1. SDTW Formal
Xuehan Ye

2. Background

3. Point Radio-map pos signal feature (x1,y1) AP1:r1
(mean RSS strength of [point1,AP1] in 2 mins)
AP2:r2
...
APn:rn
(x2,y2)
(xm,ym)
pos
signal feature
(x1,y1)
AP1:r1
(mean RSS strength of [point1,AP1] in 2 mins)
AP2:r2
...
APn:rn
(x2,y2)
pos
signal feature
(x1,y1)
AP1:r1
(mean RSS strength of [point1,AP1] in 2 mins)
AP2:r2
...
APn:rn
wait about 2 mins
collect RSS signals

4. Point Radio-map pos signal feature (x1,y1) AP1:r1
(mean RSS strength of [point1,AP1] in 2 mins)
AP2:r2
...
APn:rn
(x2,y2)
(xm,ym)
predicted
pos
ground truth
The difference between the online vector and this offline vector is smallest.

5. Point Radio-map
Disadvantages: 1) laborious efforts to calibrate a fine-grained radio-map 2) the locating result inaccuracy and not robust problems due to random signal strength (RSS) noises.
collect RSS signatures along indoor paths
utilize sequence matching to enhance the location robustness

6. Sequence Radio-map
Problems: 1) indoor path combinational explosion 2) random RSS loss during movement 3) moving speed and direction disparity during online and offline phases
An undirected graph model (Trace-graph) for efficiently calibrating and storing sequence-type radio-map.
Collaborative filter for data clean
An efficient sub-sequence dynamic time warping (SDTW) algorithm.

7. Offline Train

8. RSS Sequence Collection
pos
signal feature
[x1,x2;
y1,y2]
AP1:[r11,r12,r13,r14,r15]
(r11 is not the mean value)
AP2:[r21,r22,r23,r24,r25]
...
APn:[rn1,rn2,rn3,rn4,rn5]
pos:[x1+(x2-x1)/5*0,...,x1+(x2-x1)/5*4;
y1+(y2-y1)/5*0,...,y1+(y2-y1)/5*4]
[x3,x4;
y3,y4]
AP1:[r11,r12,r13,r14,r15,r16]
AP2:[r21,r22,r23,r24,r25,r26]
APn:[rn1,rn2,rn3,rn4,rn5,rn6]
pos:[x1+(x2-x1)/6*0,...,x1+(x2-x1)/6*5;
y1+(y2-y1)/6*0,...,y1+(y2-y1)/6*5]
pos
signal feature
[x1,x2;
y1,y2]
AP1:[r11,r12,r13,r14,r15]
(r11 is not the mean value)
AP2:[r21,r22,r23,r24,r25]
...
APn:[rn1,rn2,rn3,rn4,rn5]
pos:[x1+(x2-x1)/5*0,...,x1+(x2-x1)/5*4;
y1+(y2-y1)/5*0,...,y1+(y2-y1)/5*4]

9. Collaborative Filtering
1st fill
fixed time's fill
2nd fill T1 T2 T3 T4 T5
AP1
-85
-82
-83
-81
AP2
-70
-72
-74
-77
-78
...
APn
-50
-48
-49
pos
signal feature
[x1,x2;
y1,y2]
AP1:[r11,r12,r13,r14,r15]
(r11 is not the mean value)
AP2:[r21,r22,r23,r24,r25]
...
APn:[rn1,rn2,rn3,rn4,rn5]
pos:[x1+(x2-x1)/5*0,...,x1+(x2-x1)/5*4;
y1+(y2-y1)/5*0,...,y1+(y2-y1)/5*4]
[x3,x4;
y3,y4]
AP1:[r11,r12,r13,r14,r15,r16]
AP2:[r21,r22,r23,r24,r25,r26]
APn:[rn1,rn2,rn3,rn4,rn5,rn6]
pos:[x1+(x2-x1)/6*0,...,x1+(x2-x1)/6*5;
y1+(y2-y1)/6*0,...,y1+(y2-y1)/6*5] T1 T2 T3 T4 T5
AP1
-85
-82
-83
-81
AP2
-70
-72
-74
empty
-78
...
APn
-50
-48
-49 T1 T2 T3 T4 T5
AP1
-85
-83
-82
-81
AP2
-70
-72
-74
-78
...
APn
-50
-49
if |-84-(-80)|>=T
empty
if |-84-(-80)|<T
Avg
-82
T1 T2 T3 T4 T5
AP1 -85,[ ],-83,-82,-81
AP2 -70,-72,-74,[ ],-78
...
APn ,[ ],-49,-50
-80?
-84?
polynomial fitting in row
a weighted interpolation based on the APs' similarity in column

