Lecture 6
Resolutionand
Generalized Inverses
SyllabusLecture 01 Describing Inverse ProblemsLecture 02 Probability and Measurement Error, Part 1Lecture 03 Probability and Measurement Error, Part 2 Lecture 04 The L2 Norm and Simple Least SquaresLecture 05 A Priori Information and Weighted Least SquaredLecture 06 Resolution and Generalized InversesLecture 07 Backus-Gilbert Inverse and the Trade Off of Resolution and VarianceLecture 08 The Principle of Maximum LikelihoodLecture 09 Inexact TheoriesLecture 10 Nonuniqueness and Localized AveragesLecture 11 Vector Spaces and Singular Value DecompositionLecture 12 Equality and Inequality ConstraintsLecture 13 L1 , L∞ Norm Problems and Linear ProgrammingLecture 14 Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15 Nonlinear Problems: Newton’s Method Lecture 16 Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals Lecture 17 Factor AnalysisLecture 18 Varimax Factors, Empircal Orthogonal FunctionsLecture 19 Backus-Gilbert Theory for Continuous Problems; Radon’s ProblemLecture 20 Linear Operators and Their AdjointsLecture 21 Fréchet DerivativesLecture 22 Exemplary Inverse Problems, incl. Filter DesignLecture 23 Exemplary Inverse Problems, incl. Earthquake LocationLecture 24 Exemplary Inverse Problems, incl. Vibrational Problems
Purpose of the Lecture
Introduce the idea of a Generalized Inverse,the Data and Model Resolution Matrices
and the Unit Covariance Matrix
Quantify the spread of resolution and the size of the covariance
Use the maximization of resolution and/or covariance as the guiding principle for solving inverse problems
Part 1
The Generalized Inverse,
the Data and Model Resolution Matrices
and the Unit Covariance Matrix
all of the solutions
of the formmest = Md + v
mest = Md + vlet’s focus on
this matrix
mest = G-gd + vrename it the “generalized
inverse”and use the symbol G-g
(let’s ignore the vector v for a moment)
Generalized Inverse G-goperates on the data to give an estimate of the
model parametersifdpre = Gmest
thenmest = G-gdobs
Generalized Inverse G-gif dpre = Gmest then mest = G-gdobs
sort of looks like a matrix inverseexcept
M⨉N, not squareandGG-g≠I and G-gG≠I
so actuallythe generalized inverse is not a
matrix inverse at all
dpre = Gmest and mest = G-gdobs
dpre = Ndobs with N = GG-g “data resolution matrix”
plug one equation into the other
Data Resolution Matrix, N
How much does diobs contribute to its own
prediction?
dpre = Ndobs
ifN=I
dipre = di
obs
dpre = dobs
diobs completely controls its own prediction
dpre dobs=
(A)
The closer N is to I, the more diobs controls its own
prediction
0 5 100
5
10
15
z
d
0 5 100
5
10
15
z
d
straight line problem
= =
i i
j
j
dpre = N dobs
only the data at the ends control their own prediction
dobs = Gmtrue and mest = G-gdobs
mest = Rmtrue with R = G-gG “model resolution matrix”
plug one equation into the other
Model Resolution Matrix, R
How much does mitrue contribute to its own estimated value?
mest = Rmtrue
ifR=I
miest = mitrue
mest = mtrue
miest reflects mitrue only
else ifR≠Imiest =
… + Ri,i-1mi-1true + Ri,imitrue + Ri,i+1mi+1true+ …
miest is a weighted averageof all the elements of mtrue
mest mtrue=
The closer R is to I, the more miest reflects only mitrue
Discrete version ofLaplace Transform
large c: d is “shallow” average of m(z)small c: d is “deep” average of m(z)
e-chiz
z
m(z)z
z
⨉
⨉e-cloz dlo
dhi
integrate
integrate
= =
i i
j
j
mest = R mtrue
the shallowest model parameters are “best resolved”
Covariance associated with the Generalized Inverse
“unit covariance matrix”divide by σ2 to remove effect of the
overall magnitude of the measurement error
unit covariance for straight line problem
model parameters uncorrelated when this term zerohappens when data are centered about the origin
Part 2
The spread of resolution and the size of the covariance
a resolution matrix has small spread if only its main diagonal has
large elements
it is close to the identity matrix
“Dirichlet” Spread Functions
a unit covariance matrix has small size if its diagonal elements are small
error in the data corresponds to only small error in the model parameters
(ignore correlations)
Part 3
minimization ofspread of resolution
and/orsize of covariance
as the guiding principlefor creating a generalized inverse
over-determined case
note that forsimple least squaresG-g = [GTG]-1GTmodel resolutionR=G-gG = [GTG]-1GTG=I
always the identify matrix
suggests that we try to minimize the spread of the data resolution matrix, N
find G-g that minimizes spread(N)
spread of the k-th row of N
now compute
first term
second term
third term is zero
putting it all together
which is justsimple least squaresG-g = [GTG]-1GT
the simple least squares solutionminimizes the spread of data resolution
and has zero spread of the model resolution
under-determined case
note that forminimum length solutionG-g = GT [GGT]-1
data resolutionN=GG-g = G GT [GGT]-1 =Ialways the identify matrix
suggests that we try to minimize the spread of the model resolution matrix, R
find G-g that minimizes spread(R)
minimization leads to
[GGT]G-g = GT
which is justminimum length solutionG-g = GT [GGT]-1
the minimum length solutionminimizes the spread of model resolution
and has zero spread of the data resolution
general case
leads to
a Sylvester Equation, so explicit solution in terms of matrices
leads to
general case
1
special case #1
0 ε2
[GTG+ε2I]G-g=GTG-g=[GTG+ε2I]-1GT
I
damped least squares
0
special case #2
1 ε2
G-g[GGT+ε2I] =GTG-g=GT [GGT+ε2I]-1
I
damped minimum length
so
no new solutions have arisen …
… just a reinterpretation of previously-derived solutions
reinterpretation
instead of solving for estimates of the model parameters
We are solving for estimates of weighted averages of the model parameters,
where the weights are given by the model resolution matrix
criticism of Direchlet spread() functions
when m represents m(x)is that they don’t capture the sense of
“being localized” very well
These two rows of the model resolution matrix have the same spread …
Rij Rijindex, j index, ji i… but the left case is better “localized”
we will take up this issue in the next lecture
Top Related