train_esn: Train an Echo State Network

Description

Train an Echo State Network (ESN) to a univariate time series. The function automatically manages data pre-processing, reservoir generation (i.e., internal states) and model estimation and selection.

Usage

train_esn(
  y,
  lags = 1,
  inf_crit = "bic",
  n_diff = NULL,
  n_states = NULL,
  n_models = NULL,
  n_initial = NULL,
  n_seed = 42,
  alpha = 1,
  rho = 1,
  tau = 0.4,
  density = 0.5,
  lambda = c(1e-04, 2),
  scale_win = 0.5,
  scale_wres = 0.5,
  scale_inputs = c(-0.5, 0.5)
)

Value

A list containing:

actual: Numeric vector containing the actual values.
fitted: Numeric vector containing the fitted values.
resid: Numeric vector containing the residuals.
states_train: Numeric matrix containing the internal states.
method: A list containing several objects and meta information of the trained ESN (weight matrices, hyperparameters, model metrics, etc.).

Arguments

y: Numeric vector containing the response variable (no missing values).
lags: Integer vector with the lag(s) associated with the input variable.
inf_crit: Character value. Information criterion used for model selection among the n_models candidate ridge fits (different random lambda values). The candidate with the smallest criterion value is selected. One of c("aic", "aicc", "bic", "hqc").
n_diff: Integer value. The nth-differences of the response variable. If n_diff = NULL, the number of differences required to achieve stationarity is determined automatically via a KPSS-test.
n_states: Integer value. The number of internal states of the reservoir. If n_states = NULL, the reservoir size is determined by min(floor(n_total * tau), 200), where n_total is the time series length.
n_models: Integer value. The maximum number of (random) models to train for model selection. If n_models = NULL, the number of models is defined as n_states*2.
n_initial: Integer value. The number of observations of internal states for initial drop out (throw-off). If n_initial = NULL, the throw-off is defined as n_total*0.05, where n_total is the time series length.
n_seed: Integer value. The seed for the random number generator (for reproducibility).
alpha: Numeric value. The leakage rate (smoothing parameter) applied to the reservoir (value greater than 0 and less than or equal to 1).
rho: Numeric value. The spectral radius for scaling the reservoir weight matrix (value often between 0 and 1, but values above 1 are possible).
tau: Numeric value. The reservoir scaling parameter to determine the reservoir size based on the time series length (value greater than 0 and less than or equal to 1).
density: Numeric value. The connectivity of the reservoir weight matrix (dense or sparse) (value greater than 0 and less than or equal to 1).
lambda: Numeric vector. Lower and upper bound of lambda range for ridge regression (numeric vector of length 2 with both values greater than 0 and lambda[1] < lambda[2]).
scale_win: Numeric value. The lower and upper bound of the uniform distribution for scaling the input weight matrix (value greater than 0, weights are sampled from U(-scale_win, scale_win)).
scale_wres: Numeric value. The lower and upper bound of the uniform distribution for scaling the reservoir weight matrix (value greater than 0, weights are sampled from U(-scale_wres, scale_wres) before applying rho and density).
scale_inputs: Numeric vector. The lower and upper bound for scaling the time series data (numeric vector of length 2 with scale_inputs[1] < scale_inputs[2], often symmetric, e.g., c(-0.5, 0.5) or c(-1, 1)).

References

Häußer, A. (2026). Echo State Networks for Time Series Forecasting: Hyperparameter Sweep and Benchmarking. arXiv preprint arXiv:2602.03912, 2026. https://arxiv.org/abs/2602.03912
Jaeger, H. (2001). The “echo state” approach to analysing and training recurrent neural networks with an erratum note. Bonn, Germany: German National Research Center for Information Technology GMD Technical Report, 148(34):13.
Jaeger, H. (2002). Tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the "echo state network" approach.
Lukosevicius, M. (2012). A practical guide to applying echo state networks. In Neural Networks: Tricks of the Trade: Second Edition, pages 659–686. Springer.
Lukosevicius, M. and Jaeger, H. (2009). Reservoir computing approaches to recurrent neural network training. Computer Science Review, 3(3):127–149.

Examples

Run this code

xdata <- as.numeric(AirPassengers)
xmodel <- train_esn(y = xdata)
summary(xmodel)