Download Spectrum Estimation in Helioseismology

Inverse Problems and Data Errors
Conference in Honor of
Douglas O. Gough
On the Occasion of his 60th
Birthday
Chateau de Mons, Caussens, France
P.B. Stark
Department of Statistics
University of California
Berkeley CA
www.stat.berkeley.edu/~stark
Acknowledgements
• Media pirated from websites of
Source: www.gong.noao.edu
– Global Oscillations Network Group
(GONG)
– Solar and Heliospheric Observer (SOHO)
Solar Oscillations Investigation
• Much joint work, incl. S. Evans (UCB), I.K.
Source: sohowww.nascom.nasa.gov
Fodor (LLNL), C.R. Genovese (CMU), D.O.Gough
(Cambridge), Y. Gu (GONG), R. Komm (GONG), T. Sekii
(Natl. Astron. Obs.of Japan), M.J. Thompson (QMW)
The Difference between Theory and Practice
• In Theory, there is no difference between
Theory and Practice.
• In Practice, there is.
Part I: Theory
Forward Problems in Statistics: Ingredients
• Measurable space X of possible data.
• Set  of possible descriptions of the world--models.
• Family P = {P : } of probability distributions on X , indexed by
models .
•Forward operator   P maps model  into a probability measure on X .
Data X are a sample from P.
P is whole story: stochastic variability in the “truth,” contamination by
measurement error, systematic error, censoring, etc.
Models
Index set  that usually has special structure.
For example,  could be a convex subset of a separable
Banach space T.
The forward mapping   P maps the index of the
model to a probability distribution for the data.
The physical significance of  generally gives   P
reasonable analytic properties, e.g., continuity.
Forward Problems in Physics and Applied Math
Composition of steps:
– transform correct description of the world into ideal, noise-free,
infinite-dimensional data (“physics”)
– censor ideal data to retain only a finite list of numbers, because can
only measure, record, and compute with such lists
– possibly corrupt the list with deterministic measurement error.
Equivalent to single-step procedure with corruption on par with
physics, and mapping incorporating the censoring.
Physical v. Statistical Forward Problems
Statistical framework for forward problems is more
general:
Forward problems of Applied Math and Physics are
instances of statistical forward problems.
Parameters
A parameter of a model θ is the value g(θ) at θ of a
continuous G-valued function g defined on 
(g can be the identity.)
Inverse Problems
Observe data X drawn from distribution Pθ for some
unknown . (Assume  contains at least two points;
otherwise, data superfluous.)
Use X and the knowledge that  to learn about ; for
example, to estimate a parameter g().
Applied Math and Statistical Perspectives
• Applied Math: recover a parameter of a pde or the solution of
an integral equation from infinitely many data, noise-free or
with deterministic error.
– Common issues: existence, uniqueness, construction, stability for
deterministic noise.
• Statistics: estimate or draw inference about parameter from
finitely many noisy data
– Common issues: identifiability, consistency, bias, variance,
efficiency, MSE, etc.
Many Connections
Identifiability--distinct parameter values yield distinct
probability distributions for the observables--similar to
uniqueness--forward operator maps at most one model into the
observed data.
Consistency--parameter can be estimated with arbitrary
accuracy as the number of data grows--related to stability of a
recovery algorithm--small changes in the data produce small
changes in the recovered model.
 quantitative connections too.
Physical Inverse Problems
• Inverse problems in applied Physics often are tackled using
applied math methods for “Ill-posed problems” (e.g.,
Tichonov regularization, analytic inversions)
• Those methods are designed to answer different questions; can
behave poorly with data (bad bias & variance)
• Inference  construction: statistical viewpoint more
appropriate for data
Elements of the Statistical View
Distinguish between characteristics of the problem, and
characteristics of methods used to draw inferences
Fundamental property of a parameter:
g is identifiable if for all η, υ  Θ,
{g(η)  g(υ)}  {Pη  Pυ}.
In most inverse problems, g(θ) = θ not identifiable, and
few linear functionals of θ are identifiable
Decision Rules
A (randomized) decision rule
δ: X  M1(A)
x  δx(.),
is a measurable mapping from the space X of possible data to
the collection M1(A) of probability distributions on a separable
metric space A of actions.
