.TH "multim2d" 2 "July 5th 1997" "Fractales Group" "Scilab Function"
.so ../sci.an
.SH NAME
multim2d - multinomial 2d measure synthesis
.sp
Author: Christophe Canus
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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).
.sp
.sp
.SH Usage
\f(CR[\fPvarargout,\f(CR[\fPoptvarargout\f(CR]\fP\f(CR]\fP=binom(bx,by,p,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 
\fBp\fP : strictly positive real vector \f(CR[\fPby,bx\f(CR]\fP 
Contains the weights 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 multinomial 2d measure is completly characterized by its abscissa
base \fIbx\fP, ordonate base \fIby\fP and its weights \fIp\fP(i) (i=1 to
\fIbx\fP*\fIby\fP). The first two parameters \fIbx\fP and \fIby\fP must be
\f(CR>\fP1. The third parameter must be a vector of size equal to
\fIbx\fP*\fIby\fP. The weights \fIp\fP(i) must be \f(CR>\fP0., \f(CR<\fP1. and
their sum must be =1. (the case of \fIp\fP(i)=1/(\fIbx\fP*\fIby\fP)
corresponds to the Lebesgue measure) (i=1 to \fIbx\fP*\fIby\fP).
The fourth 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 multinomial measure
and two prefix strings for related measures ('\fIshuf\fP' for shuffled
, '\fIpert\fP' for pertubated) which can be added to the first ones to
form composed strings. For example, '\fIshufmeas\fP' 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 containing suffix string '\fImeas\fP' is given as fourth
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 fifth 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 deterministic ('\fImeas\fP'),
shuffled ('\fIshufmeas\fP') and pertubated ('\fIpertmeas\fP')
multinomial 2d measure.
When a string containing prefix '\fIshuf\fP' 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 \fIp\fP is
shuffled. This option is implemented only for the multinomial 2d
measure ('\fIshufmeas\fP').
When a string containing prefix '\fIpert\fP' 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
\fIepsilon\fP which contains the pertubation around weights. The
weights are independant random variables identically distributed
between \fIp\fP(i)-\fIepsilon\fP and \fIp\fP(i)+\fIepsilon\fP which must
be \f(CR>\fP0., \f(CR<\fP1. (i=1 to \fIbx\fP*\fIby\fP). This option is
implemented only for the multinomial 2d measure ('\fIpertmeas\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 fourth input, the partition sum
function \fIznq\fP of multifractal measure is computed as sole output
argument. In that case, the fifth input argument is a strictly positive
real (integer) vector \fIvn\fP which contains the resolutions, and the
sixth 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 multinomial 2d measure.
When string '\fIReyni\fP' is given as third 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 fifth
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 multinomial 2d
measure.
When string '\fIspec\fP' is given as fourth 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 fifth 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 multinomial 2d measure.
.SH 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 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). 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 multinomial 2d measure
is not conservative.
.SH Examples
.SH Matlab
.sp
.ft CR
.nf
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);
.fi 
.ec
.ft P
.sp
.SH Scilab
.sp
.ft CR
.nf
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);
.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
\fBbinom, sbinom, multim1d, smultim1d, smultim2d\fP (C_LAB routines).
\fBMFAS_measures, MFAS_dimensions, MFAS_spectra\fP (Matlab and/or Scilab demo scripts).