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Globally Continuous and Non-Markovian Crowd Activity Analysis from Videos-Supplement

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Supplementary Material of Globally Continuous and
Non-Markovian Activity Analysis from Videos
He Wang1,2?and Carol O’Sullivan1,3
1Disney Research Los Angeles??, United States of America
2University of Leeds, United Kingdom
realcrane@gmail.com
3Trinity College Dublin, Ireland
carol.osullivan@scss.tcd.ie
1 Generative Process of STHDP
We first again show the STHDP model in Figure 1.
Fig. 1. STHDP model Fig. 2. Model used for sampling.
The generative process of Figure 1 is explained as follows:
1. Sample a corpus-level time base distribution, e|λGEM(λ)
2. Sample a corpus-level word base distribution, v|ωGEM(ω)
3. For each corpus-level word topic k:
(a) Sample a distribution over words, βk|ηDirichlet(η)
(b) Sample a word-topic-specific distribution over time topics, γk|e, ζ DP (ζ , e)
4. For each time topic l:
(a) Sample a distribution over time, αl|ΓNormal-Inverse-Gamma(Γ)
5. For each document d:
(a) Sample a distribution over topics, πd|v, σ DP (σ, v)
(b) For each word n:
i. Sample a word topic indicator, zdn|πdπd
ii. Sample a word wdn|βzdnM ult(βzdn)
iii. Sample a time topic indicator, odn|zdn, γ γzdn
iv. Sample a time word tdn|αodnN ormal(αodn)
?Corresponding Author, ORCID-ID:orcid.org/0000-0002-2281-5679
?? This work is mostly done by the authors when they were with Disney Research Los Angeles.
2 He Wang and Carol O’Sullivan
2 Gibbs Sampling for STHDP
Based on Figure 2, we only explain the modified Chinese Restaurant Franchise scheme
here, as we fix the word HDP while running Chinese Restaurant Franchise (CRF) on
the time HDP as in [1]. Following the naming convention in CRF, word topics and time
topics are called word dishes and time dishes. Word documents are called restaurants
and the set of time stamps associated with one word topic is called a time restaurant. A
list of auxiliary variables are given in Table 1. Also, we use superscript to exclude data
samples. For instance, zji
jmeans the set of all table indicators in restaurant j excluding
wji . Bold fonts means the whole set of some quartile, for instance, lmeans all time dish
indices. We also use dots as summation. m·kis the number of word tables serving dish
k.
Table 1. Variables in CRF
vwa word in the vocabulary
Vwthe size of the vocabulary
wji the ith word in restaurant j
tji the ith time word in restaurant j
nji the number of words in restaurant j at table i
nj·the number of words in restaurant j
zji the table indicator of the ith word in restaurant j
kji the dish indicator of the ith word table in restaurant j
mjk the number of word tables in restaurant j serving dish k
mj·the number of word tables in restaurant j
Kthe number of word dishes
sji the number of time words in time restaurant j at time
table i
sj·the number of time words in time restaurant j
djl the number of time tables in time restaurant j serving
time dish l
dj·the number of tables in time restaurant j
oji the table indicator of the ith time word in time restaurant
j
lji the time dish indicator of the ith table in time restaurant
j
Sampling Word Tables The full conditional of a table indicator, zji , for a word, wji,
given all other words is:
p(zji =z, wji, tj i|zj i,wji,tji ,k,oji ,l) =
p(zji =z|zj i)
p(wji |tji, zj i =z, kj z =k, wji ,zji ,k,tji,oj i,l)
p(tji |zji =z, kj z =k, tji ,oji ,l)
(1)
Globally Continuous and Non-Markovian Activity Analysis from Videos 3
where
p(zji =z|zj i)(njz if z is an existing table
δotherwise (2)
p(wji |tji, zj i =z,kj z =k, wji ,zji ,k,tji,oji,l)
fkz(wji )if z exists
m·kfk(wji )else if k exists
ω fknew (wji )otherwise
(3)
where fis the conditional density of wji given all other variables. p(tji|zj i =z, kj z =
k, tji ,oji ,l)is the extra term from the time HDP that needs special treatment. If, for
every word, we do sampling conditioned on its time word in the time HDP , it is very
slow. So we marginalize over all time tables in the time restaurant.
p(tji |zji =z, kj z =k, tji ,oji ,l) =
dj·
X
o=1
p(oji =o|zj i =z, kjz =k, tji,oj i)
p(tji |oji =o, ljo =l, l)
(4)
p(oji =o|zj i =z, kjz =k,tji,oj i)
(sji if o exists
ζotherwise
(5)
p(tji |oji =o, ljoj i =l, l)
glo(tji )if o exists
d·lgl(tji )else if lexists
ε glnew (tji )otherwise
(6)
where gis the posterior predictive distribution of a Gaussian, a t-Distribution.
Sampling Word Dishes Sampling a word topic for a word table zin restaurant j, with
the associated words wjz and time words tjz, follows:
p(kjz =k, wj z ,tjz |wjz ,
tjz ,zjz ,kjz ,ojz,lj z)
(mjz
·kp(wjz |•)p(tjz |•)if k exists
ω p(wjz |•)p(tjz |•)otherwise
(7)
where means all the other variables the distribution is conditioned on. p(wjz|•) =
fk(wjz ). To fully compute p(tjz|•) = p(tj z|kz=k , ojz,lj z)is too expensive be-
cause table zmight have many words. So we randomly sample a number of them to
compute ˆp(tjz |•)as an approximation, which can be computed by Equation 4.
4 He Wang and Carol O’Sullivan
3 Additional Results
We show some additional patterns in the Forum dataset and TrainStation dataset in
Figure 3 and Figure 4.
Fig. 3. Additional Patterns in Forum dataset
References
1. Teh, Y.W., Jordan, M.I., Beal, M.J., Blei, D.M.: Hierarchical Dirichlet Processes. J. Am. Stat.
Assoc. 101(476) (2006) 1566–1581
Globally Continuous and Non-Markovian Activity Analysis from Videos 5
Fig. 4. Additional Patterns in TrainStationAdditional dataset
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  • Y W Teh
  • M I Jordan
  • M J Beal
  • D M Blei
Teh, Y.W., Jordan, M.I., Beal, M.J., Blei, D.M.: Hierarchical Dirichlet Processes. J. Am. Stat. Assoc. 101(476) (2006) 1566-1581