A non-randomized decision rule is a randomized decision rule
that, to each x X, assigns a unit point mass at some value
Estimators
An estimator of a parameter g(θ) is a decision rule for
which the space A of possible actions is the space G of
possible parameter values.
ĝ is common notation for an estimator of g(θ).
Usually write non-randomized estimator as a G-valued
function of x instead of a M1(G)-valued function.
Comparing Estimators
Infinitely many estimators.
Which one to use?
The best one!
But what does best mean?
Loss and Risk
• Formulate as 2-player game: Nature v. Statistician.
• Nature picks θ from Θ. θ is secret.
• Statistician picks δ from a set D of decision rules. δ is secret.
• Generate data X from Pθ, apply δ.
• Statistician pays loss l (θ, δ(X)). l should be dictated by
scientific context, but…
• Risk is expected loss: r(θ, δ) = Eθl (θ, δ(X))
• Good estimator has small risk, but what does small mean?
Strategy
Rare that a single estimator has smallest risk for every
.
• Estimator is admissible if not dominated by another.
• Minimax estimator minimizes supr (θ, δ) over dD
• Bayes estimator minimizes  r (θ, δ) π(dθ) over dD

for a given prior probability distribution p on .
• Duality: minimax is Bayes for least favorable prior
Common Risk: Mean Distance Error (MDE)
Let dG denote the metric on G.
MDE at θ of estimator ĝ of g is
MDEθ(ĝ, g) = Eθ [d(ĝ, g(θ))].
When metric derives from norm, MDE is called mean
norm error (MNE).
When the norm is Hilbertian, (MNE)2 is called mean
squared error (MSE).
Bias
When G is a Banach space, can define bias at θ of ĝ:
biasθ(ĝ, g) = Eθ [ĝ - g(θ)]
(when the expectation is well-defined).
• If biasθ(ĝ, g) = 0, say ĝ is unbiased at θ (for g).
• If ĝ is unbiased at θ for g for every θ, say ĝ is unbiased for
g. If such ĝ exists, g is unbiasedly estimable.
• If g is unbiasedly estimable then g is identifiable.
More Notation
Let T be a separable Banach space, T * its normed dual.
Write the pairing between T and T *
<•, •>: T * x T  R.
Linear Forward Problems
A forward problem is linear if
• Θ is a subset of a separable Banach space T
• For some fixed sequence (κj)j=1n of elements of T*,
X = (Xj) j=1n, where
Xj = <κj, θ> + εj, θΘ, and
ε = (εj)j=1n
is a vector of stochastic errors whose distribution does not
depend on θ (so X = Rn).
Linear Forward Problems, contd.
• The functionals {κj} are the “representers” or “data kernels”
• The distribution Pθ is the probability distribution of X.
Typically, dim(Θ) = ; at the very least, n < dim(Θ), so
estimating θ is an underdetermined problem.
Define
K : T  Rn
Θ  (<κj, θ>)j=1n .
Abbreviate forward problem by X = Kθ + ε, θ  Θ.
Linear Inverse Problems
Use X = Kθ + ε, and the knowledge θ  Θ to estimate
or draw inferences about g(θ).
Probability distribution of X depends on θ only through
Kθ, so if there are two points
θ1, θ2  Θ such that Kθ1 = Kθ2 but
g(θ1)g(θ2),
then g(θ) is not identifiable.
Backus-Gilbert++: Necessary conditions
Let g be an identifiable real-valued parameter. Suppose  θ0Θ, a
symmetric convex set Ť  T, cR, and ğ: Ť  R such that:
1. θ0 + Ť  Θ
2. For t Ť, g(θ0 + t) = c + ğ(t), and ğ(-t) = -ğ(t)
3. ğ(a1t1 + a2t2) = a1ğ(t1) + a2ğ(t2),
t1, t2  Ť, a1, a2  0, a1+a2 = 1, and
4. supt  Ť | ğ(t)| <.
Then  1×n matrix Λ s.t. the restriction of ğ to Ť is the restriction of Λ.K to
Ť.
Backus-Gilbert++: Sufficient Conditions
Suppose g = (gi)i=1m is an Rm-valued parameter that can be
written as the restriction to Θ of Λ.K for some m×n matrix Λ.
