---
tags: formula
title: STLCSS and Similarity Formula
---

$0 < \epsilon_1 < 1$
$0 < \epsilon_2 < 1$

$LCSS_{\delta,\epsilon_1,\epsilon2,t,l}(S_p,S_p)$ denoted as $M(S_p,S_q)$

$M(S_p,S_q) =
\cases{
0, \ \ \ \ if\ \ S_p=\emptyset\ \ or\ \ S_q=\emptyset \\
\\
local+M(S_p-S_{pn},\ S_q-S_{qn}), \ \ \ \ if\ a\ \le \epsilon_1\\
\\
Max\cases{\dfrac{\epsilon_1-a}{\epsilon_1-\epsilon_2}+M(S_p-S_{pn},\ S_q-S_{qn})\\\\
M(S_p-S_{pn},\ S_q),\ \ \ \ \ \ \ \ \ \ \ \ if\ \epsilon_1<a\le\epsilon_2 \\\\
M(S_p,\ S_q-S_{qn})}\\
\\
Max\cases{M(S_p-S_{pn},\ S_q)\ \ \ \ \ \ \ if\ \epsilon_2<a\\\\
M(S_p,\ S_q-S_{qn})
} \\
}$

where

$local$ = $w1$ x $d(S_{p},S_{q})$ + $w2$ x $|t_{pn}-t_{qm}|$
$a$ = $d(S_p, S_q)$
$0< LCSS_{\delta,\epsilon_1,\epsilon_2,t,l}(S_p,S_q)= M(S_p,S_q)\le min(n,m)$

Similarity is below,

$Sim(\delta,\epsilon_1,\epsilon_2,t,l,S_p,S_q) = \dfrac{\alpha_1\times LCSS_{\delta,\epsilon_1,\epsilon2,t,l}(S_p,S_p)+\alpha_2\times |(t_{pn}-t_{p1})-(t_{qm}-t_{q1})|}{min(n,m)}$

---
--- ADJUSTMENT ---
Define:
$Spatial_{Max}$ --> 地圖上最長距離
$Temporal_{Max}$ --> 有興趣的時間長度

$local$ = $w1$ x $\dfrac{d(S_{p},S_{q})}{Spatial_{Max}}$ + $w2$ x $\dfrac{|t_{pn}-t_{qm}|}{Temporal_{Max}}$

$a$ = $\dfrac{d(S_p, S_q)}{Spatial_{Max}}$

$Sim(\delta,\epsilon_1,\epsilon_2,t,l,S_p,S_q) = \alpha_1\times \dfrac{LCSS_{\delta,\epsilon_1,\epsilon2,t,l}(S_p,S_p)}{min(n,m)}+\alpha_2\times \dfrac{|(t_{pn}-t_{p1})-(t_{qm}-t_{q1})|}{Temporal_{max}}$

$M(S_p,S_q) =
\cases{
0, \ \ \ \ if\ \ S_p=\emptyset\ \ or\ \ S_q=\emptyset \\
\\
local+M(S_p-S_{pn},\ S_q-S_{qn}), \ \ \ \ if\ a\ \le \epsilon_1\\
\\
Max\cases{\dfrac{\epsilon_1-a}{\epsilon_1-\epsilon_2}+M(S_p-S_{pn},\ S_q-S_{qn})\\\\
M(S_p-S_{pn},\ S_q),\ \ \ \ \ \ \ \ \ \ \ \ if\ \epsilon_1<a\le\epsilon_2 \\\\
M(S_p,\ S_q-S_{qn})}\\
\\
Max\cases{M(S_p-S_{pn},\ S_q)\ \ \ \ \ \ \ if\ \epsilon_2<a\\\\
M(S_p,\ S_q-S_{qn})
} \\
}$

--Correction or called trying---

$local$ = $w1$ x (1 - $\dfrac{d(S_{p},S_{q})}{Spatial_{Max}}$) + $w2$ x (1 - $\dfrac{|t_{pn}-t_{qm}|}{Temporal_{Max}}$)

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
$Spatial_{Max}$ = 3.0

--Adjustment--
$M(S_p,S_q) =
\cases{\\
0, \ \ \ \ if\ \ S_p=\emptyset\ \ or\ \ S_q=\emptyset \\
\\
Max\cases{local+M(S_p-S_{pn},\ S_q-S_{qn}) \\\\
M(S_p-S_{pn},\ S_q),\ \ \ \ \ \ \ if\ a\ \le \epsilon_1\\\\
M(S_p,\ S_q-S_{qn})
} \\
\\
Max\cases{\dfrac{\epsilon_1-a}{\epsilon_1-\epsilon_2}+M(S_p-S_{pn},\ S_q-S_{qn})\\\\
M(S_p-S_{pn},\ S_q),\ \ \ \ \ \ \ \ \ \ \ \ if\ \epsilon_1<a\le\epsilon_2 \\\\
M(S_p,\ S_q-S_{qn})}\\
\\
Max\cases{M(S_p-S_{pn},\ S_q)\ \ \ \ \ \ \ if\ \epsilon_2<a\\\\
M(S_p,\ S_q-S_{qn})
} \\
}$

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
$Spatial_{Max}$ = 3.0

ST-LCSS score : 8.789181365735503
similarity : 0.9765757073039448
$Spatial_{Max}$ : 6.708203932499369

-- weighting formula

$a_i =
\cases{\\
1,\ \ \ \ \ if\ d(Sp_i, Sq_i)\le\epsilon1\\
\\
\dfrac{\epsilon_1-d(Sp_i, Sq_i)}{\epsilon_1-\epsilon_2} ,\ \ \ \ \ if\ \epsilon1<d(Sp_i, Sq_i)\le\epsilon_2\\\\
0 ,\ \ \ \ \ if\ \epsilon_2<d(Sp_i, Sq_i)\\\\
\\
}$

$Score$ = $\frac{\sum\limits_{i = 0}^{\alpha+\beta}{w_i.a_i}}{\sum\limits_{i = 0}^{\alpha+\beta}{a_i}}$

$w_i =
\cases{
2^\frac{i}{\alpha} ,\ \ \ \ \ if \ i \le \alpha\\
2^{1+\frac{i}{\beta}} ,\ \ \ \ \ if \ i > \alpha
}$