fips.Posterior.reduced_chi2#

property Posterior.reduced_chi2: float#

chi2 divided by the number of observations; near 1 when the covariances are right.

chi2 is \(d^T G^{-1} d\), with \(d = z - H x_0 - c\) the innovation and \(G = H S_0 H^T + S_z / \gamma\) its covariance (innovation_cov). When the covariances are right, \(d \sim N(0, G)\), so chi2 follows a chi-squared distribution with \(m\) degrees of freedom: its mean is \(m\) and its standard deviation \(\sqrt{2m}\), so this is 1 within about \(\sqrt{2/m}\). It is the first test of chi2_tests(), which gives its p-value (Rodgers, 2000, chapter 12). Used, for example, by Kunik et al. (2019), https://doi.org/10.1525/elementa.375.

References

Rodgers, C. D. (2000). Inverse Methods for Atmospheric Sounding: Theory and Practice. World Scientific.