Algorithm 2.

Turning point refinement (step 2)

TPSet = Sort(PeakSet $\cup$ TroughSet);                           $▹$ Indexing begins

from 1

Initialize: CurrentPeakIndex = 2;                        $▹$ Begin the peak ratio

refinement

While CurrentPeakIndex $\le$ Length(TPSet) - 2 do

$i$ = CurrentPeakIndex, $t_1$ = TPSet($i$), $t_3$ = TPSet($i+2$);

if $\frac{\hat{x}(t_3)}{\hat{x}(t_1)}\ge \delta$ then

CurrentPeakIndex = $i+2$;

else if $\frac{\hat{x}(t_3)}{\hat{x}(t_1)}<\delta$ and $i+2$ = Length(TPSet) then

Remove $t_3$ from PeakSet;

TPSet=Sort(PeakSet $\cup$ TroughSet);

else if $\frac{\hat{x}(t_3)}{\hat{x}(t_1)}<\delta$ and $i+2$ Length(TPSet) then

$t_2$ = TPSet($i+1$), $t_4$ = TPSet($i+3$);

if $\hat{x}(t_2)\le \hat{x}(t_4)$ then

Remove $t_4$ from TroughSet;

else

Remove $t_2$ from TroughSet;

Remove $t_3$ from PeakSet;

TPSet = Sort(PeakSet $\cup$ TroughSet);

Initialize: CurrentIndex = 1;                $▹$ Begin the log-grad refinement

while CurrentIndex $<$ Length(TPSet) do

$i$ = CurrentIndex, $t_0$ = TPSet($i$), $t_1$ = TPSet($i+1$);

if $|\log -\mathrm{grad}(t_0,t_1)|<ϵ$ then                                    $▹$ See Equation (3)

Remove $t_0$ and $t_1$ from both TroughSet and PeakSet;

TPSet = Sort(PeakSet $\cup$ TroughSet);

else

CurrentIndex = $i+1$;

Output PeakSet and TroughSet.