Fig 1.
A set of distributions (black line) and polynomial approximations (dashed line) to level 10 for the common distributions: Normal, Weibull, Log-Normal, Bimodal.
The polynomial distributions were computed from numerically integrated moments on the interval a = 0, b = 5.
Fig 2.
A Weibull distribution and polynomial distributions for certain number of terms N.
The polynomial distributions were computed from numerically calculated moments on the interval a = 0, b = 5. In the polynomial distributions, note the wiggle of and negative values when the PDF is close to zero.
Fig 3.
Examples of Gram-Charlier series expansions of a Weibull distribution (left) and a Normal distribution (right).
Fig 4.
Histograms of household electricity use data with Gram–Charlier PDF (left), Weibull PDF (right) and polynomial PDF (right) fit.
The polynomial distributions were computed from numerically calculated moments on the interval a = 100, b = 5000. A Weibull or Log-Normal PDF is usually a typical choice for this type of data [2].
Fig 5.
A plot of approximate solutions to the SCE for constant kernel (A) and for linear kernel (B).
The approximation was based on the moment transform and polynomial degree N = 8.