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The Methodology of the MaximumLikelihood Approach
Estimation, detection, and exploration of seismicevents
Anna Maly and Patrick Pircher
SPSCTU Graz
21.01.2013
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Inhalt
1 Introduction
2 Data Model
3 Parameter Estimation
4 Signal Detection
5 Calculation of Test Treshold tm
6 Summary
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Introduction
Seismic Signals are transient (duration shorterthan observation)
Explosions and Earthquakes give rise to anumber of different waves. (e.g. pressurewaves, shear waves, or surface waves)
Waves follow different paths.
At an array of sensors one can observe theso-called phases of the different types of waves.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Goal
Wave Parameter Estimation: Find the mostlikely parameters for the incoming data.(Parameters: Slowness, Velocity, Azimuth,Elevation)
Compute Test Statistic: Probability of beinga signal.
Signal Detection: Comparing the teststatistic with a threshold.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Maximum Likelihood Estimation
The MLE for a scalar parameter is defined tobe the value of ϑ that maximizes p(x;ϑ) for xfixed.
The maximization produces a ϑ that is afunction of x.
Log-Likelihood:
l(ϑ|A) = lnL(ϑ|A) (1)
Excellent statistical performance androbustness.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Data Model
Plane Wave Model
Array outputs sampled and STFT
Nonlinear regression model:
Xl(ω) = H(ω, ϑ)Sl(ω) + Ul(ω) (2)
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Xl(ω) = H(ω, ϑ)Sl(ω) + Ul(ω) (3)
Sl(ω) . . . Fourier-transformed signal vectorUl(ω) . . . Noise vectorH(ω, ϑ) = [d1(ω) . . .dM(ω)] . . . Transfer functiondi(ω) . . . Steering vectorϑ = [ξ1, . . . , ξM ] . . . Nonlinear wave parametersξi . . . Slowness Vector
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
The little lMulti-Taping
l . . . from STFT:
Xl(ω) =1√T
T−1∑t=0
wl(t)x(t)e−jωt (4)
l = 1,. . . ,L, where wl(t)s are orthonormal windowfunctions. We use the multi taping technique.x(t) is divided into K snapshots of duration T each.These are Fourier transformed using L orthogonalwindows depending on snapshot duration T and theselected analysis bandwidth.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Parameter Estimation
Properties: normal distribution, independency
Minimize U:
1
L
L∑l=1
J∑j=1
|Xl(ωj))−H(ωj , ϑ)Sl(ωj)|2 (5)
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Leads to the broadband log-likelihood function:
L(ϑ,S, ν) = −LL∑
l=1
J∑j=1
[Nlog(ν(ωj)) +1
ν(ωj)
(Xl(ωj))−H(ωj , ϑ)Sl(ωj))H
(Xl(ωj)−H(ωj , ϑ)Sl(ωj))]
ν(ωj). . . Noise level
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Signal and noise parameters are separable
Dependence on Sl(ωj) and ν(ωj) can beremoved ⇒ Replacing unknown signal andnoise parameters by their ML estimates at fixedand unknown wave parameters
L(ϑ) = −J∑
j=1
logtr [(I− P(ωj , ϑ))Cx(ωj)] (6)
P(ωj , ϑ) . . . Projection Matrix onto the columnspace of the transfer matrix H(ωj , ϑ)
Cx(ωj) = 1L
∑Lj=1X
l(ωj)Xl(ωj)
H . . . Sample spectralmatrix
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Signal DetectionNeyman-Pearson Theorem
N(µ,σ2) . . . Gaussian PDF
Example: Two hypotheseses:H0 : µ = 0. . . Null HypothesisH1 : µ = 1. . . Alternative Hypothesis
Determine if µ = 0 or µ = 1 based on a singleobservation x [0].
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
It’s all about the threshold.
−4 −3 −2 −1 0 1 2 3 4 5 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
PDFs
x[0]
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Type 1 Error:Decide H1, but H0 is true. ⇒ P(H1;H0)
Type 2 Error:Decide H0, but H1 is true.⇒ P(H0;H1)
Probability of Detection:PD = P(H1;H1)
Probability of False Alarm:PFA = P(H1;H0) = α
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Neyman-Pearson Theorem / Likelihood Ratio
TestTo maximize PD for a given PFA = α decide H1 if
L(x) =p(x;H1)
p(x;H0)> γ (7)
where the threshold γ is found from
PFA =
∫{x :L(x)>γ}
p(x;H0)dx = α (8)
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
In our case: Testing the null hypothesis Hm againstalternative hypothesis Am. (m = 1 . . .Mmax →number of signals/sources emphasizes matrixdimension and number of parameters associatedwith the assumed model)For m = 1,
H1: Data contains only noise.A1: Data contains at least one signal.
For m = 2, . . . ,Mmax ,Hm: Data contains at most (m − 1)
signals.Am: Data contains at least m signals.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Starting from H1, the test decides if a signal ispresent.
No Signal → Procedure stops
Signal detected → Procedure goes to next step
Continues until Mmax is reached
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
p(x;Am)
p(x;Hm)⇒ Tm = supL(ϑA)− supL(ϑH) (9)
Tm ≥ tm ⇒ Signal detected, reject Hm,Tm < tm ⇒ retain Hm
Tm . . . Central F-distributed for Null HypothesisFrequencies independent, so sum over Tm possible
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Calculation of Test Threshold
The threshold tm is chosen to keep aprespecified false alarm level at αm.
Fm(·) . . . Null distribution of Tm
tm = F−1m (αm)
Broadband case → No closed-form expression
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Broadband case: Several ways to determine tm:
Normal Approximation
Cornish Fisher Expansion
Sequentially rejectiveBonferroni-Holmprocedure
False discovery rate (FDR) through theBenjamini-Hochberg procedure
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
Normal Approximation:
tm ≈ µm +σm√J
Φ−1(αm) (10)
Φ−1(αm). . . inverse of standard normal distributionat αm
Calculation of mean and variance can be found in“Hypothesis testing for geoacoustic environmentalmodels using likelihood ratio,” from C. F.Mecklenbrauker, P. Gerstoft, J. F. Bohme, and P.-J.Chung.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary
SummaryINPUT: Fourier transformed data, maximal numberof signals Mmax , initial value for estimated numberof signals M = 0FOR m = 1, . . . ,Mmax
wave parameter estimation
compute test statistic
signal detection
ENDOUTPUT: estimated wave parameters and detectednumber of signals.
Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach
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