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Train an Echo State Network

Source: R/train_esn.R
train_esn.Rd

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)
)

Arguments

y

Numeric vector containing the response variable.

lags

Integer vector with the lag(s) associated with the input variable.

inf_crit

Character value. The information criterion used for variable selection inf_crit = c("aic", "aicc", "bic", "hqc").

n_diff

Integer vector. The nth-differences of the response variable.

n_states

Integer value. The number of internal states of the reservoir. If n_states = NULL, the reservoir size is determined by tau*n_total, 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 sequence 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_res, scale_res) 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)).

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.).

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.

See also

Other base functions: forecast_esn(), is.esn(), is.forecast_esn(), is.tune_esn(), plot.esn(), plot.forecast_esn(), plot.tune_esn(), print.esn(), summary.esn(), summary.tune_esn(), tune_esn()

Examples

xdata <- as.numeric(AirPassengers)
xmodel <- train_esn(y = xdata)
summary(xmodel)
#> 
#> --- Inputs -----------------------------------------------------
#> n_obs        = 144
#> n_diff       = 1
#> lags         = 1
#> 
#> --- Reservoir generation ---------------------------------------
#> n_states     = 57
#> alpha        = 1
#> rho          = 1
#> density      = 0.5
#> scale_inputs = [-0.5, 0.5]
#> scale_win    = [-0.5, 0.5]
#> scale_wres   = [-0.5, 0.5]
#> 
#> --- Model selection --------------------------------------------
#> n_models     = 114
#> df           = 16.68
#> lambda       = 0.0457