text: )mk( gnihtroN A. Measurand 7048 The sensitivity of a radio telescope is quantiﬁed by the ratio of its effective aperture area (A in m2) to the system noise 7047 temperature (T in K): sys A 7046 A/T = e . (1) T MWA Tiles sys AAVS 0.5 7045 466 467 468 469 This ﬁgure-of-merit is a key term in the expression for Easting (km) the telescope output signal-to-noise (S/N) ratio; many readers will recognize that it is similar and is convertible to G/T Fig. 2. A map showing the AAVS0.5 location (blue circle) and MWA telescope tiles (black cross) in Easting/Northing for Map Grid Australia UTM which is common in telecommunications. The antenna beam zone 50. pattern may be obtained by measuring A/T along the apparent trajectory of a suitable astronomical source. This assumes that (cid:90) (cid:90) the sky noise does not change T appreciably during the sys observation; in practice, this is a reasonable approximation. − B. Measurement Method V = I(l, m)e j2π(u12l+v12m)dldm, (3) 12 This subsection presents a simpliﬁed version of how A/T is obtained in radio interferometry. Interested readers are referred where V is the complex visibility on the baseline between 12 to standard textbooks such as [6] and [7] for more complete antennas 1 and 2, I(l, m) is the sky brightness distribution, u, v discussions. Let us assume an interferometer involving 3 are the baseline vector components normalized to wavelength single-polarized antennas (or antenna arrays as the case may and l, m are the direc√tion cosines with respect to uˆ, vˆ of a unit − − be) observing a point source with ﬂux S (in W/m2/Hz or vector sˆ = luˆ+mvˆ+ 1 l2 m2wˆ where wˆ is a unit vector Jy) in its phase center (in this context “phase center” refers normal to the uv plane (i.e., points to the “phase center”). to that position in the sky for which the interferometer is Simplifying further(cid:90)to the 1-D case, (3) becomes intended to point to by applying relative phase shifts between its elements). The measured cross-correlation products are: − V = I(l)e j2π(u12l)dl, (4) 12 ∗ V12 = a˜1a˜2∗S where l = sin θ(cid:48) and θ(cid:48) is the angle between wˆ and sˆ. We V = a˜ a˜ S 13 1 ∗3 recognize V12 and I(l) as a Fourier transform pair. Hence, V23 = a˜2a˜3S, (2) structures with large l extent are mostly visible for small u12. From MWA observing experience, with its small-diameter (3.3 m) tiles, correlated Galactic noise is measurable on base- where a˜ is referred to as complex “gain” (voltage quantity) lines of length < 30λ and should therefore be excluded.2 containing a combination of both antenna and receiver elec- It can be shown (see Ch. 9 of [6]) that after calibration of tronic gains. Assuming S is known, the complex “gains” may the measured cross-correlation products, the standard deviation be solved from these 3 equations. This is typically done by in ﬂux density for each baseline is inversely proportional to solving the amplitudes and phases separately [6]. the square root of the(cid:115)products of the A/T for the AUTs in To obtain antenna gains, amplitude-only solutions are suf- question: ﬁcient. Note that this method is analogous to the well- known “three-antenna” measurement technique [8]. With N − ≈ 2 elements, N(N 1)/2 pair-wise combinations (“baselines”) ∆Sij k2(A/T) (A/T) Bt , (5) are available, leading to an overdetermined system solvable i j acc with a least-squares method. As shown in Fig. 2, the AAVS 0.5 where k is Boltzmann’s constant, B is the bandwidth and is co-located with the Murchison Wideﬁeld Array (MWA) t is the accumulation time. Thus, by measuring ∆S for all radio telescope [9], [10] to take advantage of this multi- acc baselines for which the Galactic noise is uncorrelated, A/T baseline arrangement (with N = 128) to minimize calibration for each AUT may be solved. uncertainties. Each of the MWA’s 128 “tiles” is a 16-antenna III. CHARACTERIZATION RESULTS array of bow-tie dipoles. By selecting interferometry baselines larger than a certain To measure A/T through cross-correlation, we point the minimum, the dominant spatially-extended Galactic noise is interferometer towards Hydra A, a compact calibrator source no longer correlated and “bright” point-like sources are de- 2Some readers will recognize a similar situation in a highly multipath tectable. This may be illustrated by examining the spatial environment: the signal envelope is uncorrelated at locations separated in coherence function due to a distributed source (Ch. 1 in [6]): distance by >∼ 1λ [11]. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )mk(

text: gnihtroN

text: A.

text: Measurand

text: The

text: sensitivity

text: of

text: a

text: radio

text: telescope

text: is

text: quantiﬁed

text: by

text: the

text: ratio

text: of

text: its

text: effective

text: aperture

text: area

text: (A

text: in

text: to

text: the

text: system

text: noise

text: temperature

text: (T

text: in

text: K):

text: sys

text: A

label: equation

text: A/T

text: e

text: T

text: MWA

text: Tiles

text: sys

text: AAVS

text: This

text: ﬁgure-of-merit

text: is

text: a

text: key

text: term

text: in

text: the

text: expression

text: for

text: Easting

text: (km)

text: the

text: telescope

text: output

text: signal-to-noise

text: (S/N)

text: ratio;

text: many

text: readers

text: will

text: recognize

text: that

text: it

text: is

text: similar

text: and

text: is

text: convertible

text: to

text: G/T

text: Fig.

text: A

text: map

text: showing

text: the

text: location

text: (blue

text: circle)

text: and

text: MWA

text: telescope

text: tiles

text: (black

text: cross)

text: in

text: Easting/Northing

text: for

text: Map

text: Grid

text: Australia

text: UTM

text: which

text: is

text: common

text: in

text: telecommunications.

text: The

text: antenna

text: beam

text: zone

text: pattern

text: may

text: be

text: obtained

text: by

text: measuring

text: A/T

text: along

text: the

text: apparent

text: trajectory

text: of

text: a

text: suitable

text: astronomical

text: source.

text: This

text: assumes

text: that

text: the

text: sky

text: noise

text: does

text: not

text: change

text: T

text: appreciably

text: during

text: the

text: sys

text: observation;

text: in

text: practice,

text: this

text: is

text: a

text: reasonable

text: approximation.

text: B.

text: Measurement

text: Method

text: V

text: I(l,

text: m)e

text: This

text: subsection

text: presents

text: a

text: simpliﬁed

text: version

text: of

text: how

text: A/T

text: is

text: obtained

text: in

text: radio

text: interferometry.

text: Interested

text: readers

text: are

text: referred

text: where

text: V

text: is

text: the

text: complex

text: visibility

text: on

text: the

text: baseline

text: between

text: to

text: standard

text: textbooks

text: such

text: as

text: and

text: for

text: more

text: complete

text: antennas

text: and

text: I(l,

text: m)

text: is

text: the

text: sky

text: brightness

text: distribution,

text: u,

text: v

text: discussions.

text: Let

text: us

text: assume

text: an

text: interferometer

text: involving

text: are

text: the

text: baseline

text: vector

text: components

text: normalized

text: to

text: wavelength

text: single-polarized

text: antennas

text: (or

text: antenna

text: arrays

text: as

text: the

text: case

text: may

text: and

text: l,

text: m

text: are

text: the

text: direc√tion

text: cosines

text: with

text: respect

text: to

text: uˆ,

text: vˆ

text: of

text: a

text: unit

text: be)

text: observing

text: a

text: point

text: source

text: with

text: ﬂux

text: S

text: (in

text: or

text: vector

text: sˆ

text: luˆ+mvˆ+

text: where

text: wˆ

text: is

text: a

text: unit

text: vector

text: Jy)

text: in

text: its

text: phase

text: center

text: (in

text: this

text: context

text: “phase

text: center”

text: refers

text: normal

text: to

text: the

text: uv

text: plane

text: (i.e.,

text: points

text: to

text: the

text: “phase

text: center”).

text: to

text: that

text: position

text: in

text: the

text: sky

text: for

text: which

text: the

text: interferometer

text: is

text: Simplifying

text: the

text: case,

text: becomes

text: intended

text: to

text: point

text: to

text: by

text: applying

text: relative

text: phase

text: shifts

text: between

text: its

text: elements).

text: The

text: measured

text: cross-correlation

text: products

text: are:

text: V

text: I(l)e

text: where

text: l

text: sin

text: and

text: is

text: the

text: angle

text: between

text: wˆ

text: and

text: sˆ.

text: We

text: V

text: a˜

text: a˜

text: S

text: ∗3

text: recognize

text: and

text: I(l)

text: as

text: a

text: Fourier

text: transform

text: pair.

text: Hence,

text: structures

text: with

text: large

text: l

text: extent

text: are

text: mostly

text: visible

text: for

text: small

text: From

text: MWA

text: observing

text: experience,

text: with

text: its

text: small-diameter

text: m)

text: tiles,

text: correlated

text: Galactic

text: noise

text: is

text: measurable

text: on

text: base-

text: where

text: a˜

text: is

text: referred

text: to

text: as

text: complex

text: “gain”

text: (voltage

text: quantity)

text: lines

text: of

text: length

text: and

text: should

text: therefore

text: be

text: containing

text: a

text: combination

text: of

text: both

text: antenna

text: and

text: receiver

text: elec-

text: It

text: can

text: be

text: shown

text: (see

text: Ch.

text: of

text: that

text: after

text: calibration

text: of

text: tronic

text: gains.

text: Assuming

text: S

text: is

text: known,

text: the

text: complex

text: “gains”

text: may

text: the

text: measured

text: cross-correlation

text: products,

text: the

text: standard

text: deviation

text: be

text: solved

text: from

text: these

text: equations.

text: This

text: is

text: typically

text: done

text: by

text: in

text: ﬂux

text: density

text: for

text: each

text: baseline

text: is

text: inversely

text: proportional

text: to

text: solving

text: the

text: amplitudes

text: and

text: phases

text: separately

text: the

text: square

text: root

text: of

text: of

text: the

text: A/T

text: for

text: the

text: AUTs

text: in

text: To

text: obtain

text: antenna

text: gains,

text: amplitude-only

text: solutions

text: are

text: suf-

text: question:

text: ﬁcient.

text: Note

text: that

text: this

text: method

text: is

text: analogous

text: to

text: the

text: well-

text: known

text: “three-antenna”

text: measurement

text: technique

text: With

text: N

text: elements,

text: N(N

text: pair-wise

text: combinations

text: (“baselines”)

text: ∆Sij

text: (A/T)

text: Bt

text: are

text: available,

text: leading

text: to

text: an

text: overdetermined

text: system

text: solvable

text: i

text: j

text: acc

text: with

text: a

text: least-squares

text: method.

