Quickest Change Detection in Multiple On-Off ... - Semantic Scholar
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
1
Quickest Change Detection in Multiple On-Off Processes Qing Zhao,
Jia Ye
Department of Electrical and Computer Engineering University of California, Davis, CA 95616
Supported by NSF and ARL-CTA.
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
2
Quickest Change Detection Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XT0 XT0 +1
Quickest Detection: min
E[(Td − T0)+] {z } |
subject to
Detection Delay I
XTd
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
Tradeoff: Detection delay vs. detection reliability.
Pr[Td < T0] ≤ ζ {z } |
Reliability Constraint
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
3
Quickest Change Detection Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
Quickest Detection: min
XTd
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
XT0 XT0 +1
E[(Td − T0)+] {z } |
subject to
Detection Delay
Pr[Td < T0] ≤ ζ {z } |
Reliability Constraint
2 Bayesian:
Shiryaev’61, Borovkov’98, Tartakovsky&Veeravalli’05.
2 Minimax:
CUSUM (Page’54, Lorden’71).
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
4
Application in Cognitive Radio Sensing starts
Opp. arises
Sensing ends
PSfrag replacements Opp. ends
Tx starts
T0
0 I
Primary Transmission
Td
t
Measurements: {X1, X2, . . . , XT0−1} are i.i.d with distribution f0(x); {XT0 , XT0 +1, . . . }
are i.i.d with distribution f1(x).
I
Stopping Time: At time t = Td, the user declares an opportunity.
I
Quickest Detection: min
E[(Td − T0)+] {z } |
Detection Delay
subject to
Pr[Td < T0] ≤ ζ {z } |
Interference Constraint
t
t
t
to switch: loss of data vs. avoiding bad realizations.
2 Whether
to declare: delay vs. reliability.
t
2 Whether
switch, or declare? 2 Continue,
Tradeoffs: I
Quickest Detection of Idle Periods in Multiple On-Off Processes: I
5 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection in Multiple On-Off Processes
PSfrag replacements
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Climbing The Corporate Ladder Keep climbing?
Or look for a greener pasture?
6
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
7
Outline
I
Quickest change detection in a single stochastic process 2
I
Shiryaev’s algorithm
Quickest detection in multiple on-off processes 2
A decision-theoretic formulation
2
The optimal detection rule: a threshold policy
I
Simulation examples
I
Conclusion and work in progress
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
8
Quickest Change Detection: Classic Bayesian Formulation Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XT0 XT0 +1
i.i.d.∼ f0(x)
···
i.i.d. ∼ f1(x)
Bayesian Formulation: I
Priori distribution of change point T0: geometric Pr[T0 = 0] = λ0 Pr[T0 = k] = (1 − λ0)p(1 − p)k−1 , ∀k > 0,
XTd
···
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
9
Shiryaev’s Algorithm Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XTd
A sufficient statistic: a posterior probability that change has occurred λt , Pr [T0 ≤ t|X1 , X2, . . . , Xt].
I
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
XT0 XT0 +1
Shiryaev’s detection rule: Td = inf{t : λt ≥ ηd}
I
Detection threshold ηd: determined by the reliability constraint ζ .
I
Setting ηd = 1 − ζ is asymptotically optimal as ζ → 0.
t
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mI =
!
t #
PSfrag replacements
1 pI 1 pB
mB =
10 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection In Multiple On-Off Processes
t
I Fraction of idle time: λ0 = mIm+m . B
I
Idle period: geometrically distributed with mean mI = p1I .
I
t
Busy period: geometrically distributed with mean mB = p1B .
I
A large number of independent homogeneous on-off processes.
