The Bootstrap Paradigm in Signal Processing: Estimation ... 2009/Bootstrap_Signal... · The...

The Bootstrap in Signal Processing

Abdelhak M Zoubir

The Bootstrap Paradigm in Signal Processing:Estimation, Detection and Model Selection

Abdelhak M Zoubir

S i g n a l P r o c e s s i n g G r o u p

Signal Processing Group

Technische Universitat Darmstadt

E-Mail: [email protected]

URL: www.spg.tu-darmstadt.deMay 12, 2010 | SPG TUD | c© A.M. Zoubir | 1 SPG

Contents

◮ Introduction

◮ What can I use the bootstrap for?◮ How does the bootstrap work?◮ History of the bootstrap

◮ The independent data bootstrap

◮ An example for independent data

◮ The bootstrap principle for dependent data

◮ The moving blocks bootstrap

◮ Hypothesis Testing with the Bootstrap

◮ Signal Detection with the Bootstrap

May 12, 2010 | SPG TUD | c© A.M. Zoubir | 2 SPG

Contents

◮ Frequency domain bootstrap methods for dependent data

◮ The multiplicative regression approach

◮ Two real-life examples of the bootstrap

◮ Confidence bounds for spectra of combustion engine knock data◮ Micro-Doppler analysis

◮ Concluding remarks


Introduction (Cont’d)

◮ The term bootstrap is often associated with The Baron von Munchhausen.

Courtesy of Prof. Dr. Karl Heinrich Hofmann, TU Darmstadt and Head of Group Algebra, Geometry and Functional Analysis


Introduction (Cont’d)

◮ This analogy may suggest that the bootstrap is able to perform the

impossible.

◮ The bootstrap is not a magic technique that provides a panacea for all

statistical inference problems!

◮ The bootstrap is a powerful tool that can substitute tedious or often

impossible theoretical derivations with computational calculations.

◮ There are situations where the bootstrap fails and care is required [Young

(1994)].


What can I use the bootstrap for?

◮ The bootstrap is a computational tool for statistical inference.

◮ It can be used for:

◮ Estimation of statistical characteristics such as bias, variance, distribution

function of estimators and thus confidence intervals.◮ Hypothesis tests, for example for signal detection, and◮ model selection.

◮ When can I use the bootstrap?

When I know little about the statistics of the data and/or I have only

a few data so that asymptotic theory does not hold.


The Bootstrap Idea

Model

Unknown EstimatedProbability

Model

^

REAL WORLD

21X=(x , x , ..., x )F

Probability

θ

n

^

F^ x*=(x* , x* , ..., x* )1 2

θ*=s( *)x

Observed Data Bootstrap Sample

x=s( )Bootstrap replicationStatistic of interest

BOOTSTRAP WORLD

n

Basic idea: simulate the probability mechanism of the real world by substituting the

unknowns with estimates derived from the data.


A Brief History Recount

◮ It was by marrying the power of Monte Carlo approximation with an

exceptionally broad view of the sort of problem that bootstrap methods

might solve, that [Efron (1979)] famously vaulted earlier resampling ideas out

of the arena of sampling methods and into the realm of a universal statistical

methodology.

◮ Arguably, the prehistory of the bootstrap encompasses pre-1979

developments of Monte Carlo methods for sampling.

◮ Important aspects of bootstrap roots lie in methods for spatial sampling in

India in the 1920’s [Hubback (1946), Mahalanobis (1946)], even before

Quenouille’s and Tukey’s work on the jackknife [Quenouille (1949, 1956),

Tukey (1958)].


The independent data bootstrap


The Bootstrap for independent Data

Step 1. Conduct the experiment to obtain the random sample

X = {X1, X2, ... , Xn} and find the estimator θ from X .

Step 2. Construct the empirical distribution Fθ, which puts equal mass 1/n

at each observation X1 = x1, X2 = x2, ... , Xn = xn.

Step 3. From the selected Fθ, draw a sample X ∗ = {X ∗

1 , X ∗

2 , ... , X ∗

n },called the bootstrap (re)sample.

Step 4. Approximate the distribution of θ by the distribution of θ∗ derived

from X ∗.


