complex corelation and p-value

To find the complex correlation and its p-value between two complex vectors.

The complex correlation coefficient is a measure of the relationship between two complex-valued datasets. It generalizes the Pearson correlation coefficient to handle the real and imaginary components of complex signals.

Let w1, w2 be two complex vectors.

w1 = u1+i*v1

w2 = u2+i*v2

Where:

• u1,v1: Real and imaginary parts of the vector w1.
• u2,v2: Real and imaginary parts of the vector w2.

The complex corelation between w1, w2 is igven by:

Formula:

The complex correlation ρ is given by:

Where:

• ⟨⋅⟩: Mean or expected value.

Interpretation of complex corelation ρ:

1. Magnitude Analysis: The magnitude of ρ, given by |ρ|, indicates the strength of the correlation between the two vectors:

If : The two signals are perfectly correlated (linear relationship exists).

If : The two signals are uncorrelated.

Intermediate values ()): Indicates partial correlation, where higher values suggest stronger correlation.

2. Phase Analysis: The phase of ρ, given by:

represents the phase shift between the two signals.

If : The signals are in phase (no delay).

If : Signal 2 leads Signal 1.

If : Signal 1 leads Signal 2.

Statistical Significance (p-value)

The statistical significance is calculated using Monte Carlo simulations.

If p-value < 0.05, the complex correlation ρ is statistically significant.

If p-value >= 0.05, the observed correlation might have occurred by chance.

Reference: Kundu, P. K. 1975. Ekman veering observed near the ocean bottom. J. Phys. Oceanogr. 6, 238-242.