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Signal Example — ICA 1: Matrix Math (U = WX) ​

Before introducing the geometric complexities of the Cocktail Party problem or trying to optimize statistical functions, we must define exactly what an "Unmixing Matrix" technically is.

This is Part 1 of the ICA series. It provides the most fundamental mathematical prerequisite for understanding component analysis.

The Mechanism of   ​

In the Independent Component Analysis equation: ​

$$\mathbf{X}$$
is your block of EEG data, where each row is an electrode.

  • is the Unmixing Matrix ICA wants to find.

  • is the resulting independent components.

Many people assume this is an abstract equation, but it is actually a physical machine driven by Matrix Multiplication.

Matrix multiplication works column-by-column across time. At a single timepoint , we can take a "snapshot" of the data as a single column vector. We multiply the Rows of the unmixing matrix by that Column of data (taking the dot product) to calculate the resulting component outputs for that exact instant in time.

The Interactive Sandbox ​

This demonstration physicalizes this exact "Row-by-Column" mechanism using a larger,   data architecture (3 EEG channels, 3 extracted components).

Instead of watching a time-series play out automatically, we have frozen time. Using the Time Cursor slider, you manually select a single timepoint . At the top of the screen, you will see a massive, live Matrix Math execution board:

text
[ W_11  W_12  W_13 ]       [ X_1(t) ]       [ (W_11*X_1) + (W_12*X_2) + (W_13*X_3) ]       [ U_1(t) ]
                   ×                  =                                            =  
[ W_21  W_22  W_23 ]       [ X_2(t) ]       [ (W_21*X_1) + (W_22*X_2) + (W_23*X_3) ]       [ U_2(t) ]
                   ×                  =                                            =
[ W_31  W_32  W_33 ]       [ X_3(t) ]       [ (W_31*X_1) + (W_32*X_2) + (W_33*X_3) ]       [ U_3(t) ]

  1. Watch the inputs: As you drag the time cursor across the raw EEG waveforms, the 3 values in the column vector instantly update.

  2. Watch the math: The center panel actively calculates the cross-multiplication across all 3 variables.

  3. Watch the output: The resulting component numbers pop out on the right.

  4. Draw the wave: Those exact numbers are structurally plotted as the newest dots on the right-hand graphs.

By dragging the Time Cursor left-to-right, you are literally executing matrix multiplication frame-by-frame, effectively "drawing" the resulting Component wave yourself.

Experiment ​

You can click and drag the numbers inside the   matrix to change their weight. Change to see how it forces the math board to multiply differently, instantly altering the final drawn component wave.

Code ​

julia
using EegFun
EegFun.signal_example_ica_math()