Early Drift Detection Method

Abstract

An emerging problem in Data Streams is the detection of concept drift. This problem is aggravated when the drift is gradual over time. In this work we deflne a method for detecting concept drift, even in the case of slow gradual change. It is based on the estimated distribution of the distances between classiflcation errors. The proposed method can be used with any learning algorithm in two ways: using it as a wrapper of a batch learning algorithm or implementing it inside an incremental and online algorithm. The experimentation results compare our method (EDDM) with a similar one (DDM). Latter uses the error-rate instead of distance-error-rate.

Full text

Early Drift Detection Method
?
Manuel Baena-Garc´ıa
1
, Jos´e del Campo-
´
Avila
1
, Ra´ul Fidalgo
1
, Albert Bifet
2
,
Ricard Gavald`a
2
, and Rafael Morales-Bueno
1
1
Departamento de Lenguajes y Ciencias de la Computaci´on
E.T.S. Ingenier´ıa Inform´atica. Universidad de M´alaga, Spain
{mbaena, jcampo, rfm, morales}@lcc.uma.es
2
Universitat Polit`ecnica de Catalunya, Spain
{abifet, gavalda}@lsi.upc.edu
Abstract. An emerging problem in Data Streams is the detection of
concept drift. This problem is aggravated when the drift is gradual over
time. In this work we define a method for detecting concept drift, even in
the case of slow gradual change. It is based on the estimated distribution
of the distances between classification errors. The proposed method can
be used with any learning algorithm in two ways: using it as a wrapper
of a batch learning algorithm or implementing it inside an incremental
and online algorithm. The experimentation results compare our method
(EDDM) with a similar one (DDM). Latter uses the error-rate instead
of distance-error-rate.
1 Introduction
Many approaches in machine learning assume that training data has been gen-
erated from a stationary source. This assumption is likely to be false if the data
has been collected over a long period of time, as it is often the case: it is likely
that the distribution that generates the examples changes over time. In this case,
change detection becomes a necessity. Real applications examples are user mod-
elling, monitoring in biomedicine and industrial processes, fault detection and
diagnosis, safety of complex systems, etc.
We are interested in drift detection over data streams. Data streams are
unb ounded sequence of examples received at so high a rate that each one can
b e read at most once [1]. So, we can not store all the examples in memory, only
partially at maximum, and we can not spend so much time pro cessing data, due
to the high rate that it arrives.
In this work we present Early Drift Detection Method, a method to detect
concept drift, that gets good results with slow gradual changes. Abrupt changes
are easier to detect by current methods, but difficulties arise with the slow grad-
ual changes. Our method uses the distance between classification errors (number
of examples between two classification errors) to detect change, instead of using
?
This work has been partially supported by the FPI program and the MOISES-TA
project, number TIN2005-08832-C03, of the MEC, Spain.

errors classifications (as it is done in previous work [2]). It detects change faster,
without increasing the rate of false positives, and it is able to detect slow gradual
changes.
The pap er is organised as follows. The next section presents related work
in detecting concept drifting. In section 3 we present our method, Early Drift
Detection Method. In Section 4 we evaluate the method and Section 5 concludes
the pap er and present future work.
2 Related Work
To deal with change over time, most previous work has been classified observing
if they use full, partial or not examples memory.
The partial memory methods used variations of the sliding-window idea: at
every moment, one window (or more) containing the most recently read examples
is kept, and only those are considered relevant for learning. A critical p oint in any
such strategy is the choice of a window size. The easiest strategy is deciding (or
asking the user for) a window size W and keeping it fixed through the execution
of the algorithm (see e.g. [3–5]). In order to detect change, one can keep a
reference window with data from the past, also of some fixed size, and decide
that change has occurred if some statistical test indicates that the distributions
in the reference and current windows differ.
Another approach, using no examples memory, only aggregates, applies a
“decay function” to examples so that they become less important over time [6].
Other approach to detect concept drift monitors the values of three perfor-
mance indicators [7]: accuracy, recall and precision over time. Then they are
compared with a confidence interval of standard sample errors for a moving av-
erage value (using the last M batches) of each particular indicator. The key idea
is to select the window size so that the estimated generalisation error on new
examples is minimised. This approach uses unlabelled data to reduce the need
for labelled data, it doesn’t require complicated parameterization and it works
effectively and efficiently in practise.
A new method to detect changes in the distribution of the training examples
monitors the online error-rate of the algorithm [2]. In this method learning takes
place in a sequence of trials. When a new training example is available, it is
classified using the current model. The method controls the trace of the online
error of the algorithm. For the actual context they define a warning level, and
a drift level. A new context is declared, if in a sequence of examples, the error
increases reaching the warning level at example k
w
, and the drift level at example
k
d
. They take this is as an indication of a change in the distribution of the
examples. It uses 6σ ideas, well known in quality theory.
Our method it is based on the latter, but taking into account distances be-
tween classification errors as it is presented in the next section.

