pwm.beta2alpha: Conversion of Beta to Alpha Probability-Weighted Moments (PWMs) or Alpha to Beta PWMs
Description
Conversion of “beta” (the well known ones) to “alpha” probability-weighted moments (PWMs) by pwm.beta2alpha or alpha to beta PWMs by pwm.alpha2beta. The relations between the \(\alpha\) and \(\beta\) PWMs are
$$\alpha_r = \sum^r_{k=0} (-1)^k {r \choose k} \beta_k\mbox{,}$$
and
$$\beta_r = \sum^r_{k=0} (-1)^k {r \choose k} \alpha_k\mbox{.}$$
Lastly, note that the \(\beta\) are almost exclusively used in the literature. Because each is a linear combination of the other, they are equivalent in meaning but not numerically.
Usage
pwm.beta2alpha(pwm)
pwm.alpha2beta(pwm)
Value
If \(\beta_r \rightarrow \alpha_r\) (pwm.beta2alpha), a vector of the \(\alpha_r\). Note that convention is the have a \(\alpha_0\), but this is placed in the first index i=1 vector. Alternatively, if \(\alpha_r \rightarrow \beta_r\) (pwm.alpha2beta), a vector of the \(\beta_r\).
Arguments
pwm
A vector of alpha or beta probability-weighted moments depending on which related function is called.