Basi Ideas and Theory
Inverse Problems
Conlusions
Inverse Methods For Time Series
Thomas Shores
Department of Mathematis
University of Nebraska
April 27, 2006
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The suspets: Freshwater opepod, freshwater turtle, female yst
nematode, pea aphid.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Outline
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Outline
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Referenes
1
2
3
4
5
6
7
R. Aster, B. Borhers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005.
H. Caswell, Matrix Population Models, 2nd Ed., Sinaur Assoiates,
Sunderland, MA, 2001.
K. Gross, B. Craig and W. Huthison, Bayesian estimation of a
demographi matrix model from stage-frequeny data, Eology
83(12), 32853298, 2002.
J. Mithell, Population eology and life histories of the freshwater
turtles Chrysemys Pita and Sternotherus Odoratus in an urban
lake, Herpetologial Monographs (2), 4061, 1989.
S Twombly, Comparative demography and population dynamis of
two oexisting opepods in a Venezuelan oodplain lake, Limnol.
Oeanogr. 39(2), 234247, 1994.
S Twombly and C. Burns, Exuvium analysis: A nondestrutive
method of analyzing opepod growth and development, Limnol.
Oeanogr. 41(6), 13241329, 1996.
S. Wood, Inverse problems and strutured-population dynamis,
Strutured-population Models in Marine, Terrestrial and Freshwater
Systems (Ed. S. Tuljapurkar and J. Caswell), Chapman and Hall,
New York, 555586, 1997.
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
States, Age and Stage
About States:
Assume at any one time members of a time varying population
s mutually exlusive states, indiated
i , whene population vetor is
are lassied into one of
by the index
n(
t ) = (n1 (t ) , . . . , ns (t )) ≡ [n1 (t ) , . . . , ns (t )]T
All individuals experiene the same environment.
The eets of the population on the environment an be
written as the sum of the ontributions of the individuals.
Candidates: age, size, instars, genders, geographial loales
(pathes) and and meaningful ombination of these states.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
States, Age and Stage
About States:
Assume at any one time members of a time varying population
s mutually exlusive states, indiated
i , whene population vetor is
are lassied into one of
by the index
n(
t ) = (n1 (t ) , . . . , ns (t )) ≡ [n1 (t ) , . . . , ns (t )]T
All individuals experiene the same environment.
The eets of the population on the environment an be
written as the sum of the ontributions of the individuals.
Candidates: age, size, instars, genders, geographial loales
(pathes) and and meaningful ombination of these states.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
States, Age and Stage
About States:
Assume at any one time members of a time varying population
s mutually exlusive states, indiated
i , whene population vetor is
are lassied into one of
by the index
n(
t ) = (n1 (t ) , . . . , ns (t )) ≡ [n1 (t ) , . . . , ns (t )]T
All individuals experiene the same environment.
The eets of the population on the environment an be
written as the sum of the ontributions of the individuals.
Candidates: age, size, instars, genders, geographial loales
(pathes) and and meaningful ombination of these states.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
States, Age and Stage
About States:
Assume at any one time members of a time varying population
s mutually exlusive states, indiated
i , whene population vetor is
are lassied into one of
by the index
n(
t ) = (n1 (t ) , . . . , ns (t )) ≡ [n1 (t ) , . . . , ns (t )]T
All individuals experiene the same environment.
The eets of the population on the environment an be
written as the sum of the ontributions of the individuals.
Candidates: age, size, instars, genders, geographial loales
(pathes) and and meaningful ombination of these states.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
States, Age and Stage
About States:
Assume at any one time members of a time varying population
s mutually exlusive states, indiated
i , whene population vetor is
are lassied into one of
by the index
n(
t ) = (n1 (t ) , . . . , ns (t )) ≡ [n1 (t ) , . . . , ns (t )]T
All individuals experiene the same environment.
The eets of the population on the environment an be
written as the sum of the ontributions of the individuals.
Candidates: age, size, instars, genders, geographial loales
(pathes) and and meaningful ombination of these states.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Linear Form of Classi Examples
Projetion Model:
t + 1) = At n (t )
n(
where
At = [ai ,j ] is s × s projetion matrix.
Note: oeients ould vary with time. If not, this is a
stationary (or time-invariant) model and we simply write
At = A.
Coeients ould even be non-loal: e.g., birth rates ould be
dependent on a arrying apaity of environment. Ditto other
forms of reruitment.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Linear Form of Classi Examples
Projetion Model:
t + 1) = At n (t )
n(
where
At = [ai ,j ] is s × s projetion matrix.
Note: oeients ould vary with time. If not, this is a
stationary (or time-invariant) model and we simply write
At = A.
Coeients ould even be non-loal: e.g., birth rates ould be
dependent on a arrying apaity of environment. Ditto other
forms of reruitment.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Linear Form of Classi Examples
Projetion Model:
t + 1) = At n (t )
n(
where
At = [ai ,j ] is s × s projetion matrix.
Note: oeients ould vary with time. If not, this is a
stationary (or time-invariant) model and we simply write
At = A.
Coeients ould even be non-loal: e.g., birth rates ould be
dependent on a arrying apaity of environment. Ditto other
forms of reruitment.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Leslie Model (~1945)
Population is divided into disrete age groups, resulting in
projetion matrix




