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The Gaussian class

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

TrueSkillThroughTime.Gaussian — Type

The Gaussian class is used to define the prior beliefs of the agents' skills (and for internal computations).

We can create objects by passing the parameters in order or by mentioning the names. 

Gaussian(mu::Float64=MU, sigma::Float64=SIGMA)
Gaussian(;mu::Float64=MU, sigma::Float64=SIGMA)
  • mu is the mean of the Gaussian distribution
  • sigma is the standar deviation of the Gaussian distribution

The default value of MU and SIGMA are

N06 = ttt.Gaussian()
Gaussian(mu=0.0, sigma=6.0)

Others ways to create Gaussian objects

N01 = ttt.Gaussian(sigma = 1.0)
N12 = ttt.Gaussian(1.0, 2.0)
Ninf = ttt.Gaussian(1.0,Inf)
println("mu: ", N01.mu, ", sigma: ", N01.sigma)
mu: 0.0, 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 *

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

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

Base.:* — Function
*(N::Gaussian, M::Gaussian)
+(k::Float64, M::Gaussian)
julia> N06 * N12Gaussian(mu=0.9, sigma=1.897367)
julia> N12 * 5.0Gaussian(mu=5.0, sigma=10.0)
julia> N12 * NinfGaussian(mu=1.0, sigma=2.0)

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 $\sigma_{\div}^2 = (\frac{1}{\sigma_1^2} - \frac{1}{\sigma_2^2})^{-1}$.

Base.:/ — Function
/(N::Gaussian, M::Gaussian)
julia> N12 / N06Gaussian(mu=1.125, sigma=2.12132)
julia> N12 / NinfGaussian(mu=1.0, sigma=2.0)

Addition +

  • $\iint \delta(t=x + y) \mathcal{N}(x|\mu_1, \sigma_1^2)\mathcal{N}(y|\mu_2, \sigma_2^2) dxdy = \mathcal{N}(t|\mu_1+\mu_2,\sigma_1^2 + \sigma_2^2)$
Base.:+ — Function
+(N::Gaussian, M::Gaussian)
julia> N06 + N12Gaussian(mu=1.0, sigma=6.324555)

Substraction -

  • $\iint \delta(t=x - y) \mathcal{N}(x|\mu_1, \sigma_1^2)\mathcal{N}(y|\mu_2, \sigma_2^2) dxdy = \mathcal{N}(t|\mu_1-\mu_2,\sigma_1^2 + \sigma_2^2)$
Base.:- — Function
-(N::Gaussian, M::Gaussian)
julia> N06 - N12Gaussian(mu=-1.0, sigma=6.324555)

Others methods

isapprox

Base.isapprox — Function
isapprox(N::Gaussian, M::Gaussian, atol::Real=0)
julia> N06-N12 == ttt.Gaussian(mu=-1.0, sigma=6.324555)false
julia> ttt.isapprox(N06-N12, ttt.Gaussian(mu=-1.0, sigma=6.324555), 1e-6)true

forget