[HELP] Computation of Q2 and cR

[antoinecollet5]

@jonghyunharrylee

I am working on a deep refactoring of the code that I am hosting here for now: https://github.com/antoinecollet5/pyrtid/blob/master/pyrtid/inverse/solvers/pcga/pcga.py
I will push the code when it is ready and entirely tested.

I have a general question regarding the computation of Q2 and cR for model validation.

I guess the main reference is Kitanidis PK. Introduction to Geostatistics: Applications in Hydrogeology. Cambridge University Press; 1997.

If I understand correctly, Q2 is given by (6.67):

$$ Q_2 = \dfrac{1}{n-p} \sum^{n}_{k=p+1} \dfrac{\delta_k^2}{\sigma_k^{2}} $$

And we target $Q_2 = 1$

Whereas cR which we seek as small as possible is given by 4.30:

$$ cR = Q_2 \exp \left( \dfrac{1}{n-p} \sum^{n}_{k=p+1} \ln\left(\sigma_k^{2}\right) \right) $$

Question 1:

In the code, $Q_2$ is defined as

Q2_all[:, i : i + 1] = np.dot(b[:n].T, xi) / (n - p)

which I understand as

$$ Q_2 = \dfrac{1}{n-p}\left(\mathbf{y} - h(\mathbf{\overline{s}}) + \mathbf{H}\overline{\mathbf{s}}\right)^{\mathrm{T}} \overline{\boldsymbol{\xi}} $$

Where is this expression coming from ? Do you have any reference explaining the derivation ?

Question 2, how is cR derived ? Or more specifically $\sigma^{2}$ ? In the code, tt is only implemented if $\mathbf{R}$ is a single element and I am not sure how to define the variance of residuals in the context.

Thank you in advance for your reply.

Best

Antoine