CGDT-CESN: Correlation-Guided and Dynamics-Trained Clustered Echo State Network for Multivariate Time-Series Prediction

Hariharan Srinivasan; Masoom Bhargava; Sayan Gupta

cs.LGOct 15, 2026

CGDT-CESN: Correlation-Guided and Dynamics-Trained Clustered Echo State Network for Multivariate Time-Series Prediction

Authors: Hariharan Srinivasan, Masoom Bhargava, Sayan Gupta

Organizations: The Uncertainty Lab, Department of Applied Mechanics & Biomedical Engineering, Indian Institute of Technology Madras, India - 600036 · Center for Complex Systems & Dynamics IIT Madras, India - 600036

Abstract

Accurate prediction of multivariate time series requires models that simultaneously capture variable-specific dynamics and cross-variable interactions. In this work, a Correlation-Guided and Dynamics-Trained Clustered Echo State Network (CGDT-CESN) is proposed, which comprises a modular reservoir architecture with each cluster distinctly specializing in one of the variables in the multivariate time series. An innovative reservoir design is proposed in which inter-cluster connections are enabled by empirically observed correlations among variables, providing a data-driven coupling mechanism. Apart from conventional readout training, inter-cluster connection weights informed by correlations are trained, which in turn allows the reservoir to capture time-varying cross-variable dependencies without sacrificing computational efficiency and further improves forecasting accuracy. The prediction performance of CGDT-CESN is found to be superior to that of the conventional ESN when evaluated on two fundamentally different datasets: Twitter information cascade evolution and solar wind forecasting. Also, the improvement in performance is preserved across multiple prediction horizons, even when presented with a low volume of data. By unifying cluster-specific specialization with learned inter-variable coupling, CGDT-CESN offers a principled, data-driven route to multivariate time series prediction.

Figures & tables

Figure 1: Architecture of Correlation-Guided and Dynamics Trained Clustered Echo state Network Illustrated for $K=4$ clusters corresponding to the four features.

Figure 2: Convergence test of the reservoir states with respect to the Uncorrelated and correlated noise sources before and after training.

Figure 3: Largest Lyapunov Exponent (LLE) computed for varying spectral radius ( $\rho$ ) to verify the ESP for uncorrelated and correlated input signals before and after training the reservoir.

Dataset	Cascades
Twitter15	1,490
Twitter16	818
Combined	2,308

Table 1: Summary of datasets and cascade labels [ 25 ]

Figure 4: Time evolution of cascade with multiple evolving variables reflecting different aspects of a cascade such as size, breadth and depth.

Measure	Mathematical Form	Meaning
Cascade Size	$S(n)=\|Z(n)\|$	Total number of activated nodes in the cascade up to time $n$ . Measures the overall spread.
Cascade Depth	$D(n)=\max\limits_{z\in Z(n)}d(z_{0},z)$	Maximum shortest-path distance from the source node $z_{0}$ to any node in the cascade. Measures the propagation depth.
Cascade Breadth	$B(n)=\max\limits_{k}\|L_{k}(n)\|$	Maximum number of nodes present at any propagation level $k$ . Measures the widest parallel spread.
Structural Virality	$Z_{s}(n)=\dfrac{1}{\|Z(n)\|(\|Z(n)\|-1)}\sum\limits_{i\neq j}d(z_{i},z_{j})$	Average pairwise shortest-path distance among all nodes. Measures how structurally viral the cascade is.

Table 2: Time-dependent cascade descriptors and their mathematical definitions.

Hyperparameter	ESN	CG-CESN	CGDT-CESN
Reservoir size ( $N_{\text{res}}$ )	200	–	–
Neurons per cluster ( $N_{c}$ )	–	50	50
Number of clusters ( $K$ )	–	4	4
Spectral radius ( $\rho$ )	0.9	0.9	0.9
Leak rate ( $\alpha$ )	0.3	0.3	0.3
Reservoir connectivity ( $p_{\text{rc}}$ )	0.3	–	–

Table 3: Hyperparameters used for different reservoir computing models

Figure 5: Validation convergence of the CGDT- C ESN model for different training data fractions ( $10\%$ , $50\%$ , and $100\%$ ) over 50 Winter refinement epochs. The curves represent the mean validation NRMSE averaged over 10 independent runs, while the error bars denotes standard deviation. A rapid decrease in validation error is observed during the initial epochs, followed by convergence and saturation beyond approximately 10 epochs, indicating that additional refinement iterations provide only marginal improvement in predictive performance.

