Theoretical Background¶

The Spectral Kurtosis (SK) estimator is a powerful statistical tool for detecting non-Gaussian features in time-frequency data, such as radio-frequency interference (RFI) or transient signals.
This section summarizes the theoretical framework behind the Generalized Spectral Kurtosis (GSK) formulation, following the derivations presented in Nita & Gary (2010, PASP 122, 595–607).

1. Definition¶

For a sequence of complex spectral power measurements \( P_i \) (with \( i = 1, \ldots, M \)), the Spectral Kurtosis estimator is defined as

\[ \widehat{S}_K = \frac{M+1}{M-1} \left( \frac{M \sum_i P_i^2}{\left( \sum_i P_i \right)^2} - 1 \right). \]

This dimensionless quantity has an expected value of 1 for purely Gaussian signals and deviates from unity when non-Gaussian components (such as RFI or bursts) are present.

2. Generalized Formulation¶

The Generalized Spectral Kurtosis (GSK) estimator extends the above definition to the case where power estimates are averaged over \( N \) independent spectra, each derived from \( M \) accumulations:

\[ \widehat{S}_K(M, N, d) = \frac{M N d + 1}{M N d - 1} \left( \frac{M N \, S_2}{S_1^2} - 1 \right), \]

where:

\( S_1 = \sum_{j=1}^N P_j \) — the sum of averaged powers
\( S_2 = \sum_{j=1}^N P_j^2 \) — the sum of squared averaged powers
\( d \) — the shape parameter of the underlying Gamma distribution (for complex data, \( d = 1 \))

The generalized form reduces to the classical SK when \( N = 1 \) and \( d = 1 \).

3. Statistical Properties¶

Under Gaussian statistics, the power estimates \( P_j \) follow a Gamma distribution with shape parameter \( M d \) and scale parameter \( \theta \).
In this case:

\[ \mathbb{E}[\widehat{S}_K] = 1, \qquad \mathrm{Var}[\widehat{S}_K] = \frac{4}{M N (M N d + 1)}. \]

These relationships show that increasing \( M \) or \( N \) reduces the estimator variance, improving the reliability of SK-based detection.

4. Thresholds and Detection¶

The SK statistic follows an approximately scaled Gamma distribution, allowing the determination of probability-of-false-alarm (PFA) thresholds for detecting deviations from Gaussianity.

Given a target \( \mathrm{PFA} \) (e.g., \( 10^{-3} \)), one can define two-sided thresholds \( [S_{K,\mathrm{low}}, S_{K,\mathrm{high}}] \) such that:

\[ P(S_K < S_{K,\mathrm{low}}) = P(S_K > S_{K,\mathrm{high}}) = \frac{\mathrm{PFA}}{2}. \]

Values of \( S_K \) lying outside this interval indicate statistically significant departures from Gaussian noise.

The pyGSK package computes these thresholds using direct integration of the Gamma distribution and provides one- or two-sided options.

5. Renormalized SK Estimator¶

Finite-sample bias can lead to slight deviations of the SK mean from unity.
To correct for this, the renormalized estimator is defined as:

\[ \widehat{S}_K^\mathrm{(R)} = \frac{\widehat{S}_K}{\mathbb{E}[\widehat{S}_K]}, \]

ensuring \( \mathbb{E}[\widehat{S}_K^\mathrm{(R)}] = 1 \) for Gaussian inputs.
This form is implemented in the CLI command renorm-sk-test.

6. Practical Interpretation¶

\( \widehat{S}_K \approx 1 \): data consistent with Gaussian noise
\( \widehat{S}_K > S_{K,\mathrm{high}} \): impulsive or non-stationary power excess (e.g., RFI burst)
\( \widehat{S}_K < S_{K,\mathrm{low}} \): power deficit or coherence (e.g., sinusoidal contamination)

Thus, SK provides an adaptive, non-parametric detection statistic independent of absolute power levels.

7. References¶

Nita, G. M., & Gary, D. E. (2010). The generalized spectral kurtosis estimator. MNRAS Letters, 406(1), L60–L64. https://doi.org/10.1111/j.1745-3933.2010.00882.x

Next Steps¶

Learn to use the CLI tools in cli_guide.md
For implementation details, see dev_guide.md