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The meaning ofDistances in Spectral Analysis
46th IEEE Conference on Decision & Control
Plenary presentation
Tryphon Georgiou
Electrical & Computer EngineeringUniversity of Minnesota
Meaning of distances
• maximal separation (L∞)• energy-like content (L2)• integral of flow-rate (L1)
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Power spectra
⇒
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Periodogram, Blackman-Tukey, Levinson, Durbin, Burg, . . .
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Spectral analysis
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+
=
. . . u(0), u(1), . . .+
...
u(k) =∫ejkθdX(θ) E{u(k)u(k + `)} =
∫ej`θf (θ)dθ
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Signals vs. power densities
time-signals
(u1 − u2) “error signal”
power distributions
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(f1 − f2) is not a “signal”
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Communications
Speech analysis/coding
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Medical diagnostics
Noninvasive temperature sensing
Temperature field
with E. Ebbini & A.N. AminiIn IEEE Trans. on Biomedical Engineering, 2005
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Medical diagnostics Radar (SAR)
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−1
100
101
102
http://www.sandia.gov/radar/images/3dsar.gif
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Quantitative analysis
↗
different
methods
↘
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How can we compare power spectra?
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Quantitative analysis
→
same
method
→
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How can we compare power spectra?
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How can we compare power spectra?
Question:what is a natural notion of distancebetween power spectral densities?
quantify uncertainty
signal classification, detect structural changes
system identification, tune algorithms, sensor technology
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Plan of the talkMetrics based on
prediction theory (Szego, Kolmogorov)
parallels with information geometry (Fisher, Rao)
transport geometry (Monge-Kantorovich)
Case studies & applications Prediction-basedgeometry
Informationgeometry
Transportgeometry
applications
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Setting. . . u−1, u0, u1, u2, . . . . . . u−1, u0, u1, u2, . . .
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f1(θ) f2(θ)
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What is it we would like to have?
distance(
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, 0 0.5 1 1.5 2 2.5 310
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10−2 )
f1(θ), f2(θ)
• metric• meaningful & natural
candidates?Kullback-Leibler, Bregman, Itakura-Saito, Makhoul,..
convex functionalsperceptual qualities
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Linear prediction
One-step-ahead prediction: upresent − upresent|past
with upresent|past :=∑
pastαkuk
E{|upresent − upresent|past|2} = variance of prediction error
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Szego’s theorem
One-step-ahead prediction:
least error variance = exp
{1
2π
∫log f (θ)dθ
}G. Szego
it is a geometric mean. . .
exp{13
(log f1 + log f2 + log f3)} = 3√f1f2f3
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Degradation of prediction error variance
Use f2 to design a predictor (assuming uf2,time).
Then compare how this performs on uf1,time against the optimal based on f1.
degraded variance︷ ︸︸ ︷E{|uf1,present −
∑past
af2,past uf1,past|2}− optimal variance
optimal variance≥ 0
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Degradation of prediction variance
degraded variance︷ ︸︸ ︷E{|uf1,present −
∑past
af2,past uf1,past|2}
optimal variance=
arithmetic mean of(f1
f2
)geometric mean of
(f1
f2
)
=
(1
2π
∫(f1
f2)dθ)
exp(
12π
∫log(f1
f2)dθ)
arithmetic over geometric mean (≥ 1)
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Riemannian metric
f1 = f,f2 = f + ∆
degraded variance︷ ︸︸ ︷E{|uf1,present −
∑past
af2,past uf1,past|2}− optimal variance
optimal variance'
δ(f, f + ∆) =1
2π
∫ (∆
f
)2
dθ −(
1
2π
∫ (∆
f
)dθ
)2
variance-like: (mean square) - (arithmetic-mean)2
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Geodesics
Paths fr (r ∈ [0, 1]) between f0, f1 of minimal length∫ 1
0
√δ (fr, fr+dr)
each point represents a different power spectral density
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Geodesics
The geodesics are exponential families:
fr = f0
(f1
f0
)r
, r∈[0, 1]
= exp{ (1− r) log (f0) + r log (f1) }
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morphing
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Geodesic distance: metric
The path-length is
d(f0, f1) :=
√1
2π
∫ π
−π
(log
f1
f0
)2
dθ −(
1
2π
∫ π
−πlog
(f1
f0
)dθ
)2
variance-like distance on logarithms: (mean square) - (arithmetic-mean)2
scale-insensitive, “shape” recognizer
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100
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−2.5
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0
0.5
1
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2.5
In IEEE Trans. on Signal Processing, Aug. 2007 log f1
f0= log(f1)− log(f0)
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Prediction-basedgeometry
Informationgeometry
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Information geometry – parallels
f ; p : probability density
I = Ep{(∂λ log pλ)2} δλ2
Fisher information metric
I =∑∆2
p
R. Fisher
C.R. Rao
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Information geometry – parallels
Expected “message-length increase”:
H(p1|p0) =(−∑
p1 log(p0))−(−∑
p1 log(p1))
Fisher information metricp0 = pp1 = p + ∆
I =∑∆2
p
R. Leibler
S. Kullback
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Information geometry – parallels
Geodesics: great circles
p 7→ √p ∈ Sphere p(1)p(2)p(3)
7→√p(1)√p(2)√p(3)
Geodesic distance: ArclengthBattacharyya distance
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Information vs. prediction-based
∑∆2
pvs.
∫ (∆
f
)2
−(∫
∆
f
)2
p 7→ √p vs. f 7→ log f
great circles vs. logarithmic families
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Information geometry – parallels
Ability to differentiate decreases p(1)p(2)p(3)
7→ M
p(1)p(2)p(3)
Chentsov’s theorem:
Stochastic maps are contractive
under Fisher metric
and
Fisher metric is the uniqueRiemannian metric with this property
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What is the analog for power spectra?
addititive noise
f 7→ f + fnoise
multiplicative noise
f 7→ f ? fnoise
continuity of moments (second-order statistics)
f 7→ integrals of f
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Prediction-basedgeometry
Informationgeometry
Transportgeometry
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Transport geometry
Monge-Kantorovich problemminimize cost of transferring mass∫
cost(x→ y) × mass(dx, dy)
G. Monge
L. Kantorovich
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Transport for power spectra
Transport-based metric
distances do not increase
under additive noiseand multiplicative noise
with power ≤ 1
+ continuity of statistics
metric = min (cost of transport(f0, f1) + normalization)
with Johan Karlsson (KTH) & Mir Shahrouz Takyar
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Prediction-basedgeometry
Transportgeometry
applications
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Fitting geodesics
K.F. Gauss
Least squares: The theory of motion of heavenly bodies, Gauss, K.F.
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Tracking with geodesics
with Xianhua Jiang
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Voice & sounds
John Weissmuller’s MGM Tarzan Yell
http://www.complxmind.com
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Images & more
Geometric active contours
∂∂tCurve = ∇Curve metric(finside, foutside)
with Romeil Sandhu and Allen Tannenbaum
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Images & more
with Romeil Sandhu and Allen Tannenbaum
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Concluding thoughtsMetricsin spectral analysis
• Operational significance
• Effect of natural transformations
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Thank you for your attention
thanks toXianhua Jiang Johan Karlsson Romeil Sandhu Mir Shahrouz Takyar
Allen Tannenbaum & Anders Lindquist
National Science Foundation, AFOSR, and Hermes-Luh endowment
New Orleans, December 2007 40
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