Implementation of the Bootstrap Procedure

Measured Data

Conduct

Experiment- - Compute

Statistic-

xθ(x)

?Resample

Bootstrap Data

- - Compute

Statistic-

x∗1

θ∗1

- - Compute

Statistic-

x∗2

θ∗2

.

.

.

.

.

.

- - Compute

Statistic-

x∗B

θ∗B

Generate

EDF,

FΘ

(θ)

-

-θ

6FΘ

(θ)

Algorithm to compute the approximative distribution function of the estimator θ.


Resampling i.i.d. Data (Cont’d)

(6 colors) (<6 colors)

(<6 colors) (<6 colors)

Original data X Bootstrap resample X ∗

1

Bootstrap resample X ∗

2 Bootstrap resample X ∗

B

. . .


Example: Variance Estimation

Consider the problem of finding the variance σ2θ

of an estimator θ of θ, based on

the random sample X = {X1, ... , Xn} from the unknown distribution Fθ.

◮ If tractable, one may derive an analytic expression for σ2θ.

◮ For example, for X1, ... , Xn identically and independently Gaussian distributed

as N (µ, σ2) and θ = µ,

θ = µ =1

n

n∑

i=1

Xi , σ2µ =

σ2

n.

◮ Alternatively, one may use asymptotic arguments [Serfling (1980)] to

compute an estimate σ2θ

for σ2θ, in which situation the validity conditions for

the above may not be fulfilled.


Example: Variance Estimation of µ

X = {X1, ... , Xn}

X ∗

1 X ∗

2 X ∗

B

µ∗

1 µ∗

2 µ∗

B

σ∗2µ = 1

B−1

∑B

b=1

(

µ∗

b − 1B

∑B

b=1 µ∗

b

)2

Observations

Resamples

Bootstrap statistics

Variance estimation


The dependent data bootstrap


The Dependent Data Bootstrap

◮ If a model such as AR is available for the probability mechanism generating

stationary observations, one could bootstrap independent residuals.

◮ If no plausible model exists, then one can resort to dependent data bootstrap

methodologies.

◮ Strong mixing processes, i.e., loosely speaking, processes for which

observations far apart (in time) are almost independent [Rosenblatt (1985)],

for example, satisfy the weak dependence condition.

◮ The moving blocks bootstrap [Kunsch (1989), Liu & Singh (1992), Politis &

Romano (1992,1994)] has been proposed for bootstrapping weakly dependent

data.


The Moving Blocks Bootstrap

0 10 20 30 40 50 60 70 80 90 100−5

0

5

Sample number

Am

plit

ud

e

Bootstrap Data0 10 20 30 40 50 60 70 80 90 100

−5

0

5

Am

plit

ud

e

Resampled Blocks

54321

0 10 20 30 40 50 60 70 80 90 100−5

0

5

Am

plit

ud

e

Observed Data

12 345

Randomly select blocks of the original data (top) and concatenate them together

(centre) to form a resample (bottom). Here, the block size is l = 20 and n = 100.


Signal detection with the bootstrap


Hypothesis Testing with the Bootstrap

◮ Consider a random sample X = {X1, ... , Xn} observed from an unspecified

probability distribution F . Let θ be an unknown parameter of F .

◮ We wish to test the hypothesis

H : θ = θ0 against K : θ > θ0,

where θ0 is some known constant.

◮ Let θ be an estimator of θ and σ2 an estimator of the variance σ2 of θ.

◮ Define the statistic

T =θ − θ0

σ.


Hypothesis Testing (Cont’d)

Step 1. Resampling. Draw X ∗, with replacement, from X .

Step 2. Compute Bootstrap Statistics. Calculate

T ∗ =θ∗ − θ

σ∗,

where θ∗ and σ∗ are versions of θ and σ computed from X ∗.

Step 3. Repetition. Repeat Steps 1 and 2 to obtain T ∗

1 , ... , T ∗

B .

Step 4. Sorting.Rank T ∗

1 , ... , T ∗

B into T ∗

(1) ≤ · · · ≤ T ∗

(B). Reject H if

T ≥ T ∗

(q), where q = ⌊(B + 1)(1 − α)⌋.


