The Gaussian class

The Gaussian class does most of the computation of the packages.

The default value are MU = 0.0 and SIGMA = 6.0

>>> N06 = ttt.Gaussian()
>>> N06
N(mu=0.000, sigma=6.000)

Others ways to create Gaussian objects

>>> N01 = ttt.Gaussian(sigma = 1.0)
>>> N12 = ttt.Gaussian(1.0, 2.0)
>>> Ninf = ttt.Gaussian(1.0,ttt.inf)
>>> N01.mu
0.0
>>> N01.sigma
1.0

The class overwrites the addition +, subtraction -, product *, and division / to compute the marginal distributions used in the TrueSkill Through Time model.

Product *

  • \(\mathcal{N}(x|\mu_1,\sigma_1^2)\mathcal{N}(x|\mu_2,\sigma_2^2) \propto \mathcal{N}(x|\mu_{*},\sigma_{*}^2)\)

with \(\frac{\mu_{*}}{\sigma_{*}^2} = \frac{\mu_1}{\sigma_1^2} + \frac{\mu_2}{\sigma_2^2}\) and \(\sigma_{*}^2 = (\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2})^{-1}\).

>>> N06 * N12
N(mu=0.900, sigma=1.897)
>>> N12 * Ninf
N(mu=1.000, sigma=2.000)

Division /

  • \(\mathcal{N}(x|\mu_1,\sigma_1^2)/\mathcal{N}(x|\mu_2,\sigma_2^2) \propto \mathcal{N}(x|\mu_{\div},\sigma_{\div}^2)\)

with \(\frac{\mu_{\div}}{\sigma_{\div}^2} = \frac{\mu_1}{\sigma_1^2} - \frac{\mu_2}{\sigma_2^2}\) and \(\sigma_{\div}^2 = (\frac{1}{\sigma_1^2} - \frac{1}{\sigma_2^2})^{-1}\).

>>> N12 / N06
N(mu=1.125, sigma=2.121)
>>> N12 / Ninf
N(mu=1.000, sigma=2.000)

Addition +

  • \(\iint \delta(t=x + y) \mathcal{N}(x|\mu_1, \sigma_1^2)\mathcal{N}(y|\mu_2, \sigma_2^2) dxdy = \mathcal{N}(t|\mu_1+\mu_2,\sigma_1^2 + \sigma_2^2)\)

>>> N06 + N12
N(mu=1.000, sigma=6.325)

Substraction -

  • \(\iint \delta(t=x - y) \mathcal{N}(x|\mu_1, \sigma_1^2)\mathcal{N}(y|\mu_2, \sigma_2^2) dxdy = \mathcal{N}(t|\mu_1-\mu_2,\sigma_1^2 + \sigma_2^2)\)

>>> N06 - N12
N(mu=-1.000, sigma=6.325)

Others methods

>>> N06-N12 == ttt.Gaussian(mu=-1.0, sigma=6.324555)
False
>>> (N06-N12).isapprox(ttt.Gaussian(mu=-1.0, sigma=6.324555), 1e-6)
True
>>> N12.forget(gamma=1, t=1)
N(mu=1.000, sigma=2.236)