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3.12 Probability-related functions

3.12.1 Normal cumulative distribution functions

The error function erf is related to the standard normal cumulative distribution function \(\Phi\) by scaling. See section normal distribution for the general normal cumulative distribution function (and its complement).

R erf(T x)
error function, also known as the Gauss error function, of x
Available since 2.0, vectorized in 2.13

R erfc(T x)
complementary error function of x
Available since 2.0, vectorized in 2.13

R Phi(T x)
standard normal cumulative distribution function of x
Available since 2.0, vectorized in 2.13

R inv_Phi(T x)
Return the value of the inverse standard normal cdf \(\Phi^{-1}\) at the specified quantile x. The details of the algorithm can be found in (Wichura 1988). Quantile arguments below 1e-16 are untested; quantiles above 0.999999999 result in increasingly large errors.
Available since 2.0, vectorized in 2.13

R Phi_approx(T x)
fast approximation of the unit (may replace Phi for probit regression with maximum absolute error of 0.00014, see (Bowling et al. 2009) for details)
Available since 2.0, vectorized in 2.13

3.12.2 Other probability-related functions

real binary_log_loss(int y, real y_hat)
Return the log loss function for for predicting \(\hat{y} \in [0,1]\) for boolean outcome \(y \in \{0,1\}\). \[ \mathrm{binary\_log\_loss}(y,\hat{y}) = \begin{cases} -\log \hat{y} & \text{if } y = 0\\ -\log (1 - \hat{y}) & \text{otherwise} \end{cases} \]
Available since 2.0

R binary_log_loss(T1 x, T2 y)
Vectorized implementation of the binary_log_loss function
Available since 2.25

real owens_t(real h, real a)
Return the Owen’s T function for the probability of the event \(X > h\) and \(0<Y<aX\) where X and Y are independent standard normal random variables. \[ \mathrm{owens\_t}(h,a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]
Available since 2.25

R owens_t(T1 x, T2 y)
Vectorized implementation of the owens_t function
Available since 2.25

References

Bowling, Shannon R., Mohammad T. Khasawneh, Sittichai Kaewkuekool, and Byung Rae Cho. 2009. “A Logistic Approximation to the Cumulative Normal Distribution.” Journal of Industrial Engineering and Management 2 (1): 114–27.

Wichura, Michael J. 1988. “Algorithm AS 241: The Percentage Points of the Normal Distribution.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 37 (3): 477–84. http://www.jstor.org/stable/2347330.