Slide for ATD --- Tiansong Wang

Multi-Sensor Change-Point Detection Schemes


Tiansong Wang, Michael Baron

The University of Texas at Dallas

Mar 12, 2014

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1 Change-point Detection with Multiple Sensors

2 Change-point Detection Based on Bonferroni AlgorithmThe Boundaries for Mean DelayThe Approximation for Mean Delay

3 Change-point Detection Based on Holm-Bonferroni StepwiseAlgorithm

The Approximation for Mean Delay

4 Acknowledgments

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Change-point Detection with Multiple Sensors

Multi-sensor Change Detection

In a multi-sensor system, there are d sensors. Let (x(j)k )1≤k≤n

be a sequence of observed independent random variables with

density f (x(j)k |θ

(j)) from the j th sensor. Before the change

time νj , the density parameter θ(j) is equal to θ(j)0 . After the

change, the parameter is equal to θ(j)1 , j = 1, ..., d .

Goals: to detect the first change-point of the system,ν = min νj , with a fixed rate of false alarms α; discover whichchannels experienced a change; control the rate of falsealarms, mean detection delay, probability of misdetection andwrong classification.

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CUSUM Stopping Rule

Let the stopping rule for a multi-sensor system be

T = minj=1...d

(Tj)

Tj is the CUSUM stopping rule for the j th sensor.

Tj = min{n : W

(j)n ≥ hj

}W

(j)n = S

(j)n −min0≤i≤n S

(j)i is CUSUM, where S

(j)n is the sum

of log likelihood ratios for the j th sensor.

hj is the threshold for the j th sensor, which can be roughlyestimated by the rate of false alarms αj ,

hj ≈ − logαj

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Three Types Errors

Define a decision variable δj for the j th sensor, j = 1, ..., d

δj =

{1, if claim a change is detected in sensor j0, if claim there doesn’t exist a change in sensor j

Type I error in the kth sensor occurs if

{T < min νj} ∩ {δk = 1}

This is a false alarm in the kth sensor before anychange-points occur.

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Three Types Errors

Type II error in the kth sensor occurs if

{T ≥ min νj} ∩ {δk = 0} ∩ {νk ≤ T}

This is not a false alarm, but a change in the kth sensor ismissed at time T .

Type III error in the kth sensor occurs if

{T ≥ min νj} ∩ {δk = 1} ∩ {νk > T}

There exist changes in multi-sensor system and we stopped,but for a wrong reason. The kth sensor raised a false alarm.

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Examples

Smoke detectors

Border patrols monitoring traffic across the border

Cyber security, analysis of textual streams

Quality and process control with multidimensionalmeasurements

Etc.

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Change-point Detection Based on Bonferroni Algorithm

Change Detection Based on Bonferroni Algorithm

In a d-sensor system, the Bonferroni procedure controls therate of false alarms at α. Let αj be the probability of falsealarms for the j th sensor, where

∑αj = α. Then we get the

corresponding thresholds h1, ..., hd for each sensor.

The stopping rule for multi-sensor system is

T = min

n :⋃

j=1...d

(W

(j)n ≥ hj

)where W

(j)n is the CUSUM value for the j th sensor and

hj ≈ − logαj .

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The Boundaries for Mean Delay

The Boundary for Mean Delay for Normal case

Assume that before the change, sensor j follows f (x |θ(j)0 ) =

Norm(θ(j)0 , σ2); after the change, it follows f (x |θ(j)1 ) =

Norm(θ(j)1 , σ2).

— Firstly, we find the upper boundary of Mean Delay (MD)

MD ≤+∞∑k=0

[P(T1 > k)]d

= min

(hjKj

)+

+∞∑hj/Kj+1

{exp

{−1

2σ2(θ

(j)1 )2 −

M2k

2σ2+ Mkθ

(j)1

}}kd

= c1 · log d ++∞∑

hj/Kj+1

exp {c2 · k − c3 · k−1 + c4}

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The Boundary for Mean Delay for Normal case

where Kj is Kullback information number; Mk =hj+kψ(θ

(j)1 )−kψ(θ

(j)0 )

k(θ(j)1 −θ(j)0 )

;

ci , i = 1, 2, 3, 4 are constant coefficients.

— Secondly, assume that

Z(j)i = log

f (x |θ(j)1 )

f (x |θ(j)0 )

follow≡ Norm(θz , σ2z )

We find that the lower boundary for Mean Delay is

MD ≈+∞∑t=0

(P(Tj) > t)d

≥ c5 − c6θz

· e

c5σ

(c5 − 1)θz√

2π · log d

where c5 and c6 are constant coefficients.10 / 25




Graph of the Upper Boundary for Mean Delay

0 20 40 60 80 10068

70

72

74

76

78

80

82

84

86

88

d:# of channels

Mea

n De

lay

The Upper Bound for MD

Example: Normal(0,1) changes to Normal(0.5,1);α = 0.01

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Graph of the Lower Boundary for Mean Delay

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

d: # of sensors

Mean

Dela

yThe Lower Bound for MD

This agrees with the theoretical results.

Conclusion: a larger number of sensors does not necessarilyimply higher sensitivity, if we control the rate of false alarms.Eventually, the mean delay increases logarithmically.

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By Wald’s approximation, average run length function for thej th sensor is

ARL(θ(j)) =1

Eθ(j)(z(j)k )

(hj +

e−ω(j)0 hj

ω(j)0

− 1

ω(j)0

)

where ω(j)0 is a root of Eθ(j)(e

−ω(j)0 z

(j)k ) = 1.

