$0<{ϵ}_1<1$

$0<{ϵ}_2<1$

$LCS{S}_{\delta ,{ϵ}_1,ϵ2,t,l}(S_p,S_p)$ denoted as

$M(S_p,S_q)$

$M(S_p,S_q)=\{\begin{array}{l}0,if{S}_p=\mathrm{\varnothing }or{S}_q=\mathrm{\varnothing }\\ \\ local+M(S_p-S_{pn},S_q-S_{qn}),ifa\le {ϵ}_1\\ \\ Max\{\begin{array}{l}{\displaystyle \frac{{ϵ}_1-a}{{ϵ}_1-{ϵ}_2}}+M(S_p-S_{pn},S_q-S_{qn})\\ \\ M(S_p-S_{pn},S_q),if{ϵ}_1<a\le {ϵ}_2\\ \\ M(S_p,S_q-S_{qn})\end{array}\\ \\ Max\{\begin{array}{l}M(S_p-S_{pn},S_q)if{ϵ}_2<a\\ \\ M(S_p,S_q-S_{qn})\end{array}\end{array}$

where

$local$ =

$w1$ x

$d(S_p,S_q)$ +

$w2$ x

$|t_{pn}-t_{qm}|$

$a$ =

$d(S_p,S_q)$

$0<LCS{S}_{\delta ,{ϵ}_1,{ϵ}_2,t,l}(S_p,S_q)=M(S_p,S_q)\le min(n,m)$

Similarity is below,

$Sim(\delta ,{ϵ}_1,{ϵ}_2,t,l,S_p,S_q)={\displaystyle \frac{{\alpha }_1\times LCS{S}_{\delta ,{ϵ}_1,ϵ2,t,l}(S_p,S_p)+{\alpha }_2\times |(t_{pn}-t_{p1})-(t_{qm}-t_{q1})|}{min(n,m)}}$

---

–- ADJUSTMENT –-
Define:

$Spatia{l}_{Max}$ –> 地圖上最長距離

$Tempora{l}_{Max}$ –> 有興趣的時間長度

$local$ =

$w1$ x

${\displaystyle \frac{d(S_p,S_q)}{Spatia{l}_{Max}}}$ +

$w2$ x

${\displaystyle \frac{|t_{pn}-t_{qm}|}{Tempora{l}_{Max}}}$

$a$ =

${\displaystyle \frac{d(S_p,S_q)}{Spatia{l}_{Max}}}$

$Sim(\delta ,{ϵ}_1,{ϵ}_2,t,l,S_p,S_q)={\alpha }_1\times {\displaystyle \frac{LCS{S}_{\delta ,{ϵ}_1,ϵ2,t,l}(S_p,S_p)}{min(n,m)}}+{\alpha }_2\times {\displaystyle \frac{|(t_{pn}-t_{p1})-(t_{qm}-t_{q1})|}{Tempora{l}_{max}}}$

$M(S_p,S_q)=\{\begin{array}{l}0,if{S}_p=\mathrm{\varnothing }or{S}_q=\mathrm{\varnothing }\\ \\ local+M(S_p-S_{pn},S_q-S_{qn}),ifa\le {ϵ}_1\\ \\ Max\{\begin{array}{l}{\displaystyle \frac{{ϵ}_1-a}{{ϵ}_1-{ϵ}_2}}+M(S_p-S_{pn},S_q-S_{qn})\\ \\ M(S_p-S_{pn},S_q),if{ϵ}_1<a\le {ϵ}_2\\ \\ M(S_p,S_q-S_{qn})\end{array}\\ \\ Max\{\begin{array}{l}M(S_p-S_{pn},S_q)if{ϵ}_2<a\\ \\ M(S_p,S_q-S_{qn})\end{array}\end{array}$

–Correction or called trying–-

$local$ =

$w1$ x (1 -

${\displaystyle \frac{d(S_p,S_q)}{Spatia{l}_{Max}}}$) +

$w2$ x (1 -

${\displaystyle \frac{|t_{pn}-t_{qm}|}{Tempora{l}_{Max}}}$)

10/18

---

result:

	pattern	(0,0)	(1,0)	(2,0)	(3,0)
observed	0	0	0	0	0
(0,0)	0	1	0.66	0.33	0
(1,0)	0	0.66	2	1.33	0.66

stlcss_score = 0.66
similarity = 0.333

$Spatia{l}_{Max}$ = 3.0

–Adjustment–

$M(S_p,S_q)=\{\begin{array}{l}\\ 0,if{S}_p=\mathrm{\varnothing }or{S}_q=\mathrm{\varnothing }\\ \\ Max\{\begin{array}{l}local+M(S_p-S_{pn},S_q-S_{qn})\\ \\ M(S_p-S_{pn},S_q),ifa\le {ϵ}_1\\ \\ M(S_p,S_q-S_{qn})\end{array}\\ \\ Max\{\begin{array}{l}{\displaystyle \frac{{ϵ}_1-a}{{ϵ}_1-{ϵ}_2}}+M(S_p-S_{pn},S_q-S_{qn})\\ \\ M(S_p-S_{pn},S_q),if{ϵ}_1<a\le {ϵ}_2\\ \\ M(S_p,S_q-S_{qn})\end{array}\\ \\ Max\{\begin{array}{l}M(S_p-S_{pn},S_q)if{ϵ}_2<a\\ \\ M(S_p,S_q-S_{qn})\end{array}\end{array}$

result:

	pattern	(0,0)	(1,0)	(2,0)	(3,0)
observed	0	0	0	0	0
(0,0)	0	1	1	1	1
(1,0)	0	1	2	2	2

stlcss_score = 2.0
similarity = 1.0

$Spatia{l}_{Max}$ = 3.0

ST-LCSS score : 8.789181365735503
similarity : 0.9765757073039448

$Spatia{l}_{Max}$ : 6.708203932499369

– weighting formula

$a_i=\{\begin{array}{l}\\ 1,ifd(S{p}_i,S{q}_i)\le ϵ1\\ \\ {\displaystyle \frac{{ϵ}_1-d(S{p}_i,S{q}_i)}{{ϵ}_1-{ϵ}_2}},ifϵ1<d(S{p}_i,S{q}_i)\le {ϵ}_2\\ \\ 0,if{ϵ}_2<d(S{p}_i,S{q}_i)\\ \\ \end{array}$

$Score$ =

$\frac{\sum _{i=0}^{\alpha +\beta }{w}_i.a_i}{\sum _{i=0}^{\alpha +\beta }{a}_i}$

$w_i=\{\begin{array}{l}{2}^{\frac{i}{\alpha }},ifi\le \alpha \\ 2^{1+\frac{i}{\beta }},ifi>\alpha \end{array}$