A Recursive Two-Sided Change Detection Algorithm

0.12 MB·

by  James Falt

A Recursive Two-Sided Change Detection
Algorithm
James Falt and Steven D. Blostein
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Abstract
J

0
2 The problemof detectingan abruptchangein a sequenceof independentandidentically distributed

] (IID) random variables is addressed. Sequentially received samples are IID both before and after a
T
S single unknown change time. Unlike previous approaches to change detection that assume a known
.
h
t probability density function (PDF) at the start, the problem here is to detect a change between two
a
m givenPDFs in eitherdirection,meaningthatthenumberof hypothesesto be trackedisalways twice the
[
number of samples received. A Bayesian multiple hypothesis approach is proposed and shown to have

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v the following properties: (i) unlike previoustests that require a threshold, the minimum cost hypothesis
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4 istrackedthroughtime,(ii)underanexponentialdelaycostfunctionandsuitableparameterchoices,the
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6 proposedprocedure’sprobabilityofdetectingachangeintheincorrectdirectionasymptoticallyvanishes
.
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0 withtime,(iii)themethodisrecursivewithconstantcomputationperunittime,and(iv)theprobabilities
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1 offalsealarmandincorrectdetectionmaybedirectlytradedoffwith averagedelay.Performanceresults
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v using simulation confirm the derived properties and also reveal that the average delay, after a transient
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X
period corresponding to when the starting state is uncertain, approaches that of the optimal one-sided
r
a
test, CUSUM.
I. INTRODUCTION
Let Xn = {X |i = 1,2,··· ,n} be n independent random variables observed sequentially.
1 i
Each of these random variables are known to follow one of two possible known distributions,
The authors are with Dept. of Electrical and Computer Eng., Queen’s University, Kingston, ON, Canada. (email:
[email protected]). This work was supported by the Natural Sciences and Engineering Research Council of Canada
Discovery Grant 41731.
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which are described by probability density functions (PDF) f and f . While observing Xn, the
0 1 1
objective is to determine, as quickly as possible, whether a single change, either from f tof ,
0 1
or from f to f has occurred at some unknown discrete time m ∈ Z+ without prior knowledge
1 0
of the initial state, i.e., whether the PDF of X is f or f .
1 0 1
An application of particular interest is cognitive radio, where f may represent the PDF of
0
an observed discrete-time signal in the unoccupied, or ‘OFF’ state of a communication channel,
typically represented as zero-mean Gaussian background noise with variance σ2, whereas f
n 1
represents the occupied, or ‘ON’ state of a channel, where the same background Gaussian noise
is observed in addition to an unknown signal occupying the band. Here, the unknown signal is
modelled stochastically as normally distributed with variance σ2, and is thus detectable from the
s
additional observed energy of each sample. In contrast to previous formulations, the initial state
corresponding the PDF at the start of observation X is unknown. In addition, it is assumed that
1
conditioned on either f or f , the samples are assumed to be independent.
0 1
Previously, in the case where the initial state is known, detection of a single change in
distribution in an independent sequence of random variables has been well studied, including
major developments that took place in [1], [2], [3], [4], as well as the substantial number of
references that can be found in the recent book [5]. Recently, the problem of detecting multiple
changes in sequences, termed ON-OFF processes has been addressed in [6], where the structure
of an optimum test is found by formulating an optimization problem using partially observable
Markov processes. However, similar to these previous approaches the initial state is assumed
known. It has been observed previouslythat quickest detection performance is drastically affected
if the assumed initial state is incorrect [7] [8]. Finally, as is the case throughout the literature,
the general problem of threshold selection to guarantee an exact probability of false alarm for
these above quickest detection methods is generally intractable.
In Section II, the two-sided change detection problem is formulated using Bayes decision
theory. Section III derives recursive expressions for the Bayes risk, suitable for problems in-
volving causal (on-line) observation. While design of the test threshold is also intractable here, a
parameterization of the cost function is proposed and studied in Section IV. Results are described
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in Section V which reinforce the insights provided in Section IV as well as make comparisons
to CUSUM, which has been shown to be optimal for one-sided quickest detection [4] [5].
II. SEQUENTIAL TWO SIDED CHANGE DETECTION
At time n, without knowledge of the starting PDF (of X ), a sequence of independent random
1
variables, Xn ≡ X ,X ,···X , may assume 2n possible joint distributions corresponding to all
1 1 2 n
possible single abrupt changes in distribution from f to f and vice versa. Classically, for a
0 1
fixed number of samples, the problem may be formulated in terms of M-ary hypothesis testing
[9]. Two of these 2n sequences, corresponding to no change, are described by the sequence Xn
i
where all X ,1 ≤ i ≤ n are distributed according to f , where j ∈ {0,1}. The other 2(n−1)
i j
possible sequences are described by the sequence Xn where X , 1 ≤ i < m, are distributed
1 i
according to f , and the rest of the X , m ≤ i ≤ n, are distributed according to f , where
j i k
{j,k} ∈ {0,1} or {1,0}. The notation adopted to represent the 2n possible sequences is given
by:
H = {X ∼ f |1 ≤ i ≤ n} (1)
1 i 0
H = {X ∼ f |1 ≤ i ≤ n} (2)
1 i 1
X ∼ f |1 ≤ i < j
i 0
H = 2 ≤ j ≤ n (3)
j
 
