deeptime.markov.tools.estimation.error_perturbation

deeptime.markov.tools.estimation.error_perturbation(C, S)

Error perturbation for given sensitivity matrix.

Parameters:

• C ((M, M) ndarray) – Count matrix

• S ((M, M) ndarray or (K, M, M) ndarray) – Sensitivity matrix (for scalar observable) or sensitivity tensor for vector observable

Returns:

X – error-perturbation (for scalar observables) or covariance matrix (for vector-valued observable)

Return type:

float or (K, K) ndarray

Notes

Scalar observable

The sensitivity matrix \(S=(s_{ij})\) of a scalar observable \(f(T)\) is defined as

\[S= \left(\left. \frac{\partial f(T)}{\partial t_{ij}} \right \rvert_{T_0} \right) \]

evaluated at a suitable transition matrix \(T_0\).

The sensitivity is the variance of the observable

\[\mathbb{V}(f)=\sum_{i,j,k,l} s_{ij} \text{cov}[t_{ij}, t_{kl}] s_{kl} \]

Vector valued observable

The sensitivity tensor \(S=(s_{ijk})\) for a vector valued observable \((f_1(T),\dots,f_K(T))\) is defined as

\[S= \left( \left. \frac{\partial f_i(T)}{\partial t_{jk}} \right\rvert_{T_0} \right) \]

evaluated at a suitable transition matrix \(T_0\).

The sensitivity is the covariance matrix for the observable

\[\text{cov}[f_{\alpha}(T),f_{\beta}(T)] = \sum_{i,j,k,l} s_{\alpha i j} \text{cov}[t_{ij}, t_{kl}] s_{\beta kl} \]