10. Collaborative Filtering
a weighted interpolation based on the APs' similarity in column T1 T2 T3 T4 T5
AP1
-85
wait to be filled
-83
-82
-81
AP2
-70
-72
-74
-78
...
APn
-50
-49 T1 T2 T3 T4 T5
AP1
-85
-83
-82
-81
AP2
-70
-72
-74
-78
...
APn
-50
-49
-80
top-K most similar APs
AP1(2)=(S(AP1,AP2)*(-72)+S(AP1,APn)*(-50)+...)
/(S(AP1,AP2)+S(AP1,APn)+...)
S(AP1,APn)=(AP1*APn)/(||AP1||*||APn||)
S(AP1,AP2)=(AP1*AP2)/(||AP1||*||AP2||)
top-K most similar rows of AP1

11. Construct Trace-graph3
path segment 1 2
split point 4

12. Construct Trace-graph
vertex
signal feature
connected vertex 1
AP1:[r11,...,r1m1]
(r11 is not the mean value)
AP2:[r21,...,r2m1]
...
APn:[rn1,...,rnm1]
pos:[x1+(x2-x1)/m1*0,...,x1+(x2-x1)/m1*(m1-1);
y1+(y2-y1)/m1*0,...,y1+(y2-y1)/m1*(m1-1)]
{2,3,4} e1
represents adjacency of path segment 3 v2
path segment 1 2 4

13. Online Locating

14. Online RSS Sequence Collection
moving window length T1 T2 T3 T4 T5
AP1
-88
-86
-87
AP2
-61
-62
-68
-64
-65
...
APn
-70
-73
-69
-72 T1 T2 T3 T4 T5 T6
AP1
-88
-86
-87
AP2
-61
-62
-68
-64
-65
-66
...
APn
-70
-73
-69
-72
maintain moving window
e.g. l=5 (time points)

15. Candidate Vertex Set Extraction
AP1
-88
-86
-87
AP2
-61
-62
-68
-64
-65
...
APn
-70
-73
-69
-72
vertex
signal feature
connected vertex 1
AP1:[r11,...,r1m1]
(r11 is not the mean value)
AP2:[r21,...,r2m1]
...
APn:[rn1,...,rnm1]
pos:[x1+(x2-x1)/m1*0,...,x1+(x2-x1)/m1*(m1-1);
y1+(y2-y1)/m1*0,...,y1+(y2-y1)/m1*(m1-1)]
{2,3,4} 2 3 4
vertex
signal feature
connected vertex
efficient AP list 1
AP1:[r11,...,r1m1]
(r11 is not the mean value)
AP2:[r21,...,r2m1]
...
APn:[rn1,...,rnm1]
pos:[x1+(x2-x1)/m1*0,...,x1+(x2-x1)/m1*(m1-1);
y1+(y2-y1)/m1*0,...,y1+(y2-y1)/m1*(m1-1)]
{2,3,4}
EAL1 2 3
EAL2 4
EAL3
e.g.efficient AP list(EAL0):{AP1,AP2,AP4,AP8...}
if J>T, vertex is added into cadidate vertex set(CVS).
To evaluate AP list similarity, Jaccard similarity score is used.
J(EAL0,EAL1)=||EAL0∩EAL1||/|||EAL0|
If APi can be detected, it is a efficient AP.