I
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Detection Time
PSfrag replacementst )
Reliability Constraint
} {z l=1
| } {z l=1
|
Switch TS (L − 1)
Ts(l) + Td(L)) = busy] ≤ ζ Pr[SL( s.t. Ts(l) + Td(L)] min E[
)
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TS (2)
L−1 X
L−1 X
Switch
Channel 1 Channel 2 Channel L − 1 t Channel L Switch TS (1)
11 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection In Multiple On-Off Processes
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
12
A POMDP Formulation I
State Transition: S/λ0
S/1 − λ0
PSfrag replacements S/λ0
C/pB
C/1 − pB
0
1
(Busy)
(Idle) S/1 − λ0
C/1 − pI
C/pI D/1
D/1
∆
(Absorbing)
1
I
Cost: 2
Switch or Continue: 1
2
Declare during a busy period: γ
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
13
A POMDP Formulation
I
A Sufficient Statistic: the information state (belief) λt = Pr[Zt = idle|X1 , X2, . . . , Xt] mI λ0 = mI + m B
I
Update of the Information State λt =
I T (λ|x):
(
T (λ0|x)
a(t − 1) = S, Xt = x
T (λt−1 |x) a(t − 1) = C, Xt = x
.
updated information state based on the new measurement x. ¯ B )f1(x) (λ¯ pI + λp T (λ|x) = ¯ B )f1(x) + (λpI + λ¯ ¯ pB )f0(x) . (λ¯ pI + λp ∆
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
14
A POMDP Formulation
I
Channel switching and change detection policy π :
λt ∈ [0, 1] I
=⇒
a(t) ∈ {S, C, D}, for each time t.
Quickest change detection:
π ∗ = arg min Eπ [ π
∞ X
mI Rπ(λt) | λ0 = ], mB + mI | {z } t=0 Cost
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
15
Quickest Change Detection: Value Functions I V (λt):
the minimum expected total cost when the current info. state is λt. V (λt) = min{ VC (λt) , | {z }
VS (λt) , | {z }
VD (λt) }. | {z }
Continue Switch Declare
I VC (λt):
the minimum expected total cost if continue. VC (λt) = 1 +
I VS (λt):
x Pr[
P (x; λ ) V (T (λt|x))dx | {z t} observe x under λt ]
the minimum expected total cost if switch. VS (λt) = 1 +
I VD (λt):
Z
Z
x Pr[
P (x; λ ) V (T (λ0|x))dx = VC (λ0) | {z 0} observe x under λ0 ]
the minimum expected total cost if declare. VD (λt) = (1 − λt)γ.
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
16
Quickest Change Detection: A Threshold Policiy I VD (λt)
is linear.
is concave and monotonically decreasing if m1 + m1 ≤ 1. B I I I VS (λt) = VC (λ0), where λ0 = m m+m . I B I VC (λt)
γ
PSfrag replacements
VD (λt)
VS (λt)
VC (λt)
0
λ0
ηd
1
λt
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
17
Quickest Change Detection: A Threshold Policiy V (λt) = min{ VS (λt) , | {z }
VC (λt) , | {z }
VD (λt) }. | {z }
Switch Continue Declare
PSfrag replacements
γ VD (λt) VS (λt)
VC (λt)
Switch 0
ηs = λ 0
Continue
Declare ηd
1
λt
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Simulation Examples
I f0(x), f1(x):
Gaussian with zero mean and different variances.
I SN R = 10dB .
I ηd = 1 − ζ .
18
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
19
Simulation Example: Geometric Distribution Increase both mB and mI while keeping λ0 fixed 60
Single−Channel Strategy Multi−Channel Strategy
50 Average Detection Time
I
40 30 20 10 0 0
50
100 Average Busy Time
150
200
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
20
Simulation Example: Arbitrary Distributions Busy period: truncated Pareto distribution with increasing tail index 500 450
Single−Channel Strategy Multi−Channel Strategy
400 Average Detection Time
I
350 300 250 200 150 100 50 0 400
500
600
700 800 900 Average Busy Time mB
1000
1100
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
21
Conclusion and Work in Progress PSfrag replacements
Quickest Detection in Multiple On-Off Processes: γ
VD (λt) VS (λt)
Switch 0
Declare
Continue
ηs = λ 0
ηd
VC (λt)
1
λt
Work in Progress: 2 Asymptotic 2 Minimax
optimality for arbitrary distributions and non-i.i.d. data.
formulation.
1
Quickest Change Detection in Multiple On-Off Processes Qing Zhao,
Jia Ye
Department of Electrical and Computer Engineering University of California, Davis, CA 95616
Supported by NSF and ARL-CTA.
