDI, SST in parts of the Southern Ocean appears to play some role.For Node 1, no significant correlation was found between the eastern tropical and southern Pacific and the time series of Node 1, which possibly contributes to its difference with conventional SIOD mode (figure 4).
Considering different time periods, figure 4 indicates differences in the correlation patterns between global SST and the time series of Nodes 1, 2, and the SDI.For example, in the earlier periods (1900-1989), no robust relationship was found between the SIOD variability and the eastern tropical Pacific Ocean.The relationship rather emerged in the late analysis period (1990-2019).Also, the spatial extent of the correlation between the tropical Indian Ocean and the SIOD significantly increased in the 1990-2019 period.This is an indication of the dynamic nature of the oceanic and atmospheric factors that interact with the SIOD mode.Therefore, an advanced non-linear model, such as the autoencoders that capture the intricate complex variabilities of the SIOD over time might be required for a more holistic characterization of the SIOD mode.
Figure 5 shows the time series of Nodes 1, 2, and the SDI.There are observable differences in the time series variability (figure 5 A threshold of greater than 1.5 standard deviations of the time series of the SIOD indices was used to define strong SIOD years.From Table A1 the SDI does not discriminate between some SIOD events associated with the two non-linear ANN SIOD patterns represented by Nodes 1 and 2. This might result in an erroneous forecast of the onset and regional impact of the SIOD event. Figure 6 shows the circulation and temperature anomaly during strong positive and negative SIOD events (see table A1 for a list of years) for Nodes 1 and 2, the positive and negative phases of the encoded SIOD patterns show asymmetry such that the positive and the negative phases are not mirror images separated by the sign.As a result, the conventional definition of the SIOD appears to align more with its positive phase (figure 6).During the positive phase, the circulation anomaly associated with the SIOD pattern is characterized by the enhanced easterly anomaly in the subtropical Indian Ocean.During the negative phase of Nodes 1 and 2, the reverse dipole SST patterns imply that westerly anomalies dominate the subtropical Indian Ocean.Specifically, under Node 1, the Indian Ocean is anomalously warmer (figure 6(b)) and associated with warmer temperatures in Western Australia.
Further, the linear relationship between global precipitation and temperature and Nodes 1, 2, and the SDI were examined using linear regression.Figures 7 and 8 show the results.Indeed, besides Australia where the relationship is strongest, the SIOD might impact temperature and precipitation in other parts of the globe.These include the northwestern parts of South America, parts of Africa, North America, and southern Asia (figures 7 and 8).By visual examination, there are differences in the spatial extent of the linear relationship between Nodes 1 and 2, and global temperature and precipitation, supporting that they are different non-linear patterns of SIOD with differences in their impact on regional climates around the globe (cf figure 4).The global spatial pattern of Node 1 in figures 7 and 8 has a 0.97 congruence match with Nodes 2 and the SDI, while SDI has a 0.99 congruence match with Node 2 both for temperature and precipitation.However, by visual examination of figures 6 and 7, there are observable dissimilarities between the associating patterns of Node 2 and SDI.In southern Asia for example, the spatial extent of the linear relationship between temperature and Node 2 is higher compared to the SDI.Also, Node 2 has a stronger linear association with precipitation in the northern portion of Australia.