Hypothesis Testing: Variance Estimation

◮ If an estimator σ2 is unavailable, use the bootstrap or the jackknife to

estimate it.

◮ The jackknife [Miller (1974)] is based on the sample delete-one observation

at a time, X (i) = {X1, X2, ... , Xi−1, Xi+1, ... , Xn}, i = 1, 2, ... , n.

◮ For each ith jackknife sample, calculate the ith jackknife estimate θ(i) of θ,

i = 1, ... , n and compute

σ2JACK =

n − 1

n

n∑

i=1

(

θ(i) − 1

n

n∑

i=1

θ(i)

)2

,

which is less expensive than the bootstrap if n < B.

◮ We proceed similarly for estimating σ∗2.


The Bootstrap Procedure

Measured Data

Conduct

Experiment- - Compute

Statistic-

x

T

?Resample

Bootstrap Data

- - Compute

Statistic-

x∗1

T∗1

- - Compute

Statistic-

x∗2

T∗2

.

.

.

.

.

.

- - Compute

Statistic-

x∗B

T∗B

Sort T∗1 , ... , T∗

B

to get

T∗(1)

≤ ... ≤ T∗(B)

Set λ = T∗(q)

,

q = ⌈(1 − α)(B + 1)⌉

Decision-�


The Signal Detection Problem

+Transmitter Receiver- -?Signal s

Noise W

X = θs + W

Test the hypotheses:

H : θ = 0

K : θ > 0

Receiver Structure

T (X) T ≷ λ- - KH


Signal Detection with the Matched Filter

Ps-x ×- T

1

σ√

s′s6

s

- -

T =s′x

σ√

s′s=

s′Psx

σ√

s′s∼ N

(

θ√

s′s

σ, 1

)

, Ps = ss′/s′s.


Signal Detection with the CFAR Matched Filter

×

·

·

Ps-

x

- ?

I−Ps ‖·‖2

×

-

6

s′√

s′s

-

- - - √6

? -

1/(n − 1)

≷ λ

T =s′Psx

√σ2s′s

p

x′(I − Ps)x/(n − 1)σ2∼ tn−1

„

θ√

s′s

σ

«


Limitations of the Matched Filter

◮ The matched filter and the CFAR matched filter are designed (and optimal)

for Gaussian interference.

◮ Although they show high probability of detection in the non-Gaussian case,

they are unable to maintain the preset level of significance for small sample

sizes.

◮ The matched filter fails in the case where the interference/noise is

non-Gaussian and the data size is small.

◮ The goal is to develop techniques which require little in the way of modelling

and assumptions.


The Bootstrap in Signal Detection: Motivation

◮ The Bootstrap can provide solutions for detecting signals in non-Gaussian

interference, e.g., high-resolution radar and radar at low grazing angles.

◮ The bootstrap is particularly attractive when only a limited number of

samples is available for signal detection.

◮ The bootstrap is also attractive when detection is associated with estimation,

e.g., estimation of the number of signals.

◮ The bootstrap has been successfully applied to signal detection in areas such

as wireless communications, landmine detection, radar, sonar and biomedical

engineering.


The Signal Detection Problem

+Transmitter Receiver- -?Signal s

Noise W

X = θs + W

Test the hypotheses:

H : θ = 0

K : θ > 0

Receiver Structure

T (X) T ≷ λ- - KH


The Bootstrap Matched Filter (Cont’d)

ComputeStatistic

ComputeStatistic

ComputeStatistic

DataMeasured

x

Decision

Resample &

BootstrapData

ComputeStatistic

Estimateθ

Reconstruct

θ

T∗1

T∗2

T∗B

Sort T∗1 , ... , T∗

B

to get

T∗(1)

≤ ... ≤ T∗(B)

q = ⌊(1 − α)(B + 1)⌋

set T∗(q)

,

T

w = x − θs

ConductExperiment

x∗B

x∗2

x∗1


Performance of the Bootstrap Matched Filter

◮ The bootstrap matched filter is consistent, ie, Pd → 1 as n → ∞.

◮ One can show [Ong & Zoubir (2000)] that

Pf = α + Op

(

n−1)

(

α + O(

n−1/2)

for the matched filter)

for|st | < ∞, t ∈ Z

.

