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Table 2 An iterative calculation of \(IR_{i}^{(k)}(X_{i},{\mathcal {M}}^{*})\) for a signal Xi of Table 1

From: Ensemble patch transformation: a flexible framework for decomposition and filtering of signal

i\(\dots \)− 3− 2− 10123\(\dots \)
\(\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i})\)=−1\(\dots \)− 1− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)11\(\dots \)
 =0\(\dots \)− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)111\(\dots \)
 =1\(\dots \)− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)1111\(\dots \)
\(IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})=\text {median}\left (\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i})\right)\)\(\dots \)− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)111\(\dots \)
\(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\)\(\dots \)h−3h−2\(h_{-1}- \tfrac {2}{3} \)\(h_{0}+\tfrac {2}{3}\)h1h2h3\(\dots \)
\(\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\)=−1\(\dots \)00\(-\tfrac {2}{9}\)00\(\tfrac {2}{9}\)0\(\dots \)
 =0\(\dots \)0\(-\tfrac {2}{9}\)00\(\tfrac {2}{9}\)00\(\dots \)
 =1\(\dots \)\(-\tfrac {2}{9}\)00\(\tfrac {2}{9}\)000\(\dots \)
\({\mathcal {M}}^{*}(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\)\(\dots \)0000000\(\dots \)
\(IR_{i}^{(2)}(X_{i},{\mathcal {M}}^{*})\)\(\dots \)− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)111\(\dots \)
\(IR_{i}^{(3)}(X_{i},{\mathcal {M}}^{*})\)\(\dots \)− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)111\(\dots \)
   
\(IR_{i}(X_{i},{\mathcal {M}}^{*})\)\(\dots \)− 1− 1\(-\tfrac {1}{3}\)\(\tfrac {1}{3}\)111\(\dots \)