3 Drift Detection Methods
We consider that the examples arrive one at a time, but it would be easy to
assume that the examples arrive in bundles. In online learning approach, the
decision model must make a prediction when an example becomes available. Once
the prediction has been made, the system can learn from the example (using the
attributes and the class) and incorporate it to the learning model. Examples can
b e represented using pairs (
−→
x , y) where
−→
x is the vector of values for different
attributes and y is the class label. Thus, i-th example will be represented by
(
−→
x
i
, y
i
). When the current model makes a prediction (y
0
i
), it can be correct
(y
i
= y
0
i
) or not (y
i
6= y
0
i
).
3.1 DDM: Drift Detection Method [2]
There are approaches that pay attention to the number of errors produced by
the learning model during prediction. The drift detection method (DDM) pro-
p osed by Gama et al. [2] uses a binomial distribution. That distribution gives
the general form of the probability for the random variable that represents the
numb er of errors in a sample of n examples. For each point i in the sequence
that is being sampled, the error rate is the probability of missclassifying (p
i
),
with standard deviation given by s
i
=
p
p
i
(1 − p
i
)/i. They assume (as states the
PAC learning model [8]) that the error rate of the learning algorithm (p
i
) will
decrease while the number of examples increases if the distribution of the exam-
ples is stationary. A significant increase in the error of the algorithm, suggests
that the class distribution is changing and, hence, the actual decision model is
supposed to be inappropriate. Thus, they store the values of p
i
and s
i
when
p
i
+ s
i
reaches its minimum value during the process (obtaining p
min
and s
min
).
And it checks when the following conditions triggers:
– p
i
+s
i
≥ p
min
+2·s
min
for the warning level. Beyond this level, the examples
are stored in anticipation of a possible change of context.
– p
i
+ s
i
≥ p
min
+ 3 · s
min
for the drift level. Beyond this level the concept
drift is supposed to be true, the model induced by the learning method is
reset and a new model is learnt using the examples stored since the warning
level triggered. The values for p
min
and s
min
are reset too.
This approach has a good behaviour detecting abrupt changes and gradual
changes when the gradual change is not very slow, but it has difficulties when
the change is slowly gradual. In that case, the examples will be stored for long
time, the drift level can take too much time to trigger and the examples memory
can be exceeded.
3.2 EDDM: Early Drift Detection Method
The method that we propose in this paper, called Early Drift Detection Method
(EDDM), has been developed to improve the detection in presence of gradual

