butools.moments.JFactorialMomsFromJMoms¶

butools.moments.JFactorialMomsFromJMoms()¶

Matlab:	jfm = JFactorialMomsFromJMoms(jm)
Mathematica:	jfm = JFactorialMomsFromJMoms[jm]
Python/Numpy:	jfm = JFactorialMomsFromJMoms(jm)

Returns the lag-1 joint factorial moments given the lag-1 joint raw moments.

The lag-1 joint raw moments are: \(m_{i,j}=E(\mathcal{X}^i\mathcal{Y}^j)\)

The factorial moments are: \(f_{ij}=E(\mathcal{X}(\mathcal{X}-1)\cdots(\mathcal{X}-i+1)\mathcal{Y}(\mathcal{Y}-1)\cdots(\mathcal{Y}-j+1))\)

Parameters:	jm : matrix, shape (M,M) The matrix of joint raw moments. The entry in row i and column j is \(m_{i,j},i\geq 1,j\geq 1\).
Returns:	jfm : matrix, shape (M,M) The matrix of joint factorial moments. The entry in row i and column j is \(f_{i,j},i\geq 1,j\geq 1\).

References

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

Examples

For Matlab:

>>> MM = [0.7, 2., 3., 4.; 5., 6., 7., 8.; 9., 10., 11., 12.];
>>> JFmoms = JFactorialMomsFromJMoms(MM);
>>> disp(JFmoms);
          0.7          1.3         -1.6          3.8
          4.3         -0.3          0.6         -1.8
         -4.6          0.6         -1.2          3.6
>>> Jmoms = JMomsFromJFactorialMoms(JFmoms);
>>> disp(Jmoms);
          0.7            2            3            4
            5            6            7            8
            9           10           11           12
>>> err = norm(Jmoms-MM);
>>> disp(err);
     0

For Mathematica:

>>> MM = {{0.7, 2., 3., 4.},{5., 6., 7., 8.},{9., 10., 11., 12.}};
>>> JFmoms = JFactorialMomsFromJMoms[MM];
>>> Print[JFmoms];
{{0.7, 1.3, -1.5999999999999996, 3.8000000000000007},
 {4.3, -0.30000000000000004, 0.5999999999999996, -1.8000000000000007},
 {-4.6, 0.6000000000000001, -1.1999999999999993, 3.6000000000000014}}
>>> Jmoms = JMomsFromJFactorialMoms[JFmoms];
>>> Print[Jmoms];
{{0.7, 2., 3., 4.},
 {5., 6., 7., 8.},
 {9., 10., 11., 12.}}
>>> err = Norm[Jmoms-MM];
>>> Print[err];
0.

For Python/Numpy:

>>> MM = ml.matrix([[0.7, 2., 3., 4.],[5., 6., 7., 8.],[9., 10., 11., 12.]])
>>> JFmoms = JFactorialMomsFromJMoms(MM)
>>> print(JFmoms)
[[ 0.7  1.3 -1.6  3.8]
 [ 4.3 -0.3  0.6 -1.8]
 [-4.6  0.6 -1.2  3.6]]
>>> Jmoms = JMomsFromJFactorialMoms(JFmoms)
>>> print(Jmoms)
[[  0.7   2.    3.    4. ]
 [  5.    6.    7.    8. ]
 [  9.   10.   11.   12. ]]
>>> err = la.norm(np.array(Jmoms)-MM)
>>> print(err)
0.0