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docs: documentation of covariance extended.

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Fabian Joswig 2022-05-16 10:55:51 +01:00
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pyerrors/obs.py
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@ -1342,7 +1342,7 @@ def correlate(obs_a, obs_b):
def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs): def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
r'''Calculates the covariance matrix of a set of observables. r'''Calculates the error covariance matrix of a set of observables.
The gamma method has to be applied first to all observables. The gamma method has to be applied first to all observables.
@ -1353,7 +1353,7 @@ def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
visualize : bool visualize : bool
If True plots the corresponding normalized correlation matrix (default False). If True plots the corresponding normalized correlation matrix (default False).
correlation : bool correlation : bool
If True the correlation instead of the covariance is returned (default False). If True the correlation matrix instead of the error covariance matrix is returned (default False).
smooth : None or int smooth : None or int
If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue If smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue
smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the smoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the
@ -1362,8 +1362,11 @@ def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
Notes Notes
----- -----
The covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite The error covariance is defined such that it agrees with the squared standard error for two identical observables
$$v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags. $$\operatorname{cov}(a,a)=\sum_{s=1}^N\delta_a^s\delta_a^s/N^2=\Gamma_{aa}(0)/N=\operatorname{var}(a)/N=\sigma_a^2$$
in the absence of autocorrelation.
The error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite
$$\sum_{i,j}v_i\Gamma_{ij}(0)v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i,j}v_i\delta_i^s\delta_j^s v_j=\frac{1}{N}\sum_{s=1}^N\sum_{i}|v_i\delta_i^s|^2\geq 0\,,$$ for every $v\in\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.
For observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements. For observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.
$$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$ $$\tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}$$
This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors). This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).