# msmtools.estimation.error_perturbation¶

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

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