A=



Here
Fi
F1 F2
P1 0
0
P2
Fs −1 Fs
···
···
..
.
..
.
0
.
.
.
0
Ps −1
0
0




.



is the per-apita fertility of age lass
survival rate of age lass
j.
i
and
≤ Pj ≤ 1.
Clearly 0
Pj
is the
Linearity or stationarity not required.
Example:

A=
e −bN
0
e −bN
.3
0
0
0
0.5
0
5

,
where
N = n 1 + n2 .
Leslie Model (~1945)
Population is divided into disrete age groups, resulting in
projetion matrix




A=



Here
Fi
F1 F2
P1 0
0
P2
Fs −1 Fs
···
···
..
.
..
.
0
.
.
.
0
Ps −1
0
0




.



is the per-apita fertility of age lass
survival rate of age lass
j.
i
and
≤ Pj ≤ 1.
Clearly 0
Pj
is the
Linearity or stationarity not required.
Example:

A=
e −bN
0
e −bN
.3
0
0
0
0.5
0
5

,
where
N = n 1 + n2 .
Leslie Model (~1945)
Population is divided into disrete age groups, resulting in
projetion matrix




A=



Here
Fi
F1 F2
P1 0
0
P2
Fs −1 Fs
···
···
..
.
..
.
0
.
.
.
0
Ps −1
0
0




.



is the per-apita fertility of age lass
survival rate of age lass
j.
i
and
≤ Pj ≤ 1.
Clearly 0
Pj
is the
Linearity or stationarity not required.
Example:

A=
e −bN
0
e −bN
.3
0
0
0
0.5
0
5

,
where
N = n 1 + n2 .
Leslie Model (~1945)
Population is divided into disrete age groups, resulting in
projetion matrix




A=



Here
Fi
F1 F2
P1 0
0
P2
Fs −1 Fs
···
···
..
.
..
.
0
.
.
.
0
Ps −1
0
0




.



is the per-apita fertility of age lass
survival rate of age lass
j.
i
and
≤ Pj ≤ 1.
Clearly 0
Pj
is the
Linearity or stationarity not required.
Example:

A=
e −bN
0
e −bN
.3
0
0
0
0.5
0
5

,
where
N = n 1 + n2 .
Lefkovith Model (~1962)
Population is divided into disrete stages, resulting in a very general
projetion matrix
A = [ai ,j ]s ,s .
Here one hooses a projetion interval
(t , t + 1)
representing
the time states of the model.
Linearity or stationarity not required.
ai ,j represents a rate (or probability) of passage
from stage i to stage j . (Hene ai ,j ≥ 0.)
The matrix A is equivalent to a (direted) life yle graph for
The entry
the population.
Lefkovith Model (~1962)
Population is divided into disrete stages, resulting in a very general
projetion matrix
A = [ai ,j ]s ,s .
Here one hooses a projetion interval
(t , t + 1)
representing
the time states of the model.
Linearity or stationarity not required.
ai ,j represents a rate (or probability) of passage
from stage i to stage j . (Hene ai ,j ≥ 0.)
The matrix A is equivalent to a (direted) life yle graph for
The entry
the population.
Lefkovith Model (~1962)
Population is divided into disrete stages, resulting in a very general
projetion matrix
A = [ai ,j ]s ,s .
Here one hooses a projetion interval
(t , t + 1)
representing
the time states of the model.
Linearity or stationarity not required.
ai ,j represents a rate (or probability) of passage
from stage i to stage j . (Hene ai ,j ≥ 0.)
The matrix A is equivalent to a (direted) life yle graph for
The entry
the population.
Lefkovith Model (~1962)
Population is divided into disrete stages, resulting in a very general
projetion matrix
A = [ai ,j ]s ,s .
Here one hooses a projetion interval
(t , t + 1)
representing
the time states of the model.
Linearity or stationarity not required.
ai ,j represents a rate (or probability) of passage
from stage i to stage j . (Hene ai ,j ≥ 0.)
The matrix A is equivalent to a (direted) life yle graph for
The entry
the population.
Lefkovith Model (~1962)
Population is divided into disrete stages, resulting in a very general
projetion matrix
A = [ai ,j ]s ,s .
Here one hooses a projetion interval
(t , t + 1)
representing
the time states of the model.
Linearity or stationarity not required.
ai ,j represents a rate (or probability) of passage
from stage i to stage j . (Hene ai ,j ≥ 0.)
The matrix A is equivalent to a (direted) life yle graph for
The entry
the population.
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Tensor Notation
Tensor (Kroneker, diret) Produt:
Given
m × n matrix A = [ai ,j ] and p × q matrix B = [bi ,j ], the
A and B is the mp × nq blok matrix
tensor produt of