16. Candidate Vertex Set Extraction
signal feature
connected vertex 2
AP1:[r11,...,r1m2]
(r11 is not the mean value)
AP2:[r21,...,r2m2]
...
APn:[rn1,...,rnm2]
pos:[x1+(x2-x1)/m2*0,...,x1+(x2-x1)/m2*(m2-1);
y1+(y2-y1)/m2*0,...,y1+(y2-y1)/m2*(m2-1)]
{3,4} 3
SF3
{2,4} 4
SF4
{2,3} 3 3 1 2 2 4 4

17. Candidate Sequence Set Generation
vertex set
signal feature
[2,3]
AP1:[r11,...,r1m2,r11',...,r1m3']
(r11 is not the mean value)
AP2:[r21,...,r2m2,r21',...,r2m3']
...
APn:[rn1,...,rnm2,rn1',...,rnm3']
pos:[x1+(x2-x1)/m2*0,...,x1+(x2-x1)/m2*(m2-1),
x1'+(x2'-x1')/m3*0,...,x1'+(x2'-x1')/m3*(m3-1);
y1+(y2-y1)/m2*0,...,y1+(y2-y1)/m2*(m2-1),
y1'+(y2'-y1')/m3*0,...,y1'+(y2'-y1')/m3*(m3-1)]
[2,4]
[SF2,SF4]
[3,4]
[SF3,SF4] 3 3 2 2 4 4

18. A complicated example In implement, length = 3 or 4.
1) often is longer than moving window
2) reduce computational complexity
Get trace-graph WarpMap.
Green vertexes have AP
list similar with that in the
moving window are added
to candidate vertex set (CVS).
Candidate paths of length
3 are generated from CVS
to form candidate sequence set (CSS).

19. Subsequence Dynamic Time Warping
X:=(x1,x2...xN) NA*N-matrix
Y:=(y1,y2...yM) NA*M-matrix
NA is the length of predefinited AP set
size
d(n,m):=d(xn,ym)=norm(xn-yn)
xn is a vector, whose length is NA.
Other functions can also be used for cost calculation, such as cos delta=1-[dot(xn,ym)/(||xn||*||ym||)].

20. Subsequence Dynamic Time Warping
Accumulated cost matrix
Xt: one sequence in collected online moving window
Ys: one sequence in CSS
The matched end point between Xt and Ys is determined by Ds:
Set the boundary conditions of warping distance matrix::
Then the warping distance matrix is produced as:

21. Subsequence Dynamic Time Warping
Forward SDTW
Reverse SDTW
if D1<=D1' pos=forward pos
else pos=reverse pos
vertex set
reverse signal featur
reverse SDTW distance
rb*
ra*
[3,2]
AP1:[r1m3',...,r11',r1m2,...,r11]
AP2:[r2m3',...,r21',r2m2,...,r21]
...
APn:[rnm3',...,rn1',rnm2,...,rn1']
pos:[x1+(x2-x1)/m3*(m3-1),...,x1+(x2-x1)/m3*0,
x1'+(x2'-x1')/m2*(m2-1),...,x1'+(x2'-x1')/m2*0;
y1+(y2-y1)/m3*(m3-1),...,y1+(y2-y1)/m3*0,
y1'+(y2'-y1')/m2*(m2-1),...,y1'+(y2'-y1')/m2*0]
D1'
b1'
a1'
[4,2]
[SF2,SF4]'
D2'
b2'
a2'
[4,3]
[SF3,SF4]'
D3'
b3'
a3'
vertex set
signal feature
SDTW distance b* a*
[2,3]
AP1:[r11,...,r1m2,r11',...,r1m3']
(r11 is not the mean value)
AP2:[r21,...,r2m2,r21',...,r2m3']
...
APn:[rn1,...,rnm2,rn1',...,rnm3']
pos:[x1+(x2-x1)/m2*0,...,x1+(x2-x1)/m2*(m2-1),
x1'+(x2'-x1')/m3*0,...,x1'+(x2'-x1')/m3*(m3-1);
y1+(y2-y1)/m2*0,...,y1+(y2-y1)/m2*(m2-1),
y1'+(y2'-y1')/m3*0,...,y1'+(y2'-y1')/m3*(m3-1)] D1 b1 a1
[2,4]
[SF2,SF4] D2 b2 a2
[3,4]
[SF3,SF4] D3 b3 a3
CSS
[a1...b1]
D1=min{D1,D2,D3}
forward pos(1:2,b1)
reverse pos(1:2,b1')

22. Online RSS Sequence Collection
reverse predicted path
predicted
pos
forward predicted path

23. Experiment

24. Experiment

25. Experiment The number of AP is very important.
1) delete similar AP by MAC address, such as Hulu, DIVI
2) remain a part of APs whose RSS strength is relatively strong

26. Experiment

27. Thank you!
Xuehan Ye