From help-request at octave dot org Thu Feb 10 21:32:43 2005
Subject: Abar = <w,v>, scalar product of right and left eigenvectors 	corresponding to la1 when I choose w1=1 and v1=1
From: "Henry F. Mollet" <mollet at pacbell dot net>
To: Octave_post <help at octave dot org>
Date: Thu, 10 Feb 2005 19:43:09 -0800

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What is the Abar below that I=B9m defining in 4 different ways below, in
matrix algebra or in the theory of the discrete z-transform  (analogue of
the Laplace transform for continuous functions). Perhaps 3) is best to tell
me if there is a mathematical special name for this. 4) might be good in
terms matrix algebra. Could be that the Leslie matrix is so special, more
like a sparse matrix with mostly zeros, that no name was coined for this
special case.

1) Abar =3D sum [x la^(-x) l(x) m(x)] =3D sum [x la^(-x) Fj Product (Pj)  sum
over all x =3Dages, product from j=3D1 to x-1;
Gives the mean age of the reproducing females.
2) Abar =3D <w,v>, scalar product of right and left eigenvector when I choose
w1=3D1 and v1=3D1;
3) Abar =3D - la [d(phi)/d(la)] at la =3D la1; where phi =3D phi(la) is the
characteristic function;
 if I put r =3Dln(la) (instantaneous population growth rate), then Abar =3D -
[d(phi)/d(r)] at r =3D r1.
4) AbarMatrix =3D Fstar*Nstar^2; Abar=3DAbarMatrix(1,1); (see complete script
below);
I suspect that this has been derived before by a matrix algebra
mathematician. I derived it almost by trial and error.
Henry

N.B. More explanation if needed
Note that the Leslie matrix (A =3D F + T) is a special matrix with non-zero
element only on the first row (no other restrictions) and the first
sub-diagonal (have to be <1). It can be used for species that have only one
way to produce offspring which leads to an  age-stuctured life-history tabl=
e
and applies to most animals. Everything works also for a stage-based model
where we also have diagonal elements but T(1,1) has to be zero (ok to have
F(1,1)) because we need a =B3starter=B2 age-class for the =B3w1=3D1 an v1=3D1=B2 bit to
be defined clearly. While Abar can be calculated for species with different
types of offspring (many plants) it can no longer be interpreted as the mea=
n
age of the reproducing females and is therefore not a useful quantity.

For a Leslie matrix, the =B3special =B3 scalar product <w,v> when w1=3D1 and v1 =3D
1  has an easily understood meaning and is called Abar. Abar is the mean ag=
e
of the reproducing parents at the stable age distribution. Abar can also be
calculated as Abar =3D - la1 d(phi)/d(la), where phi =3D phi (la) is the
characteristic function. If we set the characteristic function to one we ge=
t
the characteristic equation (Euler sum =3D1), whose solution  will produce
all the solution including the largest and real one, namely lambda1 (la1)

Given a Leslie matrix A =3D F + T, where
F is the discounted fertility matrix with non-zero elements on the first ro=
w
only and no restrictions on their value, and
T is a transmission matrix with non-zero elements on the first sub-diagonal
only which have to be smaller than one are the survival rates, I can show
(and others have done this before) that
Abar =3D <W,V> when I choose w1=3D1 and v1 =3D 1 to calculate the scalar product.

The Leslie matrix can be represented in a life cycle graph with a node for
each age class. Arcs back to first node, from age at first to last
reproduction represent discounted fertilities on the first row  and  arrows
between notes represent survival rates on the first subdiagonal. If the lif=
e
cycle graph is z-transformed by dividing everything by la1, then the
age-structure (=3D right eigenvector) corresponding to la1 can be read off th=
e
graph starting out with w1 =3D 1, w2 =3D P1/la1 and so forth. If the arrows are
reversed (corresponds to transposition of the A-matrix) then the
reproductive value (=3D left eigenvectors) can be read off the graph. It=B9s a
bit more complicated but we start again with v1 =3D 1 but then v2 has the mos=
t
terms and it=B9s easier to start with the last one which is v(last) =3D F(1,
last)/la1. The characteristic function can also be read off this graph by
summing over all the loops. This works also for stage-based models with
matrix elements on the diagonal but the resulting self-loops have to be
reduced first based on the signal-flow graphs methods introduced by Mason.

