Information Technology Reference
In-Depth Information
2.1 Novelty Detection in Time Series
Considering the problem of NDinTS it is common to perceive the source of data as a
dynamical system with unknown dynamics. The input data are available in the form
of series of values gathered in consecutive time moments.
Def. 10.
A
(univariate) discrete time series
X
is a series of values generated by some
dynamical system in consecutive time moments labeled with natural numbers.
X: x
0
, x
1
,… , x
N
.
Sliding Window Procedure.
To reduce the problem to a time-insensitive variant a so
called sliding window procedure is used. This procedure has three integer parameters:
window length
m
; observation delay
τ
and offset
Δ
and as a result it produces a set of
observations.
Def. 11.
An
observation
x
t,m,
τ
is a vector of
m
consecutive values of a series of every
τ
-th value taken from
X
starting from moment
t
:
x
t,m,
τ
=
df
(x
t
, x
t+
τ
,
…, x
t+(m-1)
τ
)
The offset parameter does not influence the observation itself, but defines how far the
“window” is moved to generate another observation. If
x
t,m,
τ
, is the current
observation, then the next observation is
x
t+
Δ
,m,
τ
.
Novelty.
A novelty can be informally defined as every observation in the tested time
series
B
that is surprising given to the fact that
B
has been generated by the same
system as some exemplary series
A
. The concept of “surprise” is being formulated in
different ways in literature, depending on the considered problem and approach. The
most commonly used approach relies on the reduction of problem to the time-
insensitive version.
Def 12.
A
set of available observations
Obs
X,m,
τ
is a set of every
m
-sized observations
with a delay of
τ
in time series
X
:
{
}
U
1
Obs
=
x
X
,
m
,
τ
df
t
,
m
,
τ
0
≤
t
≤
N
−
m
+
We can then define the NDinTS as a ND problem in which the problem space is a
space of all
m
-sized observations and input data
S
=
Obs
A
,
m
,
2.2 Evaluation of Results
The estimated classification mapping introduces a separation of the problem space
P
onto two distinct subspaces
P
NDS+
,
P
NDS-
, where
P
NDS+
=
df
{e
∈
P|classify_est(NDS, e)
= novel}
and
P
NDS-
=
df
{e
P|classify_est(NDS, e) = normal}
. The optimal result is the
one in which this separation is equal the one introduced by
classify
, so the following
must be met:
Cond. 1.
P
NDS-
= P
-
(which is equivalent to
P
NDS+
= P
+
)
Def. 13.
A
perfect novelty detection system
NDS*
is a novelty detection system for
which the condition 1 is true.
∈