text: As

text: shown

text: in

text: Fig.

text: the

text: AAVS

text: where

text: k

text: is

text: Boltzmann’s

text: constant,

text: B

text: is

text: the

text: bandwidth

text: and

text: is

text: co-located

text: with

text: the

text: Murchison

text: Wideﬁeld

text: Array

text: (MWA)

text: t

text: is

text: the

text: accumulation

text: time.

text: Thus,

text: by

text: measuring

text: ∆S

text: for

text: all

text: radio

text: telescope

text: to

text: take

text: advantage

text: of

text: this

text: multi-

text: acc

text: baselines

text: for

text: which

text: the

text: Galactic

text: noise

text: is

text: uncorrelated,

text: A/T

text: baseline

text: arrangement

text: (with

text: N

text: to

text: minimize

text: calibration

text: for

text: each

text: AUT

text: may

text: be

text: solved.

text: uncertainties.

text: Each

text: of

text: the

text: MWA’s

text: “tiles”

text: is

text: a

text: III.

text: CHARACTERIZATION

text: RESULTS

text: array

text: of

text: bow-tie

text: dipoles.

text: By

text: selecting

text: interferometry

text: baselines

text: larger

text: than

text: a

text: certain

text: To

text: measure

text: A/T

text: through

text: cross-correlation,

text: we

text: point

text: the

text: minimum,

text: the

text: dominant

text: spatially-extended

text: Galactic

text: noise

text: is

text: interferometer

text: towards

text: Hydra

text: A,

text: a

text: compact

text: calibrator

text: source

text: no

text: longer

text: correlated

text: and

text: “bright”

text: point-like

text: sources

text: are

text: de-

text: readers

text: will

text: recognize

text: a

text: similar

text: situation

text: in

text: a

text: highly

text: multipath

text: tectable.

text: This

text: may

text: be

text: illustrated

text: by

text: examining

text: the

text: spatial

text: environment:

text: the

text: signal

text: envelope

text: is

text: uncorrelated

text: at

text: locations

text: separated

text: in

text: coherence

text: function

text: due

text: to

text: a

text: distributed

text: source

text: (Ch.

text: in

text: distance

text: by

text: 1λ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 Robust Hypothesis Testing with -Divergence α Go¨khan Gu¨l, Student Member, IEEE, Abdelhak M. Zoubir, Fellow, IEEE, Abstract— A robust minimax test for two composite hy- are partially known or are affected by some secondary physical potheses, which are determined by the neighborhoods of two effects that go unmodeled [2]. nominal distributions with respect to a set of distances - called − Imprecise knowledge of F or F leads, in general, to per- α divergence distances, is proposed. Sion’s minimax theorem 0 1 formance degradation and a useful approach is to extend the is adopted to characterize the saddle value condition. Least F favorable distributions, the robust decision rule and the robust known model by accepting a set of distributions , under each H i likelihood ratio test are derived. If the nominal probability dis- hypothesis , that are populated by probability distributions i tributions satisfy a symmetry condition, the design procedure is G , which are at the neighborhood of the nominal distribution shown to be simpliﬁed considerably. The parameters controlling i F based on some distance D [1]. Under some mild conditions the degree of robustness are bounded from above and the bounds i are shown to be resulting from a solution of a set of equations. on D, it can be shown that the best (error minimizing) decision The simulations performed evaluate and exemplify the theoretical rule δˆ for the worst case (error maximizing) pair of probability ∈ F × F derivations. distributions (Gˆ , Gˆ ) accepts a saddle value. 0 1 0 1 Therefore, such a test design guarantees a certain level of Index Terms—Detection, hypothesis testing, robustness, least favorable distributions, minimax optimization, likelihood ratio detection at all times. This type of optimization is known test. as minimax optimization and the corresponding worst case distributions (Gˆ , Gˆ ) are called least favorable distributions 0 1 (LFD)s [3]. I. INTRODUCTION The literature in this ﬁeld is unfortunately not rich. One of Decision theory has been an active ﬁeld of research ben- the earliest and probably the most crucial work goes back eﬁting from contributions from several disciplines, such as to Huber, who proposed a robust version of the probability − economics, engineering, mathematics, or statistics. A decision ratio test for the (cid:15) contamination and total variation classes maker (or a detector) chooses a course of action from several of distributions [4]. He proved the existence of least favorable possibilities. A detector is said to be optimal or to be giving distributions and showed that the corresponding robust test the best decision for a particular problem if the decision rule of was a censored version of the nominal likelihood ratio for interest minimizes (or maximizes) a well deﬁned cost function, both uncertainty classes. In a later work, Huber and Strassen − e.g., the error probability (or the probability of detection) [1]. extended the (cid:15) contamination neighborhood to a larger class, In addition to the fact that decision theory is truly an in- which includes ﬁve different distances as special cases [5]. terdisciplinary subject of research, there are many areas of It was also shown that the robust test resulting from this engineering, where decision theory ﬁnds applications, e.g., new neighborhood was still a censored likelihood ratio test. radar, sonar, seismology, communications and biomedicine. Although it was found to be less engineering oriented by Levy For some applications, such as image and speech classiﬁ- [1], the largest classes for which similar conclusions have been − cation or pattern recognition, interest is in a statistical test made was for the 2 alternating capacities proposed by Huber that performs well on average. However, for safety oriented and Strassen [6]. applications such as seismology or forest ﬁre detection, as well Another approach for robust hypothesis testing was proposed as for biomedical applications such as early cancer detection by Dabak and Johnson based on the fact that the choice from magnetic resonance images or X-ray images, interest of measures deﬁning the contamination neighborhoods was is in maximizing the worst case performance because the arbitrary [7]. They chose the relative entropy (KL-divergence) consequences of an incorrect decision can be severe [1]. because it is a natural distance between probability measures In general, any practical application of decision theory can and therefore a natural way to deﬁne the contamination be formulated as a hypothesis testing problem. For binary neighborhoods. Somewhat surprisingly, the robust test which hypothesis testing, it is assumed that under each hypothesis minimizes the KL-divergence between the LFDs obtained H ∈ , the received data y = (y , . . . , y ) Ω follows a par- from the closed balls with respect to the relative entropy i 1 n ticular distribution F corresponding to a density function f , distance was not a clipped likelihood ratio test, but a nominal ∈ { } i i i 0, 1 . A decision rule δ partitions the whole observation likelihood ratio test with a modiﬁed threshold. It was noted space Ω into non-overlapping regions corresponding to each that their approach was not robust for all sample sizes but hypothesis. The optimality of the decision rule δ depends on when Kullback’s theorem is valid, that is for a large number the correctness of the assumption that the data y follows F . of observations [7]. The difference in the robust tests for i − However, in many practical applications either F and/or F (cid:15) contamination and relative entropy neighborhoods lies in 0 1 the fact that all the densities in the class of distributions based G. Gu¨l and A. M. Zoubir are with the Signal Processing Group, Institute on relative entropy are absolutely continuous with respect of Telecommunications, Technische Universitarmstadt, 64283, Darmstadt, to the nominal distributions, but not for the case of the Germany (e-mail: ggul@spg.tu-darmstadt.de; zoubir@spg.tu-darmstadt.de) − Manuscript received April 19, 2005; revised January 11, 2007. (cid:15) contamination class.

label: paragraph

text: Robust

label: title

text: Hypothesis

text: Testing

text: with

text: -Divergence

text: α

text: Go¨khan

label: author

text: Gu¨l,

text: Student

text: Member,

text: IEEE,

text: Abdelhak

text: M.

text: Zoubir,

text: Fellow,

text: IEEE,

text: Abstract—

text: A

label: abstract

text: robust

text: minimax

text: test

text: for

text: two

text: composite

text: hy-

text: are

text: partially

text: known

text: or

text: are

text: affected

text: by

text: some

text: secondary

text: physical

text: potheses,

text: which

text: are

text: determined

text: by

text: the

text: neighborhoods

text: of

text: two

text: effects

text: that

text: go

text: unmodeled

text: nominal

text: distributions

text: with

text: respect

text: to

text: a

text: set

text: of

text: distances

text: called

text: Imprecise

text: knowledge

text: of

text: F

text: or

text: F

text: leads,

text: in

text: general,

text: to

text: per-

text: α

text: divergence

text: distances,

text: is

text: proposed.

text: Sion’s

text: minimax

text: theorem

text: formance

text: degradation

text: and

text: a

text: useful

text: approach

text: is

text: to

text: extend

text: the

text: is

text: adopted

text: to

text: characterize

text: the

text: saddle

text: value

text: condition.

text: Least

text: F

text: favorable

text: distributions,

text: the

text: robust

text: decision

text: rule

text: and

text: the

text: robust

text: known

text: model

text: by

text: accepting

text: a

text: set

text: of

text: distributions

text: under

text: each

text: H

text: i

text: likelihood

text: ratio

text: test

text: are

text: derived.

text: If

text: the

text: nominal

text: probability

text: dis-

text: hypothesis

text: that

text: are

text: populated

text: by

text: probability

text: distributions

text: i

text: tributions

text: satisfy

text: a

text: symmetry

text: condition,

text: the

text: design

text: procedure

text: is

text: G

text: which

text: are

text: at

text: the

text: neighborhood

text: of

text: the

text: nominal

text: distribution

text: shown

text: to

text: be

text: simpliﬁed

text: considerably.

text: The

text: parameters

text: controlling

text: i

text: F

text: based

text: on

text: some

text: distance

text: D

text: Under

text: some

text: mild

text: conditions

text: the

text: degree

text: of

text: robustness

text: are

text: bounded

text: from

text: above

text: and

text: the

text: bounds

text: i

text: are

text: shown

text: to

text: be

text: resulting

text: from

text: a

text: solution

text: of

text: a

text: set

text: of

text: equations.

text: on

text: D,

text: it

text: can

text: be

text: shown

text: that

text: the

text: best

text: (error

text: minimizing)

text: decision

text: The

text: simulations

text: performed

text: evaluate

text: and

text: exemplify

text: the

text: theoretical

text: rule

text: δˆ

text: for

text: the

text: worst

text: case

text: (error

text: maximizing)

text: pair

text: of

text: probability

text: F

text: F

text: derivations.

text: distributions

text: (Gˆ

text: Gˆ

text: accepts

text: a

text: saddle

text: value.

text: Therefore,

text: such

text: a

text: test

text: design

text: guarantees

text: a

text: certain

text: level

text: of

text: Index

text: Terms—Detection,

text: hypothesis

text: testing,

text: robustness,

text: least

text: favorable

text: distributions,

text: minimax

text: optimization,

text: likelihood

text: ratio

text: detection

text: at

text: all

text: times.

