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Christoph F. Mecklenbräuker (TU Wien)

Joint work with

Pei-Jung Chung (Univ. Edinburgh)Dirk Maiwald (Atlas Elektronik)Nicolai Czink (FTW)Bernard H. Fleury (Aalborg Univ. and FTW)

Model Identification for Wireless Propagationwith Control of the False Discovery Rate

Advanced Lectures in Wireless Communications

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Motivation

Tx Channel Rx

!

ˆ C

Risk for over-estimation C

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Motivation

What is interference depends on your knowledge of the channel

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Uniform Linear ArrayULA-8

Uniform Circular ArrayUCA-15

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Some paths explained

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Problem Formulation (1)

Tx

Rx

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• An array of n antennas receives m broadbandwavefronts impinging at unknown delays anddirections hidden in additive Gaussian noise(spatially and temporally white).

• Goal: Determine the number of signals mbased on the array output and the associatedparameters.

Problem Formulation (2)

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Double-directional model

Transfer function in 3-D case: DoA, DoD, delay

!=

"""

=P

p

mTjl

djk

dj

pmlk

ppp

cx1

)1(2

cos)1(2

cos)1(2

,, eee#

$%

&

$'

&

$

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• Array output x(k)(t) for the kth snapshot isshort-time Fourier transformed

• For large T, we can approximately model thearray output in frequency domain

where the columns of the transfer matrix Hmodel plane waves.

Data Model

!"

=

"=

1

0

)()(e)()(

1)(

T

t

tjkktxtw

TX ##

)()();()()()()(!!"!!

kkk

USX += H

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Data Model Statistics• Linear data model

• Data statistics conditioned on the signal

)()();()()()()(!!"!!

kkk

USX += H

!

X(k )| S

(k )~ N

C(HS

(k )," 2

I)

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Data Model Statistics• Linear data model

• Data statistics conditioned on the signal

)()();()()()()(!!"!!

kkk

USX += H

!

X(k )| S

(k )~ N

C(HS

(k )," 2

I)

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Conditional Data ModelLog-likelihood

• Data statistics conditioned on the signal

!

fX (x) =1

" N# N ($)exp %

1

#($)|| x %H($;&)S(k )($) ||2

'

( )

*

+ ,

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Wavefront Detection using aMultiple Hypotheses Test

for m = 1, 2, ... M

Hypothesis Hm: Array output contains at most(m−1) wavefronts hidden in the noise

Alternative Am: Array output contains at least mwavefronts hidden in the noise

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Test for model order selection• Generalized Likelihood Ratio Test• Equivalent to the Wald Test proposed

by Steven Kay 1993 for parametricmodel order selection

H1

H2

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Test for model order selection• Generalized Likelihood Ratio Test• Equivalent to the Wald Test proposed

by Steven Kay 1993 for parametricmodel order selection

H2

H3

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• Generalized Likelihood Ratio Test• Equivalent to the Wald Test proposed

by Steven Kay 1993 for parametricmodel order selection

H3

H4

Test for model order selection

Image: Wikipedia

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• Based on the likelihood ratio, we obtain thetest statistic

where

Generalized Likelihood Ratio Test

!=

==K

k

j

k

j

k

j XXK 1

)()()()(

1)(ˆˆ """RR

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Traditional formulation

Evaluate test statistic Tm(θ) from data andcompare with pre-computed thresholdvalue

!

Tm

<?

tm

!

tm:= F

Tm

"1(1"#

m)

Inverse of cumulativedistribution function isneeded

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Formulation with p-values

Evaluate empirial significance value (=p-value) for test statistic Tm(θ) from data and compare with the specified false-alarm probability

!

Tm

<?

tm

!

tm:= F

Tm

"1(1"#

m)

!

pm <?

1"#m

!

pm := FTm (Tm )

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Test distribution• Under hypothesis Hm, the statistic Fm(ωj;θ ) is

Fn1,n2-distributed where the degrees of freedomn1, n2 are given by

n1 = K ( 2 + dim(θm) )n2 = K ( 2n − 2m − dim(θm−1))

• Narrowband (J = 1): GLRT is equivalent to F-test [Shumway 1983].

• Broadband (J > 1): test distribution is unknown.

^

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Where we are now in the talk

• At this point of the talk, we have a tool forcomputing (estimating) the p-values for all thehypotheses.

