Package 'echos'

Title: Echo State Networks for Time Series Modeling and Forecasting
Description: Provides a lightweight implementation of functions and methods for fast and fully automatic time series modeling and forecasting using Echo State Networks (ESNs).
Authors: Alexander Häußer [aut, cre, cph] (ORCID: <https://orcid.org/0009-0000-5419-8479>)
Maintainer: Alexander Häußer <[email protected]>
License: GPL-3
Version: 1.0.4
Built: 2026-06-15 15:55:12 UTC
Source: https://github.com/cran/echos

Help Index

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. The function is a wrapper for train_esn() and intended to be used in combination with fabletools::model().

Usage

ESN(formula, ...)

Arguments

formula

A model formula used by the fabletools model specification interface. The left-hand side identifies the univariate response variable to be modeled. The current implementation does not interpret additional right-hand-side terms, because echos currently supports univariate ESN models without exogenous regressors. The formula interface is retained for consistency with fabletools::model() and to allow future extensions, such as exogenous variables or richer model specifications.
...

Further arguments passed to train_esn(), including ESN architecture, pre-processing, and model-selection parameters such as lags, n_diff, n_states, n_models, alpha, rho, tau, density, lambda, scale_win, scale_wres, and scale_inputs.

Value

An object of class ESN.

See Also

Other tidy functions: filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value))

Filter ESN models

Description

Filter an object of class mdl_df ("mable") to include ESN models only, i.e., other models like ARIMA or ETS are excluded from the mable.

Usage

filter_esn(object)

Arguments

object

An object of class mdl_df, containing an ESN model.

Value

An object of class mdl_df in long-format.

See Also

Other tidy functions: ESN(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
filter_esn()

Extract fitted values from a trained ESN

Description

Extract fitted values from a trained ESN as tsibble.

Usage

## S3 method for class 'ESN'
fitted(object, ...)

Arguments

object

An object of class mdl_df, containing an ESN model.
...

Currently not in use.

Value

Fitted values extracted from the object.

See Also

Other tidy functions: ESN(), filter_esn(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
fitted()

Forecast an Echo State Network

Description

Forecast an Echo State Network (ESN) from a trained model via recursive forecasting. Forecast intervals are generated by simulating future sample path based on a moving block bootstrap of the residuals and estimating the quantiles from the simulations.

Usage

forecast_esn(
  object,
  n_ahead = 18,
  levels = c(80, 95),
  n_sim = 100,
  n_seed = 42
)

Arguments

object

An object of class esn. The result of a call to train_esn().
n_ahead

Integer value. The number of periods for forecasting (i.e. forecast horizon).
levels

Numeric vector. The levels of the forecast intervals (in percent), e.g., c(80, 95). Values must lie between 0 and 100.
n_sim

Integer value. The number of simulated future paths used to compute forecast intervals via a moving block bootstrap of the (demeaned) in-sample residuals. If NULL, no intervals are computed.
n_seed

Integer value. The seed for the random number generator (for reproducibility).

Value

A list containing:

  • point: Numeric vector containing the point forecasts.

  • interval: Numeric matrix containing the forecast intervals.

  • sim: Numeric matrix containing the simulated future sample path.

  • std: Numeric vector with standard deviations.

  • levels: Integer vector. The levels of the forecast intervals.

  • actual: Numeric vector containing the actual values.

  • fitted: Numeric vector containing the fitted values.

  • n_ahead: Integer value. The number of periods for forecasting (forecast horizon).

  • model_spec: Character value. The model specification as string.

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: is.esn(), is.forecast_esn(), is.tune_esn(), plot.esn(), plot.forecast_esn(), plot.tune_esn(), print.esn(), summary.esn(), summary.tune_esn(), train_esn(), tune_esn()

Examples

xdata <- as.numeric(AirPassengers)
xmodel <- train_esn(y = xdata)
xfcst <- forecast_esn(xmodel, n_ahead = 12)
plot(xfcst)

Forecast an Echo State Network

Description

Forecast an Echo State Network (ESN) from a trained model via recursive forecasting. Forecast intervals are generated by simulating future sample path based on a moving block bootstrap of the residuals and estimating the quantiles from the simulations. The function is a wrapper for forecast_esn() and intended to be used in combination with fabletools::model().

