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binom(2) Scilab Function binom(2)
NAME
binom - binomial measure synthesis
Author: Christophe Canus
This C_LAB routine synthesizes a large range of pre-multifractal measures
related to the binomial measure paradigm (deterministic, shuffled, pertu-
bated, and mixing of two binomials: lumping and sum) and computes linked
theoretical functions (partition sum function, Reyni exponents function,
generalized dimensions, multifractal spectrum).
Usage
[varargout,[optvarargout]]=binom(p0,str,varargin,[optvarargin])
Input parameters
o p0 : strictly positive real scalar Contains the weight of the bino-
mial.
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 binomial measure is completly characterized by its weight p0. This
first parameter must be >0. and <1. (the case of p0=.5 corresponds to the
Lebesgue measure). The second 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 binomial measure and a lot of possibly composed prefix strings for
related measures ('shuf' for shuffled, 'pert' for pertubated, 'lump' for
lumping , 'sum' for sum, 'sumpert' for sum of pertubated, and so on) which
can be added to the first ones to form composed strings. For example,
'lumppertmeas' is for the synthesis of the lumping of 2 pertubated binomial
pre-multifractal measures and 'sumspec' is for the computation of the mul-
tifractal spectrum of the sum of two binomials. Note that all combinaisons
of strings are not implemented yet. When a string containing suffix string
'meas' is given as second input, a pre-multifractal measure mu_n (first
output argument) is synthesized on the dyadic intervals I_n (second
optional output argument) of the unit interval. In that case, the third
input argument is a strictly positive real (integer) scalar n which con-
tains the resolution of the pre-multifractal measure. The size of the out-
put real vectors mu_n (and I_n if used) is equal to 2n (so be aware the
stack size ;-)). This option is implemented for the deterministic ('meas'),
shuffled ('shufmeas') and pertubated ('pertmeas') binomial, and also for
the mixing (lumping or sum) of two deterministic ('lumpmeas' and 'summeas')
or pertubated ('lumppertmeas' and 'sumpertmeas') binomial measures. When a
string containing prefix 'shuf' 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 p0 and 1-p0. This option is implemented only for the
binomial measure ('shufmeas'). When a string containing prefix 'pert' 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 epsilon which contains the pertubation around weights. The
weight is an independant random variable identically distributed between
p0-epsilon and p0+epsilon which must be >0., <1. This option is implemented
only for the binomial measure ('pertmeas') and the mixing (lumping and sum)
of two binomial measures ('lumppertmeas' and 'sumpertmeas'). When replac-
ing suffix string 'meas' with suffix string 'cdf', respectively suffix
string 'pdf', the cumulative distribution function F_n, respectively the
probability density function p_n, related to this pre-multifractal measure
is computed (first output argument). When string 'part' is given as second
input, the partition sum function znq of multifractal measure is computed
as sole output argument. In that case, the third input argument is a
strictly positive real (integer) vector vn which contains the resolutions,
and the fourth 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 binomial measure.
When string 'Reyni' is given as second 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 third input argument is a real vector q which
contains the measure's exponents. The size of the output real vector tq is
equal to size(q)). This option is implemented only for the binomial meas-
ure. When a string containing suffix string 'spec' is given as second
input, the multifractal spectrum f_alpha (second output argument) is syn-
thesized on the Hoelder exponents alpha (first output argument). In that
case, the third input argument is a strictly positive real (integer) scalar
N which contains the number of Hoelder exponents. The size of both output
real vectors alpha and f_alpha is equal to N. This option is implemented
only for the binomial measure ('spec') and the mixing (lumping and sum) of
two binomial measures ('lumpspec' and sumspec').
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.
Examples
Matlab
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);
Scilab
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);
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 sbinom,
multim1d, multim2d, smultim1d, smultim2d (C_LAB routines). MFAS_measures,
MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).
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