qspec.analyze.linear_nd_monte_carlo  (  x ,  cov = None ,  axis = None ,  optimize_cov = False ,  n_samples = None ,  n_accepted = None ,  optimize_sampling = True ,  return_samples = False ,  method = 'py' ,  report = False ,  ** kwargs  )[source]

Maximum likelihood Monte-Carlo sampling of a straight line through points $\vec{\mu}_i\in\mathbb{R}^n$ with covariances $\mathbf{\Sigma}_i\in\mathbb{R}^{n\times n}$, assuming $n$-dimensional multivariate normal distributions $\mathcal{N}(\vec{\mu}_i, \mathbf{\Sigma}_i)$. The algorithm is described in the supplementary material of [Gebert et al., Phys. Rev. Lett. 115, 053003 (2015)].

Parameters:

xndarray | Iterable

The data vectors $\vec{\mu}_i$. Must have shape (k, n), where k is the number of data points and n is the number of dimensions of each point.

covndarray | Iterable

The covariance matrices $\mathbf{\Sigma}_i$ of the data vectors. Must have shape (k, n, n). Use covariance_matrix to construct covariance matrices. If None, samples are generated until n_accepted samples get accepted.

axisint

The index of the vector component of the n-dimensional vectors that are fixed for fitting. This is required since a straight in n dimensions is fully described by 2 * (n - 1) parameters. If None, the best axis is determined from the data, and the direction vector of the straight is normalized.

optimize_covbool

If True, the origin vector of the straight is optimized to yield the smallest covariances.

n_samplesint

The number of samples generated for each data point. If None and method == 'cpp', samples are generated until n_accepted samples get accepted.

n_acceptedint

The number of samples to be accepted for each data point. Only available if method == 'cpp'.

optimize_samplingbool

Whether to optimize the data sampling for acceptance efficiency.

return_samplesbool

Whether to also return the generated points $\vec{p}_i$ with shape (n_samples, k ,n).

methodstr

The method to generate the collinear points. Can be one of {'py', 'cpp'}. The 'py' version is faster but only allows to specify n_samples. The 'cpp' version is slower but allows to specify both n_accepted and n_samples.

reportbool

Whether to print the result of the fit.

kwargsNone

Additional keyword arguments to be passed to the chosen method. 'py': {}, 'cpp': {seed: None}.

Returns:

(popt, pcov, p)(ndarray, ndarray, Optional[ndarray])

The optimized parameters and their covariances. If return_samples == True, also the generated points $\vec{p}_i$ are returned. The resulting shapes are (2 * n, ), (2 * n, 2 * n) and (n_samples, k, n).