Then
1. g is identifiable.
2. If E[ε] = 0, Λ.X is an unbiased estimator of g.
3. If, in addition, ε has covariance matrix Σ = E[εεT], the
covariance matrix of Λ.X is Λ.Σ.ΛT whatever be Pθ.
Corollary: Backus-Gilbert
Let T be a Hilbert space; let Θ = T ; let g  T = T * be a
linear parameter; and let {κj}j=1n T *.
The parameter g(θ) is identifiable iff g = Λ.K for some
1×n matrix Λ.
In that case, if E[ε] = 0, then ĝ = Λ.X is unbiased for g.
If, in addition, ε has covariance matrix Σ = E[εεT], then
the MSE of ĝ is Λ.Σ.ΛT.
Consistency in Linear Inverse Problems
• Xi = i + i, i=1, 2, 3, …
 subset of separable Banach
{i} * linear, bounded on 
{i} iid 
•  consistently estimable w.r.t. weak topology iff
{Tk}, Tk Borel function of X1, . . . , Xk s.t.
, >0,  *, limk P{|Tk -  |>} = 0
Importance of the Error Distribution
• µ a prob. measure on ; µa(B) = µ(B-a), a 
dμ a  dμ b
• Hellinger distance δ(a, b)  12
• Pseudo-metric on **:
k
{ (
)}
2 1/ 2
1
Dk (t1 , t 2 )  {  δ 2 (t1κ i , t 2 κ i )}1/ 2
k i 1
• If restriction to  converges to metric compatible with weak
topology, can estimate  consistently in weak topology.
• For given sequence of functionals {i}, µ rougher 
consistent estimation easier.
Example: Linear Combinations of Splitting Kernels
Cuts through kernels for rotation:
A: l=15, m=8. B: l=28, m=14. C: l=28, m=24.
D: two targeted combinations: 0.7R, 60o; 0.82R, 30o
Thompson et al., 1996. Science 272, 1300-1305.
Estimated rotation rate as a function of depth
at three latitudes.
Source: SOHO-SOI/MDI website
Part II: Practice
Linear Forward Problem for Rotation
ν lmn   κ
(r , θ) r dr dθ
lmn
Note language change: θ is now latitude, Ώ is the model.
Relationship assumes eigenfunctions and radial structure
known. Observational errors usually are assumed to be zeromean independent normal random variables with known
variances.
Data Reduction
Harvey et al., 1996. Science 272, 1284-1286
GONG Data Pipeline
1.
2.
3.
4.
5.
6.
7.
8.
Read tapes from sites.
Correct for CCD characteristics
Transform intensities to Doppler velocities
Calibrate velocities using daily calibration
images
Find image geometry and modulation transfer
function (atmospheric effects, lens dirt,
instrument characteristics, ...)
High-pass filter to remove steady flows
Remap images to heliographic coordinates,
interpolate, resample, correct for line-of-sight
Transform to spherical harmonics: window,
Legendre stack in latitude, FFT in longitude
9.
10.
11.
12.
13.
14.
Adjust spherical harmonic coefficients for
estimated modulation transfer function
Merge time series of spherical harmonic
coefficients from different stations; weight for
relative uncertainties
Fill data gaps of up to 30 minutes by ARMA
modeling
Take periodogram of time series of spherical
harmonic coefficients
Fit parametric model to power spectrum by
iterative approximate maximum likelihood
Identify quantum numbers; report frequencies,
linewidths, background power, and uncertainties
More Steps
Time series of spherical
harmonic coefficients
Spectra of time series, and fitted
parametric models
Top: GONG website. Bottom: Hill et al., Science, 1996.
Effect of Gaps
• Don’t observe process of interest.
• Observe process × window
• Fourier transform of data is FT of
process, convolved with FT of
window.
• FT of window has many large
sidelobes
• Convolution causes energy to “leak”
from distant frequencies into any
particular band of interest.
95% duty cycle window
Power spectrum of window
Tapering
• Want simplicity of periodogram, but less leakage
• Traditional approach—multiply data by a smooth
“taper” that vanishes where there are no data
• Smoother tapersmaller sidelobes, but more local
smearing (loss of resolution)
• Pose choosing taper as optimization problem
Optimal Tapering
• What taper minimizes “leakage” while maximizing
resolution?