text: This

text: type

text: of

text: optimization

text: is

text: known

text: test.

text: as

text: minimax

text: optimization

text: and

text: the

text: corresponding

text: worst

text: case

text: distributions

text: (Gˆ

text: Gˆ

text: are

text: called

text: least

text: favorable

text: distributions

text: (LFD)s

text: I.

text: INTRODUCTION

text: The

text: literature

text: in

text: this

text: ﬁeld

text: is

text: unfortunately

text: not

text: rich.

text: One

text: of

text: Decision

text: theory

text: has

text: been

text: an

text: active

text: ﬁeld

text: of

text: research

text: ben-

text: the

text: earliest

text: and

text: probably

text: the

text: most

text: crucial

text: work

text: goes

text: back

text: eﬁting

text: from

text: contributions

text: from

text: several

text: disciplines,

text: such

text: as

text: to

text: Huber,

text: who

text: proposed

text: a

text: robust

text: version

text: of

text: the

text: probability

text: economics,

text: engineering,

text: mathematics,

text: or

text: statistics.

text: A

text: decision

text: ratio

text: test

text: for

text: the

text: contamination

text: and

text: total

text: variation

text: classes

text: maker

text: (or

text: a

text: detector)

text: chooses

text: a

text: course

text: of

text: action

text: from

text: several

text: of

text: distributions

text: He

text: proved

text: the

text: existence

text: of

text: least

text: favorable

text: possibilities.

text: A

text: detector

text: is

text: said

text: to

text: be

text: optimal

text: or

text: to

text: be

text: giving

text: distributions

text: and

text: showed

text: that

text: the

text: corresponding

text: robust

text: test

text: the

text: best

text: decision

text: for

text: a

text: particular

text: problem

text: if

text: the

text: decision

text: rule

text: of

text: was

text: a

text: censored

text: version

text: of

text: the

text: nominal

text: likelihood

text: ratio

text: for

text: interest

text: minimizes

text: (or

text: maximizes)

text: a

text: well

text: deﬁned

text: cost

text: function,

text: both

text: uncertainty

text: classes.

text: In

text: a

text: later

text: work,

text: Huber

text: and

text: Strassen

text: e.g.,

text: the

text: error

text: probability

text: (or

text: the

text: probability

text: of

text: detection)

text: extended

text: the

text: contamination

text: neighborhood

text: to

text: a

text: larger

text: class,

text: In

text: addition

text: to

text: the

text: fact

text: that

text: decision

text: theory

text: is

text: truly

text: an

text: in-

text: which

text: includes

text: ﬁve

text: different

text: distances

text: as

text: special

text: cases

text: terdisciplinary

text: subject

text: of

text: research,

text: there

text: are

text: many

text: areas

text: of

text: It

text: was

text: also

text: shown

text: that

text: the

text: robust

text: test

text: resulting

text: from

text: this

text: engineering,

text: where

text: decision

text: theory

text: ﬁnds

text: applications,

text: e.g.,

text: new

text: neighborhood

text: was

text: still

text: a

text: censored

text: likelihood

text: ratio

text: test.

text: radar,

text: sonar,

text: seismology,

text: communications

text: and

text: biomedicine.

text: Although

text: it

text: was

text: found

text: to

text: be

text: less

text: engineering

text: oriented

text: by

text: Levy

text: For

text: some

text: applications,

text: such

text: as

text: image

text: and

text: speech

text: classiﬁ-

text: the

text: largest

text: classes

text: for

text: which

text: similar

text: conclusions

text: have

text: been

text: cation

text: or

text: pattern

text: recognition,

text: interest

text: is

text: in

text: a

text: statistical

text: test

text: made

text: was

text: for

text: the

text: alternating

text: capacities

text: proposed

text: by

text: Huber

text: that

text: performs

text: well

text: on

text: average.

text: However,

text: for

text: safety

text: oriented

text: and

text: Strassen

text: applications

text: such

text: as

text: seismology

text: or

text: forest

text: ﬁre

text: detection,

text: as

text: well

text: Another

text: approach

text: for

text: robust

text: hypothesis

text: testing

text: was

text: proposed

text: as

text: for

text: biomedical

text: applications

text: such

text: as

text: early

text: cancer

text: detection

text: by

text: Dabak

text: and

text: Johnson

text: based

text: on

text: the

text: fact

text: that

text: the

text: choice

text: from

text: magnetic

text: resonance

text: images

text: or

text: X-ray

text: images,

text: interest

text: of

text: measures

text: deﬁning

text: the

text: contamination

text: neighborhoods

text: was

text: is

text: in

text: maximizing

text: the

text: worst

text: case

text: performance

text: because

text: the

text: arbitrary

text: They

text: chose

text: the

text: relative

text: entropy

text: (KL-divergence)

text: consequences

text: of

text: an

text: incorrect

text: decision

text: can

text: be

text: severe

text: because

text: it

text: is

text: a

text: natural

text: distance

text: between

text: probability

text: measures

text: In

text: general,

text: any

text: practical

text: application

text: of

text: decision

text: theory

text: can

text: and

text: therefore

text: a

text: natural

text: way

text: to

text: deﬁne

text: the

text: contamination

text: be

text: formulated

text: as

text: a

text: hypothesis

text: testing

text: problem.

text: For

text: binary

text: neighborhoods.

text: Somewhat

text: surprisingly,

text: the

text: robust

text: test

text: which

text: hypothesis

text: testing,

text: it

text: is

text: assumed

text: that

text: under

text: each

text: hypothesis

text: minimizes

text: the

text: KL-divergence

text: between

text: the

text: LFDs

text: obtained

text: H

text: the

text: received

text: data

text: y

text: (y

text: y

text: Ω

text: follows

text: a

text: par-

text: from

text: the

text: closed

text: balls

text: with

text: respect

text: to

text: the

text: relative

text: entropy

text: i

text: n

text: ticular

text: distribution

text: F

text: corresponding

text: to

text: a

text: density

text: function

text: f

text: distance

text: was

text: not

text: a

text: clipped

text: likelihood

text: ratio

text: test,

text: but

text: a

text: nominal

text: i

text: i

text: i

text: A

text: decision

text: rule

text: δ

text: partitions

text: the

text: whole

text: observation

text: likelihood

text: ratio

text: test

text: with

text: a

text: modiﬁed

text: threshold.

text: It

text: was

text: noted

text: space

text: Ω

text: into

text: non-overlapping

text: regions

text: corresponding

text: to

text: each

text: that

text: their

text: approach

text: was

text: not

text: robust

text: for

text: all

text: sample

text: sizes

text: but

text: hypothesis.

text: The

text: optimality

text: of

text: the

text: decision

text: rule

text: δ

text: depends

text: on

text: when

text: Kullback’s

text: theorem

text: is

text: valid,

text: that

text: is

text: for

text: a

text: large

text: number

text: the

text: correctness

text: of

text: the

text: assumption

text: that

text: the

text: data

text: y

text: follows

text: F

text: of

text: observations

text: The

text: difference

text: in

text: the

text: robust

text: tests

text: for

text: i

text: However,

text: in

text: many

text: practical

text: applications

text: either

text: F

text: and/or

text: F

text: contamination

text: and

text: relative

text: entropy

text: neighborhoods

text: lies

text: in

text: the

text: fact

text: that

text: all

text: the

text: densities

text: in

text: the

text: class

text: of

text: distributions

text: based

text: G.

text: Gu¨l

text: and

text: A.

text: M.

text: Zoubir

text: are

text: with

text: the

text: Signal

text: Processing

text: Group,

text: Institute

text: on

text: relative

text: entropy

text: are

text: absolutely

text: continuous

text: with

text: respect

text: of

text: Telecommunications,

text: Technische

text: Universitarmstadt,

text: Darmstadt,

text: to

text: the

text: nominal

text: distributions,

text: but

text: not

text: for

text: the

text: case

text: of

text: the

text: Germany

text: (e-mail:

text: ggul@spg.tu-darmstadt.de;

text: zoubir@spg.tu-darmstadt.de)

text: Manuscript

text: received

text: April

text: revised

text: January

text: contamination

text: class.


text: 4 − C. Problem deﬁnition with Φ = Φ lρ 1. 0 1 Based on (10), the minimax optimization problem (8) can Theorem III.1. The least favorable densities be solved considering the Karush-Kuhn-Tucker (KKT) multi-  pliers. Hence, the problem formulation can be restated as z(ll,llul;α,ρ)f0, l <≤ρll ≤ (cid:90) gˆ = Φ (l, l , l ; α, ρ)f , ρl l ρl , (15) Maximization: gˆ = arg sup P (δ, g ) 0 0 l u 0 l u 0 g0∈G0 F 0 zk(l(ll,ll,ulu;α)l,uρ)f0, l > ρlu s.t. g0 > 0, Υ(g0) = R g0 dµ = 1  (cid:90) gˆ1 = arg gs1u∈pG1 PM(δ, g1) gˆ = Φz(ll(,llu,1l;α,,ρl)f; α1,, ρ)f , lρl<≤ρlll ≤ ρl , (16) 1 1 l u 1 l u s.t. g1 > 0, Υ(g1) = R g1 dµ = 1 z(kll(,lllu,l;αu),ρ)f1, l > ρlu and the robust decision rule Minimization: δˆ = arg mδ∈i∆n PE(δ, gˆ0, gˆ1). (11)  wdehcIinicshio(1anr1er)u,cloethuδep2rleedawreithtwthoe smeipnairmatiezatmioanxipmroizbaletimonthproroubglhemthse,δˆ = 0(l,lα−1−(k(ll,lu)llulα)−α1−(1l/)ρ(l)/1ρ−)α1−−α1+k(ll,lu)α−1−1, ρl l<l ≤ρlll ≤ ρlu , − 1, l > ρl III. ROBUST DETECTION WITH α DIVERGENCE u (17) The following theorem provides a solution for (11), which is  composed of the least favorable densities gˆ0 and gˆ1, the robust implying the robust likelihood ratio function decision rule δˆ, the robust likelihood ratio function ˆl = gˆ /gˆ 1 0 in parametric forms, as well as two non-linear equations from l−1l, l < ρl which the parameters can be obtained. Before the statement gˆ l ≤ l ≤ ˆl = 1 = ρ, ρl l ρl (18) ol f≤th1e ≤thelor<em∞, l.eFt ullrthanerdmlou(cid:82)reb,elettwo real numbers with 0 < gˆ0 lu−1l, l >l ρlu u l u (cid:82) − I (l l )f dµ provide a unique solution to (11). Furthermore, the param- (cid:90)k(ll, lu) = I1(lu −(cid:90)ll)f00dµ, (12) eters ll and lu ca(cid:32)n be (cid:90)determined (cid:90)by solving 3 (cid:90) z(cid:18)(l , l ; α, ρ) = f dµ + k(l , l ) f dµ+ (cid:19) z(l , l 1; α, ρ)α (cid:90)llα I f0(cid:33)dµ + I Φ(cid:48)0(ll, lu; α, ρ)αf0dµ l u I 1 l u I 1 l u 1 2 1 3 k(l , l )α−1(lα−1 − lα−1) α−1 1 + (k(l , l )l )α f dµ = x(α, ε ) (19) I lα−1 − (k(l , l )ll )uα−1 + (lk(l , l )uα−1 − 1)(l/ρ)α−1 f1dµ, l u u I3 0 0 2 l l u u l u (13) and (cid:32) (cid:90) (cid:90) where I1 := {y : l(y) < ρll} ≡ {y : ˆl(y) < ρ} z(ll, lu1; α, ρ)(cid:90)α I f(cid:33)1dµ + I Φ(cid:48)1(ll, lu; α, ρ)αf1dµ I { ≤ ≤ } ≡ { } 1 2 := y : ρl l(y) ρl y : ˆl(y) = ρ 2 l u I3 := {y : l(y) > ρlu} ≡ {y : ˆl(y) > ρ} (14) + k(ll, lu)α I f1dµ = x(α, ε1) (20) 3 (cid:48) − and where Φ (l , l ; α, ρ) = z(l , l ; α, ρ)Φ , and x(α, ε) = 1 − j l u l u j α(1 α)ε. Φ (l, l , l ; α, ρ) = 1 l u (cid:18) 1 · (cid:19) A proof of Theorem III.1 is given in three stages. In the maximization stage, the Karush-Kuhn-Tucker (KKT) multipli- z(l , l ; α, ρ) l u ers are used to determine the parametric forms of the LFDs, − − k(ll, lu)−α−1(llα−1 − luα−−1)− − α−1 1 gˆ0 and gˆ1, and the robust likelihood ratio function ˆl. In the lα 1 (k(l , l )l )α 1 + (k(l , l )α 1 1)(l/ρ)α 1 minimization stage, the LFDs and the robust decision rule l l u u l u G G δˆ are made explicit. Finally, in the optimization stage, four 2In general arg sup may not always be achieved since 0 and 1 are non- parameters that are needed to design the test are reduced to compact sets in the topologies induced by the α-divergence distance. In this paper, existence of gˆ0 and gˆ1 is due to the KKT solution of the minimax two parameters without loss of generality. optimization problem, which is introduced in Section . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C.

