Table 2 Competing criterion functions in the existing literature
From: An RMT-based generalized Bayesian information criterion for signal enumeration
Criterion | Noise type | Function |
|---|---|---|
BN-AIC | Gaussian | \(\hat{k} = \arg \min \limits _{k}(m - k) n \log \frac{\frac{1}{m-k}\sum ^m_{i =k+1}{\hat{\lambda }}_{i}}{\prod ^m_{i = k+1}{\hat{\lambda }}_i^{1/(m-k)}} + 2Ck\big (m + 1 - \frac{k + 1}{2}\big )\) |
\(C = 2 + 0.001 \times \log n\) | ||
RMT-AIC | Gaussian | \(\hat{k} = \arg \min \limits _{k}\frac{n^2}{2} \bigg [\frac{(m - k) \sum ^m_{i =k+1}{\hat{\lambda }}_i^2}{\big ( \sum ^m_{i =k+1}{\hat{\lambda }}_i \big )^2} - \big (1 + \frac{m}{n}\big )\bigg ]^2 + 2(k + 1)\) |
BIC-Variant | Gaussian | \(\begin{aligned} \hat{k} &= \arg \min \limits _{k}2n(m - k) \log \frac{\frac{1}{m-k}\sum ^m_{i =k + 1}{\hat{\lambda }}_i}{\prod ^m_{i = k + 1}{\hat{\lambda }}_i^{1/(m-k)}} + mk\log (2n) \\ & \quad- m\sum ^k_{i = 1}\log \frac{{\hat{\lambda }}_i}{\frac{1}{m-k}\sum ^m_{i =k+1}{\hat{\lambda }}_i} \end{aligned}\) |
EEE | Gaussian non-Gaussian | \(\begin{aligned} \hat{k} & = \arg \min \limits _{k}\frac{-1}{m - k}\sum ^m_{j = k + 1}\log \big (\frac{1}{m - k}\sum ^m_{i = k + 1}K_G({\hat{\lambda }}_j - {\hat{\lambda }}_i)\big ) \\ & \quad- + \frac{1}{m - k + 1}\sum ^M_{j = k}\log \Big (\frac{1}{m - k +1}\sum ^M_{i = k}K_G({\hat{\lambda }}_j -{\hat{\lambda }}_i)\Big ) \end{aligned}\) |
where \(K_G(x) = \frac{1}{\sqrt{2\pi }} e^{-\frac{x^2}{2}}\) | ||
GBIC-SP | Gaussian non-Gaussian | \(\hat{k} = \arg \min \limits _{k}\frac{(n-1)^2}{4}({\hat{T}}^{(k)} - 1)^2 + (k + 1)\log n\) |
where \({\hat{T}}^{(k)}\) is the test statistic for sphericity (see (13) in [21]) |