A⊗B =



a1,1 B a1,2 B
a2,1 B a2,2 B
.
.
.
···
···
..
am,1B am,2B
.
···
a1,n B
a2,n B
.
.
.
am , 1 B



.

There are lots of algebra laws, e.g.,
A ⊗ (β B + γ C ) = β A ⊗ B + γ A ⊗ C and interesting
A ⊗ B are just the produts of
eigenvalues of A and B , et., et., that we won't need.
(
properties, e.g., eigenvalues of
Matlab knows tensors:
C = kron(A,B)
does the trik.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Tensor Notation
Tensor (Kroneker, diret) Produt:
Given
m × n matrix A = [ai ,j ] and p × q matrix B = [bi ,j ], the
A and B is the mp × nq blok matrix
tensor produt of

A⊗B =



a1,1 B a1,2 B
a2,1 B a2,2 B
.
.
.
···
···
..
am,1B am,2B
.
···
a1,n B
a2,n B
.
.
.
am , 1 B



.

There are lots of algebra laws, e.g.,
A ⊗ (β B + γ C ) = β A ⊗ B + γ A ⊗ C and interesting
A ⊗ B are just the produts of
eigenvalues of A and B , et., et., that we won't need.
(
properties, e.g., eigenvalues of
Matlab knows tensors:
C = kron(A,B)
does the trik.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
The Lefkowitz Model
Matrix Theory
Tensor Notation
Tensor (Kroneker, diret) Produt:
Given
m × n matrix A = [ai ,j ] and p × q matrix B = [bi ,j ], the
A and B is the mp × nq blok matrix
tensor produt of

A⊗B =



a1,1 B a1,2 B
a2,1 B a2,2 B
.
.
.
···
···
..
am,1B am,2B
.
···
a1,n B
a2,n B
.
.
.
am , 1 B



.

There are lots of algebra laws, e.g.,
A ⊗ (β B + γ C ) = β A ⊗ B + γ A ⊗ C and interesting
A ⊗ B are just the produts of
eigenvalues of A and B , et., et., that we won't need.
(
properties, e.g., eigenvalues of
Matlab knows tensors:
C = kron(A,B)
does the trik.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Tensors as Bookkeeper
Ve or Co(onatenate) Notion:
Given
m × n matrix A = [ai ,j ] = [a1 , a2 , . . . , an ] as a row of
olumns,

ve (
A) = 



a1
a2
.
.
.
an



.

There are more algebra laws, e.g.,
A + β B ) = α ve (A) + β ve (B ).
ve (α
Key Bookkeeping Property:
AXB ) = B T ⊗ A
ve (
Matlab knows ve. The ommand
ve (
X).
x = ve(X)
does the trik.
Tensors as Bookkeeper
Ve or Co(onatenate) Notion:
Given
m × n matrix A = [ai ,j ] = [a1 , a2 , . . . , an ] as a row of
olumns,

ve (
A) = 



a1
a2
.
.
.
an



.

There are more algebra laws, e.g.,
A + β B ) = α ve (A) + β ve (B ).
ve (α
Key Bookkeeping Property:
AXB ) = B T ⊗ A
ve (
Matlab knows ve. The ommand
ve (
X).
x = ve(X)
does the trik.
Tensors as Bookkeeper
Ve or Co(onatenate) Notion:
Given
m × n matrix A = [ai ,j ] = [a1 , a2 , . . . , an ] as a row of
olumns,

ve (
A) = 



a1
a2
.
.
.
an



.