concept drift. At the same time, it keeps a good performance with abrupt concept
drift. The basic idea is to consider the distance between two errors classification
instead of considering only the number of errors. While the learning method is
learning, it will improve the predictions and the distance between two errors
will increase. We can calculate the average distance between two errors (p
0
i
) and
its standard deviation (s
0
i
). What we store are the values of p
0
i
and s
0
i
when
p
0
i
+2· s
0
i
reaches its maximum value (obtaining p
0
max
and s
0
max
). Thus, the value
of p
0
max
+ 2 · s
0
max
corresponds with the point where the distribution of distances
b etween errors is maximum. This point is reached when the model that it is
b eing induced best approximates the current concepts in the dataset.
Our method defines two thresholds too:
– (p
0
i
+ 2 · s
0
i
)/(p
0
max
+ 2 · s
0
max
) < α for the warning level. Beyond this level,
the examples are stored in advance of a possible change of context.
– (p
0
i
+ 2 · s
0
i
)/(p
0
max
+ 2 · s
0
max
) < β for the drift level. Beyond this level
the concept drift is supposed to be true, the model induced by the learning
method is reset and a new model is learnt using the examples stored since
the warning level triggered. The values for p
0
max
and s
0
max
are reset too.
The method considers the thresholds and searches for a concept drift when a
minimum of 30 errors have happened (note that it could appear a large amount
of examples between 30 classification errors). After occurring 30 classification
errors, the method uses the thresholds to detect when a concept drift happens.
We have selected 30 classification errors because we want to estimate the dis-
tribution of the distances b etween two consecutive errors and compare it with
future distributions in order to find differences. Thus, p
0
max
+ 2 · s
0
max
represents
the 95% of the distribution. For the experimental section, the values used for α
and β have been set to 0.95 and 0.90. These values have been determined after
some exp erimentation.
If the similarity between the actual value of p
0
i
+2· s
0
i
and the maximum value
(p
0
max
+ 2 · s
0
max
) increase over the warning threshold, the stored examples are
removed and the method returns to normality.
4 Experiments And Results
In this section we describe the evaluation of the proposed method: EDDM.
The evaluation is similar to the one proposed in [2]. We have used three dis-
tinct learning algorithms with the drift detection methods: a decision tree and
two nearest-neighbourhood learning algorithms. These learning algorithms use
different representations to generalise examples. The decision tree uses DNF
to represent generalisation of the examples. The nearest-neighbourhood learn-
ing algorithms uses examples to describe the induced knowledge. We use the
weka implementation [9] of these learning algorithms: J48[10] (C4.5, decision
tree), IB1[11] (nearest-neighbourho od, it is not able to deal with noise) and
NNge[12] (nearest-neighbourhood with generalisation). We have used four artifi-
cial datasets previously used in concept drift detection [13], a new data set with

very slow gradual change and a real-world problem [14]. As we want to know
how our algorithms work in different conditions, we have chosen those artificial
datasets that have several different characteristics - abrupt and gradual drift,
presence and absence of noise, presence of irrelevant and symbolic attributes,
numerical and mixed data descriptions.
4.1 Artificial Datasets
The five artificial datasets used later (Subsections 4.2 and 4.3) are briefly de-
scribed. All the problems have two classes and each class is represented by 50%
of the examples in each context. To ensure a stable learning environment within
each context, the positive and negative examples in the training set are alter-
nated. The number of examples in each concept is 1000, except in Sine1g that
have 2000 examples in each concept and 1000 examples to transit from one
concept to another.
– SINE1. Abrupt concept drift, noise-free examples. The dataset has two rele-
vant attributes. Each attribute has values uniformly distributed in [0, 1] . In
the first concept, points that lie below the curve y = sin(x) are classified as
p ositive, otherwise they are labelled as negative. After the concept change
the classification is reversed.
– CIRCLES. Gradual concept drift, noise-free examples. The examples are la-
b elled according to a circular function: if an example is inside the circle, then
its label is positive, otherwise is negative. The gradual change is achieved by
displacing the centre of the circle and growing its size. This dataset has four
contexts defined by four circles:
Center (0.2,0.5) (0.4,0.5) (0.6,0.5) (0.8,0.5)
Radius 0.15 0.2 0.25 0.3
– GAUSS. Abrupt concept drift, noisy examples. The examples are labelled ac-
cording to two different but overlapped gaussian density functions (N([0, 0], 1)
and N([2, 0], 4)). The overlapping can be considered as noise. After each con-
text change, the classification is reversed.
– MIXED. Abrupt concept drift, boolean noise-free examples. Two boolean
attributes (v, w) and two numeric attributes (x, y). The examples are clas-
sified positive if at least two of the three following conditions are satisfied:
v, w, y < 0.5 + 0.3 sin(3 πx). After each concept change the classification is
reversed.
– SINE1G. Very slow gradual drift, noise-free examples. This dataset remains
the same as Sine1, but the concept drift is made by gradually choosing
of examples from the old and the new concept. So, there is a transition
time between concepts. The probability of selecting an example from the old
concept becomes lower gradually and the probability of selecting an example
from the new concept becomes higher when the transition time is ended.

