comyx.stats.metrics#
Module Summary#
Functions#
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Computes the ergodic rate of the system. |
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Computes the probability of inter-related SNRs. |
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Computes the probability of the received SNR being less than the threshold. |
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Computes the outage probability of the system using the Q-function. |
Reference#
- comyx.stats.metrics.get_ergodic_rate(k: float, m: float, theta: float, omega: float) float[source]#
Computes the ergodic rate of the system.
\[\begin{split}\mathcal{R(k,m,\theta,\Omega)}=\frac{1}{{\log (2) \Gamma (k+m) B(k,m)}}{G_{3,3}^{3,2}\left(\frac{\Omega }{\theta }|\begin{array}{c}0,1-m,1 \\0,0,k \\\end{array}\right)}\end{split}\]- Parameters:
k – Shape parameter of the numerator Gamma distribution.
m – Shape parameter of the denominator Gamma distribution.
theta – Scale parameter of the numerator Gamma distribution.
omega – Scale parameter of the denominator Gamma distribution.
- Returns:
The ergodic rate of the system.
- comyx.stats.metrics.get_outage_clt(k_a: float, m_a: float, theta_a: float, omega_a: float, k_b: float, m_b: float, theta_b: float, omega_b: float, lambda_a: float, lambda_b: float) float[source]#
Computes the probability of inter-related SNRs.
More specifically, it computes the probability of SNRs being greater than one threshold, but less than another.
\[\begin{split}Pr(\lambda_{a}\gt\lambda_{th}, \lambda_{b}\lt\gamma_{th})=\frac{1}{k_b \Gamma\left(m_a\right) B\left(k_b,m_b\right)}{\left(\frac{10^{\gamma /10} \Omega _b}{\theta_b}\right){}^{k_b} {_2F_1\left(k_b,k_b+m_b;k_b+1;-\frac{10^{\gamma /10} \Omega_b}{\theta _b}\right)}} \\ {\left(\Gamma \left(m_a\right)-\Gamma\left(k_a+m_a\right) \left(\frac{10^{\lambda /10} \Omega_a}{\theta _a}\right){}^{k_a} {_2\tilde{F}_1\left(k_a,k_a+m_a;k_a+1;-\frac{10^{\lambda /10}\Omega _a}{\theta _a}\right)}\right)}\end{split}\], where \(\lambda_{a}=10\ln(x)\), with \(x \sim \beta'(k_a, m_a, \theta_a / \Omega_a)\) and \(\lambda_{b}=10\ln(y)\), with \(y \sim \beta'(k_b, m_b, \theta_b / \Omega_b)\).
- Parameters:
k_a – Shape parameter of the numerator Gamma distribution of lambda_a.
m_a – Shape parameter of the denominator Gamma distribution of lambda_a.
theta_a – Scale parameter of the numerator Gamma distribution of lambda_a.
omega_a – Scale parameter of the denominator Gamma distribution of lambda_a.
k_b – Shape parameter of the numerator Gamma distribution of lambda_b.
m_b – Shape parameter of the denominator Gamma distribution of lambda_b.
theta_b – Scale parameter of the numerator Gamma distribution of lambda_b.
omega_b – Scale parameter of the denominator Gamma distribution of lambda_b.
lambda_a – First threshold of the received SNR.
lambda_b – Second threshold of the received SNR.
- Returns:
The outage probability of the system.
- comyx.stats.metrics.get_outage_lt(k: float, m: float, theta: float, omega: float, lambda_th: float) float[source]#
Computes the probability of the received SNR being less than the threshold.
\[Pr(\lambda_{r}\lt\lambda_{th})=\frac{1}{{k B(k,m)}}{\left(\frac{10^{\lambda_{th} /10} \Omega}{\theta }\right)^k{_2F_1\left(k,k+m;k+1;-\frac{10^{\lambda_{th} /10} \Omega }{\theta}\right)}}\], where \(\lambda_{r}=10\ln(x)\), with \(x \sim \beta'(k, m, \theta / \Omega)\).
- Parameters:
k – Shape parameter of the numerator Gamma distribution.
m – Shape parameter of the denominator Gamma distribution.
theta – Scale parameter of the numerator Gamma distribution.
omega – Scale parameter of the denominator Gamma distribution.
lambda_th – Threshold of the received SNR.
- Returns:
The outage probability of the system.
- comyx.stats.metrics.get_outage_q(Pr: NDArrayFloat, threshold: float) NDArrayFloat[source]#
Computes the outage probability of the system using the Q-function.
The Q-function is defined as:
\[Q(x)=\frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{-\frac{u^{2}}{2}} d u\]- Parameters:
Pr – The received power of the system.
threshold – The threshold of the received power.
- Returns:
An array containing the outages of the system.