notesum.ai

Published at November 25

stat.ML

cs.AI

q-fin.ST

Released Date: November 25, 2024

Authors: Jiaan Han¹, Junxiao Chen¹, Yanzhe Fu¹

Aff.: ¹Department of Statistics, University of Michigan

Algorithm 1 CatNet - Effective Feature Selection in LSTM

Input: Fixed FDR level

q

(x_{i},y_{i})

i=1,\ldots,n

with

x_{i}\in\mathbb{R}^{p}

y_{i}\in\mathbb{R}^{1}

Output:

\hat{S}_{1}

1. for

i=1

p

, do: A. Generate

Z_{j}\sim\mathcal{N}(0,I_{n})

. B. Calculate

c_{j}=\arg\min_{c}I_{j}=\arg\min_{c}\sum_{\tau=0}^{k}w_{\tau}\text{HSIC}_{\tau% }(X,Y)

. C. Construct mirrored pair

(X_{j}^{+},X_{j}^{-})=(X_{j}+c_{j}Z_{j},X_{j}-c_{j}Z_{j})

. D. Train LSTM with

X^{(j)}=(X_{j}^{+},X_{j}^{-},X_{-j})

as the input data. E. Compute the SHAP values for

X_{j}^{+},X_{j}^{-}

F. Take the derivatives of SHAP values

L(x_{j}^{-})=[\phi_{j}^{1-},\,\phi_{j}^{2-},\,\dots,\,\phi_{j}^{n-}],\quad% \newline L(x_{j}^{+})=[\phi_{j}^{1+},\,\phi_{j}^{2+},\,\dots,\,\phi_{j}^{n+}]

as the feature importance vector G. Compute statistic

M_{j}=\frac{\langle L(x_{j}^{+}),\,L(x_{j}^{-})\rangle}{(\|L(x_{j}^{+})\|_{2}% \ *\,\|L(x_{j}^{-})\|_{2})}\cdot(\|L(x_{j}^{+})\|_{1}\,\vee\,\|L(x_{j}^{-})\|_% {1})

. end for 2. Calculate

T=\min\{t>0:\hat{\text{FDP}}(t)\leq q\}

. 3. Return

\hat{S}_{1}\leftarrow\{j:M_{j}\geq T\}