Mean delay can be calculated, as hj goes to infinity,

MDj = ARL(θ(j)1 ) = 1

Eθ(j)1

(z(j)k )

(hj + e

−ω(j)0

hj

ω(j)0

− 1

ω(j)0

)≈ hj

Kj

where Kj = Eθ(j)1

(z(j)k ) is called Kullback information number.

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Graph of CUSUM Procedure

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Mean Delay of Multi-sensor System

Based on the proof of the upper boundary for mean delay, weprove that

Theorem 1

Assume that νj = 0, (j = 1, ..., d) and hj →∞ (equivalent toαj → 0), mean delay for a multi-sensor system based on Bonferronialgorithm is

MD ≈ minj=1...d

(hjKj

)

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The Proof of Theorem 1

Firstly, we study the first and the second sensors of multi-sensorsystem.

Assume that

W1: the CUSUM process of the first sensoraaaaaaaaaaaaaW2: the CUSUM process of the second sensoraaaaaaaaaaaK1: the Kullback information number of the first sensoraaK2: the Kullback information number of the second sensorh1: the threshold of the first sensoraaaaaaaaaaaaaaaaaaah2: the threshold of the second sensoraaaaaaaaaaaaaaaaa

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Graph of CUSUM Procedure for Two Sensors

Line W2/W1: the path of points (W1, W2);Line K2/K1: the straight line with the slope K2

K1;

Line h2/h1: the straight line with the slope h2h1

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Line:W2/W1 goes up along with the Line:K2/K1. So we canstudy Line:K2/K1 instead of Line:W2/W1.

For this example, the slope of Line:h2/h1 is smaller than ofLine:K2/K1, that is,

h2h1

<K2

K1⇒ h2

K2<

h1K1

We see that W2 procedure passes through its threshold first.So we obtain that the mean delay of two sensors

MD{1,2} ≈h2K2≈ min

{h2K2,h1K1

}

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Secondly, we study the second sensor and the third sensor ofmulti-sensor system. Similarly, we can prove that the mean delayof this two sensors

MD{2,3} ≈ min

{h2K2,h3K3

}Then the mean delay of this three sensors is

MD{1,2,3} ≈ min

{h1K1,h2K2,h3K3

}Thirdly, we repeat the above step to study the other sensors. Sowe can prove the mean delay for the multi-sensor system is

MD ≈ min

(hjKj

)19 / 25


Change-point Detection Based on Holm-Bonferroni Stepwise Algorithm

Holm-Bonferroni Stepwise Algorithm (improvement ofBonferroni)

In a d-sensor system, at any time, order the CUSUMprocesses,

W[1]t ≤W

[2]t ≤ ... ≤W

[d ]t

and the corresponding thresholds,

h1 ≤ h2 ≤ ... ≤ hd

where hj ≈ − log(αj ), which is smaller (tighter) thanBonferroni’s threshold, − log(αd ).

Stopping rule is

T = min

t :⋃

1≤j≤d

(W

[j]t ≥ hj

)20 / 25



Stopping Boundaries for the Stepwise Algorithm

-

6

��

��

��

��

��

��

��

��

��

��

��

��

�

0W (1)

n

W (2)n

h2

h1

h1 h2

��

Figure: Continue-samplingregion and stopping boundariesfor the stepwise algorithm forchange-point detectionwith d = 2 channels.

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Decision Boundaries for the Stepwise Algorithm

-

6Change in the2nd channel

Change inboth channels

Change in the1st channel

Change in the1st channel

Change in the2nd channel

��

��1

��

��

0

W(1)T

W(2)T

h2

h1

h1 h2 22 / 25




The Approximation Mean Delay

Suppose J = set of channels with change-points ⊆ {1, ..., d}For j ∈ J, E (W

[j]t ) ≥ tK (θ

[j]1 , θ

[j]0 ) > 0

For j /∈ J, E (W[j]t ) ≈ 0 because K (θ

[j]0 , θ

[j]1 ) < 0

For large t, we expect W(j)t >W

(k)t for ∀j ∈ J and ∀k /∈ J, then

we obtain

W[1]t ≤ ... ≤W

[d−|J|]t ≤W

[d−|J|+1]t ≤ ... ≤W

[d ]t

Order Kj = K (θ(j)1 , θ

(j)0 ) for j ∈ J

Let K[j]:J be the j th smallest Kullback information number in set J;among thresholds hj = log(j)− log(α), take |J| highesthd−|J|+1, ..., hd .

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Theorem 2

MD ≈ minj=1,...,|J|

hd−|J|+j

K[j]:J

[Proof]For the case d = 2,Case I: if J = {1}, then E (Wt) ≈ (tK1, 0),

MD ≈ h2K1

= minj=1

hd−|J|+1

K[j]:J

Case II: if J = {2}, then E (Wt) ≈ (0, tK2),

MD ≈ h2K2

= minj=2

hd−|J|+1

K[j]:J

Case III: if J = {1, 2}, then E (Wt) ≈ (tK1, tK2),

MD ≈ minj=1,2

hd−|J|+1

K[j]:J24 / 25


Acknowledgments

Acknowledgments

Research is based upon work supported by the National ScienceFoundation under Grant No. 1322353.

Thank you!

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Slide for ATD --- Tiansong Wang

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