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.TH "smultim2d __SUBTITLE__=stochastic multinomial 2d measure synthesis" 2 "July 5th 1997" "Fractales Group" "Scilab Function"
.so ../sci.an
.SH NAME
smultim2d __SUBTITLE__=stochastic multinomial 2d measure synthesis -
.sp
Author: Christophe Canus
.sp
This C_LAB routine synthesizes two types of pre-multifractal
stochastic measures related to the multinomial 2d measure (uniform law
and lognormal law).
.sp
.sp
.SH Usage
\f(CR[\fPvarargout,\f(CR[\fPoptvarargout\f(CR]\fP\f(CR]\fP=sbinom(bx,by,str,varargin,\f(CR[\fPoptvarargin\f(CR]\fP)
.SH Input parameters
.RS
.TP
o
\fBbx\fP : strictly positive real (integer) scalar
Contains the abscissa base of the multinomial.
.TP
o
\fBby\fP : strictly positive real (integer) scalar
Contains the ordonate base of the multinomial.
.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 stochastic multinomial 2d measure is completly characterized by
its abscissa base \fIbx\fP, ordonate base \fIby\fP. These first two
parameters must be \f(CR>\fP1.
The third parameter \fIstr\fP is a variable string used to determine
the desired type of output. There are four suffix strings ('\fImeas\fP'
for measure, '\fIcdf\fP' for cumulative distribution function
q, '\fIpdf\fP' for probability density function, '\fIspec\fP' for
multifractal spectrum) and a two prefix strings for the type of
stochastic measure ('\fIunif\fP' for uniform and '\fIlogn\fP' for
lognormal) which must added to the first ones to form composed. For
example, '\fIunifmeas\fP' is for the synthesis of a uniform law
multinomial 2d pre-multifractal measure and '\fIlognspec\fP' is for the
computation of the multifractal spectrum of a lognormal multinomial 2d
measure.
When a string containing suffix string '\fImeas\fP' is given as third
input, a pre-multifractal measure \fImu_n\fP (first output argument) is
synthesized on the \fIbx\fP-adic and \fIby\fP-adic intervals \fII_nx\fP
and \fII_ny\fP (second and third optional output argument) of the unit
square. In that case, the fourth 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 matrix \fImu_n\fP
is equal to \fIbx\fP\(ha\fIn\fP*\fIby\fP\(ha\fIn\fP and the one of the output
real vectors \fII_nx\fP and \fII_ny\fP (if used) is equal to
\fIbx\fP\(ha\fIn\fP and \fIby\fP\(ha\fIn\fP (so be aware the stack size
;-)). This option is implemented for the uniform law ('\fIunifmeas\fP')
and the lognormal law ('\fIlognmeas\fP') multinomial 2d measures.
When a string containing prefix '\fIunif\fP' is given as third input,
the synthesis or the computation is made for a uniform law multinomial
2d measure. In that case, the optional fourth input argument is a
strictly positive real scalar \fIepsilon\fP which contains the
pertubation around weight .5. The weight is an independant random
variable identically distributed between \fIepsilon\fP and
1-\fIepsilon\fP which must be \f(CR>\fP0., \f(CR<\fP1. The default value for
\fIepsilon\fP is 0.
When a string containing prefix '\fIlogn\fP' is given as third input,
the synthesis or the computation is made for a lognormal law multinomial 2d
measure. In that case, the optional fifth input argument is a
strictly positive real scalar \fIsigma\fP which contains the standard
deviation of the lognormal law.
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).
.SH Algorithm details
For the uniform and lognormal law multinomial 2d, 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 2d measure is not conservative.
.SH Examples
.SH Matlab
.sp
.ft CR
.nf
n=5;
bx=2;
by=3;
% synthesizes a pre-multifractal uniform Law multinomial 2d measure
[mu_n,I_nx,I_ny]=smultim2d(bx,by,'unifmeas',n);
mesh(I_nx,I_ny,mu_n);
s=1.;
% synthesizes the cdf of a pre-multifractal lognormal law multinomial 2d measure
F_n=smultim2d(bx,by,'logncdf',n,s);
mesh(I_nx,I_ny,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal uniform law multinomial 2d measure
p_n=smultim2d(bx,by,'unifpdf',n,e);
mesh(I_nx,I_ny,p_n);
.fi
.ec
.ft P
.sp
.SH Scilab
.sp
.ft CR
.nf
n=5;
bx=2;
by=3;
// synthesizes a pre-multifractal uniform Law multinomial 2d measure
[mu_n,I_nx,I_ny]=smultim2d(bx,by,'unifmeas',n);
mesh(I_nx,I_ny,mu_n);
s=1.;
// synthesizes the cdf of a pre-multifractal lognormal law multinomial 2d measure
F_n=smultim2d(bx,by,'logncdf',n,s);
mesh(I_nx,I_ny,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal uniform law multinomial 2d measure
p_n=smultim2d(bx,by,'unifpdf',n,e);
mesh(I_nx,I_ny,p_n);
.fi
.ec
.ft P
.sp
.SH References
\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.
\fB"Limit Lognormal Multifractal Measures"\fP, Benoit
B. MandelBrot. \fIIn 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 Pergamon,
1990 pages 309-340\fP.
.SH See also
\fBbinom, sbinom, multim1d, multim2d, smultim1d\fP (C_LAB routines).
\fBMFAS_measures, MFAS_dimensions, MFAS_spectra\fP (Matlab and/or Scilab demo scripts).
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