\begin{algorithm}
\caption{Func-LiNGAM (Can be regarded as vector-based DirectLiNGAM but with FPCA preprocessing.)}
\label{algo1}
\begin{algorithmic}[1]
\State \textbf{Input:} Each function has $W$ time points, then construct $Wp$-dimensional random vector ${f}$ ($W$: Full-time points) for $p$ functions, a set of its variable subscripts $U$ and a $Wp \times n$ data
matrix as $F$, initialize an ordered list of functions $K=\emptyset$ and $m:=1$; 
\State \textbf{Output:} Adjacent Matrix $\hat{T}\in\mathbb{R}^{p\times p}$
\State Use FPCA for finite approximating each random vector to make their dimensions from $Wp$ to $Mp$, where $M$ is the number of principal components.
\Repeat
\State 
(a) Perform least squares regressions of the approximating random vector $\hat{f}_{i}\in\mathbb{R}^M$ on  $\hat{f}_{j}\in\mathbb{R}^M$ for all $ i \in U \backslash K(i \neq j) $ and compute the residual vectors $ \mathbf{r}^{(j)} $ and the residual data matrix $ \mathbf{R}^{(j)} $ from the data matrix $ {F} $ for all $ j \in U \backslash K $. Find a variable $ \hat{f}_{m} $ that is most independent of its residuals:
$$
\hat{f}_{m}=\arg \min _{j \in U \backslash K} MI\left(\hat{f}_{j} ; U \backslash K\right),
$$
where $ MI $ is the independence measure such as mutual information or other measures.
\State (b) Append $ m $ to the end of $ K $.
\State (c) Let  $\hat{\mathbf{f}}:=\mathbf{r}^{(m)}, \hat{F}:=\mathbf{R}^{(m)} $.
\Until{$p-1$ subscripts are appended to $K$}
\State Append the remaining variable to the end of $ K $.
\State Construct a strictly lower triangular matrix $\hat{T} $ by following the order in $ K $, and estimate the connection strengths $ \hat{T}_{i j} $ by using least squares regression in this paper.
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Func-LiNGAM (Can be regarded as vector-based DirectLiNGAM but with FPCA preprocessing.)}
\begin{algorithmic}
[1]
\State \textbf{Input:} Each function has $W$ time points, then construct $Wp$-dimensional random vector ${f}$ ($W$: Full-time points) for $p$ functions, a set of its variable subscripts $U$ and a $Wp \times n$ data
matrix as $F$, initialize an ordered list of functions $K=\emptyset$ and $m:=1$;
\State \textbf{Output:} Adjacent Matrix $\hat{T}\in\mathbb{R}^{p\times p}$
\State Use FPCA for finite approximating each random vector to make their dimensions from $Wp$ to $Mp$, where $M$ is the number of principal components.
\Repeat
\State
(a) Perform least squares regressions of the approximating random vector $\hat{f}_{i}\in\mathbb{R}^M$ on $\hat{f}_{j}\in\mathbb{R}^M$ for all $ i \in U \backslash K(i \neq j) $ and compute the residual vectors $ \mathbf{r}^{(j)} $ and the residual data matrix $ \mathbf{R}^{(j)} $ from the data matrix $ {F} $ for all $ j \in U \backslash K $. Find a variable $ \hat{f}_{m} $ that is most independent of its residuals:
$$
\hat{f}_{m}=\arg \min _{j \in U \backslash K} MI\left(\hat{f}_{j} ; U \backslash K\right),
$$
where $ MI $ is the independence measure such as mutual information or other measures.
\State (b) Append $ m $ to the end of $ K $.
\State (c) Let $\hat{\mathbf{f}}:=\mathbf{r}^{(m)}, \hat{F}:=\mathbf{R}^{(m)} $.
\Until{$p-1$ subscripts are appended to $K$}
\State Append the remaining variable to the end of $ K $.
\State Construct a strictly lower triangular matrix $\hat{T} $ by following the order in $ K $, and estimate the connection strengths $ \hat{T}_{i j} $ by using least squares regression in this paper.
\end{algorithmic}
\end{algorithm}
{
    "title": "functional linear non-gaussian acyclic model for causal discovery",
    "id": "2401.09641",
    "abstract": "in causal discovery, non-gaussianity has been used to characterize the complete configuration of a linear non-gaussian acyclic model (lingam), encompassing both the causal ordering of variables and their respective connection strengths. however, lingam can only deal with the finite-dimensional case. to expand this concept, we extend the notion of variables to encompass vectors and even functions, leading to the functional linear non-gaussian acyclic model (func-lingam). our motivation stems from the desire to identify causal relationships in brain-effective connectivity tasks involving, for example, fmri and eeg datasets. we demonstrate why the original lingam fails to handle these inherently infinite-dimensional datasets and explain the availability of functional data analysis from both empirical and theoretical perspectives. {we establish theoretical guarantees of the identifiability of the causal relationship among non-gaussian random vectors and even random functions in infinite-dimensional hilbert spaces.} to address the issue of sparsity in discrete time points within intrinsic infinite-dimensional functional data, we propose optimizing the coordinates of the vectors using functional principal component analysis. experimental results on synthetic data verify the ability of the proposed framework to identify causal relationships among multivariate functions using the observed samples. for real data, we focus on analyzing the brain connectivity patterns derived from fmri data.",
    "categories": "cs.lg math.st q-bio.nc stat.me stat.th",
    "doi": "",
    "created": "2024-01-17",
    "updated": "",
    "authors": [
        "tian-le yang",
        "kuang-yao lee",
        "kun zhang",
        "joe suzuki"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.09641"
}
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