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hereafter we discuss only sampling-based summaries ... information-based summaries aim at suppressing this bias .... sharp steps in the bursts ..... Page 67 ...
information-based data stream summary
Fabrice Clérot, Pascal Gouzien Orange Labs
France Telecom Group
agenda
2
data-stream summary
information-based summary
performance with a constant memory constraint
performance with a time-varying memory constraint
related work : reglo
including weights in the compression rate
conclusion
references
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
agenda
3
data-stream summary
information-based summary
performance with a constant memory constraint
performance with a time-varying memory constraint
related work : reglo
including weights in the compression rate
conclusion
references
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
data stream
an infinite sequence of events
event : – event.timestamp = t ∈ T – event.data = X ∈ X – data space X unspecified at this point – denoted X(t) but not necessarily a "time series"
"minimal" assumptions – events are observed in increasing timestamp order – events are not lost
4
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
generic data stream summary [collectif midas]
a data structure designed so as to keep "as much information as possible" on the stream under – memory usage constraints – computation time constraints – for the on-line maintenance of the summary – for the off-line answer to queries
5
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
generic data stream summary [collectif midas]
"as much information as possible" : – allow the computation of the (approximate) answer to any query on the past of the stream – allow the (approximate) density estimation of TxX would reach the target – allow the computation of error bounds on the answer to any query (with respect to the memory and cpu constraints)
6
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
generic data stream summary
hereafter we discuss only sampling-based summaries – computationaly fast – memory usage constraint naturally addressed – resample if necessary – samples are ok for approximate density estimation – elements kept in the sample are given a weight inversely proportional to the sampling rate they experienced – query processing on a sample is as fast as on the original data
examples – random sampling – reservoir sampling [vitter] – streamsamp [csernel et al], [gabsi et al]
7
Orange Labs - Research & Development - IBDSS – 17/11/2010
two main characteristics : • the volume of the sample • the weight of the elements of the sample as a function of • the current time (or size of the stream seen so far) : t • the "temporal depth" of the query : τ 8
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
sampling-based summaries [gabsi, phd thesis] t-τ
0
query t present
random sampling reservoir sampling streamsamp
9
volume
sample weight
(t)
(t, τ)
O(t)
constant
constant O(Log(t))
Orange Labs - Research & Development - IBDSS – 17/11/2010
O(t) independent of τ O(τ) independent of t
France Telecom Group
sampling-based summaries
streamsamp tries get the best on both worlds – slow increase in volume – sample weight increase with respect of the age of the sample measured from present time, independent of the duration of the stream
but streamsamp has a strong deterministic bias against the past
information-based summaries aim at suppressing this bias – sample where sampling degrades the signal as little as possible
10
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
sampling-based summaries t-τ
0
query t present
random sampling reservoir sampling streamsamp information based 11
volume
sample weight
(t)
(t, τ)
O(t)
constant
constant O(Log(t)) tunable
Orange Labs - Research & Development - IBDSS – 17/11/2010
O(t) independent of τ O(τ) independent of t optimal
France Telecom Group
agenda
12
data-stream summary
information-based summary
performance with a constant memory constraint
performance with a time-varying memory constraint
related work : reglo
including weights in the compression rate
conclusion
references
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
principle in the case of a constant memory constraint
summary is built from S windows of F elements – volume is limited to S*F elements – windows are samples of a time period on the stream – windows are ordered accordingly
W5
W4
W3
W2
W1
0
13
W0 t
incoming stream events are stored in an input window
when the input window W0 is full, room is made by merging two adjacent windows
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
merging of two adjacent windows W1
W2
stratified sampling with respect to the sample weights in each window – F*w1/(w1+w2) elements from W1 – F*w2/(w1+w2) elements from W2
these elements form a new window W* of size F
the sample weight of each sample in the resulting window is the sum of the sample weights – w* = w1 + w2
W* 14
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
merging strategy W(i)
W(i+1)
classifier classification performance
merge undistinguishable windows !
two adjacent windows are considered as two different labels
use a classifier to learn the label from the data
the worse the classification performance, the more undistinguishable the windows are – merge the window pair with the minimum classification performance
Perf(i) 15
Orange Labs - Research & Development - IBDSS – 17/11/2010
France Telecom Group
merging strategy W(i)
W(i+1)
Perf(i) is unchanged if W(i) and W(i+1) are not merged
when W(0) is full – compute Perf(0) – find j = argmini=0…S-1 Perf(i) – merge W(j) and W(j+1) into W* – W(j+1) W* – for i
