Chi-squared variable

This variable describes the chi-squared deviation

\[\chi^2 = \sum_{i=1}^K \frac{\left(t_i - y_i\right)^2}{\eta_i^2}.\]

between K outputs \(y_i\) of a surrogate and K targets \(t_i\) scaled by the target uncertainties \(\eta_i\).

If the target uncertainties are not independent but correlated, it is also possible to specify the covariance matrix \(G\) of the targets. In this case, the chi-squared deviation is given as

\[\chi^2 = \sum_{i,j=1}^K (t_i - y_i) G_{i j}^{-1} (t_j - y_j).\]

If the predictions of the surrogate, \(y_i\), follow a Gaussian distribution, the variable follows a generalized chi-squared distribution.

name (str)

The name of the variable under which it can be addressed by other variables or objectives. The name must be distinct from any surrogate name.

Default: "v"

input (str)

The name of a surrogate model or a multi-output variable.

Default: This value has no default and must be provided

target_vector (cell{float})

Vector of target values \(t_i\).

Default: Vector of zeros.

uncertainty_vector (cell{float})

Vector of target uncertainties \(\eta_i\) such that \(\chi^2 = \sum_{i=1}^K \frac{(t_i - y_i)^2}{\eta_i^2}\).

Default: Vector of ones.

covariance_matrix (cell{cell{float}})

Covariance matrix \(G\).

Default: Diagonal identity matrix.

approximate (bool)

If true, the generalized chi-squared variable with different uncertainties of the predictions of each channel K is approximated by a chi-squared variable with averaged uncertainties. This allows to analytically compute probability densities and any acquisition function that is directly based on the variable.

Default: true

force_MC_integration (bool)

If true, the posterior of the variable is based on Monte-Carlo samples. In this case one can avoid to determine a Gaussian distribution of function values, which might be numerically unstable, if the correlation between the inputs is strong.

Default: false

effective_DOF (float)

Number of effective degrees of freedom (DOF) used for stochastic variable of the chi-squared distribution. This number roughly indicates how many output channels of the forward model are statistically independent.

Default: If not specified, the value us determined automatically.

Note

If approximate is false, this parameter has no effect.

effective_DOF_bounds (cell{float})

The number of effective degrees of freedom (DOF) is determined by a maximum likelihood estimate within the given lower and upper bounds.

Default: {10.0,50.0}

Note

If approximate is false or the effective_DOF is set manually, this parameter has no effect.