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.TH "binom" 2 "July 5th 1997" "Fractales Group" "Scilab Function"
.so ../sci.an
.SH NAME
binom - binomial measure synthesis
.sp
Author: Christophe Canus
.sp
This C_LAB routine synthesizes a large range of pre-multifractal
measures related to the binomial measure paradigm (deterministic,
shuffled, pertubated, and mixing of two binomials: lumping and sum)
and computes linked theoretical functions (partition sum function,
Reyni exponents function, generalized dimensions, multifractal
spectrum).
.sp
.sp
.SH Usage
\f(CR[\fPvarargout,\f(CR[\fPoptvarargout\f(CR]\fP\f(CR]\fP=binom(p0,str,varargin,\f(CR[\fPoptvarargin\f(CR]\fP)
.SH Input parameters
.RS
.TP
o
\fBp0\fP : strictly positive real scalar
Contains the weight of the binomial.
.TP
o
\fBstr\fP : string
Contains the type of ouput.
.TP
o
\fBvarargin\fP : variable input argument
Contains the variable input argument.
.TP
o
\fBoptvarargin\fP : optional variable input arguments
Contains optional variable input arguments.
.RE
.SH Output parameters
.RS
.TP
o
\fBvarargout\fP : variable output argument
Contains the variable output argument.
.TP
o
\fBoptvarargout\fP : optional variable output argument
Contains an optional variable output argument.
.RE
.SH Description
.SH Parameters
The binomial measure is completly characterized by its weight \fIp0\fP. This
first parameter must be \f(CR>\fP0. and \f(CR<\fP1. (the case of \fIp0\fP=.5
corresponds to the Lebesgue measure).
The second parameter \fIstr\fP is a variable string used to determine
the desired type of output. There are six suffix strings ('\fImeas\fP'
for measure, '\fIcdf\fP' for cumulative distribution function,
, '\fIpdf\fP' for probability density function, '\fIpart\fP' for partition
sum function, '\fIReyni\fP' for Reyni exponent function , '\fIspec\fP'
for multifractal spectrum) for the deterministic binomial measure and
a lot of possibly composed prefix strings for related measures
('\fIshuf\fP' for shuffled, '\fIpert\fP' for pertubated, '\fIlump\fP' for
lumping , '\fIsum\fP' for sum, '\fIsumpert\fP' for sum of pertubated,
and so on) which can be added to the first ones to form composed
strings. For example, '\fIlumppertmeas\fP' is for the synthesis of the
lumping of 2 pertubated binomial pre-multifractal measures
and '\fIsumspec\fP' is for the computation of the multifractal spectrum of
the sum of two binomials. Note that all combinaisons of strings are
not implemented yet.
When a string containing suffix string '\fImeas\fP' is given as second
input, a pre-multifractal measure \fImu_n\fP (first output argument) is
synthesized on the dyadic intervals \fII_n\fP (second optional output
argument) of the unit interval. In that case, the third input argument
is a strictly positive real (integer) scalar \fIn\fP which contains the
resolution of the pre-multifractal measure. The size of the output
real vectors \fImu_n\fP (and \fII_n\fP if used) is equal to 2\(ha\fIn\fP (so
be aware the stack size ;-)). This option is implemented for the
deterministic ('\fImeas\fP'), shuffled ('\fIshufmeas\fP') and pertubated
('\fIpertmeas\fP') binomial, and also for the mixing (lumping or sum)
of two deterministic ('\fIlumpmeas\fP' and '\fIsummeas\fP') or
pertubated ('\fIlumppertmeas\fP' and '\fIsumpertmeas\fP')
binomial measures.
When a string containing prefix '\fIshuf\fP' is given as second input,
the synthesis is made for a shuffled binomial measure. At each level
of the multiplicative cascade and for all nodes of the corresponding
binary tree, the weight is chosen uniformly among \fIp0\fP and
1-\fIp0\fP. This option is implemented only for the binomial measure
('\fIshufmeas\fP').
When a string containing prefix '\fIpert\fP' is given as second input, the
synthesis is made for a pertubated binomial measure. In that case, the
fourth input argument is a strictly positive real scalar \fIepsilon\fP which
contains the pertubation around weights. The weight is an independant
random variable identically distributed between \fIp0\fP-\fIepsilon\fP
and \fIp0\fP+\fIepsilon\fP which must be \f(CR>\fP0., \f(CR<\fP1. This option
is implemented only for the binomial measure ('\fIpertmeas\fP') and the
mixing (lumping and sum) of two binomial measures ('\fIlumppertmeas\fP'
and '\fIsumpertmeas\fP').
