Change-Point Detection Using the Conditional Entropy of Ordinal Patterns

. 2018 Sep 14;20(9):709. doi: 10.3390/e20090709

Algorithm 1 Detecting at most one change-point

Input: sequence

π = {(π (k))}_{k = t_{start}}^{t_{end}}

of ordinal patterns of order d, nominal probability

α

of false alarm
Output: estimate of a change-point

{\hat{t}}^{*}

if change-point is detected, otherwise return 0.

1:
functionDetectSingleCP( $π$ , $α$ )
2:
$T_{\min}$ ← $(d + 1)! (d + 1)$ ;
3:
if $t_{end} - t_{start} < 2 T_{\min}$ then
4:
return 0; ▷ sequence is too short, no change-point can be detected
5:
end if
6:
${\hat{t}}^{*}$ ← $\underset{t = t_{start} + T_{\min}, \dots, t_{end} - T_{\min}}{arg max} CEofOP (t; π)$ ;
7:
$N_{boot}$ ← $⌊ \frac{5}{α} ⌋$ ; ▷ number of bootstrap samples for computing threshold
8:
for $l = 1, 2, \dots, N_{boot}$ do ▷ computing threshold by reshuffling
9:
$ξ$ ← randomly shuffled blocks of length $(d + 1)$ from $π$ ;
10:
$c_{j}$ ← $\underset{t = t_{start} + T_{\min}, \dots, t_{end} - T_{\min}}{arg max} CEofOP (t; ξ)$ ;
11:
end for
12:
$c_{j}$ ← Sort( $c_{j}$ ); ▷ sort the maximal values of $CEofOP$ for bootstrap samples in decreasing order
13:
h ← $c_{⌊ α N_{boot} ⌋}$
14:
if $S ({\hat{t}}^{*}; π) < h$ then
15:
return 0;
16:
else
17:
return ${\hat{t}}^{*}$ ;
18:
end if
19:
end function