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multim2d(2) Scilab Function multim2d(2)
NAME
multim2d - multinomial 2d measure synthesis
Author: Christophe Canus
This C_LAB routine synthesizes a large range of pre-multifractal measures
related to the multinomial 2d measure (deterministic, shuffled, pertubated)
and computes linked theoretical functions (partition sum function, Reyni
exponents function, generalized dimensions, multifractal spectrum).
Usage
[varargout,[optvarargout]]=binom(bx,by,p,str,varargin,[optvarargin])
Input parameters
o bx : strictly positive real (integer) scalar Contains the abscissa
base of the multinomial.
o by : strictly positive real (integer) scalar Contains the ordonate
base of the multinomial.
o p : strictly positive real vector [by,bx] Contains the weights of
the multinomial.
o str : string Contains the type of ouput.
o varargin : variable input argument Contains the variable input argu-
ment.
o optvarargin : optional variable input arguments Contains optional
variable input arguments.
Output parameters
o varargout : variable output argument Contains the variable output
argument.
o optvarargout : optional variable output argument Contains an
optional variable output argument.
Description
Parameters
The multinomial 2d measure is completly characterized by its abscissa base
bx, ordonate base by and its weights p(i) (i=1 to bx*by). The first two
parameters bx and by must be >1. The third parameter must be a vector of
size equal to bx*by. The weights p(i) must be >0., <1. and their sum must
be =1. (the case of p(i)=1/(bx*by) corresponds to the Lebesgue measure)
(i=1 to bx*by). The fourth parameter str is a variable string used to
determine the desired type of output. There are six suffix strings ('meas'
for measure, 'cdf' for cumulative distribution function , 'pdf' for proba-
bility density function, 'part' for partition sum function, 'Reyni' for
Reyni exponent function , 'spec' for multifractal spectrum) for the deter-
ministic multinomial measure and two prefix strings for related measures
('shuf' for shuffled , 'pert' for pertubated) which can be added to the
first ones to form composed strings. For example, 'shufmeas' is for the
synthesis of a shuffled multinomial 2d pre-multifractal measure. Note that
all combinaisons of strings are not implemented yet. When a string con-
taining suffix string 'meas' is given as fourth input, a pre-multifractal
measure mu_n (first output argument) is synthesized on the bx-adic and by-
adic intervals I_nx and I_ny (second and third optional output argument) of
the unit square. In that case, the fifth input argument is a strictly posi-
tive real (integer) scalar n which contains the resolution of the pre-
multifractal measure. The size of the output real matrix mu_n is equal to
bxn*byn and the one of the output real vectors I_nx and I_ny (if used) is
equal to bxn and byn (so be aware the stack size ;-)). This option is
implemented for the deterministic ('meas'), shuffled ('shufmeas') and per-
tubated ('pertmeas') multinomial 2d measure. When a string containing pre-
fix 'shuf' is given as fourth input, the synthesis is made for a shuffled
multinomial measure. At each level of the multiplicative cascade and for
all nodes of the corresponding binary tree, the vector of weights p is
shuffled. This option is implemented only for the multinomial 2d measure
('shufmeas'). When a string containing prefix 'pert' is given as fourth
input, the synthesis is made for a pertubated multinomial measure. In that
case, the fifth input argument is a strictly positive real scalar epsilon
which contains the pertubation around weights. The weights are independant
random variables identically distributed between p(i)-epsilon and
p(i)+epsilon which must be >0., <1. (i=1 to bx*by). This option is imple-
mented only for the multinomial 2d measure ('pertmeas'). When replacing
suffix string 'meas' with suffix string 'cdf', respectively suffix string
'pdf', the cumulative distribution function F_n, respectively the probabil-
ity density function p_n, related to this pre-multifractal measure is com-
puted (first output argument). When string 'part' is given as fourth
input, the partition sum function znq of multifractal measure is computed
as sole output argument. In that case, the fifth input argument is a
strictly positive real (integer) vector vn which contains the resolutions,
and the sixth input argument is a real vector q which contains the measure
exponents. The size of the output real matrix znq is equal to
size(q)*size(vn). This option is implemented only for the multinomial 2d
measure. When string 'Reyni' is given as third input, the Reyni exponents
function tq (and the generalized dimensions Dq if used) of the multifractal
measure is computed as first output argument (and second optional output
argument if used). In that case, the fifth input argument is a real vector
q which contains the measure's exponents. The size of the output real vec-
tor tq is equal to size(q)). This option is implemented only for the multi-
nomial 2d measure. When string 'spec' is given as fourth input, the mul-
tifractal spectrum f_alpha (second output argument) is synthesized on the
Hoelder exponents alpha (first output argument). In that case, the fifth
input argument is a strictly positive real (integer) scalar N which
contains the number of Hoelder exponents. The size of both output real vec-
tors alpha and f_alpha is equal to N. This option is implemented only for
the multinomial 2d measure.