label: section

text: Problem

text: deﬁnition

text: with

text: Φ

text: Φ

text: lρ

text: Based

text: on

text: the

text: minimax

text: optimization

text: problem

text: can

text: Theorem

text: The

text: least

text: favorable

text: densities

text: be

text: solved

text: considering

text: the

text: Karush-Kuhn-Tucker

text: (KKT)

text: multi-

text: pliers.

text: Hence,

text: the

text: problem

text: formulation

text: can

text: be

text: restated

text: as

text: l

text: <≤ρll

text: gˆ

text: Φ

text: (l,

text: l

text: l

text: α,

text: ρ)f

text: ρl

text: l

text: ρl

text: Maximization:

text: gˆ

text: arg

text: sup

text: P

text: (δ,

text: g

text: l

text: u

text: l

text: u

text: F

text: l

text: ρlu

text: s.t.

text: R

text: dµ

text: arg

text: PM(δ,

text: gˆ

text: ρ)f

text: lρl<≤ρlll

text: ρl

text: l

text: u

text: l

text: u

text: s.t.

text: R

text: dµ

text: l

text: ρlu

text: and

text: the

text: robust

text: decision

text: rule

text: Minimization:

text: δˆ

text: arg

text: mδ∈i∆n

text: PE(δ,

text: ρl

text: l<l

text: ≤ρlll

text: ρlu

text: l

text: ρl

text: III.

text: ROBUST

text: DETECTION

text: WITH

text: α

text: DIVERGENCE

text: u

text: The

text: following

text: theorem

text: provides

text: a

text: solution

text: for

text: which

text: is

text: composed

text: of

text: the

text: least

text: favorable

text: densities

text: and

text: the

text: robust

text: implying

text: the

text: robust

text: likelihood

text: ratio

text: function

text: decision

text: rule

text: δˆ,

text: the

text: robust

text: likelihood

text: ratio

text: function

text: ˆl

text: gˆ

text: /gˆ

text: in

text: parametric

text: forms,

text: as

text: well

text: as

text: two

text: non-linear

text: equations

text: from

text: l

text: ρl

text: which

text: the

text: parameters

text: can

text: be

text: obtained.

text: Before

text: the

text: statement

text: gˆ

text: l

text: l

text: ˆl

text: ρ,

text: ρl

text: l

text: ρl

text: ol

text: ≤thelor<em∞,

text: l.eFt

text: real

text: numbers

text: with

text: l

text: >l

text: ρlu

text: u

text: l

text: u

text: I

text: (l

text: l

text: )f

text: dµ

text: provide

text: a

text: unique

text: solution

text: to

text: Furthermore,

text: the

text: param-

text: lu)

text: eters

text: ll

text: and

text: lu

text: be

text: solving

text: l

text: α,

text: ρ)

text: f

text: dµ

text: k(l

text: l

text: f

text: dµ+

text: z(l

text: l

text: α,

text: ρ)α

text: I

text: I

text: lu;

text: α,

text: l

text: u

text: I

text: l

text: u

text: I

text: l

text: u

text: k(l

text: l

text: α−1

text: (k(l

text: l

text: )l

text: )α

text: f

text: dµ

text: x(α,

text: ε

text: I

text: lα−1

text: (k(l

text: l

text: )ll

text: (lk(l

text: l

text: l

text: u

text: u

text: l

text: l

text: u

text: u

text: l

text: u

text: and

text: where

text: {y

text: l(y)

text: ρll}

text: {y

text: ˆl(y)

text: ρ}

text: z(ll,

text: α,

text: I

text: I

text: lu;

text: α,

text: I

text: y

text: ρl

text: l(y)

text: ρl

text: y

text: ˆl(y)

text: ρ

text: l

text: u

text: {y

text: l(y)

text: ρlu}

text: {y

text: ˆl(y)

text: ρ}

text: k(ll,

text: lu)α

text: I

text: x(α,

text: and

text: where

text: Φ

text: (l

text: l

text: α,

text: ρ)

text: z(l

text: l

text: α,

text: ρ)Φ

text: and

text: x(α,

text: ε)

text: j

text: l

text: u

text: l

text: u

text: j

text: α)ε.

text: Φ

text: (l,

text: l

text: l

text: α,

text: ρ)

text: l

text: u

text: A

text: proof

text: of

text: Theorem

text: is

text: given

text: in

text: three

text: stages.

text: In

text: the

text: maximization

text: stage,

text: the

text: Karush-Kuhn-Tucker

text: (KKT)

text: multipli-

text: z(l

text: l

text: α,

text: ρ)

text: l

text: u

text: ers

text: are

text: used

text: to

text: determine

text: the

text: parametric

text: forms

text: of

text: the

text: LFDs,

text: k(ll,

text: luα−−1)−

text: α−1

text: and

text: and

text: the

text: robust

text: likelihood

text: ratio

text: function

text: ˆl.

text: In

text: the

text: lα

text: (k(l

text: l

text: )l

text: )α

text: (k(l

text: l

text: )α

text: minimization

text: stage,

text: the

text: LFDs

text: and

text: the

text: robust

text: decision

text: rule

text: l

text: l

text: u

text: u

text: l

text: u

text: G

text: G

text: δˆ

text: are

text: made

text: explicit.

text: Finally,

text: in

text: the

text: optimization

text: stage,

text: four

text: general

text: arg

text: sup

text: may

text: not

text: always

text: be

text: achieved

text: since

text: and

text: are

text: non-

text: parameters

text: that

text: are

text: needed

text: to

text: design

text: the

text: test

text: are

text: reduced

text: to

text: compact

text: sets

text: in

text: the

text: topologies

text: induced

text: by

text: the

text: α-divergence

text: distance.

text: In

text: this

text: paper,

text: existence

text: of

text: and

text: is

text: due

text: to

text: the

text: KKT

text: solution

text: of

text: the

text: minimax

text: two

text: parameters

text: without

text: loss

text: of

text: generality.

