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 | − 1 | 0 | 1 | 2 | 3 | \(\dots \) | |
|---|---|---|---|---|---|---|---|---|---|---|
\(\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i})\) | ℓ=−1 | \(\dots \) | − 1 | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | \(\dots \) |
ℓ=0 | \(\dots \) | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | \(\dots \) | |
ℓ=1 | \(\dots \) | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | 1 | \(\dots \) | |
\(IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})=\text {median}\left (\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i})\right)\) | \(\dots \) | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | \(\dots \) | |
\(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\) | \(\dots \) | h−3 | h−2 | \(h_{-1}- \tfrac {2}{3} \) | \(h_{0}+\tfrac {2}{3}\) | h1 | h2 | h3 | \(\dots \) | |
\(\text {Ave}_{i+\ell }^{\tau _{0}}(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\) | ℓ=−1 | \(\dots \) | 0 | 0 | \(-\tfrac {2}{9}\) | 0 | 0 | \(\tfrac {2}{9}\) | 0 | \(\dots \) |
ℓ=0 | \(\dots \) | 0 | \(-\tfrac {2}{9}\) | 0 | 0 | \(\tfrac {2}{9}\) | 0 | 0 | \(\dots \) | |
ℓ=1 | \(\dots \) | \(-\tfrac {2}{9}\) | 0 | 0 | \(\tfrac {2}{9}\) | 0 | 0 | 0 | \(\dots \) | |
\({\mathcal {M}}^{*}(X_{i} - IR_{i}^{(1)}(X_{i},{\mathcal {M}}^{*})\) | \(\dots \) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | \(\dots \) | |
\(IR_{i}^{(2)}(X_{i},{\mathcal {M}}^{*})\) | \(\dots \) | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | \(\dots \) | |
\(IR_{i}^{(3)}(X_{i},{\mathcal {M}}^{*})\) | \(\dots \) | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | \(\dots \) | |
⋮ | ||||||||||
\(IR_{i}(X_{i},{\mathcal {M}}^{*})\) | \(\dots \) | − 1 | − 1 | \(-\tfrac {1}{3}\) | \(\tfrac {1}{3}\) | 1 | 1 | 1 | \(\dots \) | |