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OPTIMAL DESIGN OF THE SHIRYAEVROBERTS CHART: GIVE YOUR SHIRYAEVROBERTS A HEADSTART Aleksey S. Polunchenko Department of Mathematical Sciences State University of New York (SUNY) at Binghamton Binghamton, New York 139026000 :

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OPTIMAL DESIGN OF THE SHIRYAEV–ROBERTS CHART: GIVE YOUR SHIRYAEV–ROBERTS A HEADSTART

Aleksey S. Polunchenko Department of Mathematical Sciences State University of New York (SUNY) at Binghamton Binghamton, New York 13902–6000 : aleksey@binghamton.edu : http://www.math.binghamton.edu/aleksey The 34th Quality and Productivity Research Conference (QPRC’2017) Department of Statistics, University of Connecticut, Storrs, CT June 14, 2017

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 1 / 19

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Quickest change-point detection problem

  • Given: A series {Xn}n≥1 of independent scalar data, collected sequentially, one at a

time.

Surveillance Starts each Xn ∝ f(x)

bc bc bc bc bc

X1 X2 X3 Xν−1 Xν

b b b b

Xν+1 Xν+2 Xν+3 Xn each Xn ∝ g(x) ≡ f(x) Change-Point (ν ≥ 0, unknown) Surveillance Continues {Xn}n≥1 independent throughout

  • Assumption 1: Xn ∝ pXn(x), where pXn(x) = µ 1

l{n>ν} +ǫn, µ = 0, and {ǫn}n1 are iid standard Gaussian random variables.

  • Assumption 2: The change-point, ν 0, is unknown, but not random (no prior

distribution).

  • Notation: from now on

a) ν = 0 is to be understood as E[Xn] = µ for all n 1, and b) ν = ∞ is to mean that E[Xn] = 0 for all n 1.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 2 / 19

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Minimax change-point detection problem

  • Notation: let T denote a generic chart (stopping time).
  • Notation: for k 0 let Pk(·) = P( · |ν = k) and Ek[ · ] = E[ · |ν = k]; in particular,

P∞(·) = P( · |ν = ∞) and E∞[ · ] = E[ · |ν = ∞]

  • Define SADD(T) = sup0≤ν<∞ ADDν(T) with ADDν(T) = Eν[T − ν|T > ν],

ν 0.

  • Define ARL(T) = E∞[T].
  • Goal: To find stopping time T such that SADD(T) is minimized within class

∆(γ) = {T : ARL(T) γ} for every γ 1.

  • Note: Except for a few special cases, this problem has not yet been solved.
  • The best available solution is a “tweaked” version of the so-called Shiryaev–Roberts

detection procedure, due to Shiryaev (1961’63) and Roberts (1966).

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 3 / 19

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The Shiryaev–Roberts (SR) chart

  • Introduce the “score-function”

Sn = µXn − µ2 2 , n = 1, 2, . . . and define Rn = (1 + Rn−1) exp{Sn}, n = 1, 2, . . . , where R0 = 0.

  • The classical SR chart calls for stopping at

SA = inf{n 1: Rn A} with A > 0, where A > 0 is a control limit selected in advance so that ARL(SA) γ for a given γ 1.

  • Moustakides, P. and Tartakovsky (2011) proposed to parameterize the starting point

R0 = r 0. One can then define the Generalized SR chart as Sr

A = inf{n 1: Rr n A}, where Rr n = (1 + Rr n−1) exp{Sn}.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 4 / 19

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The GSR chart (cont’d)

STATISTICAL INFERENCE X1, . . . , Xn MAXIMUM LIKELIHOOD APPROACH DECISION STATISTIC OF THE FORM maxk φk(X1, . . . , Xk) (GENERALIZED) BAYESIAN APPROACH DECISION STATISTIC OF THE FORM

  • k φk(X1, . . . , Xk)

Figure 1. Approaches to statistical inference.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 5 / 19

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The GSR chart: Properties

  • Observation: {Rr

n − n − r}n0 is a zero-mean P∞-martingale, i.e.,

E∞[Rr

n − n − r] = 0.

  • Consequence 1: E∞[Rr

Sr

A − Sr

A − r] = 0 so that ARL(Sr A) = E∞[Rr Sr

A] − r.

  • Consequence 2: ARL(Sr

A) as a function of r should be almost linear.

  • Moreover, it can be shown that E∞[Rr

Sr

A] → A/v as A → ∞, where v is the limiting

exponential overshoot.

  • Hence, ARL(Sr

A) is also linear in A.

  • The linearity of ARL(Sr

A) is important in designing a numerical method to compute

it.

