Signal Example — ICA 4: Blind Source Separation

Interactive demonstration of Independent Component Analysis (ICA) — 3 sources, rotation geometry, and scatter plots.

This is Part 4 of the ICA series. Start with [Part 2: What Is a Mixture?](signal_example_ica_mixture.md) or [Part 3: Mixing & Unmixing](signal_example_ica_separation.md) first.

What it shows

Feature	Description
True Sources (Row 1)	Three independent signals: S1 (oscillatory bursts, Gabor-shaped), S2 (blink artifacts), S3 (muscle burst)
EEG Recordings (Row 2)	The mixed signals as they would appear at electrodes — each is a weighted combination of all sources
Recovered Components (Row 3)	The unmixed signals, either via manual rotation sliders or the Auto-ICA (Infomax) algorithm
Scatter Plots	Three pairwise joint distributions (S1 vs S2, S1 vs S3, S2 vs S3) — see below
Excess Kurtosis	Displayed for sources and recovered components

The Scatter Plots

The scatter plots are the most informative panel:

• Independent, non-Gaussian signals → distinctive cross or T-shape (the signals are statistically independent)

• Mixed signals → correlated blob (everything jumbled together)

• Correctly unmixed → cross shape restored

Things to Try

1. Set all mixing angles > 0° and watch the scatter plots collapse from crosses into blobs.

2. Drag the unmixing sliders manually to match the mixing angles — the sources reappear.

3. Reset unmixing angles to 0°, then click Infomax — ICA automatically recovers the sources.

4. Increase the Noise slider — Infomax still works because the blink and muscle sources are highly non-Gaussian.

5. Compare manual rotation with Infomax — Infomax finds a general unmixing matrix, not just a rotation.

6. Try Extended Infomax, which handles both sub-Gaussian and super-Gaussian sources.

Controls

Control	Range	Description
Mix α (S1-S2)	0–90°	Rotation angle blending S1 and S2 into the recordings
Mix β (S1-S3)	0–90°	Rotation angle blending S1 and S3 into the recordings
Mix γ (S2-S3)	0–90°	Rotation angle blending S2 and S3 into the recordings
Unmix φ (S1-S2)	0–90°	Inverse rotation to recover S1/S2 (manual mode)
Unmix ψ (S1-S3)	0–90°	Inverse rotation to recover S1/S3 (manual mode)
Unmix χ (S2-S3)	0–90°	Inverse rotation to recover S2/S3 (manual mode)
Burst Freq	1–20 Hz	Frequency of the oscillatory source (S1)
Noise	0–1	Additive Gaussian noise on all sources
Infomax	—	Run standard Infomax ICA (Bell & Sejnowski, 1995)
Extended	—	Run Extended Infomax (Lee et al., 1999)

Under the Hood — The Mathematics (click to expand)

The Mixing Model

With N sources and N sensors, mixing is an N×N matrix multiplication. For 3 sources, we compose three pairwise 2D rotations (α, β, γ):

Mixing: R₂₃(γ) · R₁₃(β) · R₁₂(α) · S Unmixing: R₁₂ᵀ(φ) · R₁₃ᵀ(ψ) · R₂₃ᵀ(χ) · M

When φ=α, ψ=β, χ=γ → perfect recovery.

Why Non-Gaussianity?

ICA exploits the Central Limit Theorem in reverse: mixtures of independent signals are more Gaussian than the original signals. So finding the unmixing that maximises non-Gaussianity (measured by kurtosis or negentropy) recovers the independent sources.

• Excess kurtosis = 0: Gaussian distribution

• Excess kurtosis > 0: Super-Gaussian (sharp peaks, heavy tails — like blink spikes)

• Excess kurtosis < 0: Sub-Gaussian (flat-topped distributions)

The Infomax algorithm (Bell & Sejnowski, 1995) used in EegFun maximises the total non-Gaussianity of the output components.

See Also

• Part 1: Matrix Math Basics — What is an Unmixing Matrix?

• Part 2: What Is a Mixture? — Simplest introduction to the problem

• Part 3: Mixing & Unmixing — 2-source introduction to ICA

• ICA: Central Limit Theorem — Deep Dive: Why mixing creates Gaussian blobs

• Part 5: Sphering (Whitening) — Why ICA optimization is just finding an angle

• Part 6: Inside the Black Box — The optimization landscape and gradient ascent

• ICA Demo — Full ICA workflow on real EEG data

• EEGLAB: Running ICA — Practical guide to ICA in EEGLAB

• ICA for Dummies — Arnaud Delorme's accessible introduction to ICA

• Bell, A. J., & Sejnowski, T. J. (1995). An information-maximization approach to blind separation. Neural Computation, 7 (6), 1129–1159.

• Hyvärinen, A., & Oja, E. (2000). Independent component analysis: algorithms and applications. Neural Networks, 13 (4-5), 411–430.

Code

using EegFun
EegFun.signal_example_ica_geometry()