• Leakage is a bias; optimality depends on definition
• Broad-band & asymptotic yield eigenvalue problems:
optimal taper 
arg max
w n :suppw  
Aw
w
2
2
Prolate Spheroidal Tapers
• Maximize the fraction of energy in
a band [-w, w] around zero
• Analytic solution when no gaps:
– 2wT tapers nearly perfect
– others very poor
• Must choose w
• Character different with gaps
T = 1024, w = 0.004.
Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483
Minimum Asymptotic Bias Tapers
• Minimize integral of
spectrum against frequency
squared
• Leading term in asymptotic
bias
T = 1024
Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483
Sine Tapers
• Without gaps, approximate
minimum asymptotic bias
tapers
• With gaps, reorthogonalize
T = 1024.
Fodor&Stark, 2000. IEEE Trans. Sig. Proc. 48, 3472-3483.
Optimization Problems
• Prolate and minimum asymptotic bias tapers are top
eigenfunctions of large eigenvalue problems
• The problems have special structure; can be solved
efficiently (top 6 tapers for T=103,680 in < 1day)
• Sine tapers very cheap to compute
Sample Concentration of Tapers
T=1024, w = 0.004
Order
Prolate
Prolate w/ gaps
Projected prolate
0
0.9999
0.9980
0.8126
1
0.9999
0.9977
0.9704
2
0.9999
0.9785
0.9317
3
0.9999
0.9753
0.9722
4
0.9999
0.9609
0.9550
Fodor & Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483
Sample Asymptotic Bias of Tapers, T = 1024
Order
Minimum Asympt Bias
Min Bias w/ gaps
Projected Min Bias
0
0.00021
0.00437
0.73516
1
0.00090
0.02265
0.43173
2
0.00212
0.13854
0.63884
3
0.00382
0.23002
0.55921
4
0.00594
0.27385
0.45409
Fodor & Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483
Multitaper Estimation
• Top several eigenfunctions have eigenvalues close to 1.
• Eigenvalues drop to zero, abruptly for no-gap
• Estimates using orthogonal tapers are asymptotically
independent (mild conditions)
• Averaging spectrum estimates from several “good” tapers can
decrease variance without increasing bias much.
• Get rank K quadratic estimator.
Multitaper Procedure
• Compute K orthogonal tapers, each with good
concentration
• Multiply data by each taper in turn
• Compute periodogram of each product
• Average the periodograms
Special case: break data into segments
Cheapest is Fine
For simulated and real helioseismic time series of
length T=103,680, no discernable systematic
difference among 12-taper multitaper estimates using
the three families of tapers.
Use re-orthogonalized sine tapers because they are
much cheaper to compute, for each gap pattern in each
time series.
Multitaper Simulation
• Can combine with
segmenting to decrease
dependence
• Prettier than periodogram;
less leakage.
• Better, too?
T=103,680. Truth in grey. Left panels: periodogram.
Right panels: 3-segment 4-taper gapped sine taper estimate. Fodor&Stark, 2000.
Multitaper: SOHO Data
• Easier to identify mode
parameters from multitaper
spectrum
• Maximum likelihood more
stable; can identify 20% to
60% more modes (Komm et al.,
1999. Ap.J., 519, 407-421)
SOHO l=85, m=0. T = 103, 680.
Periodogram (left) and 3-segment 4 sine taper estimate
Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483
Error bars: Confidence Level in Simulation
Method
% Coverage
Method
% Coverage
Parametric chi-square
72
Blockwise bootstrap
56
Simulation from Estimate
82
Bootstrap pivot percentile
71
Jackknife
75
Pre-pivot bootstrap
78
Bootstrap normal
77
Iterated pre-pivot bootstrap
91
Bootstrap-t
78
Bootstrap percentile of percentiles
90
Bootstrap percentile
69
Bootstrap pivot percentile of percentiles
96
1,000 realizations of simulated normal mode data. 95% target confidence level.
Fodor&Stark, 2000. IEEE Trans. Sig. Proc., 48, 3472-3483