NumCosmoMathStatsDistKDE

Base class for implementing N-dimensional probability distributions with a fixed density estimator kernel.

Abstract object to reconstruct an arbitrary N-dimensional probability distribution. This object provides the complementary tools to perform a radial basis interpolation in a multidimensional function using the NcmStatsDist class.

This object sets the kernel $\phi$ to be used in the radial basis interpolation. This object also implements some calculations needed in the NcmStatsDist class, such as the covariance matrix of the whole sample and its Cholesky decomposition, the preparation of the interpolation matrix $IM$, the kernel normalization factor, and given a sample vector $\vec{x}$, the distribution evaluated in these points. Some of these calculations are explained below.

The NcmStatsDistKDE class uses one covariance matrix for all the sample points. So, given $n$ points, there is only one covariance matrix $\Sigma$ that is used for all the $i$-th kernels $\phi(|x-x_i|, \Sigma)$. After the covariance matrix is computed, the algorithm computes the Cholesky decomposition, that is \begin{align} \Sigma &= AA^T ,\end{align} where $A$ is a triangular positive defined matrix and $A^T$ is its transpose. The $A$ matrix is used in the least square squares calculation method that is called in the NcmStatsDist class.

The object also prepares the interpolation matrix to be implemented in the least-squares problem, that is, given the relation

$\left[\begin{array}{cccc} \phi\left(\left|\mathbf{x}{1}-\mathbf{x}{1}\right|\right) & \phi\left(\left|\mathbf{x}{2}-\mathbf{x}{1}\right|\right) & \ldots & \phi\left(\left|\mathbf{x}{n}-\mathbf{x}{1}\right|\right) \newline \phi\left(\left|\mathbf{x}{1}-\mathbf{x}{2}\right|\right) & \phi\left(\left|\mathbf{x}{2}-\mathbf{x}{2}\right|\right) & \ldots & \phi\left(\left|\mathbf{x}{n}-\mathbf{x}{2}\right|\right) \newline \vdots & \vdots & & \vdots \newline \phi\left(\left|\mathbf{x}{1}-\mathbf{x}{n}\right|\right) & \phi\left(\left|\mathbf{x}{2}-\mathbf{x}{n}\right|\right) & \ldots & \phi\left(\left|\mathbf{x}{n}-\mathbf{x}{n}\right|\right) \end{array}\right]\left[\begin{array}{c} \lambda_{1} \newline \lambda_{2} \newline \vdots \newline \lambda_{n} \end{array}\right]=\left[\begin{array}{c} g_{1} \newline g_{2} \newline \vdots \newline g_{n} ,\end{array}\right]$

which is explained in the NcmStatsDist class, this object prepares the first matrix for all the $n$ points in the sample, using the covariance matrix and the defined kernel. The NcmStatsDist class implements the solution for this relation and then one can compute the distribution for a given vector $\vec{x}$ using a method of the NcmStatsDist class but that is implemented in this object.

The user must provide input the values: sdk, CV_type - ncm_stats_dist_kde_new(), y - ncm_stats_dist_add_obs(), split_frac - ncm_stats_dist_set_split_frac(), over_smooth - ncm_stats_dist_set_over_smooth(), $v(x)$ - ncm_stats_dist_prepare_interp(). To see an example of how to use this object and the main functions that are called within each function, check the flowchart at the end of this documentation, where the order of the functions that should be called by the user and some of the functions that the algorithm calls.

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