Usage

## S3 method for class 'ESN'
forecast(
  object,
  new_data,
  normal = TRUE,
  n_sim = 200,
  specials = NULL,
  xreg = NULL,
  ...
)

Arguments

object

An object of class mdl_df, containing an ESN model.
new_data

Forecast horizon (n-step ahead forecast).
normal

Logical value. If TRUE, dist_normal() is used, otherwise dist_sample().
n_sim

Integer value. The number of future sample path generated during simulation.
specials

Currently not in use.
xreg

A tsibble containing exogenous variables.
...

Currently not in use.

Value

An object of class fbl_ts ("fable").

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
forecast(h = 18)

Summary of trained models during random search

Description

Return summary statistics from trained ESN models during random search as tibble.

  • model: Model identifier.

  • loglik: Log-likelihood.

  • nobs: Number of observations.

  • df: Effective degrees of freedom.

  • lambda: Regularization parameter.

  • aic: Akaike Information Criterion.

  • aicc: Corrected Akaike Information Criterion.

  • bic: Bayesian Information Criterion.

  • hqc: Hannan-Quinn Information Criterion.

  • mse: Mean Squared Error.

  • mae: Mean Absolute Error.

Usage

## S3 method for class 'ESN'
glance(x, ...)

Arguments

x

An object of class mdl_df, containing an ESN model.
...

Currently not in use.

Value

Summary statistics extracted from the object.

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
glance()

Checks if object is of class "esn"

Description

Returns TRUE if the object is of class esn.

Usage

is.esn(object)

Arguments

object

object to be tested.

Value

Logical value. If TRUE, the object is of class esn.

See Also

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

Examples

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

Checks if object is of class "forecast_esn"

Description

Returns TRUE if the object is of class forecast_esn.

Usage

is.forecast_esn(object)

Arguments

object

object to be tested.

Value

Logical value. If TRUE, the object is of class forecast_esn.

See Also

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

Examples

xdata <- as.numeric(AirPassengers)
xmodel <- train_esn(y = xdata)
xfcst <- forecast_esn(xmodel, n_ahead = 12)
is.forecast_esn(xfcst)

Checks if object is of class "tune_esn"

Description

Returns TRUE if the object is of class tune_esn.

Usage

is.tune_esn(object)

Arguments

object

object to be tested.

Value

Logical value. If TRUE, the object is of class tune_esn.

See Also

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

Examples

xdata <- as.numeric(AirPassengers)
fit <- tune_esn(
  y = xdata,
  n_ahead = 12,
  n_split = 5,
  alpha = c(0.5, 1),
  rho   = c(1.0),
  tau   = c(0.4),
  inf_crit = "bic"
)
is.tune_esn(fit)

Deprecated name for m4_monthly_subset

Description

m4_data has been renamed to m4_monthly_subset. The old name is retained for backward compatibility with earlier versions of echos, but users should prefer m4_monthly_subset in new code.

Usage

data(m4_data)

Format

See m4_monthly_subset.

See Also

m4_monthly_subset

Examples

data(m4_data)
head(m4_data)

Monthly subset of the M4 competition data

Description

A tsibble with six monthly time series from the M4 Forecasting Competition. The dataset is intended for examples, documentation, and tests in the echos package. It is not the full M4 dataset.

The dataset contains the following time series:

  • M21655 (Demographic), 1995 Jan - 2015 Mar

  • M21683 (Demographic), 2000 Jan - 2023 Apr

  • M2717 (Macro), 1996 Jan - 2016 Nov

  • M28597 (Industry), 1996 Jan - 2016 Dec

  • M42529 (Finance), 2001 Jan - 2009 Apr

  • M4813 (Macro), 1994 Apr - 2006 May

Usage

data(m4_monthly_subset)

Format

A time series object of class tsibble with 1,152 rows and 4 columns:

  • series: Unique identifier as character (key variable).