There are more algebra laws, e.g.,
A + β B ) = α ve (A) + β ve (B ).
ve (α
Key Bookkeeping Property:
AXB ) = B T ⊗ A
ve (
Matlab knows ve. The ommand
ve (
X).
x = ve(X)
does the trik.
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Formulation
The Problems:
Forward (Diret) Problem: Given a question, nd the answer.
A and present system state
t ), nd the next state of the system n (t + 1). Solution:
n (t + 1) = An (t ).
E.g., given a projetion matrix
n(
Inverse Problem: Given an answer, nd the question. E.g.,
A and present system state n (t ),
nd the previous state of the system n (t − 1). Solution:
−1
n (t − 1) = A n (t ) (maybe!)
given a projetion matrix
Parameter Identiation: a speial lass of inverse problems
that nds parameters of a model, e.g., given many system
states n (
ti ), nd the projetion matrix.
This is tougher.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Formulation
The Problems:
Forward (Diret) Problem: Given a question, nd the answer.
A and present system state
t ), nd the next state of the system n (t + 1). Solution:
n (t + 1) = An (t ).
E.g., given a projetion matrix
n(
Inverse Problem: Given an answer, nd the question. E.g.,
A and present system state n (t ),
nd the previous state of the system n (t − 1). Solution:
−1
n (t − 1) = A n (t ) (maybe!)
given a projetion matrix
Parameter Identiation: a speial lass of inverse problems
that nds parameters of a model, e.g., given many system
states n (
ti ), nd the projetion matrix.
This is tougher.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Formulation
The Problems:
Forward (Diret) Problem: Given a question, nd the answer.
A and present system state
t ), nd the next state of the system n (t + 1). Solution:
n (t + 1) = An (t ).
E.g., given a projetion matrix
n(
Inverse Problem: Given an answer, nd the question. E.g.,
A and present system state n (t ),
nd the previous state of the system n (t − 1). Solution:
−1
n (t − 1) = A n (t ) (maybe!)
given a projetion matrix
Parameter Identiation: a speial lass of inverse problems
that nds parameters of a model, e.g., given many system
states n (
ti ), nd the projetion matrix.
This is tougher.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Formulation
The Problems:
Forward (Diret) Problem: Given a question, nd the answer.
A and present system state
t ), nd the next state of the system n (t + 1). Solution:
n (t + 1) = An (t ).
E.g., given a projetion matrix
n(
Inverse Problem: Given an answer, nd the question. E.g.,
A and present system state n (t ),
nd the previous state of the system n (t − 1). Solution:
−1
n (t − 1) = A n (t ) (maybe!)
given a projetion matrix
Parameter Identiation: a speial lass of inverse problems
that nds parameters of a model, e.g., given many system
states n (
ti ), nd the projetion matrix.
This is tougher.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Nature of Inverse Problems
Question:
Given a two-way proess with answers at both ends, what makes
one diretion diret and the other inverse?
Forward problems are generally well-posed, that is, have a
solution, it is unique and it varies ontinuously with the
parameters of the problem
Inverse problems are generally ill-posed, i.e., not well-posed,
and all three possible failings our in the simple inverse
problem of solving
Ax = b for the unknown x , given A, b.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Nature of Inverse Problems
Question:
Given a two-way proess with answers at both ends, what makes
one diretion diret and the other inverse?
Forward problems are generally well-posed, that is, have a
solution, it is unique and it varies ontinuously with the
parameters of the problem
Inverse problems are generally ill-posed, i.e., not well-posed,
and all three possible failings our in the simple inverse
problem of solving
Ax = b for the unknown x , given A, b.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Nature of Inverse Problems
Question:
Given a two-way proess with answers at both ends, what makes
one diretion diret and the other inverse?
Forward problems are generally well-posed, that is, have a
solution, it is unique and it varies ontinuously with the
parameters of the problem
Inverse problems are generally ill-posed, i.e., not well-posed,
and all three possible failings our in the simple inverse
problem of solving
Ax = b for the unknown x , given A, b.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Generi Example
s×s
Postulate
projetion matrix
A for a stage strutured
population, together with data (possibly repliated and averaged)
for the states n (1) , n (2) , . . . , n (
of
A:
s + 1).
We have prior knowledge
all entries are nonnegative and ertain entries are zero.
Frame the problem as follows:
An (k ) = n (k + 1), k = 1, . . . , s .
Set M = [n (1) , n (2) , . . . , n (s )] and
P = [n (2) , n (2) , . . . , n (s + 1)].
Reast problem as AM = P = Is AN .
Tensor bookkeeping:
Is AM ) = M T ⊗ Is
A) = ve (P ) ≡ d.
T
Delete zero variables from ve (A) and olumns of M ⊗ Is .
Rename resulting vetors, matrix as m,G and rhs d. System
takes form G m = d, with G p × q , typially p > q .
ve (
ve (
A Working Example
Taken from Caswell's text, in turn from a referened paper that I
an't nd by Kaplan and Caswell-Chen: the sugarbeet nematode
Heterodera shahtii has ve stages (eggs, juvenile J2, J3, J4 and
adult.) Following data is density of nematodes (per 60 of soil) for
stages J2, J3+J4, adult, averaged over four repliates, measured
every two days:
t=0
t=1
t=2
t=3
t =4
t=5
5.32
0.33
2.41
2.06
1.70
3.16
24.84
18.16
17.14
3.25
2.08
11.23
115.50
167.16
159.25
112.87
132.62
149.62
This population leads to a population projetion matrix