  • The GSR procedure is asymptotically (as the ARL to false alarm increases) nearly

(order-three) optimal with respect to SADD(T). That is, SADD(Sr

A) = infT∈∆(γ) SADD(T) + o(1), as γ → ∞, where o(1) → 0, as γ → ∞.

  • P. & Tartakovsky (2010) and Tartakovsky & P. (2010) offer two special cases where

the GSR procedure is exactly SADD(T)-optimal.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 6 / 19

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The GSR chart: Choosing the headstart

  • Let ARL(Sr

A) = γ, where γ > 1 is fixed.

ν Eν[Sr

A − ν|Sr A > ν]

R0 = 0 0 < R0 < r∗ R0 = r∗ R0 > r∗ R0 ∼ QA(x)

bc bc bc bc bc bc

Figure 2. ADDν(Sr

A) as a function of ν and Rr 0 = r.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 7 / 19

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The GSR chart: Choosing the headstart (cont’d)

  • If we had a lowerbound for SADD(Sr

A) as a function of r, then it would make sense

to let r to be such that the difference between SADD(Sr

A) and the lowerbound is

minimized.

Theorem

If A = Aγ so that ARL(Sr

Aγ) = γ, then for every r 0 it is true that

inf

T∈∆(γ) SADD(T) SADD(Sr A),

where SADD(Sr

A)

  • r E0[Sr

A] + ∞

  • k=0

Ek[max{0, Sr

A − k}]

  • / (r + ARL(Sr

A)) .

  • See Moustakides, P. & Tartakovsky (2011), P. & Tartakovsky (2010), Tartakovsky

& P. (2010), Pollak & Tartakovsky (2009), and Shiryaev & Zryumov (2010).

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 8 / 19

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The GSR chart: Choosing the headstart (cont’d)

  • Since SADD(Sr

A) infT∈∆(γ) SADD(T) SADD(Sr A), the number

ropt = arg inf

0r<A

  • SADD(Sr

A) − SADD(Sr A)

  • is a good candidate for the point to start the GSR chart from.
  • The lowerbound can be conveniently and accurately computed via the numerical

framework proposed by P., Sokolov & Du (2014).

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 9 / 19

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Numerical results: ARL to false alarm

100 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 300 300 300 300 300 300 300 400 400 400 400 400 400 500 500 500 500 500 500 600 600 600 600 600 700 700 700 700 800 800 900 900 1000

ARL(Sr

A)

1 200 400 600 800 1000 Detection Threshold, A 200 400 600 800 1000 Headstart, Rr

0 = r

200 400 600 800 1000

(a) µ = 0.2.

100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 300 300 300 300 300 300 300 300 400 400 400 400 400 400 400 400 500 500 500 500 500 500 500 600 600 600 600 600 600 700 700 700 700 700 800 800 800 800 900 900 900 900 1000 1000 1000

ARL(Sr

A)

1 200 400 600 800 1000 Detection Threshold, A 200 400 600 800 1000 Headstart, Rr

0 = r

200 400 600 800 1000

(b) µ = 0.5. Figure 3. ARL(Sr

A) as a function of the headstart Rr 0 = r 0 and the detection threshold A > 0

for µ = {0.2, 0.5}.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 10 / 19

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Numerical results: Average Detection Delay

ADDk(Sr

A)

10 20 30 40 50 60 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 20 40 60 80 100

(a) γ = 100.

ADDk(Sr

A)

40 60 80 100 120 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 50 100 150 200 250

(b) γ = 500.

ADDk(Sr

A)

60 80 100 120 140 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 100 200 300 400 500

(c) γ = 1 000. Figure 4. ADDk(Sr

A) as a function of the headstart Rr 0 = r 0, the change-point k = 0, 1, . . .,

and the ARL to false alarm level ARL(Sr

A) = γ > 1 for µ = 0.2.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 11 / 19

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Numerical results: Average Detection Delay (cont’d)

ADDk(Sr

A)

5 10 15 20 25 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 20 40 60 80 100

(a) γ = 100.

ADDk(Sr

A)

5 10 15 20 25 30 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 20 40 60 80 100

(b) γ = 500.

ADDk(Sr

A)

10 15 20 25 30 35 Headstart, Rr

0 = r

50 100 150 200 Change-Point, k 20 40 60 80 100

(c) γ = 1 000. Figure 5. ADDk(Sr

A) as a function of the headstart Rr 0 = r 0, the change-point k = 0, 1, . . .,

and the ARL to false alarm level ARL(Sr

A) = γ > 1 for µ = 0.5.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 12 / 19

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Numerical results: Average Detection Delay (cont’d)

Change-Point, k 20 40 60 80 100 ADDk(Sr

A)

15 20 25 30 35 40 45 50 r = 0, A = 89.3 r = 37.42, A = 122.02 r = 111.6, A = 188.2 r = 21.7, A = 108.1

(a) γ = 100.

