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

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