A=
P1 0 F 3
G1 P2
G2 P3


Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Least Squares (?)
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
What we do:
Of ourse, with muh data we will almost ertainly have an
inonsistent system
G m = d.
The problem is therefore ill-posed.
Reouh the (probably) ill-posed problem
G m = d as the
optimization problem
2
m kG m − dk2 .
min
This is equivalent to solving the normal equations
GTGm = GTd
whih, ASSUMING
G
has full olumn rank, has a unique
∗
solution m .
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
What we do:
Of ourse, with muh data we will almost ertainly have an
inonsistent system
G m = d.
The problem is therefore ill-posed.
Reouh the (probably) ill-posed problem
G m = d as the
optimization problem
2
m kG m − dk2 .
min
This is equivalent to solving the normal equations
GTGm = GTd
whih, ASSUMING
G
has full olumn rank, has a unique
∗
solution m .
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
What we do:
Of ourse, with muh data we will almost ertainly have an
inonsistent system
G m = d.
The problem is therefore ill-posed.
Reouh the (probably) ill-posed problem
G m = d as the
optimization problem
2
m kG m − dk2 .
min
This is equivalent to solving the normal equations
GTGm = GTd
whih, ASSUMING
G
has full olumn rank, has a unique
∗
solution m .
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Least Squares and Working Example
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
More Least Squares:
Least squares has many pleasant statistial properties, e.g., if
the data errors are i.i.d. normal r.v.'s, then entries of m
normally distributed and
G mtrue = dtrue .
E [m∗] = mtrue , where
Given that the variane of data error is
σ2 ,
∗ are
one an form the
hi-square statisti
χ2obs = kG m − dk22 /σ 2
and this turns out to be a r.v. with a
m − n (row number of G
χ2
distribution with
minus olumn number) degrees of
freedom.
The probability of obtaining a
the observed number is the
χ2
value as large or larger than
p-value p =
is a uniformly distributed r.v.
R∞
f
χ2obs χ2
(x ) dx
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
whih
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
More Least Squares:
Least squares has many pleasant statistial properties, e.g., if
the data errors are i.i.d. normal r.v.'s, then entries of m
normally distributed and
G mtrue = dtrue .
E [m∗] = mtrue , where
Given that the variane of data error is
σ2 ,
∗ are
one an form the
hi-square statisti
χ2obs = kG m − dk22 /σ 2
and this turns out to be a r.v. with a
m − n (row number of G
χ2
distribution with
minus olumn number) degrees of
freedom.
The probability of obtaining a
the observed number is the
χ2
value as large or larger than
p-value p =
is a uniformly distributed r.v.
R∞
f
χ2obs χ2
(x ) dx
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
whih
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
More Least Squares:
Least squares has many pleasant statistial properties, e.g., if
the data errors are i.i.d. normal r.v.'s, then entries of m
normally distributed and
G mtrue = dtrue .
E [m∗] = mtrue , where
Given that the variane of data error is
σ2 ,
∗ are
one an form the
hi-square statisti
χ2obs = kG m − dk22 /σ 2
and this turns out to be a r.v. with a
m − n (row number of G
χ2
distribution with
minus olumn number) degrees of
freedom.
The probability of obtaining a
the observed number is the
χ2
value as large or larger than
p-value p =
is a uniformly distributed r.v.
R∞
f
χ2obs χ2
(x ) dx
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
whih
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
More Least Squares:
Least squares has many pleasant statistial properties, e.g., if
the data errors are i.i.d. normal r.v.'s, then entries of m
normally distributed and
G mtrue = dtrue .
E [m∗] = mtrue , where
Given that the variane of data error is
σ2 ,
∗ are
one an form the
hi-square statisti
χ2obs = kG m − dk22 /σ 2
and this turns out to be a r.v. with a
m − n (row number of G
χ2
distribution with
minus olumn number) degrees of
freedom.
The probability of obtaining a
the observed number is the
χ2
value as large or larger than
p-value p =
is a uniformly distributed r.v.
R∞
f
χ2obs χ2
(x ) dx
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
whih
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Wood's Quadrati Programming Method
Why ast aspersions on least squares? Inter alia, the good features
don't apply: take a look at
G.