% Octave script for the calculation of  Abar      (A.14)
% 1. Calculation of dominant eigenvalue la1 of projection matrix A =3D F + T
lambda1 =3D max(eig(A));
% 2. Calculation of Generation Time (GT) and mu1 from the fundamental matri=
x
N
% Note that here =8CT=B9 is used for Transition Matrix T and Generation Time is
abbreviated with GT
N=3Dinv(eye(length(T))-T);                                              % N =3D
(I - T)-1 (Caswell 2001, p. 118)
R=3DF*N;            =20
% Caswell (2001, p. 126, eq. 5.64)
Ro=3Dmax(eig(R))                                                            %
Ro is dominant eigenvalue of R-matrix
GenerationTime=3Dlog(Ro)/log(lambda1)                         % GT =3D
ln(Ro)/ln(la1) =3D ln(Ro)/r
Mu1Matrix=3DF*N^2./Ro;                                                 % Afte=
r
(Cochran and Ellner 1992, Table 2)
Mu1=3DMu1Matrix(1,1)                                                     % m1
is the (1,1) element of the m1-matrix
% 3. Calculation of  Abar from new matrices F* and N*
Fstar=3Dlambda1*F;                                                         %
F* =3D l1 * F
Nstar=3D inv(max(eig(A))*eye(length(T))-T);                    % Replacing
identiy matrix I with la1*I
AbarMatrix =3D Fstar*Nstar^2;                                         % This
study
Abar=3DAbarMatrix(1,1)                                                    %
Abar is the (1,1) element of the Abar-matrix=20