X ∼ f |j ≤ i ≤ n
 i 1 
 X ∼ f |1 ≤ i < j 
i 1
H = 2 ≤ j ≤ n. (4)
j
 
X ∼ f |j ≤ i ≤ n
 i 0 
Suppose for now that n is fixed. To formulate a decision rule to select among the 2n possible
 
hypotheses, we adopt a Bayesian framework based on selecting the hypothesis with minimum
risk according to Bayesian (2n)-ary hypothesis testing. Under equally likely prior probabilities of
change time, a traditional Bayesian hypothesis testing scheme minimizes the posterior risk asso-
ciated with each of the 2n hypotheses. However, when an infinite-duration sequence is causally
observed, the number of hypotheses grows with n, which results in increasing computational and
memory resources. To alleviate this, a time-recursive version of Bayes risk computation testing
is first formulated.
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In a Bayesian formulation, the posterior probabilities of hypothesis H being true given Xn =
j 1
xn for 1 < j < n, which we denote as P(H |Xn = xn), as well as the posteriors for hypothesis
1 j 1 1
H , for 1 < j < n need to be tracked. Using Bayes’ Rule,
j
P(Xn = xn|H )P(H )
P(H |Xn = xn) = 1 1 j j (5)
j 1 1 P(Xn = xn)
1 1
where P(Xn = xn|H ) is the likelihood of observing Xn = xn given that H is true, P(H ) is
1 1 j 1 1 j j
the prior probability of H , and P(Xn = xn) is the joint probability of observing Xn = xn for
j 1 1 1 1
the n samples received. From independence conditioned on H , the likelihood function
j
n f (x ) if j = 1,
i=1 0 i
P(Xn = xn|H ) =  (6)
1 1 j
Qj−1f (x ) n f (x ) otherwise.
i=1 0 i i=j 1 i
A similar expression can be written forQP(Xn = xnQ|H ),1 ≤ j ≤ n. This Bayes formulation
 1 1 j
uses the following notation for costs and prior probabilities:
L ≡ cost of choosing H when H is true
kj k j
L ≡ cost of choosing H when H is true
kj k j
L ≡ cost of choosing H when H is true
kj k j
L ≡ cost of choosing H when H is true
kj k j
π ≡ prior probability that H is true
j j
π ≡ prior probability that H is true.
j j
For simplicity purposes, we assume symmetrical costs: L = L and L = L for 1 ≤ j ≤
kj kj kj kj
n, 1 ≤ k ≤ n. The conditional risk in choosing hypothesis H , 1 ≤ k ≤ n, given Xn = xn,
k 1 1
n
Rn = (π L P(H |Xn = xn)+π L P(H |Xn = xn)), (7)
k j kj j 1 1 j kj j 1 1
j=1
X
and the conditional risk in choosing H given Xn = xn,
k 1 1
n
Rn = π L P(H |Xn = xn)+π L P(H |Xn = xn). (8)
k j kj j 1 1 j kj j 1 1
j=1
X
It is desired, at every n, to determine the hypothesis with minimum risk, i.e.,
argmin Rn, Rn, min Rn, min Rn . (9)
n 1 1 k k
2≤k≤n 2≤k≤n
(cid:26) (cid:27)
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Thefollowingterminologywillbeadoptedin thesequel:fornsuch that Rn (orRn)has minimum
1 1
risk, there is no detection of a change. If H (or H ) has minimumrisk, at some n < m, detection
k k
occurs too soon, i.e., a false alarm is declared. On the other hand, if n > m there is detection
delay. If H , for 2 ≤ k ≤ n becomes the least risk hypothesis, and H is true, 2 ≤ j ≤ n, or
k j
vice-versa, an incorrect detection is said to occur at time n.
III. PROPOSED BAYES COST STRUCTURE
We assume equally likely a priori hypotheses and omit constant factor 1/2n affecting all
hypotheses equally. From (7) we have
n
Rn = (L P(H |Xn = xn)+L P(H |Xn = xn)). (10)
1 1j j 1 1 1j j 1 1
j=1
X
Next, we adopt a specific cost structure to account for delay, incorrect detection and false alarm
so that the risks may be meaningfully compared across time. Exponentially increasing costs in
choosing H when H or H ,j = 1,2,...n is true are assumed:
1 j j
0 if j = 1
L ≡ (11)
1j 
an−j+1 otherwise,