• That‘s acceptable because, we don‘t knowthe exact distribution of the broadband GLRTtest statistic. (J being a small integer > 1)

• Now, let‘s talk about the type of errors, wecan commit.

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PCE, FWE, FDR definitions

m hypothesis are assumed to be known in advance, R is observable, U, V, S, T are unobservable

Control of type-one errorsPCE = E(V/m) Per Comparison Error RateFWE = P(V≥1) Familywise Error RateFDR = E(V/R) False Discovery Rate

Ref.[1]

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Control of thefalse discovery rate (FDR)

• Traditional approach controls familywiseerror-rate (FWE).

• When the number of hypotheses is large thanthe power of Bonferroni-type procedures issubstantially reduced.

• Benjamini and Hochberg proposed to controlFDR instead of FWE in 1995.

• FDR is defined as the expected proportion oferroneously rejected hypotheses

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Benjamini-Hochberg proc.• When the test statistics corresponding to the true

null hpotheses are independent, the followingprocedure controls the FDR at level q

• Sort the p-values: p(1), p(2) , ..., p(M)• Find k = max { m : p(m) ≤ mq/M }

• Reject all H(1), H(2), ... , H(k).(if no such k exists then don‘t reject any hypothesis)

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Benjamini-Hochberg proc.• Sort the p-values: p(1), p(2) , ..., p(M)• Find k = max { m : p(m) ≤ mq/M }• Reject all H(1), H(2), ... , H(k).

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Early take-home message• The broadband test distribution under Hm is

not known.• We apply the bootstrap technique to

determine the distribution numerically.• If all null hypotheses are true then controlling

the FDR is equivalent to controlling the FWE• Simulations show that the FDR-controlling

procedure has always a higher probability ofdetection than the FWE controlling procedure.

• Reliability of the proposed test is not affectedby the gain in power.

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ULA-8 UCA-15

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Receiver‘s View onWeikendorf Site

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Closing remark

Model order selection is a problem whichis asymmetric in its risks for over- orunder-estimating the true modelstructure

Multiple hypotheses tests let you controlthe various types of errors you couldcommit

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Happy birthday

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ReferencesY. Benjamini and Y. Hochberg. Controlling the false discovery rate: a practical and powerful

approach to multiple testing. J. Roy. Statist. Soc. Ser. B, (57):289–300, 1995.R.H. Shumway. Replicated time-series regression: An approach to signal estimation and

detection. In D.R. Brillinger and P.R. Krishnaiah, editors, Handbook of Statistics, Vol. 3,pages 383–408. Elsevier Science Publishers B.V., 1983.

S. Holm. A simple sequentially rejective multiple test procedure. Scand. J. Statist., 6:65–70,1979.

E. Efron. Bootstrap method. Another look at Jacknife. The Annals of Statistics, 7:1–26, 1979.Abdelhak M. Zoubir and B. Boashash. The bootstrap and its application in signal processing.

IEEE Signal Processing Magazine, 15(1):56–76, January 1998.D. Maiwald. Breitbandverfahren zur Signalentdeckung und –ortung mit Sensorgruppen in

Seismik– und Sonaranwendungen. Dr.–Ing. Dissertation, Dept. of Electrical Engineering,Ruhr–Universität Bochum, Shaker Verlag, Aachen, 1995.

P.-J. Chung, J.F. Böhme, A.O. Hero, and C.F. Mecklenbräuker. Signal detection using amultiple hypothesis test. In Proc. Third IEEE Sensor Array and Multichannel SignalProcessing Workshop, Barcelona, Spain, July 18-21 2004.

P.-J. Chung, J.F. Böhme, C.F. Mecklenbräuker, and A.O. Hero. On signal detection using thebenjamini-hochberg procedure. In Proc. IEEE Workshop on Statistical and SignalProcessing, Bordeaux, France, July 2005.

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FDR example

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FDR example (continued)

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120 MHz.

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Bootstrap approximation:assumptions

• The test statistic Tm( θm) is the sample mean of Jsamples

• We consider T1, T2 ,..., TJ as i.i.d. samplesbecause– X(ωj) are asymptotically independent for T → ∞– Fm(ωj ;θm) are asymptotically Fn1,n2-distributed

!

Tj = log 1+n1

n2Fm (" j ;#m )

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