Moreover, it may result in negative
entries in the projetion matrix!
Fix the Negativity:
Reast the problem as onstrained optimization problem:
2
m kG m − dk2
min
subjet to onstraints
onditions like m
≥0
C m ≥ b where the onstaints ensure
and Pi + Gi ≤ 1.
Solving a least squares problem only by adding onstraints is
one kind of regularization strategy. Sometimes it works well,
but there are examples where it's awful.
Now let's run the sript WorkingExample.m.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Wood's Quadrati Programming Method
Why ast aspersions on least squares? Inter alia, the good features
don't apply: take a look at
G.
Moreover, it may result in negative
entries in the projetion matrix!
Fix the Negativity:
Reast the problem as onstrained optimization problem:
2
m kG m − dk2
min
subjet to onstraints
onditions like m
≥0
C m ≥ b where the onstaints ensure
and Pi + Gi ≤ 1.
Solving a least squares problem only by adding onstraints is
one kind of regularization strategy. Sometimes it works well,
but there are examples where it's awful.
Now let's run the sript WorkingExample.m.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Wood's Quadrati Programming Method
Why ast aspersions on least squares? Inter alia, the good features
don't apply: take a look at
G.
Moreover, it may result in negative
entries in the projetion matrix!
Fix the Negativity:
Reast the problem as onstrained optimization problem:
2
m kG m − dk2
min
subjet to onstraints
onditions like m
≥0
C m ≥ b where the onstaints ensure
and Pi + Gi ≤ 1.
Solving a least squares problem only by adding onstraints is
one kind of regularization strategy. Sometimes it works well,
but there are examples where it's awful.
Now let's run the sript WorkingExample.m.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Regularized Least Squares
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Suppose that our problem had been severely poorly onditioned or
even rank deient.
What to do?
m kG m − dk22 gets us into trouble, with
Now even the problem min
or without onstraints.
Tikhonov regularization: add a regularizing term that makes
the problem well posed:
min
Here
L=I
α
kG m − dk22 + α2 kL (m − m0 )k22 .
has to be hosen and
L is a smoothing matrix like
(zeroth order Tikhonov regularization) or matries whih
mimi disretized rst or seond derivatives (higher order
regularization. There's a Bayesian avor here, esp. if m0
6= 0.)
Note: statistiians are somewhat wary of this regularization as
that it introdues bias into model estimates.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Suppose that our problem had been severely poorly onditioned or
even rank deient.
What to do?
m kG m − dk22 gets us into trouble, with
Now even the problem min
or without onstraints.
Tikhonov regularization: add a regularizing term that makes
the problem well posed:
min
Here
L=I
α
kG m − dk22 + α2 kL (m − m0 )k22 .
has to be hosen and
L is a smoothing matrix like
(zeroth order Tikhonov regularization) or matries whih
mimi disretized rst or seond derivatives (higher order
regularization. There's a Bayesian avor here, esp. if m0
6= 0.)
Note: statistiians are somewhat wary of this regularization as
that it introdues bias into model estimates.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Suppose that our problem had been severely poorly onditioned or
even rank deient.
What to do?
m kG m − dk22 gets us into trouble, with
Now even the problem min
or without onstraints.
Tikhonov regularization: add a regularizing term that makes
the problem well posed:
min
Here
L=I
α
kG m − dk22 + α2 kL (m − m0 )k22 .
has to be hosen and
L is a smoothing matrix like
(zeroth order Tikhonov regularization) or matries whih
mimi disretized rst or seond derivatives (higher order
regularization. There's a Bayesian avor here, esp. if m0
6= 0.)
Note: statistiians are somewhat wary of this regularization as
that it introdues bias into model estimates.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Formulation and Examples
Some Methodologies
Suppose that our problem had been severely poorly onditioned or
even rank deient.
What to do?
m kG m − dk22 gets us into trouble, with
Now even the problem min
or without onstraints.
Tikhonov regularization: add a regularizing term that makes
the problem well posed:
min
Here
L=I
α
kG m − dk22 + α2 kL (m − m0 )k22 .
has to be hosen and
L is a smoothing matrix like
(zeroth order Tikhonov regularization) or matries whih