  • category: Category, e.g. Demographic or Macro, as factor.

  • index: Monthly date as yearmonth (index variable).

  • value: Observed value as numeric (measurement variable).

Details

This dataset contains a small monthly subset of the M4 Forecasting Competition data. It is included to demonstrate the tidy forecasting interface of echos with tsibble, fable, and fabletools.

Source

M4 Forecasting Competition

Examples

data(m4_monthly_subset)
head(m4_monthly_subset)

Model specification of a trained ESN model

Description

Provides a compact overview of the model specification in the format ESN({n_states, alpha, rho}, {n_models, df}).

Usage

## S3 method for class 'ESN'
model_sum(x)

Arguments

x

An object of class mdl_df, containing an ESN model.

Value

Model summary extracted from the object.

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), report.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value))

Plot internal states of a trained ESN model

Description

Plot internal states (i.e., the reservoir) of a trained ESN model as line chart.

Usage

## S3 method for class 'esn'
plot(x, ...)

Arguments

x

An object of class esn. The result of a call to train_esn().
...

Further arguments passed to matplot().

Value

Line chart of internal states.

See Also

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

Examples

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

Plot forecasts of a trained ESN model

Description

Plot point forecasts and forecast intervals, actual values of a trained ESN model. Optionally, test data (out-of-sample) and fitted values can be added to the plot.

Usage

## S3 method for class 'forecast_esn'
plot(x, test = NULL, fitted = TRUE, interval = TRUE, n_obs = NULL, ...)

Arguments

x

An object of class forecast_esn. The result of a call to forecast_esn().
test

Numeric vector. Test data, i.e., out-of-sample actual values.
fitted

Logical value. If TRUE, fitted values are added.
interval

Logical value. If TRUE, forecast intervals are added.
n_obs

Integer value. If NULL, all in-sample values are shown, otherwise only the last n_obs.
...

Further arguments passed to plot().

Value

Line chart of point forecast and actual values.

See Also

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

Examples

xdata <- as.numeric(AirPassengers)
xmodel <- train_esn(y = xdata)
xfcst <- forecast_esn(xmodel, n_ahead = 12)
plot(xfcst)

Plot forecasts from a tuned ESN object

Description

Plot actual values and the point forecasts from the best hyperparameter combination selected via tune.esn() using the selected accuracy metric. Forecasts are shown as separate line segments for each test split, with vertical dashed lines marking the starts of test windows.

Usage

## S3 method for class 'tune_esn'
plot(x, metric = "mse", ...)

Arguments

x

An object of class tune_esn. The result of a call to tune_esn().
metric

Character value. The metric used to select the best hyperparameter combination (metric = c("mse", "mae").
...

Further arguments passed to plot().

Value

Line chart of point forecast and actual values.

See Also

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

Examples

xdata <- as.numeric(AirPassengers)
fit <- tune_esn(
  y = xdata,
  n_ahead = 12,
  n_split = 5,
  alpha = c(0.5, 1),
  rho   = c(1.0),
  tau   = c(0.4),
  inf_crit = "bic"
)

plot(fit)

Print model specification of the trained ESN model

Description

Provides a compact overview of the model specification in the format ESN({n_states, alpha, rho}, {n_models, df}).

Usage

## S3 method for class 'esn'
print(x, ...)

Arguments

x

An object of class esn. The result of a call to train_esn().
...

Currently not in use.

Value

Print specification of the trained ESN model.

See Also

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

Examples

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

Provide a detailed summary of the trained ESN model

Description

Provide a detailed summary of the trained ESN model. The function is a wrapper for summary.esn().

Usage

## S3 method for class 'ESN'
report(object, ...)

Arguments

object

An object of class mdl_df, containing an ESN model.
...

Currently not in use.

Value

Print detailed model summary.