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<HTML>
<HEAD>
<TITLE>Abar =3D &lt;w,v&gt;, scalar product of right and left eigenvectors co=
rresponding to la1 when I choose w1=3D1 and v1=3D1</TITLE>
</HEAD>
<BODY>
<FONT FACE=3D"Verdana">What is the Abar below that I&#8217;m defining in 4 di=
fferent ways below, in matrix algebra or in the theory of the discrete z-tra=
nsform &nbsp;(analogue of the Laplace transform for continuous functions). P=
erhaps 3) is best to tell me if there is a mathematical special name for thi=
s. 4) might be good in terms matrix algebra. Could be that the Leslie matrix=
 is so special, more like a sparse matrix with mostly zeros, that no name wa=
s coined for this special case.<BR>
<BR>
1) Abar =3D sum [x la^(-x) l(x) m(x)] =3D sum [x la^(-x) Fj Product (Pj) &nbsp;=
sum over all x =3Dages, product from j=3D1 to x-1; <BR>
Gives the mean age of the reproducing females.<BR>
2) Abar =3D &lt;w,v&gt;, scalar product of right and left eigenvector when I =
choose w1=3D1 and v1=3D1;<BR>
3) Abar =3D - la [d(phi)/d(la)] at la =3D la1; where phi =3D phi(la) is the chara=
cteristic function;<BR>
&nbsp;if I put r =3Dln(la) (instantaneous population growth rate), then Abar =
=3D - [d(phi)/d(r)] at r =3D r1.<BR>
4) AbarMatrix =3D Fstar*Nstar^2; Abar=3DAbarMatrix(1,1); (see complete script b=
elow);<BR>
I suspect that this has been derived before by a matrix algebra mathematici=
an. I derived it almost by trial and error.<BR>
Henry<BR>
<BR>
N.B. More explanation if needed<BR>
Note that the Leslie matrix (A =3D F + T) is a special matrix with non-zero e=
lement only on the first row (no other restrictions) and the first sub-diago=
nal (have to be &lt;1). It can be used for species that have only one way to=
 produce offspring which leads to an &nbsp;age-stuctured life-history table =
and applies to most animals. Everything works also for a stage-based model w=
here we also have diagonal elements but T(1,1) has to be zero (ok to have F(=
1,1)) because we need a &#8220;starter&#8221; age-class for the &#8220;w1=3D1 =
an v1=3D1&#8221; bit to be defined clearly. While Abar can be calculated for s=
pecies with different types of offspring (many plants) it can no longer be i=
nterpreted as the mean age of the reproducing females and is therefore not a=
 useful quantity.<BR>
<BR>
For a Leslie matrix, the &#8220;special &#8220; scalar product &lt;w,v&gt; =
when w1=3D1 and v1 =3D 1 &nbsp;has an easily understood meaning and is called Ab=
ar. Abar is the mean age of the reproducing parents at the stable age distri=
bution. Abar can also be calculated as Abar =3D - la1 d(phi)/d(la), where phi =
=3D phi (la) is the characteristic function. If we set the characteristic func=
tion to one we get the characteristic equation (Euler sum =3D1), whose solutio=
n &nbsp;will produce &nbsp;all the solution including the largest and real o=
ne, namely lambda1 (la1)<BR>
<BR>
Given a Leslie matrix A =3D F + T, where <BR>
F is the discounted fertility matrix with non-zero elements on the first ro=
w only and no restrictions on their value, and<BR>
T is a transmission matrix with non-zero elements on the first sub-diagonal=
 only which have to be smaller than one are the survival rates, I can show (=
and others have done this before) that<BR>
Abar =3D &lt;W,V&gt; when I choose w1=3D1 and v1 =3D 1 to calculate the scalar pr=
oduct.<BR>
<BR>
The Leslie matrix can be represented in a life cycle graph with a node for =
each age class. Arcs back to first node, from age at first to last reproduct=
ion represent discounted fertilities on the first row &nbsp;and &nbsp;arrows=
 between notes represent survival rates on the first subdiagonal. If the lif=
e cycle graph is z-transformed by dividing everything by la1, then the age-s=
tructure (=3D right eigenvector) corresponding to la1 can be read off the grap=
h starting out with w1 =3D 1, w2 =3D P1/la1 and so forth. If the arrows are reve=
rsed (corresponds to transposition of the A-matrix) then the reproductive va=
lue (=3D left eigenvectors) can be read off the graph. It&#8217;s a bit more c=
omplicated but we start again with v1 =3D 1 but then v2 has the most terms and=
 it&#8217;s easier to start with the last one which is v(last) =3D F(1, last)/=
la1. The characteristic function can also be read off this graph by summing =
over all the loops. This works also for stage-based models with matrix eleme=
nts on the diagonal but the resulting self-loops have to be reduced first ba=
sed on the signal-flow graphs methods introduced by Mason.<BR>
<BR>
% Octave script for the calculation of &nbsp;Abar &nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;(A.14)<BR>
% 1. Calculation of dominant eigenvalue la1 of projection matrix A =3D F + T<=
BR>
lambda1 =3D max(eig(A));<BR>
% 2. Calculation of Generation Time (GT) and mu1 from the fundamental matri=
x N<BR>
% Note that here &#8216;T&#8217; is used for Transition Matrix T and Genera=
tion Time is abbreviated with GT<BR>
N=3Dinv(eye(length(T))-T); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% N =3D (I=
 - T)-1 (Caswell 2001, p. 118)<BR>
R=3DF*N; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% Caswell (2001,=
 p. 126, eq. 5.64)<BR>
Ro=3Dmax(eig(R)) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% Ro is do=
minant eigenvalue of R-matrix<BR>
GenerationTime=3Dlog(Ro)/log(lambda1) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% GT =3D ln(Ro)/ln(la1) =3D ln(Ro)/r<BR>
Mu1Matrix=3DF*N^2./Ro; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;% After (Cochran and Ellner 1992, Table 2)<BR>
Mu1=3DMu1Matrix(1,1) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;% m1 is the (1,1) element of the m1-matrix <BR>
% 3. Calculation of &nbsp;Abar from new matrices F* and N*<BR>
Fstar=3Dlambda1*F; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% F* =3D l1 * F<BR>
Nstar=3D inv(max(eig(A))*eye(length(T))-T); &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;% Replacing identiy matrix I with la1*I<BR>
AbarMatrix =3D Fstar*Nstar^2; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;% This study<BR>
Abar=3DAbarMatrix(1,1) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;% Abar is the (1,1) element of the Abar-matrix <BR>
</FONT>
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