◮ If st = cos(ωt + φ), then

Pf = α + Op

(

n−3/2)

(

α + O(

n−1)

for the matched filter)


Simulation Results (Gaussian Case)

Let st = sin(2πt/6), n = 10, B = 999 (25 for variance estimation), α = 5%, and

the number of independent runs be 5,000.

N (0, 1), SNR=7 dB

Detector Pf [%] Pd [%]

Matched Filter (MF) 5 99

CFAR MF 5 98

Boot. (σ known) 4 99

Boot. (σ unknown) 5 98


Simulation Results (Non-Gaussian Case)

Let st = sin(2πt/6), n = 10, B = 999 (25 for variance estimation), α = 5%, and

the number of independent runs be 5,000.

Wt ∼∑4

i=1 aiN (µi , σ2i ) with a = (0.5, 0.1, 0.2, 0.2), µ = (−0.1, 0.2, 0.5, 1),

σ = (0.25, 0.4, 0.9, 1.6), and SNR = -6 dB

Detector Pf [%] Pd [%]

Matched Filter (MF) 8 83

CFAR MF 7 77

Boot. (σ known) 5 77

Boot. (σ unknown) 7 79


Simulation Results (Cont’d)

10 20 30 40 50 60 70 80 90 1000

1

2

3(a)

Pfa

/ α

10 20 30 40 50 60 70 80 90 1000

1

2

3(b)

Pfa

/ α

10 20 30 40 50 60 70 80 90 1000

1

2

3(c)

Pfa

/ α

Sample size

Ratio of actual vs. nominal false alarm rate for the detection of a DC in Gaussian (top),

skewed (middle) and heavy tailed (bottom) noise.


Simulation Results (Cont’d)

10 20 30 40 50 60 70 80 90 1000

1

2

3(a)

Pfa

/ α

10 20 30 40 50 60 70 80 90 1000

1

2

3(c)

Pfa

/ α

Sample size

10 20 30 40 50 60 70 80 90 1000

1

2

3

Pfa

/ α

(b)

Ratio of actual vs. nominal false alarm rate for the detection of a sinusoid in Gaussian

(top), skewed (middle) and heavy tailed (bottom) noise.


Frequency domain bootstrap

methods for dependent data


Bootstrapping Spectral Densities

◮ Confidence interval estimation (or hypothesis testing) for spectral densities is

encountered in numerous real-life applications, e.g., car engine monitoring.

◮ Two approaches to bootstrapping spectral densities:

◮ Use a time domain approach (block bootstrap) to bootstrap time series and

estimate bootstrap spectral density estimates from the time series replicae, or◮ use a technique that applies directly in the frequency domain.

◮ Bootstrap time series not always mimic the (complicated) dependence

structure of the underlying process.

◮ Frequency domain bootstrap methods do not require to first generate time

series replicae to get periodogram resamples.


Frequency Domain Bootstrap (Cont’d)

◮ Assume Xt , t ∈ Z to be a real-valued strictly stationary process.

◮ Given X1, ... , XT , define the periodogram by

IXX (ω) =1

2πT

∣

∣

∣

∣

∣

T∑

t=1

Xte−jωt

∣

∣

∣

∣

∣

2

, −π ≤ ω ≤ π

◮ and construct the estimator of CXX (ω) by

CXX (ω) =1

T · h

n∑

k=−n

K

(

ω − ωk

h

)

IXX (ωk),

where the kernel K (·) is a known symmetric, non-negative, real-valued

function, h is its bandwidth and n = [T/2].


Frequency Domain Bootstrap (Cont’d)

◮ The periodogram evaluated at frequencies, IXX (ωk), ωk = 2πk/T ,

−n ≤ k ≤ n, is known to have a scaled χ2 distribution with two degrees of

freedom; the scaling factor being the true, but unknown spectral density.

◮ It is also known that periodogram ordinates are asymptotically for large T

nearly independent.

◮ This suggests to write the periodogram at frequency ωk as a multiplicative

regression:

IXX (ω : k) ≈ CXX (ωk) · Ek

where the errors Ek are assumed to be i.i.d.

◮ This suggests independent frequency data resampling to construct confidence

intervals for spectra.


