The modified lognormal power-law (MLP) function is a three parameter function that can be used to model data that have characteristics of a log-normal distribution and a power law behavior. It has been used to model the functional form of the initial mass function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions.
Functional form
The closed form of the probability density function of the MLP is as follows:
![{\displaystyle {\begin{aligned}f(m)={\frac {\alpha }{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-(1+\alpha )}{\text{erfc}}\left({\frac {1}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{\sigma _{0}}}\right)\right),\ m\in [0,\infty )\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the asymptotic power-law index of the distribution. Here
and
are the mean and variance, respectively, of an underlying lognormal distribution from which the MLP is derived.
Mathematical properties
Following are the few mathematical properties of the MLP distribution:
Cumulative distribution
The MLP cumulative distribution function (
) is given by:
![{\displaystyle {\begin{aligned}F(m)={\frac {1}{2}}{\text{erfc}}\left(-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)-{\frac {1}{2}}\exp \left(\alpha \mu _{0}+{\frac {\alpha ^{2}\sigma _{0}^{2}}{2}}\right)m^{-\alpha }{\text{erfc}}\left({\frac {\alpha \sigma _{0}}{\sqrt {2}}}\left(\alpha \sigma _{0}-{\frac {\ln(m)-\mu _{0}}{{\sqrt {2}}\sigma _{0}}}\right)\right)\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
We can see that as
that
which is the cumulative distribution function for a lognormal distribution with parameters μ0 and σ0.
Mean, variance, raw moments
The expectation value of
k gives the
th raw moment of
,
![{\displaystyle {\begin{aligned}\langle M^{k}\rangle =\int _{0}^{\infty }m^{k}f(m)\mathrm {d} m\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
This exists if and only if α >
, in which case it becomes:
![{\displaystyle {\begin{aligned}\langle M^{k}\rangle ={\frac {\alpha }{\alpha -k}}\exp \left({\frac {\sigma _{0}^{2}k^{2}}{2}}+\mu _{0}k\right),\ \alpha >k\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which is the
th raw moment of the lognormal distribution with the parameters μ0 and σ0 scaled by α⁄α-
in the limit α→∞. This gives the mean and variance of the MLP distribution:
![{\displaystyle {\begin{aligned}\langle M\rangle ={\frac {\alpha }{\alpha -1}}\exp \left({\frac {\sigma _{0}^{2}}{2}}+\mu _{0}\right),\ \alpha >1\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle {\begin{aligned}\langle M^{2}\rangle ={\frac {\alpha }{\alpha -2}}\exp \left(2\left(\sigma _{0}^{2}+\mu _{0}\right)\right),\ \alpha >2\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Var(
) = ⟨
2⟩-(⟨
⟩)2 = α exp(σ02 + 2μ0) (exp(σ02)/α-2 - α/(α-2)2), α > 2
Mode
The solution to the equation
= 0 (equating the slope to zero at the point of maxima) for
gives the mode of the MLP distribution.
![{\displaystyle f'(m)=0\Leftrightarrow K\operatorname {erfc} (u)=\exp(-u^{2}),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
and ![{\displaystyle K=\sigma _{0}(\alpha +1){\tfrac {\sqrt {\pi }}{2}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Numerical methods are required to solve this transcendental equation. However, noting that if
≈1 then u = 0 gives us the mode
*:
![{\displaystyle m^{*}=\exp(\mu _{0}+\alpha \sigma _{0}^{2})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Random variate
The lognormal random variate is:
![{\displaystyle {\begin{aligned}L(\mu ,\sigma )=\exp(\mu +\sigma N(0,1))\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is standard normal random variate. The exponential random variate is :
![{\displaystyle {\begin{aligned}E(\delta )=-\delta ^{-1}\ln(R(0,1))\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where R(0,1) is the uniform random variate in the interval [0,1]. Using these two, we can derive the random variate for the MLP distribution to be:
![{\displaystyle {\begin{aligned}M(\mu _{0},\sigma _{0},\alpha )=\exp(\mu _{0}+\sigma _{0}N(0,1)-\alpha ^{-1}\ln(R(0,1)))\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
References
- Basu, Shantanu; Gil, M; Auddy, Sayatan (April 1, 2015). "The MLP distribution: a modified lognormal power-law model for the stellar initial mass function". MNRAS. 449 (3): 2413–2420. arXiv:1503.00023. Bibcode:2015MNRAS.449.2413B. doi:10.1093/mnras/stv445.