Manuscript Psy03-D@1 · Psy03-D-v1.tex · section · 3e18951d

Disjoint post-switch windows

Response metrics and old-function persistence › Disjoint post-switch windows · lines 900–935

Psy03-D@1#eq:windows

Let rule-ID switches occur at

b_{1} < \dots < b_{J} .

For each switch, set $b_{J + 1} = T$ and define

W_{j} = {b_{j}, \dots, min (b_{j} + w, b_{j + 1}, T) - 1} .

These windows stop at the next ID switch and do not overlap. The old and new IDs are

r_{j}^{-} = R_{b_{j} - 1}, r_{j}^{+} = R_{b_{j}} .

For $t \in W_{j}$ , define

I_{t} = 1 {f_{r_{j}^{-}} (X_{t}) \neq = f_{r_{j}^{+}} (X_{t})},

P_{t} = I_{t} 1 {Y_{t} = f_{r_{j}^{-}} (X_{t})},

N_{t} = I_{t} 1 {Y_{t} = f_{r_{j}^{+}} (X_{t})},

and

O_{t} = I_{t} - P_{t} - N_{t} .

Then $P_{t}, N_{t}, O_{t}$ are disjoint indicators and

P_{t} + N_{t} + O_{t} = I_{t} .

Displayed formulas in this passage

unlabeled · line 903
$b_{1} < \dots < b_{J} .$
Psy03-D@1#eq:windows · line 907
$W_{j} = {b_{j}, \dots, min (b_{j} + w, b_{j + 1}, T) - 1} .$
unlabeled · line 913
$r_{j}^{-} = R_{b_{j} - 1}, r_{j}^{+} = R_{b_{j}} .$
unlabeled · line 918
$I_{t} = 1 {f_{r_{j}^{-}} (X_{t}) \neq = f_{r_{j}^{+}} (X_{t})},$
unlabeled · line 921
$P_{t} = I_{t} 1 {Y_{t} = f_{r_{j}^{-}} (X_{t})},$
unlabeled · line 924
$N_{t} = I_{t} 1 {Y_{t} = f_{r_{j}^{+}} (X_{t})},$
unlabeled · line 928
$O_{t} = I_{t} - P_{t} - N_{t} .$
unlabeled · line 932
$P_{t} + N_{t} + O_{t} = I_{t} .$