butools.moments.MomsFromFactorialMoms
=====================================

.. currentmodule:: butools.moments

.. np:function:: MomsFromFactorialMoms

    .. list-table::
        :widths: 25 150

        * - Matlab:
          - :code:`m = MomsFromFactorialMoms(fm)`
        * - Mathematica:
          - :code:`m = MomsFromFactorialMoms[fm]`
        * - Python/Numpy:
          - :code:`m = MomsFromFactorialMoms(fm)`
    
    Returns the raw moments given the factorial moments.
    
    The raw moments are: :math:`m_i=E(\mathcal{X}^i)`
    
    The factorial moments are: :math:`f_i=E(\mathcal{X}(\mathcal{X}-1)\cdots(\mathcal{X}-i+1))`
       
    Parameters
    ----------
    fm : vector of doubles
        The list of factorial moments (starting with the first
        moment)
        
    Returns
    -------
    m : vector of doubles
        The list of raw moments

    References
    ----------
    http://en.wikipedia.org/wiki/Factorial_moment    

    Examples
    ========
    For Matlab:

    >>> M = [1.3, 2.4, 6.03, 20.5, 89.5, 474.9];
    >>> fmoms = FactorialMomsFromMoms(M);
    >>> disp(fmoms);
              1.3          1.1         1.43         2.92         6.75        19.75
    >>> moms = MomsFromFactorialMoms(fmoms);
    >>> disp(moms);
              1.3          2.4         6.03         20.5         89.5        474.9
    >>> err = norm(moms-M);
    >>> disp(err);
       3.5527e-15

    For Mathematica:

    >>> M = {1.3, 2.4, 6.03, 20.5, 89.5, 474.9};
    >>> fmoms = FactorialMomsFromMoms[M];
    >>> Print[fmoms];
    {1.3, 1.0999999999999999, 1.4300000000000006, 2.919999999999998, 6.750000000000014, 19.75}
    >>> moms = MomsFromFactorialMoms[fmoms];
    >>> Print[moms];
    {1.3, 2.4, 6.03, 20.5, 89.5, 474.9}
    >>> err = Norm[moms-M];
    >>> Print[err];
    0.

    For Python/Numpy:

    >>> M = [1.3, 2.4, 6.03, 20.5, 89.5, 474.9]
    >>> fmoms = FactorialMomsFromMoms(M)
    >>> print(fmoms)
    [1.3, 1.0999999999999999, 1.4300000000000006, 2.9199999999999982, 6.7500000000000142, 19.75]
    >>> moms = MomsFromFactorialMoms(fmoms)
    >>> print(moms)
    [1.3, 2.3999999999999999, 6.0300000000000002, 20.5, 89.5, 474.89999999999998]
    >>> err = la.norm(np.array(moms)-np.array(M))
    >>> print(err)
    0.0