0
0.05
0.1
0.15
0.2
0.25
0.3
0 1000 2000 3000 4000
IB1
Prequential Error
Nr. Examples
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1000 2000 3000 4000
IB1
Prequential Error
Nr. Examples
EDDM
DDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1000 2000 3000 4000
J48
Prequential Error
CIRCLES
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1000 2000 3000 4000
J48
Prequential Error
CIRCLES
EDDM
DDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2000 4000 6000 8000 10000
Nr. Examples
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2000 4000 6000 8000 10000
Nr. Examples
EDDM
DDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2000 4000 6000 8000 10000
SINE1
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2000 4000 6000 8000 10000
SINE1
EDDM
DDM
Fig. 1. Prequential error of J48 (up) and IB1 (down) on Circles (left) and Sine1 (right)
datasets
4.2 Results On Artificial Domains: Abrupt And Gradual Drifts
The purpose of these experiments is to analyse how the proposed drift detection
method works with different learning algorithms, and compare it to the one
proposed by Gama et al. [2] (we refer to it as DDM, Drift Detection Method) in
terms of prequential [15] error (prequential error is the mean of the classification
errors obtained with the examples to be learnt). It is not the aim of this article
to compare the results of the different learning algorithms used. Figure 1 shows
the prequential error results when Circles and Sine1 datasets are faced with two
learning algorithms with the proposed drift detection method (EDDM) and with
DDM. Note that changes occur every 1000 examples.
We can observe that prequential error curves obtained by EDDM and DDM
on Sine1 dataset (an abrupt changing dataset) are almost the same. When they
deal with a gradual dataset (Circles) the results of both methods are very similar,
independently of the learning method used.
Table 1 presents the prequential error results and the total number of changes
detected by the learning algorithms with both drift detection methods at the
end of each dataset. On datasets with abrupt concept change, both algorithms
react quickly and reach low error rates. When noise is present, EDDM is more
sensitive than DDM, detecting changes and improving the performance when the
base algorithm does not support noise (i.e. IB1). This is so because after some
time (some number of examples) the base algorithms overfit and the frequency
of classification errors increases.

Table 1. Prequential error and total number of changes detected by methods
EDDM DDM
Prequential Drifts Prequential Drifts
IB1 0.0340 3 0.0343 3
Circle J48 0.0421 3 0.0449 3
NNge 0.0504 4 0.0431 4
IB1 0.1927 32 0.1888 9
Gauss J48 0.1736 22 0.1530 9
NNge 0.1763 31 0.1685 12
IB1 0.0322 9 0.0330 10
Mixed J48 0.0425 9 0.0449 11
NNge 0.0639 10 0.0562 9
IB1 0.0376 9 0.0377 9
Sine1 J48 0.0847 11 0.0637 10
NNge 0.0819 14 0.0767 11
Table 2. Prequential error and total number of changes detected by methods in Sine1g
dataset
EDDM DDM
Prequential Drifts Prequential Drifts
IB1 0.1462 50 0.2107 12
Sine1g J48 0.1350 34 0.1516 12
NNge 0.2104 28 0.2327 12
4.3 Results On Artificial Domains: Slow Gradual Drift
There are many real-world problems that have slow gradual drift. In this section
we use the Sine1g dataset to illustrate how EDDM works with this kind of
change. Figure 2 shows the prequential error curves obtained when EDDM and
DDM deal with this dataset. The plots on the left are prequential error calculated
with the whole dataset, and the plots on the right are prequential error calculated
from scratch after every concept change detected by both methods.
Although the global curves are similar, the local prequential curves show that
EDDM reacts before and more times than DDM when a slow concept drift is
present. During the transition from the previous concept to the next, EDDM
detects repeatedly concept drifts. This is so because the frequency of classifi-
cation errors continuously increase until the next concept is stable. Meanwhile,
DDM shows less sensitivity to this kind of problems, reacting later and less
than EDDM. Table 2 presents the final prequential errors and number of drifts
obtained by these two metho ds on Sine1g dataset.