 cn if j = 1
L ≡
1j 
cj−1 otherwise,

where a > 1 denotes unit cost of delay and c > 1 denotes unit cost of incorrect detection. Based

on these costs, it can be shown that (10) can be computed recursively.
Rn1[1]
n
Rn = an−j+1P(H |Xn = xn)
1 j 1 1
z }| {
j=2
X(cid:0) (cid:1)
Rn1[2]
n
+ cj−1P(H |Xn = xn)
j 1 1
z }| {
j=2
X(cid:0) (cid:1)
+cnP(H |Xn = xn). (12)
1 1 1
Rn1[3]
| {z }
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where
n
Rn[1] = an−j+1P(H |Xn = xn) ,
1 j 1 1
j=2
X(cid:0) (cid:1)
n
Rn[2] = cj−1P(H |Xn = xn) ,
1 j 1 1
j=2
X(cid:0) (cid:1)
Rn[3] = cnP(H |Xn = xn).
1 1 1 1
where the first term can be updated recursively as:
n+1
Rn+1[1] = a(n+1)−j+1P(H |Xn+1 = xn+1)
1 j 1 1
j=2
X(cid:0) (cid:1)
n
= a(n+1)−j+1P(H |Xn+1 = xn+1) +aP(H |Xn+1 = xn+1)
j 1 1 n+1 1 1
j=2
X(cid:0) (cid:1)
n
= an−j+1P(H |Xn = xn) af (x )+aP(H |Xn+1 = xn+1)
j 1 1 1 n+1 n+1 1 1
" #
j=2
X(cid:0) (cid:1)
= Rn[1]af (x )+aP(H |Xn+1 = xn+1),
1 1 n+1 n+1 1 1
Rn[2] can be updated recursively as:
1
n+1
Rn+1[2] = cj−1P(H |Xn+1 = xn+1)
1 j 1 1
j=2
X(cid:0) (cid:1)
n
= cj−1P(H |Xn+1 = xn+1) +cnP(H |Xn+1 = xn+1)
j 1 1 n+1 1 1
j=2
X(cid:0) (cid:1)
n
= cj−1P(H |Xn = xn) f (x )+cnP(H |Xn+1 = xn+1)
j 1 1 0 n+1 n+1 1 1
j=2
X(cid:0) (cid:1)
= Rn[2]f (x )+cnP(H |Xn+1 = xn+1).
1 0 n+1 n+1 1 1
and Rn[3] can be updated recursively as
1
Rn+1[3] = cn+1P(H |Xn+1 = xn+1)
1 1 1 1
= (cnP(H |Xn = xn))cf (x )
1 1 1 1 n+1
= Rn[3]cf (x ).
1 1 n+1
ShowingthattheminimumriskRn canbetrackedrecursivelyismoreinvolved,astheminimum
k
risk change time k may differ over time. Let k represent the minimum risk change time at time
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n and let k′ be the minimum risk change time corresponding to time n−1. The minimum risk
corresponding to change time k is then updated at time n as
Rn = min{ Rn, Rn }, (13)
k n k′
where from (7) under equal priors,
n
Rn = L P(H |Xn = xn)+L P(H |Xn = xn) (14)
n nj j 1 i nj j 1 1
j=1
X(cid:0) (cid:1)
and the costs of choosing H given H or H , respectively, are
n j j
b if j = 1

