  We develop a new analysis method that allows us to compare multi-dimensional
observables to a theoretical model. The method is based on unsupervised
clustering algorithms which assign the observational and simulated data to
clusters in high dimensionality. From the clustering result, a goodness of fit
(the p-value) is determined with the Fisher-Freeman-Halton test. We first show
that this approach is robust for 2D Gaussian distributions. We then apply the
method to the observed MW satellites and simulated satellites from the fiducial
model of our semi-analytic code A-SLOTH. We use the following 5 observables of
the galaxies in the analysis: stellar mass, virial mass, heliocentric distance,
mean stellar metallicity [Fe/H], and stellar metallicity dispersion
{\sigma}[Fe/H]. A low p-value returned from the analysis tells us that our
A-SLOTH fiducial model does not reproduce the mean stellar metallicity of the
observed MW satellites well. We implement an ad-hoc improvement to the physical
model and show that the number of dark matter merger trees which have p-values
> 0.01 increases from 3 to 6. This method can be extended to data with higher
dimensionality easily. We plan to further improve the physical model in A-SLOTH
using this method to study elemental abundances of stars in the observed MW
satellites.