text: optimization

text: problem,

text: which

text: is

text: introduced

text: in

text: Section

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 (cid:96) l f0, f1 (cid:96) l l Minimax Robust ∆ u Ε Ε 0, 1 (cid:96) (cid:96) Test Design g g Ρ Α 0, 1 , Ρ (cid:144) l l Fig. 1. Summary of the robust hypothesis testing scheme given by Theo- l rem III.1. I I I • Consider new domains , , 1 2 3 • Use the substitutions c1 := c3ll and c2 := c4lu (cid:144) • Divide both sides of the ﬁrst two equations by c l • Equate the resulting equations to each other via 31/c Ρll Ρlu 3 Fig. 2. Nonlinearity relating the nominal likelihood ratios to the robust leads to c4 = k(ll, lu)c3, where k(ll, lu) is as deﬁned by likelihood ratios. (12). Next, the goal is to ﬁnd a functional f s.t. Φ = 1 c f(l, l , l , α). Since Φ f ρ = Φ f , it follows that Φ = 3 l u − 0 0 1 1 0 c f(l, l , l , α)lρ 1, therefore it sufﬁces to evaluate only Φ . 3 l u 1 test, which is given by (18) is not explicitly a function of α. A step by step derivation of the functional f is given in Theorem III.1 is a generalization of [22] in the sense that Appendix . Accordingly, Φ is also fully speciﬁed in terms of → 0 as α 1 and ρ = 1, the least favorable densities gˆ and the desired parameters and functions. Inserting Φ1 (which is gˆ as well as the robust decision rule δˆ reduce to the0ones now a functional of c , l, l , l , α), c.f., (51), into the second 1 3 l u found in [22]. The ﬂexibility afforded by the generality of equation in (32) and noticing that c = k(l , l )c leads to − 4 l u 3 considering a set of distances, called the α divergence, over c = 1/z(l , l ; α, ρ), where z(l , l ; α, ρ) is as deﬁned by 3 l u l u [22] is twofold. First, the designer does not need to search for (13). Applying a similar procedure, which can be found in a suitable distance for modeling errors, and each time test for Appendix , to δˆ, c.f., (28), for the case ˆl = ρ leads to the the applicability to the engineering problem at hand, following robust decision rule δˆ as given by Theorem III.1. The least tedious steps of derivations. Instead, only the parameter α is favorable densities, gˆ and gˆ , and the robust likelihood ratio 0 1 required to be determined, which can be done over a training function ˆl are obtained similarly, by exploiting the connection data set via using a suitable search algorithm. Second, the a between the parameters c , c , c , c and l , l . The same 1 2 3 4 l u priori probabilities are not necessarily to be chosen equal. The simpliﬁcations eventually let the four equations given by (32) − proposed design with the α divergence covers both cases, in to be rewritten as the two equations stated by Theorem III.1. addition to the fact that the choice of the nominal probability As it was mentioned earlier, both gˆ and gˆ are obtained 0 1 distributions also does not require any assumption. Additional uniquely from the Lagrangian L. Hence, ˆl = gˆ /gˆ , and 1 0 constraints on the choice of nominal distributions as well as as a result, δˆ are also unique. It follows that the solution on the robustness parameters simplify the design as introduced found for (11) by the KKT multipliers approach is unique in the next section. as claimed. Theorem III.1 can be summarized as illustrated in Figure 1. In other words, for any choice of pair of nominal density functions f and f , the robustness parameters (cid:15) and (cid:15) , the 0 1 0 1 Bayesian threshold ρ and the distance parameter α, the robust design outputs the least favorable density functions gˆ and gˆ B. Simpliﬁed model with additional constraints 0 1 and the robust decision rule δˆ. Notice that gˆ and gˆ are the 0 1 scaled versions (with different scaling factors) of the nominal distributions on l < ρl and l > ρl , and in between, they In some cases, evidence that the following assumption holds l u are a composition of both nominals, since Φ and Φ are both may be available: 0 1 functionals of f and f . Interpretation of the decision rule δˆ is 0 1 similar, i.e. in the same two regions the robust decision rule is almost surely zero or one, and in between it is a randomized Assumption III.2. The nominal likelihood ratio l is monotone decision rule. The robust version of the nominal likelihood and the nominal density functions are symmetric, i.e., f (y) = − ∀ 1 ratio test is a non-linearly transformed version of the nominal f ( y) y 0 likelihood ratios as illustrated by Figure 2. It is somewhat surprising that the resulting robust likelihood ratio test is the If, additionally, the robustness parameters are set to be same for the whole family of distances that are parameterized equal, (cid:15) = (cid:15) = (cid:15) , or in other words x(α, (cid:15)) = x(α, (cid:15) ) = 0 1 0 by α. In other words, the robust version of the likelihood ratio x(α, (cid:15) ), it follows that 1 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: l

text: l

text: l

text: Minimax

text: Robust

text: u

text: Ε

text: Ε

text: Test

text: Design

text: g

text: g

text: Ρ

text: Α

text: Ρ

text: l

text: l

text: Fig.

text: Summary

label: caption

text: of

text: the

text: robust

text: hypothesis

text: testing

text: scheme

text: given

text: by

text: Theo-

text: l

text: rem

text: I

label: list

text: I

text: I

text: Consider

text: new

text: domains

text: Use

text: the

text: substitutions

text: and

text: Divide

text: both

text: sides

text: of

text: the

text: ﬁrst

text: two

text: equations

text: by

text: c

text: l

text: Equate

text: the

text: resulting

text: equations

text: to

text: each

text: other

text: via

text: Ρll

text: Ρlu

text: Fig.

text: Nonlinearity

text: relating

text: the

text: nominal

text: likelihood

text: ratios

text: to

text: the

text: robust

text: leads

text: to

text: k(ll,

text: where

text: k(ll,

text: lu)

text: is

text: as

text: deﬁned

text: by

text: likelihood

text: ratios.

text: Next,

text: the

text: goal

text: is

text: to

text: ﬁnd

text: a

text: functional

text: f

text: s.t.

text: Φ

text: c

text: f(l,

text: l

text: l

text: α).

text: Since

text: Φ

text: f

text: ρ

text: Φ

text: f

text: it

text: follows

text: that

text: Φ

text: l

text: u

text: c

text: f(l,

text: l

text: l

text: α)lρ

text: therefore

text: it

text: sufﬁces

text: to

text: evaluate

text: only

text: Φ

text: l

text: u

text: test,

text: which

text: is

text: given

text: by

text: is

text: not

text: explicitly

text: a

text: function

text: of

text: α.

text: A

text: step

text: by

text: step

text: derivation

text: of

text: the

text: functional

text: f

text: is

text: given

text: in

text: Theorem

text: is

text: a

text: generalization

text: of

text: in

text: the

text: sense

text: that

text: Appendix

text: Accordingly,

text: Φ

text: is

text: also

text: fully

text: speciﬁed

text: in

text: terms

text: of

text: as

text: α

text: and

text: ρ

text: the

text: least

text: favorable

text: densities

text: gˆ

text: and

text: the

text: desired

text: parameters

text: and

text: functions.

text: Inserting

text: Φ1

text: (which

text: is

text: gˆ

text: as

text: well

text: as

text: the

text: robust

text: decision

text: rule

text: δˆ

text: reduce

text: to

text: now

text: a

text: functional

text: of

text: c

text: l,

text: l

text: l

text: α),

text: c.f.,

text: into

text: the

text: second

text: l

text: u

text: found

text: in

text: The

text: ﬂexibility

text: afforded

text: by

text: the

text: generality

text: of

text: equation

text: in

text: and

text: noticing

text: that

text: c

text: k(l

text: l

text: )c

text: leads

text: to

text: l

text: u

text: considering

text: a

text: set

text: of

text: distances,

text: called

text: the

text: α

text: divergence,

text: over

text: c

text: l

text: α,

text: ρ),

text: where

text: z(l

text: l

text: α,

text: ρ)

text: is

text: as

text: deﬁned

text: by

text: l

text: u

text: l

text: u

text: is

text: twofold.

text: First,

text: the

text: designer

text: does

text: not

text: need

text: to

text: search

text: for

text: Applying

text: a

text: similar

text: procedure,

text: which

text: can

text: be

text: found

text: in

text: a

text: suitable

text: distance

text: for

text: modeling

text: errors,

text: and

text: each

text: time

text: test

text: for

text: Appendix

text: to

text: δˆ,

text: c.f.,

text: for

text: the

text: case

text: ˆl

text: ρ

text: leads

text: to

text: the

text: the

text: applicability

text: to

text: the

text: engineering

text: problem

text: at

text: hand,

text: following

text: robust

text: decision

text: rule

text: δˆ

text: as

text: given

text: by

text: Theorem

text: The

text: least

text: tedious

text: steps

text: of

text: derivations.

text: Instead,

text: only

text: the

text: parameter

text: α

text: is

text: favorable

text: densities,

text: gˆ

text: and

text: gˆ

text: and

text: the

text: robust

text: likelihood

text: ratio

text: required

text: to

text: be

text: determined,

text: which

text: can

text: be

text: done

text: over

text: a

text: training

text: function

text: ˆl

text: are

text: obtained

text: similarly,

text: by

text: exploiting

text: the

text: connection

text: data

text: set

text: via

text: using

text: a

text: suitable

text: search

text: algorithm.

text: Second,

text: the

text: a

text: between

text: the

text: parameters

text: c

text: c

text: c

text: c

text: and

text: l

text: l

text: The

text: same

text: l

text: u

text: priori

text: probabilities

text: are

text: not

text: necessarily

text: to

text: be

text: chosen

text: equal.

text: The

text: simpliﬁcations

text: eventually

text: let

text: the

text: four

text: equations

text: given

text: by

text: proposed

text: design

text: with

text: the

text: α

text: divergence

text: covers

text: both

text: cases,

text: in

text: to

text: be

text: rewritten

text: as

text: the

text: two

text: equations

text: stated

text: by

text: Theorem

text: addition

text: to

text: the

text: fact

text: that

text: the

text: choice

text: of

text: the

text: nominal

text: probability

text: As

text: it

text: was

text: mentioned

text: earlier,

text: both

text: gˆ

text: and

text: gˆ

text: are

text: obtained

text: distributions

text: also

text: does

text: not

text: require

text: any

text: assumption.

text: Additional

text: uniquely

text: from

text: the

text: Lagrangian

text: L.

text: Hence,

text: ˆl

text: gˆ

text: /gˆ

text: and

text: constraints

text: on

text: the

text: choice

text: of

text: nominal

text: distributions

text: as

text: well

text: as

text: as

text: a

text: result,

text: δˆ

text: are

text: also

text: unique.

text: It

text: follows

text: that

text: the

text: solution

text: on

text: the

text: robustness

text: parameters

text: simplify

text: the

text: design

text: as

text: introduced

text: found

text: for

text: by

text: the

text: KKT

text: multipliers

text: approach

text: is

text: unique

text: in

text: the

text: next

text: section.

text: as

text: claimed.

text: Theorem

text: can

text: be

text: summarized

text: as

text: illustrated

text: in

text: Figure

text: In

text: other

text: words,

text: for

text: any

text: choice

text: of

text: pair

text: of

text: nominal

text: density

text: functions

text: f

text: and

text: f

text: the

text: robustness

text: parameters

text: and

text: the

text: Bayesian

text: threshold

text: ρ

text: and

text: the

text: distance

text: parameter

text: α,

text: the

text: robust

text: design

text: outputs

text: the

text: least

text: favorable

text: density

text: functions

text: gˆ

text: and

text: gˆ

text: B.

text: Simpliﬁed

text: model

text: with

text: additional

text: constraints

text: and

text: the

text: robust

text: decision

text: rule

text: δˆ.

text: Notice

text: that

text: gˆ

text: and

text: gˆ

text: are

text: the

text: scaled

text: versions

text: (with

text: different

text: scaling

text: factors)

text: of

text: the

text: nominal

text: distributions

text: on

text: l

text: ρl

text: and

text: l

text: ρl

text: and

text: in

text: between,

text: they

text: In

text: some

text: cases,

text: evidence

text: that

text: the

text: following

text: assumption

text: holds

text: l

text: u

text: are

text: a

text: composition

text: of

text: both

text: nominals,

text: since

text: Φ

text: and

text: Φ

text: are

text: both

text: may

text: be

text: available:

text: functionals

text: of

text: f

text: and

text: f

text: Interpretation

text: of

text: the

text: decision

text: rule

text: δˆ

text: is

text: similar,

text: i.e.

text: in

text: the

text: same

text: two

text: regions

text: the

text: robust

text: decision

text: rule

text: is

text: almost

text: surely

text: zero

text: or

text: one,

text: and

text: in

text: between

text: it

text: is

text: a

text: randomized

text: Assumption

text: The

text: nominal

text: likelihood

text: ratio

text: l

text: is

text: monotone

text: decision

text: rule.