When replacing suffix string '\fImeas\fP' with suffix
string '\fIcdf\fP', respectively suffix string '\fIpdf\fP', the cumulative
distribution function \fIF_n\fP, respectively the probability density
function \fIp_n\fP, related to this pre-multifractal measure is
computed (first output argument).
When string '\fIpart\fP' is given as second input, the partition sum
function \fIznq\fP of multifractal measure is computed as sole output
argument. In that case, the third input argument is a strictly positive
real (integer) vector \fIvn\fP which contains the resolutions, and the
fourth input argument is a real vector \fIq\fP which contains the
measure exponents. The size of the output real matrix \fIznq\fP is
equal to size(\fIq\fP)*size(\fIvn\fP). This option is implemented only
for the binomial measure.
When string '\fIReyni\fP' is given as second input, the Reyni exponents
function \fItq\fP (and the generalized dimensions \fIDq\fP if used) of
the multifractal measure is computed as first output argument (and
second optional output argument if used). In that case, the third
input argument is a real vector \fIq\fP which contains the measure's
exponents. The size of the output real vector \fItq\fP is equal to
size(\fIq\fP)). This option is implemented only for the binomial
measure.
When a string containing suffix string '\fIspec\fP' is given as second
input, the multifractal spectrum \fIf_alpha\fP (second output argument)
is synthesized on the Hoelder exponents \fIalpha\fP (first output
argument). In that case, the third input argument is a strictly
positive real (integer) scalar \fIN\fP which contains the number of
Hoelder exponents. The size of both output real vectors \fIalpha\fP and
\fIf_alpha\fP is equal to \fIN\fP. This option is implemented only for
the binomial measure ('\fIspec\fP') and the mixing (lumping and sum) of
two binomial measures ('\fIlumpspec\fP' and \fIsumspec\fP').
.SH Algorithm details
For the deterministic binomial, 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 binomial,
the synthesis 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). Note that the shuffled binomial measure is not conservative.
.SH Examples
.SH Matlab
.sp
.ft CR
.nf
p0=.2;
n=10;
% synthesizes a pre-multifractal binomial measure
[mu_n,I_n]=binom(p0,'meas',n);
plot(I_n,mu_n);
% synthesizes the cdf of a pre-multifractal shuffled binomial measure
F_n=binom(p0,'shufcdf',n);
plot(I_n,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal pertubated binomial measure
p_n=binom(p0,'pertpdf',n,e);
plot(I_n,p_n);
vn=[1:1:8];
q=[-5:.1:+5];
% computes the partition sum function of a binomial measure
znq=binom(p0,'part',vn,q);
plot(-vn*log(2),log(znq));
% computes the Reyni exponents function of a binomial measure
tq=binom(p0,'Reyni',q);
plot(q,tq);
N=200;
q0=.4;
% computes the multifractal spectrum of the lumping of two binomial measures
[alpha,f_alpha]=binom(p0,'lumpspec',N,q0);
plot(alpha,f_alpha);
.fi
.ec
.ft P
.sp
.SH Scilab
.sp
.ft CR
.nf
p0=.2;
n=10;
// synthesizes a pre-multifractal binomial measure
[mu_n,I_n]=binom(p0,'meas',n);
plot(I_n,mu_n);
// synthesizes the cdf of a pre-multifractal shuffled binomial measure
F_n=binom(p0,'shufcdf',n);
plot(I_n,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal pertubated binomial measure
p_n=binom(p0,'pertpdf',n,e);
plot(I_n,p_n);
xbasc();
vn=[1:1:8];
q=[-5:.1:+5];
// computes the partition sum function of a binomial measure
znq=binom(p0,'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 binomial measure
tq=binom(p0,'Reyni',q);
plot(q,tq);
N=200;
q0=.4;
// computes the multifractal spectrum of the lumping of two binomial measures
[alpha,f_alpha]=binom(p0,'lumpspec',N,q0);
plot(alpha,f_alpha);
.fi
.ec
.ft P
.sp
.SH References
\fB"Multifractal Measures"\fP, Carl J. G. Evertsz and Benoit
B. MandelBrot. \fIIn Chaos and Fractals, New Frontiers of Science,
Appendix B. Edited by Peitgen, Juergens and Saupe, Springer Verlag,
1992 pages 921-953\fP.
\fB"A class of Multinomial Multifractal Measures with negative
(latent) values for the "Dimension" f(alpha)"\fP, Benoit
B. MandelBrot. \fIIn Fractals' Physical Origins and Properties,
Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum
Press, New York, 1989 pages 3-29\fP. .SH See also
\fBsbinom, multim1d, multim2d, smultim1d, smultim2d\fP (C_LAB routines).
\fBMFAS_measures, MFAS_dimensions, MFAS_spectra\fP (Matlab and/or Scilab demo scripts).
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