The Residual Based Bootstrap (Cont’d)

DataMeasured

x

ConductExperiment

BootstrapData

Reconstruct& Estimate


CXX

`

ωk

´

SmoothPeriodogram


Resample

Uk =IXX

`

ωk

´

CXX

`

ωk

´

Confidence

IntervalSort

k = 1, ... , M

FindPeriodogram

C∗(B)XX

`

ωk

´

C∗(2)XX

`

ωk

´

C∗(1)XX

`

ωk

´

, ...

... , C∗(B)XX

`

ωk

´

IXX

`

ωk

´

U∗k,1

U∗k,2

U∗k,B

C∗(1)XX

`

ωk

´


Two real-life example

of the bootstrap


Confidence Intervals for knock Spectra

◮ Special interest continues to a be reduction of fuel consumption in

combustion engines due to meeting the demands of new legislations on

greenhouse emission.

◮ A means for lowering fuel consumption in spark ignition engines is to increase

the compression ratio. Further increase in compression is limited by the

occurrence of knock.

◮ To attain safe operation at maximum efficiency, engine control systems which

detect knock and adapt spark timing of each cylinder separately have found

special interest.

◮ The data was collected at Robert Bosch GmbH on an engine test bed.

Measurements from a combustion engine running at 3500 rpm under

knocking conditions were recorded and sampled at 100 kHz.



(Cont’d)

0 50 100 150 200 250 300 350 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sample

Am

plitu

de (

norm

alis

ed)

In-cylinder pressure signal (high-pass filtered) of a knocking combustion cycle.



(Cont’d)

0 50 100 150 200 250 300 350 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sample

Am

plitu

de (

norm

alis

ed)

Structure-borne sound signal of a knocking combustion cycle.



(Cont’d)

0 5 10 15 20 25 30 35 40 45 50−60

−50

−40

−30

−20

−10

0

Frequency [kHz]

Spe

ctra

l den

sity

(no

rmal

ised

)[dB

]

Spectral density estimates for in-cylinder pressure (solid —) and structure-borne sound

(dashed-dotted -.-) obtained by averaging 594 periodograms.



(Cont’d)

0 5 10 15 20 25 30 35 40 45 50−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency [kHz]

Spe

ctra

l den

sity

(no

rmal

ised

) [d

B]

Estimated 95% confidence interval (dashed line - - -) for the spectral density of

in-cylinder pressure, along with the spectral density estimate for the combustion cycle

above.May 12, 2010 | SPG TUD | c© A.M. Zoubir | 45 SPG

Confidence Intervals for Knock Spectra

(Cont’d)

0 5 10 15 20 25 30 35 40 45 50−70

−60

−50

−40

−30

−20

−10

0

Frequency [kHz]

Spe

ctra

l den

sity

(no

rmal

ised

) [d

B]

Estimated 95% confidence interval for the spectral density of structure-borne sound

(dashed line - –), along with the spectral density estimate for the combustion cycle above


Micro-Doppler Analysis

◮ Doppler often arises in engineering applications due to the relative motion of

an object with respect to the measurement system.

◮ If the motion is harmonic, for example due to vibration or rotation, the

resulting signal can be well modeled by an FM process [Huang et al. (1990)].

◮ Estimation of the FM parameters may allow one to determine physical

properties such as the angular velocity and displacement of the

vibrational/rotational motion which can in turn be used for classification.

Objective: Estimate the FM parameters along with a measure of accuracy,

such as confidence intervals.


Micro-Doppler Analysis (Cont’d)

◮ Assume the following AM-FM signal model:

s(t) = a(t) exp{jϕ(t)}where a(t; α) =

∑q

k=0 αk tk and α = (α0, ... , αq) .

◮ The phase modulation for a micro-Doppler signal is described by:

ϕ(t) = −D/ωm cos(ωmt + φ) and the instantaneous frequency (IF) by:

ωi (t; β) ,dϕ(t)

dt= D sin(ωmt + φ),

where β = (D, ωm, φ) are the FM or micro-Doppler parameters.

◮ Given {x(k)}nk=1 of X (t) = s(t) + V (t), the goal is to estimate

β = (D, ωm, φ) as well as their confidence intervals.