text: The

text: robust

text: version

text: of

text: the

text: nominal

text: likelihood

text: and

text: the

text: nominal

text: density

text: functions

text: are

text: symmetric,

text: i.e.,

text: f

text: (y)

text: ratio

text: test

text: is

text: a

text: non-linearly

text: transformed

text: version

text: of

text: the

text: nominal

text: f

text: y)

text: y

text: likelihood

text: ratios

text: as

text: illustrated

text: by

text: Figure

text: It

text: is

text: somewhat

text: surprising

text: that

text: the

text: resulting

text: robust

text: likelihood

text: ratio

text: test

text: is

text: the

text: If,

text: additionally,

text: the

text: robustness

text: parameters

text: are

text: set

text: to

text: be

text: same

text: for

text: the

text: whole

text: family

text: of

text: distances

text: that

text: are

text: parameterized

text: equal,

text: or

text: in

text: other

text: words

text: x(α,

text: x(α,

text: by

text: α.

text: In

text: other

text: words,

text: the

text: robust

text: version

text: of

text: the

text: likelihood

text: ratio

text: x(α,

text: it

text: follows

text: that

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 (cid:82) lowest error rates, its performance can degrade considerably The integral R log2(g/f)(g/f)αfdµ is also positive and under uncertainties in the nominal model. The robust tests, on → ∞ → ∞ convex in α. This implies K for α . Now, assume the other hand, have slightly higher error rates, but guaranteed ∗ that g depends on α and tends to a limiting distribution g for power of the test, which indicates the trade-off between || ∗ − || → ∞ g f > 0, when α . Then, our conclusion does not performance and robustness. Finally, in the last simulation, the → ∞ → ∞ change, i.e., K for α . Since D(f, g; α) is ﬁnite, 3D boundary surface of the maximum robustness parameters → ∞ ⇒ ∗ → we require that α = g f. Consequently, from (15) is determined for α = 0.5 (45) and is shown in Figure 10. → → → and (16), gˆ f whenever l 1 and l 1 explains the This surface has a cropped rotated cone like shape, which is i i l u asymptotic of Figure 6 for any pair (f , f ). symmetric about its main diagonal, i.e., with respect to the 0 1 Based on simulation results the following are conjectured: plane (cid:15) = (cid:15) on the space ((cid:15) , (cid:15) , a). Notice that except 0 1 0 1 • For a ﬁxed (cid:15) and (cid:15) , increasing α leads to a monotone for the points on the cone like shape that intersect with the 0 1 R \{ } decrease in l and monotone increase in l on + 0, 1 . ((cid:15) , (cid:15) , a = 0) plane, all other points on ((cid:15) , (cid:15) , a = 0) that are u l 0 1 0 1 • For a ﬁxed α, increasing (cid:15) , (cid:15) or both introduces a non- plotted in blue color are un-deﬁned (rather than being valid 0 1 decrease in l , non-increase in l , or both, given that points with a = 0), implying that for those points no minimax u l (cid:15) and (cid:15) are less than their allowable maximum, cf. robust test exists. 0 1 Section . The proof of these conjectures is an open problem. f , g(cid:96) i i From (19) and (20), it is clear that given a pair ((cid:15) , (cid:15) ), 0 1 f a slight change in α changes the equations completely and 0(cid:56).20 (cid:60) 0 in general ll and lu are functions of α. In Figure 7, the f1 robust decision rule δˆ for various α values is plotted, without (cid:96) considering the dependency of l and l on α. To do this, 0.15 g0 ≈ ≈ l u ll 0.605 and lu 1.618, that are found for ρ = 1, α = 4, g(cid:96)1 (cid:15) = 0.02 and (cid:15) = 0.03, are ﬁxed constants in (17). Then, 0 { 0 } 0.10 for α = 0.01, 10, 100 , (17) is plotted. The decision rule δˆ tends to a step like function for an increasing α, whereas for a smaller α, i.e., α = 0.01, the decision rule is almost linear at 0.05 the domain of the likelihood ratio for which ˆl = 1. This result is also in agreement with the previous ﬁndings; δˆ tends to a → ∞ non-randomized likelihood ratio test for α , for which y we obtained gˆ → f and for (f , f ) optimum decision rule (cid:45)6 (cid:45)4 (cid:45)2 0 2 4 6 i i 0 1 Fig. 3. Nominal densities and the corresponding least favorable densities for is known to be a non-randomized likelihood ratio test. ρ = 1, α = 4, (cid:15)0 = 0.02 and (cid:15)1 = 0.03. In the following simulation, the simpliﬁed model (f (y) = − 0 f ( y)) is tested for mean shifted Gaussian distributions; 1 ∼ N ∼ N − (cid:96) F (µ , σ2) and F (µ , σ2) with means µ = 1, fi, gi 0 0 1 1 0 µte1st=ar1eacnhdovseanriatnocebeσ2ρ==11. ,T(cid:15)he p=ar0a.m1eatenrds o(cid:15)f t=he 0co.1m. pHoesritee, 0(cid:56).20 (cid:60) f0 0 1 our main interest is to observe the change in overlapping f1 regions of least favorable density pairs for various α. Figure 8 (cid:96) 0.15 g0 illustrates the outcome of this simulation. It can be seen that − (cid:96) the overlapping region is convex for a negative α, (α = 10) g1 almost constant for α = 0.01 and concave for a positive α, 0.10 (α = 10). For the sake of clarity only three examples of α are plotted. In Figure 9, the false alarm and miss detection probabilities 0.05 of the likelihood ratio test δ for (f , f ) are graphed and 0 1 compared with the robust test δˆ for (gˆ , gˆ ). Two different 0 1 robust parameter pairs and various signal to noise ratios y (cid:45)6 (cid:45)4 (cid:45)2 0 2 4 6 (SNR)s, i.e., SNR = 20 log(A/σ) are considered. It can be Fig. 4. Nominal densities and the corresponding least favorable densities for seen that increasing the robustness parameters increases the ρ = 1.2, α = 4, (cid:15)0 = 0.02 and (cid:15)1 = 0.03. false alarm and miss detection probabilities for all SNRs, as expected. The difference between false alarm and miss detection probabilities for the same robust test is small and it is more pronounced for low SNRs. For high SNRs the V. CONCLUSION performance of two robust tests are close to each other. The A robust version of the likelihood ratio test considering − reason is that for high SNRs maximum allowable robustness α divergence as the distance to characterize the uncertainty parameters become relatively high compared to the parameters sets has been proposed. The existence of a saddle value to the of both robust settings. Although the nominal test has the minimax optimization problem was shown by adopting Sion’s ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: lowest

text: error

text: rates,

text: its

text: performance

text: can

text: degrade

text: considerably

text: The

text: integral

text: R

text: is

text: also

text: positive

text: and

text: under

text: uncertainties

text: in

text: the

text: nominal

text: model.

text: The

text: robust

text: tests,

text: on

text: convex

text: in

text: α.

text: This

text: implies

text: K

text: for

text: α

text: Now,

text: assume

text: the

text: other

text: hand,

text: have

text: slightly

text: higher

text: error

text: rates,

text: but

text: guaranteed

text: that

text: g

text: depends

text: on

text: α

text: and

text: tends

text: to

text: a

text: limiting

text: distribution

text: g

text: for

text: power

text: of

text: the

text: test,

text: which

text: indicates

text: the

text: trade-off

text: between

text: g

text: f

text: when

text: α

text: Then,

text: our

text: conclusion

text: does

text: not

text: performance

text: and

text: robustness.

text: Finally,

text: in

text: the

text: last

text: simulation,

text: the

text: change,

text: i.e.,

text: K

text: for

text: α

text: Since

text: D(f,

text: g;

text: α)

text: is

text: ﬁnite,

text: boundary

text: surface

text: of

text: the

text: maximum

text: robustness

text: parameters

text: we

text: require

text: that

text: α

text: g

text: f.

text: Consequently,

text: from

text: is

text: determined

text: for

text: α

text: and

text: is

text: shown

text: in

text: Figure

text: and

text: gˆ

text: f

text: whenever

text: l

text: and

text: l

text: explains

text: the

text: This

text: surface

text: has

text: a

text: cropped

text: rotated

text: cone

text: like

text: shape,

text: which

text: is

text: i

text: i

text: l

text: u

text: asymptotic

text: of

text: Figure

text: for

text: any

text: pair

text: (f

text: f

text: symmetric

text: about

text: its

text: main

text: diagonal,

text: i.e.,

text: with

text: respect

text: to

text: the

text: Based

text: on

text: simulation

text: results

text: the

text: following

text: are

text: conjectured:

text: plane

text: on

text: the

text: space

text: a).

text: Notice

text: that

text: except

text: For

text: a

text: ﬁxed

text: and

text: increasing

text: α

text: leads

text: to

text: a

text: monotone

text: for

text: the

text: points

text: on

text: the

text: cone

text: like

text: shape

text: that

text: intersect

text: with

text: the

text: R

text: decrease

text: in

text: l

text: and

text: monotone

text: increase

text: in

text: l

text: on

text: a

text: plane,

text: all

text: other

text: points

text: on

text: a

text: that

text: are

text: u

text: l

text: For

text: a

text: ﬁxed

text: α,

text: increasing

text: or

text: both

text: introduces

text: a

text: non-

text: plotted

text: in

text: blue

text: color

text: are

text: un-deﬁned

text: (rather

text: than

text: being

text: valid

text: decrease

text: in

text: l

text: non-increase

text: in

text: l

text: or

text: both,

text: given

text: that

text: points

text: with

text: a

text: implying

text: that

text: for

text: those

text: points

text: no

text: minimax

text: u

text: l

text: and

text: are

text: less

text: than

text: their

text: allowable

text: maximum,

text: cf.

text: robust

text: test

text: exists.

text: Section

text: The

text: proof

text: of

text: these

text: conjectures

text: is

text: an

text: open

text: problem.

text: f

text: i

text: i

text: From

text: and

text: it

text: is

text: clear

text: that

text: given

text: a

text: pair

text: f

text: a

text: slight

text: change

text: in

text: α

text: changes

text: the

text: equations

text: completely

text: and

text: in

text: general

text: ll

text: and

text: lu

text: are

text: functions

text: of

text: α.

text: In

text: Figure

text: the

text: robust

text: decision

text: rule

text: δˆ

text: for

text: various

text: α

text: values

text: is

text: plotted,

text: without

text: considering

text: the

text: dependency

text: of

text: l

text: and

text: l

text: on

text: α.

text: To

text: do

text: this,

text: l

text: u

text: ll

text: and

text: lu

text: that

text: are

text: found

text: for

text: ρ

text: α

text: and

text: are

text: ﬁxed

text: constants

text: in

text: Then,

text: for

text: α

text: is

text: plotted.

