dressed through a split and merge criterion. However,
these methods are either too slow for online learning
(Hall and Hicks, 2005), assume that data arrives in
chunks (Song and Wang, 2005) or does not guarantee
the ﬁdelity of the resulting model (Arandjelovic and
Cipolla, 2005). Here we propose a new efﬁcient on
line method that explicitly guarantees the accuracy of
the model through a ﬁdelity criterion.
2.1 Update of the Gaussian Mixture
Model
Suppose we have already learned a precise GMM
from the observations up to time t:
p
t
(x) =
∑
N
i=1
π
t
i
g(x;µ
t
i
,C
t
i
)
∑
N
i=1
π
t
i
(1)
where each Gaussian is represented by its weight
π
t
i
, its mean µ
t
i
and its covariance C
t
i
. We then re
ceive a new data point represented by its distribution
g
t
(x;µ
t
,C
t
) and its weight π
t
. C
t
here represents the
observation noise. As suggested by Hall and Hicks
(Hall and Hicks, 2005), the new resulting GMM is
computed in two steps:
1. Concatenate – produce a model with N + 1 com
ponents by trivially combining the GMM and the
new data into a single model.
2. Simplify – if possible, merge some of the Gaus
sians to reduce the complexity of the GMM.
The GMM resulting from the ﬁrst step is simply
p
t
(x) =
∑
N
i=1
π
t−1
i
g(x;µ
t−1
i
,C
t−1
i
) + π
t
g
t
(x;µ
t
,C
t
)
∑
N
i=1
π
t−1
i
+ π
t
(2)
The goal of the second step is to reduce the complex
ity of the model while still giving a precise descrip
tion of the observations. Hall and Hicks (Hall and
Hicks, 2005) propose to group the Gaussians using
the Chernoff bound to detect overlapping Gaussians.
Different thresholds on this bound are then tested and
the most likely result is kept as the simpliﬁed GMM.
Since this method is too slow for an online process,
we use a different criterion proposed by Declercq and
Piater (Declercq and Piater, 2007) for their uncertain
Gaussian model. This model provides a quantitative
estimate λ of its ability to describe the associated data
that takes on a value close to 1 if the data distribu
tion is Gaussian and near zero if it is not. This value,
called the ﬁdelity in the sequel, is useful to decide if
we can merge two given Gaussians without drifting
from the real data distribution.
2.2 Estimating the Fidelity of a
Gaussian Model
To estimate the ﬁdelity λ of a Gaussian model, we ﬁrst
need to compute the distance between this model and
its corresponding data set. This is done with a method
inspired from the KolmogorovSmirnoff test,
D =
1

I

Z
I
ˆ
F(x) − F
n
(x)
dx, (3)
where F
n
(x) is the empirical cumulative distribution
function of the n observations,
ˆ
F(x) is the correspond
ing cumulative Gaussian distribution, and I is the in
terval within which the two functions are compared.
To simplify matters, the distance D is assumed to have
a Gaussian distribution, which leads to the pseudo
probabilistic weighting function
λ = e
−D
2
T
2
D
, (4)
where T
D
is a usersettable parameter that represents
the allowed deviation of observed data from Gaus
sianity. Whereas the sensitivity of the Kolmogorov
Smirnov test grows without bounds with n, λ provides
a bounded quantiﬁcation of the correspondence be
tween the model and the data. Therefore, this crite
rion is more appropriate for our case since we need
to estimate the correspondence of the data with the
model and not their possible convergence to a Gaus
sian distribution.
Thus, the original data are not required anymore
if we keep in memory an approximation of their cu
mulative distribution within a given interval. Since
the number of dimensions of the data space can be
large, we compute the distance D for each dimension
separately to keep the computational cost linear in the
number of dimensions. The total distance is then sim
ply the sum of these individual distances.
2.3 Simpliﬁcation of the Gaussian
Mixture Model
To decide whether two Gaussians G
i
and G
j
can
be simpliﬁed into one, we merge them together and
check if the resulting Gaussian has a ﬁdelity λ close
to one, say, exceeding a given threshold λ
+
min
= 0.95.
The resulting Gaussian is computed using the usual
equations supplemented by the combination of the cu
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