Lnj ≡  an−j if 1 < j < n (15)




0 if j = n.



 cn−1 if j = 1
L ≡
nj 
aj−1c if 1 < j ≤ n,

where a and c are as defined before, and b is defined as the cost of false alarm. Using (15),

Rnn[2]
Rnn[1] n−1
Rn = bP(H |Xn = xn)+ an−jP(H |Xn = xn)
n 1 1 1 z }|j 1 1 {
j=2
z }| { X(cid:0) (cid:1)
n
+cn−1P(H |Xn = xn)+ aj−1cP(H |Xn = xn)
1 1 1 j 1 1
Rnn[3] Xj=2 (cid:0) (cid:1)
| {z } Rnn[4]
where | {z }
n−1
Rn[1] = an−jP(H |Xn = xn)
n j 1 1
j=2
X(cid:0) (cid:1)
Rn[2] = bP(H |Xn = xn))
n 1 1 1
Rn[3] = cn−1P(H |Xn = xn)
n 1 1 1
n
Rn[4] = aj−1cP(H |Xn = xn) ,
n j 1 1
j=2
X(cid:0) (cid:1)
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where Rn[1] can be updated recursively as:
n
n
Rn+1[1] = a(n+1)−jP(H |Xn+1 = xn+1)
n+1 j 1 1
j=2
X(cid:0) (cid:1)
n−1
= a(n+1)−jP(H |Xn+1 = xn+1) +aP(H |Xn+1 = xn+1)
j 1 1 n 1 1
j=2
X(cid:0) (cid:1)
n−1
= an−jP(H |Xn = xn) af (x )+aP(H |Xn+1 = xn+1)
j 1 1 1 n+1 n 1 1
j=2
X(cid:0) (cid:1)
= Rn[1]af (x )+aP(H |Xn+1 = xn+1),
n 1 n+1 n 1 1
Rn[2] can be updated recursively as:
n
Rn+1[2] = bP(H |Xn+1 = xn+1)
n+1 1 1 1
= (bP(H |Xn = xn))f (x )
1 1 1 0 n+1
= Rn[2]f (x ),
n 0 n+1
Rn[3] can be updated recursively as:
n
Rn+1[3] = cnP(H |Xn+1 = xn+1)
n+1 1 1 1
= cn−1P(H |Xn = xn) cf (x )
1 1 1 1 n+1
= (cid:0)Rn[3]cf (x ), (cid:1)
n 1 n+1
and Rn[4] can be updated recursively as:
n
n+1
Rn+1[4] = aj−1cP(H |Xn+1 = xn+1)
n+1 j 1 1
j=2
X(cid:0) (cid:1)
n
= aj−1cP(H |Xn+1 = xn+1) +ancP(H |Xn+1 = xn+1)
j 1 1 n+1 1 1
j=2
X(cid:0) (cid:1)
n
= aj−1cP(H |Xn = xn) f (x )+ancP(H |Xn+1 = xn+1)
j 1 1 0 n+1 n+1 1 1
!
j=2
X(cid:0) (cid:1)
= Rn[4]f (x )+ancP(H |Xn+1 = xn+1).
n 0 n+1 n+1 1 1
In (13) Rn has Bayes risk
k′
n
Rkn′ = Lk′jP(Hj|X1n = xni)+Lk′jP(Hj|X1n = xn1) ,
j=1
X(cid:0) (cid:1)
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where (5) is used along with the following costs associated with choosing hypothesis H , 1 ≤
k
k < n, given H or H , respectively:
j j
b if j = 1 6= k or j > k