Figure 6: Effect of the leak rate parameter ( $\alpha$ ) on the performance of ESN ( $\bullet$ ), CG-CESN ( $\blacksquare$ ), and CGDT-CESN ( $\blacktriangle$ ) for training sample percentages of 10% (a), 50% (b), and 100% (c). The leak rate is varied over the range ([0.1,1.0]) while all other parameters, including the number of training epochs, are kept fixed.

Figure 7: Performance (mean NRMSE, $\xi$ ) of models as a function of training data proportion (where 100 % denotes the maximum training sample size corresponding to 80% of the original data set) evaluated at three prediction horizons (a) 50%, (b) 75% and (c) 100% for 50 realizations. The gray, blue and green bars correspond the ESN, CG-CESN and CGDT-CESN with the error bar depicting the standard deviation

Figure 8: Scatter plots of predicted versus ground truth values for the four cascade features: size (red), breadth (green), depth (blue), and structural virality (yellow), evaluated at prediction horizon $50\%$ , across three models (columns) and three training set sizes (rows). All variables are independently normalized to $[0,1]$ . The dashed diagonal line ( $y=x$ ) represents perfect prediction; points lying closer to this line indicate higher predictive accuracy. Row (a): training size $10\%$ (panels a1–a3); Row (b): training size $50\%$ (panels b1–b3); Row (c): training size $100\%$ (panels c1–c3). Column 1 (panels a1, b1, c1): ESN; Column 2 (panels a2, b2, c2): CG-CESN; Column 3 (panels a3, b3, c3): CGDT-CESN. The mean normalized root mean square error ( $\xi$ , averaged across all four cascade features over 50 independent realizations) is reported in the upper left corner of each subplot. Progressively tighter clustering of points around the diagonal with increasing training size reflects improved model generalization, with CGDT-CESN consistently achieving the lowest $\xi$ values, particularly under limited training data.

Model	$\xi$
ESN	$0.081\pm 0.0075$
CG-CESN	$0.067\pm 0.0025$
CGDT-CESN	$\mathbf{0.063\pm 0.0019}$

Table 4: Prediction performance (

\xi

) of the models on the solar wind dataset. Results are averaged over 50 realizations and reported as mean

\pm

standard deviation of the

\xi

Figure 9: Visualization of solar wind prediction performance on the test set consisting of variables temperature, speed and density. The panels show the ground-truth depicted as thick orange curve and predicted values of ESN (dashed blue line), CG-CESN (dashed dotted red line) and CGDT-CESN (solid green line). Prediction of CGDT-CESN agrees closely with the ground truth across all variables and also exhibits its stability through time as compared to ESN.

Model	Cascade			Solar Wind
Model	Inter-cluster Wiring	Training	Total	Inter-cluster Wiring	Training	Total
ESN	–	$9.02\pm 1.09$	$9.02\pm 1.09$	–	$0.16\pm 0.03$	$0.16\pm 0.03$
CG-CESN	$341.54\pm 13.39$	$8.72\pm 1.00$	$350.26\pm 13.59$	$0.30\pm 0.03$	$0.49\pm 0.08$	$0.79\pm 0.09$
CGDT-CESN	$341.54\pm 13.39$	$125.70\pm 9.63$	$467.24\pm 20.37$	$0.30\pm 0.03$	$0.93\pm 0.18$	$0.95\pm 0.18$

Table 5: Comparison of computational cost for training the model for the Cascade and Solar Wind datasets. Times are reported in seconds as mean

\pm

standard deviation over 10 realizations. The epochs chosen for cascade and solar wind data sets for CGDT-CESN training is 10 and 5.

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