Change-Point, k 50 100 150 200 250 ADDk(Sr

A)

60 65 70 75 80 85 90 95 100 105 r = 0, A = 445.1 r = 21.7, A = 464.1 r = 89.1, A = 524.1 r = 63.8, A = 501.6

(b) γ = 500.

Change-Point, k 100 200 300 400 500 ADDk(Sr

A)

85 90 95 100 105 110 115 120 125 130 r = 0, A = 890.1 r = 32.9, A = 919.1 r = 111.6, A = 989.1 r = 75.3, A = 956.8

(c) γ = 1 000. Figure 6. ADDk(Sr

A) as a function of the headstart Rr 0 = r 0, the change-point k = 0, 1, . . .,

and the ARL to false alarm level ARL(Sr

A) = γ > 1 for µ = 0.2.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 13 / 19

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Numerical results: Average Detection Delay (cont’d)

Change-Point, k 20 40 60 80 100 ADDk(Sr

A)

4 6 8 10 12 14 16 18 r = 0, A = 74.8 r = 10.32, A = 82.1 r = 107.4, A = 153.8 r = 25.9, A = 93.8

(a) γ = 100.

Change-Point, k 20 40 60 80 100 ADDk(Sr

A)

12 14 16 18 20 22 24 26 28 30 r = 106.1, A = 452.8 r = 25.9, A = 392.8 r = 0, A = 373.8 r = 14.4, A = 384.2

(b) γ = 500.

Change-Point, k 20 40 60 80 100 ADDk(Sr

A)

16 18 20 22 24 26 28 30 32 34 r = 132.9, A = 846.6 r = 39.2, A = 776.6 r = 16.4, A = 759.4 r = 0, A = 747.6

(c) γ = 1 000. Figure 7. ADDk(Sr

A) as a function of the headstart Rr 0 = r 0, the change-point k = 0, 1, . . .,

and the ARL to false alarm level ARL(Sr

A) = γ > 1 for µ = 0.5.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 14 / 19

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Numerical results: Maximal Average Detection Delay vs. Lowerbound

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 25 30 35 40 45 50 55 SADD(Sr

A)

Optimal Performance SADD(Sr

A)

(a) γ = 100.

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 65 70 75 80 85 90 95 100 105 SADD(Sr

A)

SADD(Sr

A)

Optimal Performance

(b) γ = 500.

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 90 95 100 105 110 115 120 125 130 SADD(Sr

A)

SADD(Sr

A)

Optimal Performance

(c) γ = 1 000. Figure 8. SADD(Sr

A) and SADD(Sr A) as functions of the headstart Rr 0 = r and the ARL to false

alarm level ARL(Sr

A) = γ > 1 for µ = 0.2.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 15 / 19

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Numerical results: Maximal ADD vs. Lowerbound (cont’d)

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 8 10 12 14 16 18 20 SADD(Sr

A)

SADD(Sr

A)

Optimal Performance

(a) γ = 100.

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 20 21 22 23 24 25 26 27 28 29 SADD(Sr

A)

SADD(Sr

A)

Optimal Performance

(b) γ = 500.

Headstart, Rr

0 = r

50 100 150 200 Performance Metric 25 26 27 28 29 30 31 32 33 34 SADD(Sr

A)

SADD(Sr

A)

Optimal Performance

(c) γ = 1 000. Figure 9. SADD(Sr

A) and SADD(Sr A) as functions of the headstart Rr 0 = r and the ARL to false

alarm level ARL(Sr

A) = γ > 1 for µ = 0.5.

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 16 / 19

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Summary

  • Qualitatively, the effect of headstarting on the performance of an SR chart is the

same as that previously observed by Lucas & Croiser for the CUSUM chart.

  • Quantitatively, giving an SR chart a specifically designed headstart makes the chart’s

performance nearly the best one can possibly get for any given level of the ARL to false alarm.

  • However, the improvement in performance is the greater, the fainter the change. To

boot, for contrast changes the optimal headstart value tends to zero (no headstart).

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 17 / 19

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Acknowledgements

  • Simons Foundation (www.simonsfoundation.org)

Collaboration Grant in Mathematics (Award # 304574) New York City, New York, USA

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 18 / 19

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Thank You!

A.S. Polunchenko (SUNY Binghamton) Optimal Design of the Shiryaev–Roberts Chart QPRC’17 @ UConn–Storrs 19 / 19