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smultim1d __SUBTITLE__=stochastic multinomial 1d measure synthesis(2)Scilab Functionsmultim1d __SUBTITLE__=stochastic multinomial 1d measure synthesis(2)
NAME
smultim1d __SUBTITLE__=stochastic multinomial 1d measure synthesis -
Author: Christophe Canus
This C_LAB routine synthesizes two types of pre-multifractal stochastic
measures related to the multinomial 1d measure (uniform law and lognormal
law) and computes linked multifractal spectrum.
Usage
[varargout,[optvarargout]]=sbinom(b,str,varargin,[optvarargin])
Input parameters
o b : strictly positive real (integer) scalar Contains the base 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 stochastic multinomial 1d measure is completly characterized by its
base b. This first parameter must be >1.
The second parameter str is a variable string used to determine the desired
type of output. There are four suffix strings ('meas' for measure, 'cdf'
for cumulative distribution function q, 'pdf' for probability density func-
tion, 'spec' for multifractal spectrum) and a two prefix strings for the
type of stochastic measure ('unif' for uniform and 'logn' for lognormal)
which must added to the first ones to form composed. For example, 'unif-
meas' is for the synthesis of a uniform law multinomial 1d pre-multifractal
measure and 'lognspec' is for the computation of the multifractal spectrum
of a lognormal multinomial 1d measure. 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 b-adic 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 bn (so be aware the
stack size ;-)). This option is implemented for the uniform law ('unif-
meas') and the lognormal law ('lognmeas') multinomial 1d measures. When a
string containing prefix 'unif' is given as second input, the synthesis or
the computation is made for a uniform law multinomial 1d measure. In that
case, the optional fourth input argument is a strictly positive real scalar
epsilon which contains the pertubation around weight .5. The weight is an
independant random variable identically distributed between epsilon and 1-
epsilon which must be >0., <1. The default value for epsilon is 0. When a
string containing prefix 'logn' is given as second input, the synthesis or
the computation is made for a lognormal law multinomial 1d measure. In that
case, the optional fourth input argument is a strictly positive real scalar
sigma which contains the standard deviation of the lognormal law. When
replacing suffix string 'meas' with suffix string 'cdf', respectively suf-
fix 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 a string containing suf-
fix string 'spec' is given as second input, the multifractal spectrum
f_alpha (second output argument) is synthesized 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 lognormal law
('lognspec') multinomial 1d measures.
Algorithm details
For the uniform and lognormal law multinomial 1d, 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 lognormal
law multinomial 1d measure is not conservative.
Examples
Matlab
n=10;
% synthesizes a pre-multifractal uniform Law multinomial 1d measure
[mu_n,I_n]=smultim1d(b,'unifmeas',n);
plot(I_n,mu_n);
s=1.;
% synthesizes the cdf of a pre-multifractal lognormal law multinomial 1d measure
F_n=smultim1d(b,'logncdf',n,s);
plot(I_n,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal uniform law multinomial 1d measure
p_n=smultim1d(b,'unifpdf',n,e);
plot(I_n,p_n);
N=200;
s=1.;
% computes the multifractal spectrum of the lognormal law multinomial 1d measure
[alpha,f_alpha]=smultim1d(b,'lognspec',N,s);
plot(alpha,f_alpha);
Scilab
n=10;
// synthesizes a pre-multifractal uniform Law multinomial 1d measure
[mu_n,I_n]=smultim1d(b,'unifmeas',n);
plot(I_n,mu_n);
s=1.;
// synthesizes the cdf of a pre-multifractal lognormal law multinomial 1d measure
F_n=smultim1d(b,'logncdf',n,s);
plot(I_n,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal uniform law multinomial 1d measure
p_n=smultim1d(b,'unifpdf',n,e);
plot(I_n,p_n);
N=200;
// computes the multifractal spectrum of the lognormal law multinomial 1d measure
[alpha,f_alpha]=smultim1d(b,'lognspec',N,s);
plot(alpha,f_alpha);
References
"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. "Limit Lognormal Mul-
tifractal Measures", Benoit B. MandelBrot. In Frontiers of Physics, Landau
Memorial Conference, Proceeding of the Tel-Aviv Meeting, 1988. Edited by
Errol Asher Gotsman, Yuval Ne'eman and Alexander Voronoi, New York Per-
gamon, 1990 pages 309-340.
See also
binom, sbinom, multim1d, multim2d, smultim2d (C_LAB routines).
MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo
scripts).
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