0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
IB1
Prequential Error
Nr. Examples
EDDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
IB1
Prequential Error
Nr. Examples
EDDM
DDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
J48
Prequential Error
Global
EDDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
J48
Prequential Error
Global
EDDM
DDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
Nr. Examples
EDDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
Nr. Examples
EDDM
DDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
Local
EDDM
0
0.2
0.4
0.6
0.8
1
0 9000 18000 27000
Local
EDDM
DDM
Fig. 2. Prequential global error (left) and local error (right) for EDDM and DDM in
Sine1g dataset
4.4 The Electricity Market Dataset
The data used in this exp eriment was first described by M. Harries [14]. The data
was collected from the Australian New South Wales Electricity Market. In this
market, the prices are not fixed and they are affected by demand and supply of
the market. A factor for the price evolution is the time evolution of the electricity
market. During the time period described in the data the electricity market was
expanded with the inclusion of adjacent areas. This produced a more elaborated
management of the supply. The production surplus of one region could be sold
in the adjacent region. A consequence of this expansion was a dampener of the
extreme prices. The ELEC2 dataset contains 45312 instances dated from May
1996 to December 1998. Each example of the dataset refers to a period of 30
minutes. Each example on the dataset has 5 fields, the day of week, the time
stamp, the NSW electricity demand, the Vic electricity demand, the scheduled
electricity transfer between states and the class label.
The class label identifies the change of the price related to a moving average
of the last 24 hours. The class level only reflect deviations of the price on a one
day average and removes the impact of longer term price trends. The interest of
this dataset is that it is a real-world dataset. We do not know when drift occurs
or if there is drift. We have considered this problem as a short term prediction:
predict the changes in the prices relative to the next 30 minutes.
In Figure 3 we present the traces of the prequential error rate of EDDM and
DDM, with the base learning algorithms, through the full ELEC2 dataset. As

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 10000 20000 30000 40000
Prequential Error
Nr. Examples
J48
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 10000 20000 30000 40000
Prequential Error
Nr. Examples
J48
EDDM
DDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 10000 20000 30000 40000
Nr. Examples
IB1
EDDM
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 10000 20000 30000 40000
Nr. Examples
IB1
EDDM
DDM
Fig. 3. Prequential error for EDDM and DDM in ELEC2 dataset
Table 3. Prequential error and total number of changes detected by methods in ELEC2
dataset
EDDM DDM
Prequential Drifts Prequential Drifts
IB1 0.1426 171 0.2764 44
Elect2 J48 0.1564 187 0.2123 10
NNge 0.1594 193 0.2110 130
can be seen, EDDM outperforms DDM, detecting concept changes earlier and
with a better sensitivity. Table 3 shows the final prequential errors and number
of drifts obtained by these two methods on the electricity market dataset.
5 Conclusion
This paper introduces a new method for detecting concept drifts based on the
distances between classification errors. This method achieves an early detection
in presence of gradual changes, even when that change is very slow. The results
that are presented have been obtained after using the method as a wrapper for
different learning algorithms, but it would b e easy to implement it in a local
way inside those algorithms. The experimental evaluation of EDDM illustrates
the advantages of using this detection method. As well as obtaining good results
detecting concept drifts, this method shows itself as a way to deal with noisy
datasets even when the base algorithm is not designed with that aim. When the
base algorithm begins to overfit, the frequency of classification errors begins to
increase and that is detected by the proposed method.

Out aim of improving EDDM involves some issues where the most important
is finding a way to determine the values for the parameters of the method (α
and β) in an automatic way.
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