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
2
Quickest Change Detection Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XT0 XT0 +1
Quickest Detection: min
E[(Td − T0)+] {z } |
subject to
Detection Delay I
XTd
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
Tradeoff: Detection delay vs. detection reliability.
Pr[Td < T0] ≤ ζ {z } |
Reliability Constraint
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
3
Quickest Change Detection Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
Quickest Detection: min
XTd
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
XT0 XT0 +1
E[(Td − T0)+] {z } |
subject to
Detection Delay
Pr[Td < T0] ≤ ζ {z } |
Reliability Constraint
2 Bayesian:
Shiryaev’61, Borovkov’98, Tartakovsky&Veeravalli’05.
2 Minimax:
CUSUM (Page’54, Lorden’71).
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
4
Application in Cognitive Radio Sensing starts
Opp. arises
Sensing ends
PSfrag replacements Opp. ends
Tx starts
T0
0 I
Primary Transmission
Td
t
Measurements: {X1, X2, . . . , XT0−1} are i.i.d with distribution f0(x); {XT0 , XT0 +1, . . . }
are i.i.d with distribution f1(x).
I
Stopping Time: At time t = Td, the user declares an opportunity.
I
Quickest Detection: min
E[(Td − T0)+] {z } |
Detection Delay
subject to
Pr[Td < T0] ≤ ζ {z } |
Interference Constraint
t
t
t
to switch: loss of data vs. avoiding bad realizations.
2 Whether
to declare: delay vs. reliability.
t
2 Whether
switch, or declare? 2 Continue,
Tradeoffs: I
Quickest Detection of Idle Periods in Multiple On-Off Processes: I
5 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection in Multiple On-Off Processes
PSfrag replacements
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Climbing The Corporate Ladder Keep climbing?
Or look for a greener pasture?
6
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
7
Outline
I
Quickest change detection in a single stochastic process 2
I
Shiryaev’s algorithm
Quickest detection in multiple on-off processes 2
A decision-theoretic formulation
2
The optimal detection rule: a threshold policy
I
Simulation examples
I
Conclusion and work in progress
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
8
Quickest Change Detection: Classic Bayesian Formulation Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XT0 XT0 +1
i.i.d.∼ f0(x)
···
i.i.d. ∼ f1(x)
Bayesian Formulation: I
Priori distribution of change point T0: geometric Pr[T0 = 0] = λ0 Pr[T0 = k] = (1 − λ0)p(1 − p)k−1 , ∀k > 0,
XTd
···
t
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
9
Shiryaev’s Algorithm Change Point T0
Declare at Td
Detection Delay
X1
X2
···
XT0−1
XTd
A sufficient statistic: a posterior probability that change has occurred λt , Pr [T0 ≤ t|X1 , X2, . . . , Xt].
I
···
i.i.d. ∼ f1(x)
i.i.d.∼ f0(x) I
···
XT0 XT0 +1
Shiryaev’s detection rule: Td = inf{t : λt ≥ ηd}
I
Detection threshold ηd: determined by the reliability constraint ζ .
I
Setting ηd = 1 − ζ is asymptotically optimal as ζ → 0.
t
!
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#
"
#
"
mI =
!
t #
PSfrag replacements
1 pI 1 pB
mB =
10 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection In Multiple On-Off Processes
t
I Fraction of idle time: λ0 = mIm+m . B
I
Idle period: geometrically distributed with mean mI = p1I .
I
t
Busy period: geometrically distributed with mean mB = p1B .
I
A large number of independent homogeneous on-off processes.
