fips.Posterior.r2

property Posterior.r2: float

The R² of the enhancement, \(\operatorname{corr}(z - c, H\hat{x})^2\).

How well the estimate fits the part of the observations the state explains: the observations \(z\) less the offset \(c\) (in a flux inversion, the enhancement over the background), against \(H\hat{x}\), which is modeled_obs less the offset.

The offset is taken out because it is given, not fitted. One that varies from observation to observation, as a background does from day to day, would otherwise raise the correlation by how well it follows the observations, whatever the fit of the state. A constant offset, or none, changes no correlation: then this is the squared correlation of the observations and modeled_obs.

It is a squared correlation, not \(1 - SS_{res} / SS_{tot}\): a fit that is off by a constant or a factor scores as well as an exact one. rmse measures the misfit itself.

See also

modeled_obs

\(H\hat{x} + c\).

rmse

The root mean square of the residuals.

Examples

Four days of one flux over a background that rises from day to day, given as the offset:

>>> import numpy as np
>>> import pandas as pd
>>> import fips
>>> state = fips.Space({"flux": pd.Index(["a"], name="cell")})
>>> days = fips.Space({"co2": pd.RangeIndex(4, name="day")})
>>> background = np.array([400.0, 402.0, 404.0, 406.0])
>>> problem = fips.Problem(
...     prior=fips.Vector(np.zeros(1), state),
...     prior_cov=fips.Covariance.diagonal(1.0, state),
...     obs=fips.Vector(background + [1.0, 0.0, 2.0, 1.0], days),
...     obs_cov=fips.Covariance.diagonal(0.5, days),
...     forward=fips.Operator(
...         np.array([[1.0], [0.5], [1.5], [0.0]]), days, state
...     ),
...     offset=fips.Vector(background, days),
... )
>>> post = fips.solve(problem)
>>> round(post.r2, 3)
0.4

With the background left in, its rise alone would make the fit look nearly perfect:

>>> z, modeled = problem.obs.values, post.modeled_obs.values
>>> round(float(np.corrcoef(z, modeled)[0, 1] ** 2), 3)
0.983