text: The

text: decision

text: rule

text: δˆ

text: tends

text: to

text: a

text: step

text: like

text: function

text: for

text: an

text: increasing

text: α,

text: whereas

text: for

text: a

text: smaller

text: α,

text: i.e.,

text: α

text: the

text: decision

text: rule

text: is

text: almost

text: linear

text: at

text: the

text: domain

text: of

text: the

text: likelihood

text: ratio

text: for

text: which

text: ˆl

text: This

text: result

text: is

text: also

text: in

text: agreement

text: with

text: the

text: previous

text: ﬁndings;

text: δˆ

text: tends

text: to

text: a

text: non-randomized

text: likelihood

text: ratio

text: test

text: for

text: α

text: for

text: which

text: y

text: we

text: obtained

text: gˆ

text: f

text: and

text: for

text: (f

text: f

text: optimum

text: decision

text: rule

text: i

text: i

text: Fig.

text: Nominal

text: densities

text: and

text: the

text: corresponding

text: least

text: favorable

text: densities

text: for

text: is

text: known

text: to

text: be

text: a

text: non-randomized

text: likelihood

text: ratio

text: test.

text: ρ

text: α

text: and

text: In

text: the

text: following

text: simulation,

text: the

text: simpliﬁed

text: model

text: (f

text: (y)

text: f

text: y))

text: is

text: tested

text: for

text: mean

text: shifted

text: Gaussian

text: distributions;

text: N

text: N

text: F

text: (µ

text: and

text: F

text: (µ

text: with

text: means

text: µ

text: fi,

text: gi

text: t=he

text: pHoesritee,

text: our

text: main

text: interest

text: is

text: to

text: observe

text: the

text: change

text: in

text: overlapping

text: regions

text: of

text: least

text: favorable

text: density

text: pairs

text: for

text: various

text: α.

text: Figure

text: illustrates

text: the

text: outcome

text: of

text: this

text: simulation.

text: It

text: can

text: be

text: seen

text: that

text: the

text: overlapping

text: region

text: is

text: convex

text: for

text: a

text: negative

text: α,

text: (α

text: almost

text: constant

text: for

text: α

text: and

text: concave

text: for

text: a

text: positive

text: α,

text: (α

text: For

text: the

text: sake

text: of

text: clarity

text: only

text: three

text: examples

text: of

text: α

text: are

text: plotted.

text: In

text: Figure

text: the

text: false

text: alarm

text: and

text: miss

text: detection

text: probabilities

text: of

text: the

text: likelihood

text: ratio

text: test

text: δ

text: for

text: (f

text: f

text: are

text: graphed

text: and

text: compared

text: with

text: the

text: robust

text: test

text: δˆ

text: for

text: (gˆ

text: gˆ

text: Two

text: different

text: robust

text: parameter

text: pairs

text: and

text: various

text: signal

text: to

text: noise

text: ratios

text: y

text: (SNR)s,

text: i.e.,

text: SNR

text: log(A/σ)

text: are

text: considered.

text: It

text: can

text: be

text: Fig.

text: Nominal

text: densities

text: and

text: the

text: corresponding

text: least

text: favorable

text: densities

text: for

text: seen

text: that

text: increasing

text: the

text: robustness

text: parameters

text: increases

text: the

text: ρ

text: α

text: and

text: false

text: alarm

text: and

text: miss

text: detection

text: probabilities

text: for

text: all

text: SNRs,

text: as

text: expected.

text: The

text: difference

text: between

text: false

text: alarm

text: and

text: miss

text: detection

text: probabilities

text: for

text: the

text: same

text: robust

text: test

text: is

text: small

text: and

text: it

text: is

text: more

text: pronounced

text: for

text: low

text: SNRs.

text: For

text: high

text: SNRs

text: the

text: V.

text: CONCLUSION

text: performance

text: of

text: two

text: robust

text: tests

text: are

text: close

text: to

text: each

text: other.

text: The

text: A

text: robust

text: version

text: of

text: the

text: likelihood

text: ratio

text: test

text: considering

text: reason

text: is

text: that

text: for

text: high

text: SNRs

text: maximum

text: allowable

text: robustness

text: α

text: divergence

text: as

text: the

text: distance

text: to

text: characterize

text: the

text: uncertainty

text: parameters

text: become

text: relatively

text: high

text: compared

text: to

text: the

text: parameters

text: sets

text: has

text: been

text: proposed.

text: The

text: existence

text: of

text: a

text: saddle

text: value

text: to

text: the

text: of

text: both

text: robust

text: settings.

text: Although

text: the

text: nominal

text: test

text: has

text: the

text: minimax

text: optimization

text: problem

text: was

text: shown

text: by

text: adopting

text: Sion’s

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: EXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC SYSTEMS 13 (cid:98) − (cid:99) − where g(n) = log n l|og| π is such that, for j > g(n), the set T j(U ) is suﬃciently stretched 2log λ n along the stable direction to start wrapping around the torus. The third summand on the right of (5.4) goes to on account of (5.1). We proceed considering the other summands, 0 starting by the ﬁrst. Contrary to the periodic setting, we now have to deal with the ∩ − (cid:54) ∅ ≤ { ∈ N ∩ − (cid:54) ∅} possibility of U T j(U ) = for j g(n). Let R = min j : U T j(U ) = . n n ∩ { } n → ∞n → n∞ By continuity of and the fact that , we have that as . T U = ζ R n n n n ≤ − − For j g(n), observe that T j(U ) is an ellipse centered at T j(ζ), stretched along the | | n | | stable direction by a factor λ j and contracted on the unstable direction by 1/ λ j. Also, R notice that, as we are dealing with a projection on the torus of a linear map in 2, the − stable and unstable directions are always lined up regardless of the position of T j(ζ). − Now, since the set T j(U ) is not long enough √(on t√he stable direction) to wrap around the n ∼ − torus, the diameter of U is 2s , where s τ/ πn. The length of T j(U ) along the n n n − | | n stable direction (which coincides with the diameter of T j(U )) is λ j s ; and the length − | |− n n of T j(U ) on the unstable direction is λ j s . So, by the geometry of the sets, the area n ∩ − n − of the intersection U T j(U ) occupies a portion of T j(U ) corresponding at most to | |−n n n a constant times λ j of its area. That is, there exists some C > 0 independent of j such that ∩ − ≤ | |− − | |− m(U T j(U )) C λ j m(T j(U )) = C λ j m(U ). n n n n ≈ In the case of the Euclidean metric we can take Then it follows that (cid:88) (cid:88) C 2/π. g(n) g(n) ∩ − | |− ≤ | |− −−−→ n m(U T j(U )) = n λ j τ/n C τ λ Rn 0. n n →∞ n j=0 j=Rn ∈ ∩ ∩ We are left to estimate the second summand of (5.4). Let x U W u and γ := W s U . − (cid:91) n ζ x x n For j > g(n), the curve T j(γ ) may be suﬃciently long to wrap around the torus. Set x U˜j := γ . n { } x (cid:112) ∈ ∩ | − | x Un Wζu : T j(γx) >1/2 Note that for the usual Euclidean metric we have |W u ∩ U \ U˜j| ≤ 2 s2 − |λ|−2j/16. ∈ | − | ζ n n n Since for x U˜j, we have T j(γ ) > 1/2 and the diameter of U is less than 2s , then n x ∈ n n there exists C > 0 such that, for all j > g(n) and all x U˜j, we have n | − ∩ | ≤ | − | T j(γ ) U C s T j(γ ) . x n n x As restricted to local stable manifolds has bounded distortion (since the derivative is T |{ ∈ − ∈ }| ≤ | | constant), it follows that y : y γ , T j(y) U C s γ . Moreover, by the x n n x -invariance of we have T m ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EXTREMAL

text: DICHOTOMY

text: FOR

text: UNIFORMLY

text: HYPERBOLIC

text: SYSTEMS

text: where

text: g(n)

text: log

text: n

text: l|og|

text: π

text: is

text: such

text: that,

text: for

text: j

text: g(n),

text: the

text: set

text: T

text: j(U

text: is

text: suﬃciently

text: stretched

text: λ

text: n

text: along

text: the

text: stable

text: direction

text: to

text: start

text: wrapping

text: around

text: the

text: torus.

text: The

text: third

text: summand

text: on

text: the

text: right

text: of

text: goes

text: to

text: on

text: account

text: of

text: We

text: proceed

text: considering

text: the

text: other

text: summands,

text: starting

text: by

text: the

text: ﬁrst.

text: Contrary

text: to

text: the

text: periodic

text: setting,

text: we

text: now

text: have

text: to

text: deal

text: with

text: the

text: N

text: possibility

text: of

text: U

text: T

text: j(U

text: for

text: j

text: g(n).

text: Let

text: R

text: min

text: j

text: U

text: T

text: j(U

text: n

text: n

text: n

text: ∞n

text: n∞

text: By

text: continuity

text: of

text: and

text: the

text: fact

text: that

text: we

text: have

text: that

text: as

text: T

text: U

text: ζ

text: R

text: n

text: n

text: n

text: n

text: For

text: j

text: g(n),

text: observe

text: that

text: T

text: j(U

text: is

text: an

text: ellipse

text: centered

text: at

text: T

text: j(ζ),

text: stretched

text: along

text: the

text: n

text: stable

text: direction

text: by

text: a

text: factor

text: λ

text: j

text: and

text: contracted

text: on

text: the

text: unstable

text: direction

text: by

text: λ

text: j.

text: Also,

text: R

text: notice

text: that,

text: as

text: we

text: are

text: dealing

text: with

text: a

text: projection

text: on

text: the

text: torus

text: of

text: a

text: linear

text: map

text: in

text: the

text: stable

text: and

text: unstable

text: directions

text: are

text: always

text: lined

text: up

text: regardless

text: of

text: the

text: position

text: of

text: T

text: j(ζ).

text: Now,

text: since

text: the

text: set

text: T

text: j(U

text: is

text: not

text: long

text: enough

text: √(on

text: t√he

text: stable

text: direction)

text: to

text: wrap

text: around

text: the

text: n

text: torus,

text: the

text: diameter

text: of

text: U

text: is

text: where

text: s

text: τ/

text: πn.

