Signal complexity measures how intricate or unpredictable a signal (e.g., EEG) is over time. High complexity often reflects rich, non-linear dynamics, while low complexity may indicate periodicity or noise.
Key Metrics for EEG Complexity
Sample Entropy (SampEn):
Quantifies the unpredictability of EEG time series.
Lower values indicate more regularity (e.g., during seizures or deep sleep).
Fractal Dimension:
Describes the self-similarity of EEG signals across different time scales.
Higher values suggest greater complexity (e.g., during cognitive tasks).
Lempel-Ziv Complexity (LZC):
Estimates the rate at which new patterns emerge in the signal.
Useful for detecting changes in brain states (e.g., transition from wakefulness to sleep).
Multiscale Entropy (MSE):
Measures complexity across multiple time scales.
Helps capture both fast and slow EEG dynamics.
Signal similarity assesses how alike two EEG signals or segments are, often used for comparing brain states, subjects, or conditions.
Key Metrics for EEG Similarity
Correlation Coefficient:
Measures linear relationship between two EEG signals.
Example: Pearson’s rrr or Spearman’s ρ.
Dynamic Time Warping (DTW):
Aligns EEG signals in time to compare their shapes, even if they are misaligned or vary in speed.
Cross-Approximate Entropy (Cross-ApEn):
Quantifies the synchrony or asynchrony between two EEG signals.
Useful for studying brain connectivity.
Phase Locking Value (PLV):
Measures the consistency of phase differences between two EEG signals.
High PLV indicates strong phase synchronization (e.g., during cognitive tasks).
Euclidean Distance:
Computes the point-by-point difference between two signals.
Simple but sensitive to amplitude and time shifts.
Applications in EEG Analysis
Complexity:
Detecting pathological states (e.g., epilepsy, Alzheimer’s).
Monitoring depth of anesthesia or cognitive load.
Similarity:
Comparing EEG patterns across subjects or conditions.
Studying brain connectivity and functional networks.
Entropy and Negentropy
Entropy
Entropy is a measure of randomness, uncertainty, or disorder in a system or signal. In information theory, entropy quantifies the amount of information contained in a signal.
For EEG signals, entropy helps assess the complexity or predictability of brain activity.
Types of Entropy in EEG Analysis
Shannon Entropy:
Measures the average unpredictability in a signal.
Formula:
\[
H(X) = -\sum_{i} p(x_i) \log p(x_i)
\]
where \(p(x_i)\) is the probability of observing \(x_i\).
Sample Entropy (SampEn):
Estimates the likelihood that similar patterns in the EEG signal remain similar when extended by one sample.
Lower values indicate more regularity (e.g., during seizures or deep sleep).
Approximate Entropy (ApEn):
Similar to SampEn but includes self-matches, which can introduce bias.
Useful for short or noisy EEG segments.
Multiscale Entropy (MSE):
Extends entropy analysis across multiple time scales.
Captures both fast and slow dynamics in EEG signals.
Why Use Entropy in EEG?
Complexity Assessment: High entropy indicates complex, irregular brain activity (e.g., during cognitive tasks).
Pathology Detection: Low entropy may reflect abnormal regularity (e.g., epilepsy or anesthesia).
State Monitoring: Tracks changes in brain states (e.g., sleep stages, cognitive load).
In NeuroAnalyzer the following entropy descriptors are available:
ent: histogram-based entropy in bits (Freedman-Diaconis binning)
Entropy of the signal is calculated using its histogram bins (number of bins is calculated using the Freedman-Diaconis formula) using the formulas p = n / sum(n) and entropy = -sum(p .* log2(p)), where p is the probability of each bin and n are bins’ weights.
Shannon entropy and log energy entropy are calculated using Wavelets.coefentropy().
Sample and normalized sample entropy and log energy entropy are calculated using ComplexityMeasures.
Negentropy
Negentropy is a measure of non-Gaussianity or structure in a signal. It quantifies how much a signal deviates from a Gaussian (normal) distribution. ICA (Independent Component Analysis), negentropy is used as a contrast function to separate independent components.
Mathematical Definition
Negentropy \(J\) of a random variable \(Y\) is defined as:
\[
J(Y) = H(Y_{\text{Gaussian}}) - H(Y)
\]
where: \(H(Y)\) is the differential entropy of \(Y\), \(Y_{\text{Gaussian}}\) is a Gaussian random variable with the same covariance as \(Y\).