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), reservoir(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
report()

Return the reservoir from a trained ESN as tibble

Description

Return the reservoir (internal states) from a trained ESN as tibble. The function works only for models of class ESN.

Usage

reservoir(object)

Arguments

object

An object of class mdl_df, containing an ESN model.

Value

A tibble containing the reservoir (internal states).

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), residuals.ESN(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
reservoir()

Extract residuals from a trained ESN

Description

Extract residuals from a trained ESN as tsibble.

Usage

## S3 method for class 'ESN'
residuals(object, ...)

Arguments

object

An object of class mdl_df, containing an ESN model.
...

Currently not in use.

Value

Residuals extracted from the object.

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), tidy.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
residuals()

Run reservoir

Description

Run reservoir creates the internal states for the ESN.

Arguments

input

Numeric matrix containing the input features
win

Numeric matrix. The input weight matrix.
wres

Numeric matrix. The reservoir weight matrix.
alpha

Numeric value. The leakage rate (smoothing parameter).

Value

states train Numeric matrix with the internal states.

Provide a detailed summary of the trained ESN model

Description

Provide a detailed summary of the trained ESN model.

Usage

## S3 method for class 'esn'
summary(object, ...)

Arguments

object

An object of class esn. The result of a call to train_esn().
...

Currently not in use.

Value

Print detailed model summary.

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.tune_esn(), train_esn(), tune_esn()

Examples

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

Provide a summary of the hyperparameter tuning

Description

Provide a summary of the tuned hyperparameters alpha, rho and tau.

Usage

## S3 method for class 'tune_esn'
summary(object, metric = "mse", ...)

Arguments

object

An object of class tune_esn. The result of a call to tune_esn().
metric

Character value. The metric used to select the best hyperparameter combination (metric = c("mse", "mae").
...

Currently not in use.

Value

Print detailed model summary.

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(), train_esn(), tune_esn()

Examples

xdata <- as.numeric(AirPassengers)
fit <- tune_esn(
  y = xdata,
  n_ahead = 12,
  n_split = 5,
  alpha = c(0.5, 1),
  rho   = c(1.0),
  tau   = c(0.4),
  inf_crit = "bic"
)

summary(fit)

Synthetic time series data

Description

A tibble with ten synthetic univariate time series. The dataset is intended for examples, documentation, and tests in the echos package. It contains deterministic wave patterns as well as stochastic benchmark processes.

The dataset contains the following time series:

  • Square Wave

  • Sawtooth Wave

  • Harmonic Wave

  • Harmonic Wave w/ Trend

  • Amplitude Modulated Wave

  • Frequency Modulated Wave

  • AR(1) Process

  • MA(2) Process

  • White Noise Process

  • Random Walk Process

Usage

data(synthetic_data)

Format

An object of class tibble with 2,000 rows and 3 columns:

  • variable: Unique identifier of the synthetic time series as character.

  • index: Time index as integer.

  • value: Observed value as numeric.

Details

Each synthetic series contains 200 observations. The deterministic series provide examples of common nonlinear and periodic patterns, while the stochastic series provide simple autoregressive, moving-average, white-noise, and random-walk benchmark processes.

Examples

data(synthetic_data)
head(synthetic_data)

Estimated coefficients

Description

Return the estimated coefficients from a trained ESN as tibble.

Usage

## S3 method for class 'ESN'
tidy(x, ...)

Arguments

x

An object of class mdl_df, containing an ESN model.
...

Currently not in use.

Value

Coefficients extracted from the object.

See Also

Other tidy functions: ESN(), filter_esn(), fitted.ESN(), forecast.ESN(), glance.ESN(), model_sum.ESN(), report.ESN(), reservoir(), residuals.ESN()

Examples

library(tsibble)
library(fable)
AirPassengers %>%
as_tsibble() %>%
model("ESN" = ESN(value)) %>%
tidy()

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)
)

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)).

Details

The implemented Echo State Network consists of an input layer, a randomly generated recurrent reservoir, and a linear readout layer. The input and reservoir weight matrices are randomly initialized and then kept fixed. Only the readout weights are estimated, using ridge regression.

