sdf.test: Nonparametric Two Sample Test for Equality of Spectral Densities

Description

Nonparametric method for testing the equality of the spectral densities of two time series of possibly different lengths. The time series are preprocessed with the discrete cosine transform and the variance stabilising transform to obtain an approximate Gaussian regression setting for the log-spectral density function. The test statistic is based on the squared L2 norm of the difference between the estimated log-spectral densities. The test returns the result, the statistic value, and the p-value. It also provides the estimated empirical quantile and null distribution under the hypothesis of equal spectral densities. An example using EEG data is included. For details see Nadin, Krivobokova, Enikeeva (2026), doi:10.48550/arXiv.2602.10774.

Author(s)

Maintainer: Ilaria Nadin ilaria.nadin@gmail.com

Authors:

• Tatyana Krivobokova

• Farida Enikeeva

• Karolina Klockmann

EEG Signals

Description

A dataset containing four EEG recordings of a patient under rest or different transcranial Direct Current Stimulations (tDCS).

Usage

eeg_data

Format

A data frame with columns:

index

Integer index of the measurement (frequency of 250 Hz).

condition

Can take four values: "initial_rest", devtools::document() "first_frontal_tDCS", "second_frontal_tDCS", "posterior_tDCS".

signal

Numeric EEG measurement.

Source

Original: Alexis Pomares, "Dissertation preprocessed EEG dataset", https://www.kaggle.com/datasets/alexispomares/dissertation-preprocessed This version has been preprocessed by selecting specific conditions and removing outliers, for details see (Nadin, Kivobokova, Enikeeva; 2026).

Examples

data(eeg_data)
head(eeg_data)
tail(eeg_data)
str(eeg_data)
X1 <- subset(eeg_data, condition == "first_frontal_tDCS")$signal

Print test

Description

Custom printing function for the class "test".

Usage

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

Arguments

Value

Invisible x.

Examples

test <- sdf.test(
  X1 = rnorm(50), X2 = rnorm(50),
  q1 = 2, method1 = "GCV", Te = 10, alpha = 0.05, N = 1000
)
print(test)
test

SDF Test

Description

Performs the test for equality of spectral densities of two time series.

Usage

sdf.test(
  X1,
  X2,
  Te,
  alpha,
  q1 = 4,
  method1 = "GCV",
  f1_true = NULL,
  q2 = q1,
  method2 = method1,
  f2_true = f1_true,
  N = 10000,
  cores = 1,
  nulldistr = NULL
)

Arguments

Value

An object of class "test", which is a list containing the following components:

• result: Logical. TRUE if the null hypothesis of equality of spectral densities is accepted by the test, FALSE otherwise.

• S: Value of the statistic.

• quantile: Empirical quantile used for the test.

• p-value: p-value of the test according to the empirical null distribution.

• nulldistr: Vector of empirical null distribution.

References

Nadin, Krivobokova, Enikeeva (2026). Nonparametric two sample test of spectral densities. https://arxiv.org/abs/2602.10774

Examples

# --- Example 1: Using random data ---
test <- sdf.test(
  X1 = rnorm(100), X2 = rnorm(80),
  q1 = 4, method1 = "GCV", Te = 20, alpha = 0.05, N = 1000
)
test

# --- Example 2: Using random data ---

X1 <- arima.sim(list(order=c(1,0,0),ar=0.5),n = 1200,rand.gen = rnorm, sd = 1)
X2 <- arima.sim(list(order=c(1,0,0),ar=0.8),n = 1000,rand.gen = rnorm, sd = 1)
sdf.test(
  X1 = X1, X2 = X2,
  q1 = 4, method1 = "GCV", Te = 176, alpha = 0.05
)


# --- Example 3: Using EEG dataset ---

data(eeg_data)
X1 <- subset(eeg_data, condition == "first_frontal_tDCS")$signal
X2 <- subset(eeg_data, condition == "second_frontal_tDCS")$signal
test <- sdf.test(
  X1 = X1, X2 = X2,
  q1 = 4, method1 = "ML", Te = 2100, alpha = 0.05, cores = 50
)
test