Lkj ≡  ak−j if 2 ≤ j < k (16)




0 if j = k ,



 ck−1 if j = 1

Lkj ≡  aj−1cn−k+1 if 1 < j < k



ak−1cn−j+1 if j ≥ k ,


where a, b, and c are as definedbefore. Using the above costs,

Rnk′[1]
k′−1
Rn = ak−jP(H |Xn = xn)
k′ z }|j 1 1 {
j=2
X(cid:0) (cid:1)
Rnk′[2]
n
+ (bP(H |Xn = xn))
j 1 1
z }| {
j=k′+1
X
Rnk′[3] Rnk′[4]
+bP(H |Xn = xn)+ck′−1P(H |Xn = xn)
1 1 1 1 1 1
z }| R{nk′[5]z }| {
k′−1
+ aj−1cn−k′+1P(H |Xn = xn)
z }| j 1 1 {
Xj=2 (cid:16) (cid:17)
n
+ cn−j+1ak′−1P(H |Xn = xn) (17)
j 1 1
jX=k′(cid:16) (cid:17)
Rnk′[6]
where | {z }
k′−1
Rn[1] = ak−jP(H |Xn = xn) (18)
k′ j 1 1
j=2
X(cid:0) (cid:1)
n
Rn[2] = (bP(H |Xn = xn)) (19)
k′ j 1 1
j=k′+1
X
Rn[3] = bP(H |Xn = xn) (20)
k′ 1 1 1
Rn[4] = ck′−1P(H |Xn = xn) (21)
k′ 1 1 1
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k′−1
Rn[5] = aj−1cn−k′+1P(H |Xn = xn) (22)
k′ j 1 1
Xj=2 (cid:16) (cid:17)
n
Rn[6] = cn−j+1ak′−1P(H |Xn = xn) . (23)
k′ j 1 1
jX=k′(cid:16) (cid:17)
where Rn[i], i = 1,2,··· ,6 can each be updated recursively. Rn[1] can be updated recursively
k′ k′
as:
k′−1
Rn+1[1] = ak−jP(H |Xn+1 = xn+1)
k′ j 1 1
j=2
X(cid:0) (cid:1)
k′−1
= ak−jP(H |Xn = xn) f (x )
j 1 1 1 n+1
!
j=2
X(cid:0) (cid:1)
= Rn[1]f (x ),
k′ 1 n+1
Rn[2] can be updated recursively as:
k′
n+1
Rn+1[2] = bP(H |Xn+1 = xn+1)
k′ j 1 1
j=k′+1
X (cid:0) (cid:1)
n
= bP(H |Xn+1 = xn+1) +bP(H |Xn+1 = xn+1)
j 1 1 n+1 1 1
j=k′+1
X (cid:0) (cid:1)
n
= (bP(H |Xn = xn)) f (x )+bP(H |Xn+1 = xn+1)
j 1 1 1 n+1 n+1 1 1
j=k′+1 !
X
= Rn[2]f (x )+bP(H |Xn+1 = xn+1),
k′ 1 n+1 n+1 1 1
Rn[3] can be updated recursively as:
k′
Rn+1[3] = bP(H |Xn+1 = xn+1)
k′ 1 1 1
= (bP(H |Xn = xn))f (x )
1 1 1 0 n+1
= Rnf (x ),
k′ 0 n+1
Rn[4] can be updated recursively as:
k′
Rn+1[4] = ck′−1P(H |Xn+1 = xn+1)
k′ 1 1 1
= ck′−1P(H |Xn = xn) f (x )
1 1 1 1 n+1
(cid:16) (cid:17)
= Rn[4]f (x ),
k′ 1 n+1