mimi disretized rst or seond derivatives (higher order
regularization. There's a Bayesian avor here, esp. if m0
6= 0.)
Note: statistiians are somewhat wary of this regularization as
that it introdues bias into model estimates.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Determination of
Formulation and Examples
Some Methodologies
α
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
To hoose an
Formulation and Examples
Some Methodologies
α:
Some of the prinipal options:
The L-urve: do a loglog plot of
look for the
α
kG mα − dk22
vs
kLmαk22
and
that gives a orner value that balanes these
two terms.
(Morozov's disrepany priniple) Choose
kG mα − dk2
α
so that the mist
is the same size as the data noise
kδdk2
GCV (omes from statistial leave-one-out ross validation):
Leave out one data point and use model to predit it. Sum
these up and hoose regularization parameter
α
that minimizes
the sum of the squares of the preditive errors
V0 (α) =
1
m X
m k =1
[k ]
α,L
Gm
k
− dk
2
.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
To hoose an
Formulation and Examples
Some Methodologies
α:
Some of the prinipal options:
The L-urve: do a loglog plot of
look for the
α
kG mα − dk22
vs
kLmαk22
and
that gives a orner value that balanes these
two terms.
(Morozov's disrepany priniple) Choose
kG mα − dk2
α
so that the mist
is the same size as the data noise
kδdk2
GCV (omes from statistial leave-one-out ross validation):
Leave out one data point and use model to predit it. Sum
these up and hoose regularization parameter
α
that minimizes
the sum of the squares of the preditive errors
V0 (α) =
1
m X
m k =1
[k ]
α,L
Gm
k
− dk
2
.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
To hoose an
Formulation and Examples
Some Methodologies
α:
Some of the prinipal options:
The L-urve: do a loglog plot of
look for the
α
kG mα − dk22
vs
kLmαk22
and
that gives a orner value that balanes these
two terms.
(Morozov's disrepany priniple) Choose
kG mα − dk2
α
so that the mist
is the same size as the data noise
kδdk2
GCV (omes from statistial leave-one-out ross validation):
Leave out one data point and use model to predit it. Sum
these up and hoose regularization parameter
α
that minimizes
the sum of the squares of the preditive errors
V0 (α) =
1
m X
m k =1
[k ]
α,L
Gm
k
− dk
2
.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
To hoose an
Formulation and Examples
Some Methodologies
α:
Some of the prinipal options:
The L-urve: do a loglog plot of
look for the
α
kG mα − dk22
vs
kLmαk22
and
that gives a orner value that balanes these
two terms.
(Morozov's disrepany priniple) Choose
kG mα − dk2
α
so that the mist
is the same size as the data noise
kδdk2
GCV (omes from statistial leave-one-out ross validation):
Leave out one data point and use model to predit it. Sum
these up and hoose regularization parameter
α
that minimizes
the sum of the squares of the preditive errors
V0 (α) =
1
m X
m k =1
[k ]
α,L
Gm
k
− dk
2
.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Other Regularization Methods
Total least squares: this method attempts to aount for the
error in the oeient matrix as well as right hand side. If
onstraints are not an issue, this method is preferable to least
squares and has some good statistial properties more
favorable than ordinary least squares.
Maximum likelihood approah: introdue a stohasti
omponent into the model
t + 1) = exp (D (t )) An (t )
n(
where
D (t ) is a diagonal matrix with a multivariate normal
distribution of mean zero and ovariane matrix
Σ.
Let p be
the vetor of parameters to be estimated and use the observed
data to obtain maximum likelihood estimates of p and
Σ.
And, of ourse, there are innitely many other statistial
methods for point estimates of individual parameters....
Other Regularization Methods
Total least squares: this method attempts to aount for the
error in the oeient matrix as well as right hand side. If
onstraints are not an issue, this method is preferable to least
squares and has some good statistial properties more
favorable than ordinary least squares.
Maximum likelihood approah: introdue a stohasti
omponent into the model
t + 1) = exp (D (t )) An (t )
n(
where
D (t ) is a diagonal matrix with a multivariate normal
distribution of mean zero and ovariane matrix
Σ.
Let p be
the vetor of parameters to be estimated and use the observed
data to obtain maximum likelihood estimates of p and
Σ.
And, of ourse, there are innitely many other statistial
methods for point estimates of individual parameters....