High negentropy: Signal is far from Gaussian (structured, non-random).
Low negentropy: Signal is close to Gaussian (random, unstructured).
Role of Negentropy in ICA
ICA aims to maximize negentropy to find independent components (ICs) in EEG signals.
Non-Gaussian components are more likely to represent meaningful sources (e.g., brain activity or artifacts).
Why Use Negentropy in EEG?
Source Separation: Helps isolate neural sources and artifacts in EEG signals.
Feature Extraction: Identifies structured, non-random patterns in brain activity.
Artifact Removal: Distinguishes between Gaussian noise and non-Gaussian artifacts (e.g., eye blinks).
Calculating negentropy, use sample entropy and do not normalize signal against its total energy:
ne =negentropy(e10; ch ="eeg", norm =false,type=:sample)plot_bar(ne[:, 1]; glabels =labels(eeg)[1:19], xlabel ="Channels", title ="Negentropy, epoch 1")
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Entropy vs. Negentropy
Feature
Entropy
Negentropy
Definition
Measure of randomness
Measure of non-Gaussianity
High Value
High randomness, unpredictability
High structure, non-randomness
Use in EEG
Assesses signal complexity
Separates independent sources
Application
Detects brain states, pathologies
ICA for artifact removal
Higuchi Fractal Dimension
The Higuchi fractal dimension is a method for measuring the fractal dimension of a time series, such as EEG signals. It quantifies the complexity and self-similarity of the signal, providing insights into its underlying dynamics.
It was introduced by Tomoyuki Higuchi in 1988 as a method to estimate the fractal dimension directly from time-series data.
Unlike other fractal dimension methods, HFD does not require phase-space reconstruction, making it computationally efficient and suitable for short and noisy time series.
Interpretation
Higher HFD (closer to 2): Indicates a more complex, irregular time series (e.g., chaotic or noisy signals).
Lower HFD (closer to 1): Indicates a more regular, smooth time series (e.g., periodic signals).
The Generalised Hurst Exponents (GHEs) is used as a measure of long-term memory of time series. It relates to the auto-correlations of the time series, and the rate at which these decrease as the lag between pairs of values increases.
The Hurst exponent is referred to as the “index of dependence” or “index of long-range dependence”. It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.
Interpretation
A value in the range 0.5–1 indicates a time series with long-term positive autocorrelation, meaning that the decay in autocorrelation is slower than exponential, following a power law; for the series it means that a high value tends to be followed by another high value and that future excursions to more high values do occur.
A value in the range 0–0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future, also following a power law.
A value of 0.5 indicates short-memory, with (absolute) auto-correlations decaying exponentially quickly to zero.
Summed similarity is a method for aggregating similarity scores across multiple items, features, or dimensions into a single value. It is commonly used in:
Signal processing (e.g., combining similarity across EEG channels or time points),
Cognitive neuroscience (e.g., measuring overall similarity between brain states),
Data mining (e.g., combining similarity scores across datasets).
Summed similarity is the sum of similarity scores between items, features, or dimensions. It provides a single scalar value representing the overall similarity across all comparisons.
Key Concepts
Concept
Description
Similarity Score
A measure of how similar two items are (e.g., correlation, cosine similarity, Euclidean distance).
Summed Similarity
The sum of all pairwise similarity scores across items or dimensions.
Aggregation
Combining multiple similarity scores into a single value.
Examples of Use Cases
Use Case
Description
EEG Signal Analysis
Compute the sum of pairwise correlations between EEG channels to measure overall connectivity.
Feature Similarity
Sum the cosine similarity between features to measure overall feature similarity.
Brain State Comparison
Sum the Euclidean distance between brain states to measure overall dissimilarity.
Time-Series Similarity
Sum the cross-correlation between time-series segments to measure overall similarity.
How to Calculate Summed Similarity
Calculating summed similarity using an exponential decay model between two signals:
Tip: Values of summed similarity are in the range [0, 1]; higher value indicates larger similarity.
Dirichlet Energy
Dirichlet energy is a mathematical concept used to measure the smoothness or variation of a function defined on a graph or manifold. In the context of EEG signal processing, it quantifies how much a signal changes across channels or time points, making it useful for:
Smoothing EEG signals,
Detecting artifacts or discontinuities,
Graph-based signal processing (e.g., EEG connectivity networks),