Before the ESN is trained, the input series is optionally differenced using n_diff; if n_diff = NULL, the number of differences is selected automatically using a KPSS-based procedure. The transformed series is scaled to scale_inputs. Lagged values specified by lags are used as inputs to the reservoir.

If n_states = NULL, the reservoir size is set to min(floor(n_total * tau), 200), where n_total is the length of the input series. If n_models = NULL, n_states * 2 candidate readout models are estimated. For these candidate models, regularization parameters are drawn from the interval specified by lambda, and the best model is selected according to inf_crit.

The most influential ESN hyperparameters are n_states, alpha, rho, density, scale_win, scale_wres, and lambda. Larger reservoirs can represent richer dynamics but increase runtime and overfitting risk. The leakage rate alpha controls the speed of reservoir state updates. The spectral radius rho affects the memory and stability of the reservoir. The density controls the sparsity of recurrent connections. The lambda interval controls the amount of ridge regularization considered during readout selection.

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)

Tune hyperparameters of an Echo State Network

Description

Tune hyperparameters of an Echo State Network (ESN) based on time series cross-validation (i.e., rolling forecast). The input series is split into n_split expanding-window train/test sets with test size n_ahead. For each split and each hyperparameter combination (alpha, rho, tau) an ESN is trained via train_esn() and forecasts are generated via forecast_esn().

Usage

tune_esn(
  y,
  n_ahead = 12,
  n_split = 5,
  alpha = seq(0.1, 1, by = 0.1),
  rho = seq(0.1, 1, by = 0.1),
  tau = c(0.1, 0.2, 0.4),
  min_train = NULL,
  ...
)

Arguments

y

Numeric vector containing the response variable (no missing values).
n_ahead

Integer value. The number of periods for forecasting (i.e. forecast horizon).
n_split

Integer value. The number of rolling train/test splits.
alpha

Numeric vector of candidate leakage rates. Smaller values produce slower reservoir state updates; larger values make the reservoir react more strongly to recent inputs.
rho

Numeric vector of candidate spectral radii. This parameter scales the recurrent reservoir weights and affects reservoir memory and stability.
tau

Numeric vector of candidate reservoir scaling values. Used to determine the reservoir size when n_states = NULL in train_esn().
min_train

Integer value. Minimum training sample size for the first split.
...

Further arguments passed to train_esn() (except alpha, rho, and tau, which are set by the tuning grid).

Details

tune_esn() performs grid-based hyperparameter tuning using expanding-window time series cross-validation. The tuning grid is formed from all combinations of alpha, rho, and tau. For each split, the model is trained on observations from the beginning of the series up to the split-specific training endpoint and evaluated on the following n_ahead observations.

For every candidate configuration and split, train_esn() is called with the corresponding alpha, rho, and tau; all other arguments supplied through ... are passed to train_esn(). Forecasts are generated with forecast_esn(), and the mean squared error (mse) and mean absolute error (mae) are stored. The accompanying summary() and plot() methods use these stored errors to identify and display the best-performing hyperparameter configuration.

Runtime increases approximately linearly with the number of grid combinations and the number of validation splits. Users should start with coarse grids and increase the grid resolution only where needed.

Value

An object of class "tune_esn" (a list) with:

  • pars: A tibble with one row per hyperparameter combination and split. Columns include alpha, rho, tau, split, train_start, train_end, test_start, test_end, mse, mae, and id.

  • fcst: A numeric matrix of point forecasts with nrow(fcst) == nrow(pars) and ncol(fcst) == n_ahead.

  • actual: The original input series y (numeric vector), returned for convenience.

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(), train_esn()

Examples

xdata <- as.numeric(AirPassengers)
fit <- tune_esn(
  y = xdata,
  n_ahead = 12,
  n_split = 5,
  alpha = c(0.5, 1),
  rho   = c(1.0),
  tau   = c(0.4),
  inf_crit = "bic"
)
summary(fit)
plot(fit)