NormalToyMC

Description

Generates a random sample distributed according to multivariate normal distribution without correlations.

Uses boost::mt19937 random generator.

The sample becomes frozen upon generation. One has to manually taint the transformation for the next sample by calling nextSample method.

Inputs

1. Average model vector ${\mu }_1$.

2. First model uncertainties vector ${\sigma }_1$.

3. Average model vector ${\mu }_2$.

4. Second model uncertainties vector ${\sigma }_2$.

5. etc.

6. etc.

Inputs are added via add(theory, sigma) method.

Outputs

1. 'toymc' — output vector $x$ of size of concatination of ${\mu }_i$.

Implementation

For the random variable vector $x$ of size $N$, distributed around $\mu$ with uncertainties $\sigma$ the p.d.f. is:

$$P(x|\mu )=\frac{1}{\sqrt{(2\pi {)}^N}\prod _i{\sigma }_i}{e}^{{\displaystyle \frac{1}{2}\sum _i\frac{(x_i-{\mu }_i{)}^2}{{\sigma }_i^2}}}.$$

One can define the vector $z$:

$$\begin{array}{rl}& z_i=\frac{x_i-{\mu }_i}{{\sigma }_i},\\ & x_i={\sigma }_i{z}_i+{\mu }_i.\end{array}$$

Since the transition Jacobian $|dx/dz|=|L|=\prod _i{\sigma }_i$ each $z_i$ is distributed normally with $\sigma =1$ with central value of $0$.

The algorithm generates normal vector $z$ and transforms it to $x_i={\sigma }_i{z}_i+{\mu }_i$.

By $\mu$ we mean the concatenation of vectors ${\mu }_i$.