Description Usage Arguments Details Value Author(s) Examples

Compute the Mahalanobis distance (MD) based on the kriging models of statistics

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`points` |
either matrix or list of points or a vector of parameters (but then considered as a single (multidimensional) point) |

`qsd` |
object of class |

`Sigma` |
either a constant variance matrix estimate or an prespecified value |

`...` |
further arguments passed to |

`cvm` |
list of fitted cross-validation models (see |

`obs` |
numeric vector of observed statistics (overwrites ' |

`inverted` |
logical, |

`check` |
logical, |

`value.only` |
only return the value of the MD |

`na.rm` |
logical, if |

`cl` |
cluster object, |

`verbose` |
if |

The function computes the Mahalanobis distance of the given statistics *T(X)\in R^p* with different options
for the approximation type of the variance matrix. The Mahalanobis distance can be used as an alternative criterion function
for estimating the unknown model parameter during the main estimation function `qle`

.

There are several options how to estimate or choose the variance matrix of the statistics *Σ*.
First, in case of a given constant variance matrix estimate '`Sigma`

', the Mahalanobis distance reads

* (T(x)-E_{θ}[T(X)])^tΣ^{-1}(T(x)-E_{θ}[T(X)]) *

and '`Sigma`

' is used as given.

As a second option, the variance matrix *Σ* can be estimated by the average approximation

*\bar{V}=\frac{1}{n}∑_{i=1}^n V_i *

based on the simulated variance matrices *V_i=V(θ_i)* of statistics over all sample points
*θ_1,...,θ_n* (see vignette).
Unless '`qsd$var.type`

' equals "`const`

" additional prediction variances are added as diagonal terms to
account for the kriging approximation error of the statistics using kriging with calculation of kriging variances
if '`qsd$krig.type`

' equal to "`var`

". Otherwise no additional variances are added. A weighted version of
these average approximation types is also available (see `covarTx`

).

As a continuous version of variance approximation we use a kriging approach (see [1]). Then

*Σ(θ) = Var_{θ}(T(X))*

denotes the variance matrix which depends on the parameter *θ\in R^q* and corresponds to the
formal function argument '`points`

'. Each time a value of the criterion function is calculated at any parameter
'`points`

' the variance matrix is estimated by the correpsonding approach either with or
without using prediction variances as explained above. Note that in this case the argument '`Sigma`

' is ignored.

Either a vector of MD values or a list of lists, where each contains the following elements:

`value` |
Mahalanobis distance value |

`par` |
parameter estimate |

`I` |
approximate variance matrix of the parameter estimate |

`score` |
gradient of MD (for constant ' |

`jac` |
Jacobian of sample mean values of statistics |

`varS` |
estimated variance matrix of ' |

and, if applicable, the following attributes:

`Sigma` |
estimate of variance matrix (if ' |

`inverted` |
whether ' |

M. Baaske

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