Algorithm details
For the deterministic multinomial, the pre-multifractal measure synthesis
algorithm is implemented is a iterative way (supposed to run faster than a
recursive one). For the shuffled or the pertubated multinomial, the syn-
thesis algorithm is implemented is a recursive way (to be able to pick up a
i.i.d. r.v. at each level of the multiplicative cascade and for all nodes
of the corresponding binary tree w.r.t. the given law). In the case of the
pertubated multinomial, the weights of each node are normalised by their
sum for the measure to remain conservative. Note that the shuffled multino-
mial 2d measure is not conservative.
Examples
Matlab
bx=2;
by=3;
p=[.05 .1; .15 .2; .24 .26];
n=5;
% synthesizes a pre-multifractal multinomial 2d measure
[mu_n,I_nx,I_ny]=multim2d(bx,by,p,'meas',n);
mesh(I_nx,I_ny,mu_n);
% synthesizes the cdf of a pre-multifractal shuffled multinomial 2d measure
F_n=multim2d(bx,by,p,'shufcdf',n);
mesh(I_nx,I_ny,F_n);
e=.049;
% synthesizes the pdf of a pre-multifractal pertubated multinomial 2d measure
p_n=multim2d(bx,by,p,'pertpdf',n,e);
mesh(I_nx,I_ny,p_n);
vn=[1:1:8];
q=[-5:.1:+5];
% computes the partition sum function of a multinomial 2d measure
znq=multim2d(bx,by,p,'part',vn,q);
plot(-vn*log(2),log(znq));
% computes the Reyni exponents function of a multinomial 2d measure
tq=multim2d(bx,by,p,'Reyni',q);
plot(q,tq);
N=200;
% computes the multifractal spectrum of a multinomial 2d measure
[alpha,f_alpha]=multim2d(bx,by,p,'spec',N);
plot(alpha,f_alpha);
Scilab
bx=2;
by=3;
p=[.05 .1; .15 .2; .24 .26];
n=5;
// synthesizes a pre-multifractal multinomial 2d measure
[mu_n,I_nx,I_ny]=multim2d(bx,by,p,'meas',n);
plot3d(I_nx,I_ny,mu_n);
// synthesizes the cdf of a pre-multifractal shuffled multinomial 2d measure
F_n=multim2d(bx,by,p,'shufcdf',n);
plot3d(I_nx,I_ny,F_n);
e=.049;
// synthesizes the pdf of a pre-multifractal pertubated multinomial 2d measure
p_n=multim2d(bx,by,p,'pertpdf',n,e);
plot3d(I_nx,I_ny,p_n);
xbasc();
vn=[1:1:8];
q=[-5:.1:+5];
// computes the partition sum function of a multinomial 2d measure
znq=multim2d(bx,by,p,'part',vn,q);
mn=zeros(max(size(q)),max(size(vn)));
for i=1:max(size(q))
mn(i,:)=-vn*log(2);
end
plot2d(mn',log(znq'));
// computes the Reyni exponents function of a multinomial 2d measure
tq=multim2d(bx,by,p,'Reyni',q);
plot(q,tq);
N=200;
// computes the multifractal spectrum of a multinomial 2d measure
[alpha,f_alpha]=multim2d(bx,by,p,'spec',N);
plot(alpha,f_alpha);
References
"Multifractal Measures", Carl J. G. Evertsz and Benoit B. MandelBrot. In
Chaos and Fractals, New Frontiers of Science, Appendix B. Edited by Peit-
gen, Juergens and Saupe, Springer Verlag, 1992 pages 921-953. "A class of
Multinomial Multifractal Measures with negative (latent) values for the
"Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins
and Properties, Proceeding of the Erice Meeting, 1988. Edited by L.
Pietronero, Plenum Press, New York, 1989 pages 3-29. .SH See also binom,
sbinom, multim1d, smultim1d, smultim2d (C_LAB routines). MFAS_measures,
MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).
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