Discussion and conclusions
The associated temperature, precipitation, and SST patterns of the traditional definition of the SIOD index as the difference in SST anomaly between the western (55 • E-65 • E, 37 • S-27 • S) and eastern (90 • E-100 • E, 28 • S-18 • S) regions of the subtropical Indian Ocean were consistent with the autoencoder ANN-based definition of the SIOD index.The advantage of the traditional index lies in its simplicity, making it easily calculable and interpretable for a wide audience.However, this study acknowledges that a deeper examination is warranted to ascertain whether this simplicity compromises crucial aspects of the SIOD.
One of the motivations for using a more intricate methodology lies in the complexity of climate modes, such as ENSO, which has been extensively studied and found to possess non-linear and asymmetric characteristics [18].As climate science advances, it becomes imperative to explore similar complexities in other climate modes like the SIOD, which may exhibit complex behaviors that the traditional index might overlook.Furthermore, the impacts of climate change on the traditional definition of the SIOD should not be underestimated.With the evolving climate, the conventional approach of defining the SIOD based solely on the difference in average SST between two ocean regions might no longer suffice.Indeed, climate change can alter the fundamental dynamics of climate modes (cf figure 4), necessitating a more adaptive approach to capture these changes accurately.
Thus, this study leverages an Autoencoder ANN to redefine the SIOD, aiming to uncover potential subtle variations within the SIOD that may have distinct implications for weather patterns.The autoencoder successfully captures the traditional SIOD definition but also suggests the existence of other non-linear aspects of the SIOD that may have subtle differences in their impact on weather.
Indeed, as the complexity of the climate system continues to evolve due to climate change, employing methodologies that can capture these intricacies becomes increasingly valuable ensuring that critical nuances that may have significant implications for climate and weather patterns are not neglected.
In conclusion, the variabilities inherent in climate phenomena such as the SIOD may warrant the utilization of advanced modeling techniques, such as ANNs.These techniques can model and characterize non-linear, high-dimensional data effectively, thereby providing a more comprehensive understanding of SIOD.While traditional definitions of SIOD have proved valuable, the dynamic and evolving nature of climate change necessitates more sophisticated tools capable of capturing the complexity of our climate system [2,16,19].Such advancements in modeling methodologies will enhance the accuracy of climate predictions and projections, thereby bolstering our preparedness and response to the pressing issue of climate change.

Figure 1 .
Figure 1.Flow chart using autoencoders artificial neural networks to derive SST modes in the subtropical Indian Ocean.

Figure 2 .
Figure 2. Epoch number versus training and validation losses in reconstructing the original SST data from the encoded data.

Figure 3 .
Figure 3. Spatial patterns of the encoded patterns with at least 0.90 congruence spatial match between the encoded patterns and the spatial pattern of the convectional SIOD pattern during 1900-2022 The black frame shows the region used in calculating the conventional SIOD index.

Figure 4 .
Figure 4.The correlation between Node1, Node2, and the SDI time series with global SST from 1900 to 2022 (first row), and for different 30 year periods between 1900 and 2019 (second to fifth row).Stippling shows grids with statistically significant correlations at a 95% confidence level.A false discovery rate was used to control for false positives due to multiple significance testing.
(a)) and the statistical distribution of the respective indices (figures 5(b) and (c)).Quantitatively, the time series of Nodes 2 and SDI has the closest congruence match of 0.92; Node 1 and SDI matches with 0.83, and Node 1 and Node 2 match with 0.72 (figure 5(c)).The congruence matches (<1.0) further indicate that while Node 2 and SDI are close, the temporal variability captured by the ANN pattern is not the same as the SDI, and Node 1 is a different non-linear aspect of the SIOD.

Figure 5 .
Figure 5. 10 year moving average of the time series of Nodes 1, 2 and SDI (a) distribution of the SIOD indices (b)and scatter plot of the SIOD indices (c).The Pearson correlation is reported in figure 5(c).

Figure 6 .
Figure 6.Circulation anomalies (a) and 2 m temperature anomalies (b) during strong positive and negative SIOD years.Anomalies were calculated with respect to the JFM climatology.Only significant values based on the block permutation test are shaded.In (a) color is SST and the vector is 10 m wind.

Figure 7 .
Figure 7. Linear regression of the time series of global temperature onto the time series of Nodes 1, 2, and SDI.Only significant values at a 95% confidence level are plotted.The false discovery rate is used to control for false positives.

Figure 8 .
Figure 8. Linear regression of the time series of global precipitation onto the time series of Nodes 1, 2, and SDI.Only significant values at a 95% confidence level are plotted.The false discovery rate is used to control for false positives.