Examples of Gram-Charlier series expansions of a Weibull distribution (left) and a Normal distribution (right).

<p>Examples of Gram-Charlier series expansions of a Weibull distribution (left) and a Normal distribution (right).</p

A Weibull distribution and polynomial distributions for certain number of terms <i>N</i>.

<p>The polynomial distributions were computed from numerically calculated moments on the interval <i>a</i> = 0, <i>b</i> = 5. In the polynomial distributions, note the wiggle of and negative values when the PDF is close to zero.</p

A set of distributions (black line) and polynomial approximations (dashed line) to level 10 for the common distributions: Normal, Weibull, Log-Normal, Bimodal.

<p>The polynomial distributions were computed from numerically integrated moments on the interval <i>a</i> = 0, <i>b</i> = 5.</p

A plot of approximate solutions to the SCE for constant kernel (A) and for linear kernel (B).

<p>The approximation was based on the moment transform and polynomial degree <i>N</i> = 8.</p

Histograms of household electricity use data with Gram–Charlier PDF (left), Weibull PDF (right) and polynomial PDF (right) fit.

<p>The polynomial distributions were computed from numerically calculated moments on the interval <i>a</i> = 100, <i>b</i> = 5000. A Weibull or Log-Normal PDF is usually a typical choice for this type of data [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174573#pone.0174573.ref002" target="_blank">2</a>].</p