Other Regularization Methods
Total least squares: this method attempts to aount for the
error in the oeient matrix as well as right hand side. If
onstraints are not an issue, this method is preferable to least
squares and has some good statistial properties more
favorable than ordinary least squares.
Maximum likelihood approah: introdue a stohasti
omponent into the model
t + 1) = exp (D (t )) An (t )
n(
where
D (t ) is a diagonal matrix with a multivariate normal
distribution of mean zero and ovariane matrix
Σ.
Let p be
the vetor of parameters to be estimated and use the observed
data to obtain maximum likelihood estimates of p and
Σ.
And, of ourse, there are innitely many other statistial
methods for point estimates of individual parameters....
Basi Ideas and Theory
Inverse Problems
Conlusions
Summary
Inverse problems arising from parameter reovery in Lefkowitz
models are ill posed, but an be managed by tools of inverse
theory suh as least squares, Tikhonov regularization and
onstrained optimization.
There are some interesting data in the literature relating to
freshwater turtles that seem to exploit purely statistial
methods. I plan to explore Lefkovith modeling in this ontext.
Regularization tools may oer new insights, partiularly in
modeling that leads to rank deient problems.
Speially, one might try to push the envelope with a
non-stationary projetion matrix. Or nonlinear one. Or takle
unknown reprodutive rates. These will likely give problems
with worse onditioned that our working example.
The role of total least squares seems to be largely unexplored
for these problems. This warrants further investigation.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Summary
Inverse problems arising from parameter reovery in Lefkowitz
models are ill posed, but an be managed by tools of inverse
theory suh as least squares, Tikhonov regularization and
onstrained optimization.
There are some interesting data in the literature relating to
freshwater turtles that seem to exploit purely statistial
methods. I plan to explore Lefkovith modeling in this ontext.
Regularization tools may oer new insights, partiularly in
modeling that leads to rank deient problems.
Speially, one might try to push the envelope with a
non-stationary projetion matrix. Or nonlinear one. Or takle
unknown reprodutive rates. These will likely give problems
with worse onditioned that our working example.
The role of total least squares seems to be largely unexplored
for these problems. This warrants further investigation.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Summary
Inverse problems arising from parameter reovery in Lefkowitz
models are ill posed, but an be managed by tools of inverse
theory suh as least squares, Tikhonov regularization and
onstrained optimization.
There are some interesting data in the literature relating to
freshwater turtles that seem to exploit purely statistial
methods. I plan to explore Lefkovith modeling in this ontext.
Regularization tools may oer new insights, partiularly in
modeling that leads to rank deient problems.
Speially, one might try to push the envelope with a
non-stationary projetion matrix. Or nonlinear one. Or takle
unknown reprodutive rates. These will likely give problems
with worse onditioned that our working example.
The role of total least squares seems to be largely unexplored
for these problems. This warrants further investigation.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series
Basi Ideas and Theory
Inverse Problems
Conlusions
Summary
Inverse problems arising from parameter reovery in Lefkowitz
models are ill posed, but an be managed by tools of inverse
theory suh as least squares, Tikhonov regularization and
onstrained optimization.
There are some interesting data in the literature relating to
freshwater turtles that seem to exploit purely statistial
methods. I plan to explore Lefkovith modeling in this ontext.
Regularization tools may oer new insights, partiularly in
modeling that leads to rank deient problems.
Speially, one might try to push the envelope with a
non-stationary projetion matrix. Or nonlinear one. Or takle
unknown reprodutive rates. These will likely give problems
with worse onditioned that our working example.
The role of total least squares seems to be largely unexplored
for these problems. This warrants further investigation.
Thomas Shores Department of Mathematis University of Nebraska
Inverse Methods For Time Series