I
t
*
*
+
*
+
+
*
*
+
+
*
+
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Detection Time
PSfrag replacementst )
Reliability Constraint
} {z l=1
| } {z l=1
|
Switch TS (L − 1)
Ts(l) + Td(L)) = busy] ≤ ζ Pr[SL( s.t. Ts(l) + Td(L)] min E[
)
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TS (2)
L−1 X
L−1 X
Switch
Channel 1 Channel 2 Channel L − 1 t Channel L Switch TS (1)
11 c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Quickest Detection In Multiple On-Off Processes
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
12
A POMDP Formulation I
State Transition: S/λ0
S/1 − λ0
PSfrag replacements S/λ0
C/pB
C/1 − pB
0
1
(Busy)
(Idle) S/1 − λ0
C/1 − pI
C/pI D/1
D/1
∆
(Absorbing)
1
I
Cost: 2
Switch or Continue: 1
2
Declare during a busy period: γ
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
13
A POMDP Formulation
I
A Sufficient Statistic: the information state (belief) λt = Pr[Zt = idle|X1 , X2, . . . , Xt] mI λ0 = mI + m B
I
Update of the Information State λt =
I T (λ|x):
(
T (λ0|x)
a(t − 1) = S, Xt = x
T (λt−1 |x) a(t − 1) = C, Xt = x
.
updated information state based on the new measurement x. ¯ B )f1(x) (λ¯ pI + λp T (λ|x) = ¯ B )f1(x) + (λpI + λ¯ ¯ pB )f0(x) . (λ¯ pI + λp ∆
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
14
A POMDP Formulation
I
Channel switching and change detection policy π :
λt ∈ [0, 1] I
=⇒
a(t) ∈ {S, C, D}, for each time t.
Quickest change detection:
π ∗ = arg min Eπ [ π
∞ X
mI Rπ(λt) | λ0 = ], mB + mI | {z } t=0 Cost
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
15
Quickest Change Detection: Value Functions I V (λt):
the minimum expected total cost when the current info. state is λt. V (λt) = min{ VC (λt) , | {z }
VS (λt) , | {z }
VD (λt) }. | {z }
Continue Switch Declare
I VC (λt):
the minimum expected total cost if continue. VC (λt) = 1 +
I VS (λt):
x Pr[
P (x; λ ) V (T (λt|x))dx | {z t} observe x under λt ]
the minimum expected total cost if switch. VS (λt) = 1 +
I VD (λt):
Z
Z
x Pr[
P (x; λ ) V (T (λ0|x))dx = VC (λ0) | {z 0} observe x under λ0 ]
the minimum expected total cost if declare. VD (λt) = (1 − λt)γ.
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
16
Quickest Change Detection: A Threshold Policiy I VD (λt)
is linear.
is concave and monotonically decreasing if m1 + m1 ≤ 1. B I I I VS (λt) = VC (λ0), where λ0 = m m+m . I B I VC (λt)
γ
PSfrag replacements
VD (λt)
VS (λt)
VC (λt)
0
λ0
ηd
1
λt
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
17
Quickest Change Detection: A Threshold Policiy V (λt) = min{ VS (λt) , | {z }
VC (λt) , | {z }
VD (λt) }. | {z }
Switch Continue Declare
PSfrag replacements
γ VD (λt) VS (λt)
VC (λt)
Switch 0
ηs = λ 0
Continue
Declare ηd
1
λt
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
Simulation Examples
I f0(x), f1(x):
Gaussian with zero mean and different variances.
I SN R = 10dB .
I ηd = 1 − ζ .
18
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
19
Simulation Example: Geometric Distribution Increase both mB and mI while keeping λ0 fixed 60
Single−Channel Strategy Multi−Channel Strategy
50 Average Detection Time
I
40 30 20 10 0 0
50
100 Average Busy Time
150
200
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
20
Simulation Example: Arbitrary Distributions Busy period: truncated Pareto distribution with increasing tail index 500 450
Single−Channel Strategy Multi−Channel Strategy
400 Average Detection Time
I
350 300 250 200 150 100 50 0 400
500
600
700 800 900 Average Busy Time mB
1000
1100
c °Qing Zhao, Jia Ye. Presentation at ITA, Jan., 2009.
21
Conclusion and Work in Progress PSfrag replacements
Quickest Detection in Multiple On-Off Processes: γ
VD (λt) VS (λt)
Switch 0
Declare
Continue
ηs = λ 0
ηd
VC (λt)
1
λt
Work in Progress: 2 Asymptotic 2 Minimax
optimality for arbitrary distributions and non-i.i.d. data.
formulation.