text: The

text: length

text: of

text: T

text: j(U

text: along

text: the

text: n

text: n

text: n

text: n

text: stable

text: direction

text: (which

text: coincides

text: with

text: the

text: diameter

text: of

text: T

text: j(U

text: is

text: λ

text: j

text: s

text: and

text: the

text: length

text: n

text: n

text: of

text: T

text: j(U

text: on

text: the

text: unstable

text: direction

text: is

text: λ

text: j

text: s

text: So,

text: by

text: the

text: geometry

text: of

text: the

text: sets,

text: the

text: area

text: n

text: n

text: of

text: the

text: intersection

text: U

text: T

text: j(U

text: occupies

text: a

text: portion

text: of

text: T

text: j(U

text: corresponding

text: at

text: most

text: to

text: |−n

text: n

text: n

text: a

text: constant

text: times

text: λ

text: j

text: of

text: its

text: area.

text: That

text: is,

text: there

text: exists

text: some

text: C

text: independent

text: of

text: j

text: such

text: that

text: m(U

text: T

text: j(U

text: C

text: λ

text: j

text: m(T

text: j(U

text: C

text: λ

text: j

text: m(U

text: n

text: n

text: n

text: n

text: In

text: the

text: case

text: of

text: the

text: Euclidean

text: metric

text: we

text: can

text: take

text: Then

text: it

text: follows

text: that

text: C

text: g(n)

text: g(n)

text: n

text: m(U

text: T

text: j(U

text: n

text: λ

text: j

text: τ/n

text: C

text: τ

text: λ

text: Rn

text: n

text: n

text: n

text: j=Rn

text: We

text: are

text: left

text: to

text: estimate

text: the

text: second

text: summand

text: of

text: Let

text: x

text: U

text: W

text: u

text: and

text: γ

text: W

text: s

text: U

text: n

text: ζ

text: x

text: x

text: n

text: For

text: j

text: g(n),

text: the

text: curve

text: T

text: j(γ

text: may

text: be

text: suﬃciently

text: long

text: to

text: wrap

text: around

text: the

text: torus.

text: Set

text: x

text: U˜j

text: γ

text: n

text: x

text: x

text: Un

text: Wζu

text: T

text: j(γx)

text: Note

text: that

text: for

text: the

text: usual

text: Euclidean

text: metric

text: we

text: have

text: |W

text: u

text: U

text: U˜j|

text: ζ

text: n

text: n

text: n

text: Since

text: for

text: x

text: U˜j,

text: we

text: have

text: T

text: j(γ

text: and

text: the

text: diameter

text: of

text: U

text: is

text: less

text: than

text: then

text: n

text: x

text: n

text: n

text: there

text: exists

text: C

text: such

text: that,

text: for

text: all

text: j

text: g(n)

text: and

text: all

text: x

text: U˜j,

text: we

text: have

text: n

text: T

text: j(γ

text: U

text: C

text: s

text: T

text: j(γ

text: x

text: n

text: n

text: x

text: As

text: restricted

text: to

text: local

text: stable

text: manifolds

text: has

text: bounded

text: distortion

text: (since

text: the

text: derivative

text: is

text: T

text: constant),

text: it

text: follows

text: that

text: y

text: y

text: γ

text: T

text: j(y)

text: U

text: C

text: s

text: γ

text: Moreover,

text: by

text: the

text: x

text: n

text: n

text: x

text: -invariance

text: of

text: we

text: have

text: T

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL { } { ≤ } N n = 0 = M u (8.1) u,0 n In order to deﬁne a point process that captures the essence of an EVL and HTS through P (8.1), we need to re-scale time using the factor given by Kac’s Theorem. v := 1/ (X > u) S R Let denote the semi-ring of subsets of + whose elements are intervals of the type [a, b), ∈ R R 0 S ∈ R for a, b +. Denote by the ring generated by ; recall that for every J there ∈ N 0 ∈ S ∈ R are k and k intervals I , . . . , I , say I = [a , b ) with a , b +, such that ∪ ∈ S1 k∈ R j j j j j 0 J = k I . For I = [a, b) and α , we set αI := [αa, αb) and I + α := [a + α, b + α). i=1 j ∈ R ∪ · · · ∪ ∪ · · · ∪ Similarly, for , we deﬁne and . J αJ := αI αI J + α := (I + α) (I + α) 1 k 1 k ∈ R Deﬁnition 1. Given J and sequences (u ) ∈N and (v ) ∈N, the rare event point process n n n n (REPP) is deﬁned by counting the number of exceedances (or hits to ) during the U(u ) ∈ R ∈ R n (re-scaled) time period . More precisely, fo(cid:88)r every , set v J J n N (J) := N (v J) = 1 . (8.2) n un n Xj>un ∈ ∩N j vnJ 0 Under dependence conditions similar to the ones previously analyzed, the REPP just de- ﬁned converges in distribution to a standard Poisson process when no clustering is involved, and converges in distribution to a compound Poisson process with intensity and a geo- ϑ metric multiplicity distribution function otherwise. For the sake of completeness, we now deﬁne what we mean by a Poisson and a compound Poisson process. (See [20] for more details.) Deﬁnition 2. Let Y , Y , . . . be an iid sequence of random variables with common exponen- 1 2 tial distribution of mean . Let be another iid sequence of random variables, 1/ϑ Z , Z , . . . 1 2 (cid:32) (cid:33) independent of the previous one, and(cid:90) with dis(cid:88)tribution function π. Given these sequences, ∈ R for , set J ∞ N(J) = 1 d Z δ , J i Y1+...+Yi i=1 where denotes the Dirac measure at . Whenever we are in this setting, we say that δ t > 0 t is a compound Poisson process of intensity and multiplicity d.f. . N ϑ π Remark 8.1. In this paper, the multiplicity will always be integer valued. This means that P ∈ N is completely deﬁned by the values , for every . Note that, if π π = (Z = k) k k 1 0 and , then is the standard Poisson process and, for every , the random π = 1 ϑ = 1 N t > 0 1 variable has a Poisson distribution of mean . N([0, t)) t

text: M.

text: CARVALHO,

text: A.

text: C.

text: M.

text: FREITAS,

text: J.

text: M.

text: FREITAS,

text: M.

text: HOLLAND,

text: AND

text: M.

text: NICOL

text: N

text: n

text: M

text: u

text: n

text: In

text: order

text: to

text: deﬁne

text: a

text: point

text: process

text: that

text: captures

text: the

text: essence

text: of

text: an

text: EVL

text: and

text: HTS

text: through

text: P

text: we

text: need

text: to

text: re-scale

text: time

text: using

text: the

text: factor

text: given

text: by

text: Kac’s

text: Theorem.

text: v

text: (X

text: u)

text: S

text: R

text: Let

text: denote

text: the

text: semi-ring

text: of

text: subsets

text: of

text: whose

text: elements

text: are

text: intervals

text: of

text: the

text: type

text: [a,

text: b),

text: R

text: R

text: S

text: R

text: for

text: a,

text: b

text: Denote

text: by

text: the

text: ring

text: generated

text: by

text: recall

text: that

text: for

text: every

text: J

text: there

text: N

text: S

text: R

text: are

text: k

text: and

text: k

text: intervals

text: I

text: I

text: say

text: I

text: [a

text: b

text: with

text: a

text: b

text: such

text: that

text: k∈

text: R

text: j

text: j

text: j

text: j

text: j

text: J

text: k

text: I

text: For

text: I

text: [a,

text: b)

text: and

text: α

text: we

text: set

text: αI

text: [αa,

text: αb)

text: and

text: I

text: α

text: [a

text: α,

text: b

text: α).

text: j

text: R

text: Similarly,

text: for

text: we

text: deﬁne

text: and

text: J

text: αJ

text: αI

text: αI

text: J

text: α

text: (I

text: α)

text: (I

text: α)

text: k

text: k

text: R

text: Deﬁnition

text: Given

text: J

text: and

text: sequences

text: (u

text: ∈N

text: and

text: (v

text: ∈N,

text: the

text: rare

text: event

text: point

text: process

text: n

text: n

text: n

text: n

text: (REPP)

text: is

text: deﬁned

text: by

text: counting

text: the

text: number

text: of

text: exceedances

text: (or

text: hits

text: to

text: during

text: the

text: U(u

text: R

text: R

text: n

text: (re-scaled)

text: time

text: period

text: More

text: precisely,

text: every

text: set

text: v

text: J

text: J

text: n

text: N

text: (J)

text: N

text: (v

text: J)

text: n

text: un

text: n

text: Xj>un

text: ∩N

text: j

text: vnJ

text: Under

text: dependence

text: conditions

text: similar

text: to

text: the

text: ones

text: previously

text: analyzed,

text: the

text: REPP

text: just

text: de-

text: ﬁned

text: converges

text: in

text: distribution

text: to

text: a

text: standard

text: Poisson

text: process

text: when

text: no

text: clustering

text: is

text: involved,

text: and

text: converges

text: in

text: distribution

text: to

text: a

text: compound

text: Poisson

text: process

text: with

text: intensity

text: and

text: a

text: geo-

text: ϑ

text: metric

text: multiplicity

text: distribution

text: function

text: otherwise.

text: For

text: the

text: sake

text: of

text: completeness,

text: we

text: now

text: deﬁne

text: what

text: we

text: mean

text: by

text: a

text: Poisson

text: and

text: a

text: compound

text: Poisson

text: process.

text: (See

text: for

text: more

text: details.)

text: Deﬁnition

text: Let

text: Y

text: Y

text: be

text: an

text: iid

text: sequence

text: of

text: random

text: variables

text: with

text: common

text: exponen-

text: tial

text: distribution

text: of

text: mean

text: Let

text: be

text: another

text: iid

text: sequence

text: of

text: random

text: variables,

text: Z

text: Z

text: independent

text: of

text: the

text: previous

text: one,

text: with

text: function

text: π.

text: Given

text: these

text: sequences,

text: R

text: for

text: set

text: J

text: N(J)

text: d

text: Z

text: δ

text: J

text: i

text: where

text: denotes

text: the

text: Dirac

text: measure

text: at

text: Whenever

text: we

text: are

text: in

text: this

text: setting,

text: we

text: say

text: that

text: δ

text: t

text: t

text: is

text: a

text: compound

text: Poisson

text: process

text: of

text: intensity

text: and

text: multiplicity

text: d.f.

text: N

text: ϑ

text: π

text: Remark

text: In

text: this

text: paper,

text: the

text: multiplicity

text: will

text: always

text: be

text: integer

text: valued.

text: This

text: means

text: that

text: P

text: N

text: is

text: completely

text: deﬁned

text: by

text: the

text: values

text: for

text: every

text: Note

text: that,

text: if

text: π

text: π

text: (Z

text: k)

text: k

text: k

text: and

text: then

text: is

text: the

text: standard

text: Poisson

text: process

text: and,

text: for

text: every

text: the

text: random

text: π

text: ϑ

text: N

text: t

text: variable

text: has

text: a

text: Poisson

text: distribution

text: of

text: mean

text: t))

text: t