\begin{algorithm}[H]
    \caption{D-Mapper}%标题
    \label{D-Mapper}%标签
    \begin{algorithmic}[1]
      \State Choose a proper filter function $f$ to project data on the real line,$f: X \rightarrow \mathbb{R}$. 
      \State Choose a component number $n$ and quantile $\alpha$.
      \State Fit projected data to a mixture model.
      \For{$i$th component of the mixture model} 
        \State Set the $\frac{\alpha}{2}$ quantile $s_i$ as the start point of the interval,
        \State Set the $1-\frac{\alpha}{2}$ quantile $e_i$ as the end point of the interval,
        \State The interval of $i$th component is $u_i = (s_i,e_i)$.
      \EndFor
      \State The collection of intervals $\mathcal{U} = (u_i) ,i=1...n,$ is a cover on projected data $f(X)$.
      \State Pull back the intervals of projected data, $f^{-1}(\mathcal{U})$.
      \State Clustering on the refined cover and build the nerve by clustering result.
    \end{algorithmic}
\end{algorithm}

\begin{algorithm}
[H]
\caption{D-Mapper}%标题
%标签
\begin{algorithmic}
[1]
\State Choose a proper filter function $f$ to project data on the real line,$f: X \rightarrow \mathbb{R}$.
\State Choose a component number $n$ and quantile $\alpha$.
\State Fit projected data to a mixture model.
\For{$i$th component of the mixture model}
\State Set the $\frac{\alpha}{2}$ quantile $s_i$ as the start point of the interval,
\State Set the $1-\frac{\alpha}{2}$ quantile $e_i$ as the end point of the interval,
\State The interval of $i$th component is $u_i = (s_i,e_i)$.
\EndFor
\State The collection of intervals $\mathcal{U} = (u_i) ,i=1...n,$ is a cover on projected data $f(X)$.
\State Pull back the intervals of projected data, $f^{-1}(\mathcal{U})$.
\State Clustering on the refined cover and build the nerve by clustering result.
\end{algorithmic}
\end{algorithm}
{
    "title": "a distribution-guided mapper algorithm",
    "id": "2401.12237",
    "abstract": "motivation: the mapper algorithm is an essential tool to explore shape of data in topology data analysis. with a dataset as an input, the mapper algorithm outputs a graph representing the topological features of the whole dataset. this graph is often regarded as an approximation of a reeb graph of data. the classic mapper algorithm uses fixed interval lengths and overlapping ratios, which might fail to reveal subtle features of data, especially when the underlying structure is complex.   results: in this work, we introduce a distribution guided mapper algorithm named d-mapper, that utilizes the property of the probability model and data intrinsic characteristics to generate density guided covers and provides enhanced topological features. our proposed algorithm is a probabilistic model-based approach, which could serve as an alternative to non-prababilistic ones. moreover, we introduce a metric accounting for both the quality of overlap clustering and extended persistence homology to measure the performance of mapper type algorithm. our numerical experiments indicate that the d-mapper outperforms the classical mapper algorithm in various scenarios. we also apply the d-mapper to a sars-cov-2 coronavirus rna sequences dataset to explore the topological structure of different virus variants. the results indicate that the d-mapper algorithm can reveal both vertical and horizontal evolution processes of the viruses.   availability: our package is available at https://github.com/shufeige/d-mapper.",
    "categories": "math.at cs.lg q-bio.qm",
    "doi": "",
    "created": "2024-01-19",
    "updated": "",
    "authors": [
        "yuyang tao",
        "shufei ge"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.12237"
}
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