◮ The estimation of the phase parameters is performed using a time-frequency

Hough transform (TFHT) [Barbarossa & Lemoine (1996), Cirillo, Zoubir &

Amin (2006)].

◮ Once the phase parameters have been estimated, the phase term is

demodulated and the amplitude parameters are estimated via least-squares.

◮ Given estimates for α and β, the residuals are obtained.

◮ We whiten the residuals using a fitted AR model. The innovations are

re-sampled, filtered and added to the estimated signal term.



◮ The results shown here are based on an experimental radar system*,

operating at carrier frequency fc = 919.82 MHz.

◮ After demodulation, the in-phase and quadrature baseband channels are

sampled at fs = 1 kHz.

◮ The radar system is directed toward a spherical object, swinging with a

pendulum motion, which results in a typical micro-Doppler signature.

◮ The PWVD of the observations is computed and shown below:

* Courtesy of Prof. Moeness Amin, Director of the Center of Advanced

Communications, Villanova University, Philadelphia, USA



Time (sec)

Fre

quen

cy (

Hz)

PWVD of the radar data

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−250

−200

−150

−100

−50

0

50

100

150

200

Time (sec)F

requ

ency

(H

z)

PWVD of the radar data

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45−250

−200

−150

−100

−50

0

50

100

150

200

(a) (b)

The PWVD of the radar data (a). The PWVD of the radar data and the

micro-Doppler signature estimated using the TFHT (b).



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.04

−0.02

0

0.02

0.04

Time (sec)

Am

plitu

de (

V)

Real (in−phase) component

DataAM−FM Fit

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.04

−0.02

0

0.02

0.04

Time (sec)

Am

plitu

de (

V)

Imaginary (quadrature) component

DataAM−FM Fit

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.015

−0.01

−0.005

0

0.005

0.01

Time (sec)

Am

plitu

de (

V)

Time−domain waveform of residuals

Re{ r(n) }Im{ r(n) }

−500 −400 −300 −200 −100 0 100 200 300 400 500−90

−80

−70

−60

−50

−40

−30

−20

f (Hz)

Pow

er (

dBW

)

Estimated spectrum of residuals

PeriodogramAR Fit

(c) (d)

The real and imaginary components of the radar signal are with their estimated

counterparts (c). The real and imaginary parts of the residuals and their spectral

estimates (d).May 12, 2010 | SPG TUD | c© A.M. Zoubir | 52 SPG


71.7 71.8 71.9 72 72.1 72.2 72.30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

D/(2π) (Hz)

Boostrap confidence interval for D

.Bootstrap dist

Initial estimat

95% CI

e

3.037 3.038 3.039 3.04 3.041 3.042 3.043 3.044 3.045 3.0460

50

100

150

200

250

300

350

400

450

ωm

/(2π) (Hz)

Boostrap confidence interval for ωm

Bootstrap dist.

Initial estimate

95% CI

(a) (b)

The bootstrap distributions and 95% confidence intervals for the FM parameters

D (a) and ωm (b) using B = 500.


Summary

◮ Many signal processing problems require the computation of quality measures

for estimators.

◮ Most techniques available assume that the size of the available set of sample

values is sufficiently large, so that “asymptotic” results can be applied.

◮ In many signal processing problems this assumption cannot be made because,

for example, the process is non-stationary and only small portions of

stationary data are considered.

◮ Bootstrap techniques are an alternative to asymptotic methods and provide

good results for many practical problems.


Summary (Cont’d)

What was not discussed in this presentation:

◮ Other variants of the bootstrap such as bootstrap bagging and bootstrap

bumping [Tibshirani & Knight (1997)].

◮ Model selection with the bootstrap [Shao & Tu (1995), Zoubir (1999)].

◮ Hypothesis testing [Hall & Titterington (1989), Hall & Wilson (1991), Hall

(1992)] and signal detection [Zoubir (2000)].

◮ Bayesian approaches to the bootstrap.


Which Method Should I Use?


The Baron von Munchhausen

Varney, Allen (2007), ”The Extraordinary Adventures of Baron Munchhausen”, in Lowder, James, Hobby Games: The Best 100, Green Ronin Publishing,

pp. 107109, ISBN 978-1-932442-96-0


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