change detection in high-dimensional datastreamsΒ Β· π0βπ1 sklπ0,π1 =klπ0,π1...
TRANSCRIPT
-
mailto:[email protected]
-
π‘
-
Spam Classification
Ε½ Δ
-
π π‘ , π‘ = π‘0, β¦ , π π‘ β βπ
π₯(π‘) ππ
π π‘ βΌ ΰ΅π0 normal dataπ1 anomalies
,
π‘
π(π‘)
β¦β¦
π1π0 π0
-
π π‘ , π‘ = π‘0, β¦ , π π‘ β βπ
π₯(π‘) ππ
π π‘ βΌ ΰ΅π0 normal dataπ1 anomalies
,
π‘
π(π‘)
β¦β¦
π1π0 π0
-
π π‘ , π‘ = 1,β¦ , π π‘ β βπ
π
π π‘ βΌ ΰ΅π0 π‘ < ππ1 π‘ β₯ π
,
{π π‘ , π‘ < π} π0 β π1
ππ β π1
π‘
π(π‘)
β¦β¦
π1π0
π
-
π π‘ , π‘ = 1,β¦ , π π‘ β βπ
π
π π‘ βΌ ΰ΅π0 π‘ < ππ1 π‘ β₯ π
,
{π π‘ , π‘ < π} π0 β π1
ππ β π1
π‘
π(π‘)
β¦β¦
π1π0
π
-
ππ β π1 β π2 β π3 β π4
ππ π1 π2 π3 π4
-
π‘
π(π‘)
β¦β¦
π1π0 π0
-
π‘
π(π‘)
β¦β¦
π1π0
-
π0 π1
-
ππ
π»0: π = π0 π»1: π = π1
Ξ π₯ =π1(π₯)
π0(π₯)
Ξ π₯ > πΎ πΎ
-
Ξ(π)
π‘
π(π‘)
β¦β¦
π1π0 π0
π‘
π¬(π)
β¦
πΎ
-
log Ξ π₯ = logπ1(π₯)
π0(π₯)= α
< 0 when π0 π₯ > π1(π₯)> 0 otherwise
π π‘ = max 0, S t β 1 + log Ξ π₯(π‘)
π(π‘) > Ξ³
-
Ξ(π)
π‘
π(π‘)
β¦β¦
π1π0
π‘
ππ‘
+
β¦
πΎ
-
π0π1
-
ππ
ππ
ππ .
ππ
-
+ β
ππ = π(π‘), π¦(π‘) , π‘ < π‘0, π₯ β βπ , π¦ β {+,β}
π¦ ππ
π¦(π)ππ¦ β π
-
ππ
ππ = π₯ π‘ , π‘ < π‘0, π₯ βΌ π0
-
π0ππ = π₯ π‘ , π‘ < π‘0, π₯ βΌ π0
π0 π < π
-
ππ
ππ = π₯ π‘ , π‘ < π‘0
ππ
-
π β
-
π β
-
π β
-
π0 π1
-
π0 π1π0 β π1 π0 β π1
-
π0 π1 π0 β π1
-
π0 π1 π0 β π1
π΄π πΏ0
-
π0 ππ
β π π‘ = log( π0(π(π‘)))
β π π‘ , π‘ = 1,β¦
π‘
π(π‘)
β¦
βππ‘
β¦
-
π0 ππ
β π π‘ = log( π0(π(π‘)))
β π π‘ , π‘ = 1,β¦
-
β³
β³ ππ
π β³
-
β³
β³ ππ
π β³
Dictionary learned from normal ECG signal (sparse representations)
-
πΌ π β³
π π‘ = π ββ³(πΌ) 2
: β³πΆ: π
-
π
πͺ 1 ,
π(π‘)
β¦ β¦π
π
-
π
π0
π(π‘)
β¦ β¦
π
π
-
π
π‘
π(π‘)
β¦β¦
π1π0
π
π
-
π‘
π(π‘)
β¦β¦
π
π1π0
π
π
-
π‘
π(π‘)
β¦β¦
π
π1π0
π
π
-
π
π0 π1
-
π
π0 π1
π
-
π
π0 π1
-
π0 ππ
β π π‘ = log( π0(π(π‘)))
β π π‘ , π‘ = 1,β¦
π‘
π(π‘)
β¦
βππ‘
β¦
-
SNR π0 β π1 =
EπβΌπ0
β(π) β EπβΌπ1
β(π)2
varπβΌπ0
β(π) + varπβΌπ1
β(π)
π0 β π1E β(π)
-
π0 β π1
sKL π0, π1 = KL π0, π1 + KL π1, π0 =
= ΰΆ± logπ0 π
π1 ππ0 π ππ + ΰΆ± log
π1 π
π0 ππ1 π ππ
sKL π0, π1π0 β π1 sKL π0, π1
π0 β π1 π1 β π0
T. Dasu, K. Shankar, S. Venkatasubramanian, K. Yi, βAn information-theoretic approach to detecting
changes in multi-dimensional data streamsβ In Proc. Symp. on the Interface of Statistics, Computing
Science, and Applications, 2006
-
π0 = π©(π0, Ξ£0) π1 π = π0(ππ + π)π β βπΓπ π β βπ
SNR π0 β π1 <πΆ
π
πΆ sKL π0, π1
-
π0 = π©(π0, Ξ£0) π1 π = π0(ππ + π)π β βπΓπ π β βπ
SNR π0 β π1 <πΆ
π
πΆ sKL π0, π1
sKL π0, π1π
π0
π0
-
π0 = π©(π0, Ξ£0) π1 π = π0(ππ + π)π β βπΓπ π β βπ
SNR π0 β π1 <πΆ
π
πΆ sKL π0, π1
-
π1 π = π0(ππ + π)
π0 +π
π0
π1
-
π1 π = π0(ππ + π)
π0 +π
π ππ
π0
π1
-
π1 π = π0(ππ + π)
π0 +π
π ππ
π0| π |
π0
π1
-
π0 = π©(π0, Ξ£0) π1 π = π0(ππ + π)π β βπΓπ π β βπ
SNR π0 β π1 <πΆ
π
πΆ sKL π0, π1
-
π0 = π©(π0, Ξ£0)
β β
-
π
π0π0 β π1
π1 π = π0 ππ + π and sKL π0, π1 = 1
π0 π1
π‘π‘
π(π‘)
π1π0
-
β π0(π) π0 π1, π0 π1
π―(π0,π1)
π― π0,π1 βΆ β
β
π‘
βππ‘
π1π0
π‘
βππ‘
-
Gaussians
β’ π1
π
Lepage log(π0(β ))
Lepage log( π0(β ))
t-test log(π0(β ))
t-test log( π0(β ))
-
Particle Wine
-
Particle Wine
β’ π1sKL π0, π1 β 1
β’ π0 π
β(β )
https://home.deib.polimi.it/carrerad/projects.html
https://home.deib.polimi.it/carrerad/projects.html
-
π