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mcfg1d(2) Scilab Function mcfg1d(2)
NAME
mcfg1d - Continuous large deviation spectrum estimation on 1d measure
Author: Christophe Canus
This C_LAB routine estimates the continuous large deviation spectrum on 1d
measure.
Usage
[alpha,fgc_alpha,[pc_alpha,epsilon_star,eta,alpha_eta_x]]=
mcfg1d(mu_n,[S_min,S_max,J],progstr,ballstr,N,epsilon,contstr,adapstr,kernstr,normstr,I_n])
Input parameters
o mu_n : strictly positive real vector [1,N_n] or [N_n,1] Contains the
1d measure.
o S_min : strictly positive real scalar Contains the minimum size.
o S_max : strictly positive real scalar Contains the maximum size.
o J : strictly positive real (integer) scalar Contains the number of
scales.
o progstr : string Contains the string which specifies the scale pro-
gression.
o ballstr : string Contains the string which specifies the type of
ball.
o N : strictly positive real (integer) scalar Contains the number of
Hoelder exponents.
o epsilon : strictly positive real vector [1,N] or [N,1] Contains the
precisions.
o contstr : string Contains the string which specifies the definition
of continuous spectrum.
o adapstr : string Contains the string which specifies the precision
adaptation.
o kernstr : string Contains the string which specifies the kernel
form.
o normstr : string Contains the string which specifies the pdf's nor-
malization.
o I_n : strictly positive real vector [1,N_n] or [N_n,1] Contains the
intervals on which the pre-multifractal 1d measure is defined.
Output parameters
o alpha : real vector [1,N] Contains the Hoelder exponents.
o fgc_alpha : real matrix [J,N] Contains the spectrum(a).
o pc_alpha : real matrix [J,N] Contains the pdf('s).
o epsilon_star : strictly positive real matrix [J,N] Contains the
optimal precisions.
o eta : strictly positive real vector [1,J]
Contains the sizes.
o alpha_eta_x : strictly positive real matrix [J,N_n] Contains the
coarse grain Hoelder exponents.
Description
Parameters
The continuous large deviation spectrum (alpha,fgc_alpha) is estimated
for J sizes eta_j and for the precision vector epsilon by taking into
account the resolution of the 1d measure mu_n. The minimum size S_min
sets the equivalent size eta_1 in the unit interval at which the first
spectrum is estimated. eta_1 is equal to S_min*eta_n where eta_n is
related to the resolution of the 1d measure (eta_n=N_n{-1} when all
intervals are of equal size else it is max(|I_n|{-1}). It must be
>=1. The default value for S_min is 1. The maximum size S_max sets the
equivalent size eta_J in the unit interval at which the last spectrum is
estimated. eta_J is equal to S_max*eta_n. It must be >=S_min. The
default value for S_max is 1. The number of scales J sets the number of
computed spectra. The bigger J is, the slower the computation is.
It must be >=1. The default value for J is 1. The scale progression
string progstr specifies the type of scale discretization. It can be
'dec' for decimated, 'log' for logarithmic or 'lin' for linear scale.
The default value for progstr is 'dec'. The ball string ballstr
specifies the type of ball B_eta(x). It can be 'asym' for asym-
metric, 'cent' for centered or 'star' for three times bigger asymmetric
ball. The default value for ballstr is 'asym'. The number N sets the
discretization of the Hoelder exponents interval. They are linearly
spaced between alpha_eta_min and alpha_eta_max which are the minimum and
maximum values of the coarse grain Hoelder exponents at size eta. The
bigger N is, the slower the computation is. It must be >=1. The default
value for N is 100. The precision vector epsilon sets the precisions at
which the spectrum is estimated. It must be of size [1,N] or
[N,1]. When no precision vector is given as input or when it is uni-
formly equal to 0, the algorithm determines the optimal precisions
vector epsilon_star. The default value for epsilon is zeros(1,N).
The continuous string contstr specifies the definition of continu-
ous spectrum. It can be equal to 'hnokern' for definition without preci-
sion and kernel or 'hkern' for definition with precision and ker-
nel. The default value for contstr is 'hkern'. The precision
adaptation string adapstr specifies the local adaptation of the preci-
sion w.r.t. the Hoelder exponents alpha. It can be equal to 'max-
dev' for maximum deviation or 'maxadaptdev' for maximum adaptive
deviation. The default value for adapstr is 'maxdev'. The kernel string
kernstr specifies the kernel. It can be equal to 'box' for boxcar, 'tri'
for triangle, 'mol' for mollifier, 'epa' for epanechnikhov or 'gau'
for gaussian kernel. The default value for kernstr is 'gau'. The normali-
zation string normstr specifies the type of pdf's normalization con-
ducted before double log-normalization. It can be equal to 'nonorm' for
no normalization conducted, 'suppdf' for normalization w.r.t the
supremum of pdf's, 'infsuppdf' for normalization w.r.t the infimum and
the supremum of pdf's. The default value for normstr is 'suppdf'. The
intervals vector I_n can be useful when the intervals on which the
pre-multifractal 1d measure is defined are not of equal size (not imple-
mented yet). The pdf of the coarse grain Hoelder exponents matrix
or vector pc_alpha, the optimal precisions matrix or vector
epsilon_star, the sizes vector eta and the coarse grain Hoelder
exponents matrix or vector alpha_eta_x can be obtained as outputs parame-
ters.
Algorithm details
The coarse Hoelder exponents are estimated on each point x of the unit
interval discretization by summing interval measures into a sliding
window of size eta containing x (which corresponds to ball B_eta(x)).
The probability density function pc_alpha is obtained by integrating hor-
izontal sections.
Examples
Matlab
% synthesis of pre-multifractal binomial measure: mu_n
% resolution of the pre-multifractal measure
n=10;
% parameter of the binomial measure
p_0=.4;
% synthesis of the pre-multifractal beiscovitch 1d measure
mu_n=binom(p_0,'meas',n);
% continuous large deviation spectrum estimation: fgc_alpha
% minimum size, maximum size & # of scales
S_min=1;S_max=8;J=4;
% # of hoelder exponents, precision vector
N=200;epsilon=zeros(1,N);
% estimate the continuous large deviation spectrum
[alpha,fgc_alpha,pc_alpha,epsilon_star]=mcfg1d(mu_n,[S_min,S_max,J],'dec','cent',N,epsilon,'hkern','maxdev','gau','suppdf');
% plot the continuous large deviation spectrum
plot(alpha,fgc_alpha);
title('Continuous Large Deviation spectrum');
xlabel('lpha');
ylabel('f_{g,\ta}^{c,\psilon}(lpha)');
Scilab
// computation of pre-multifractal besicovitch measure: mu_n
// resolution of the pre-multifractal measure
n=10;
// parameter of the besicovitch measure
p_0=.4;
// synthesis of the pre-multifractal besicovitch 1d measure
[mu_n,I_n]=binom(p_0,'meas',n);
// continuous large deviation spectrum estimation: fgc_alpha
// minimum size, maximum size & # of scales
S_min=1;S_max=8;J=4;
// # of hoelder exponents, precision vector
N=200;epsilon=zeros(1,N);
// estimate the continuous large deviation spectrum
[alpha,fgc_alpha,pc_alpha,epsilon_star]=mcfg1d(mu_n,[S_min,S_max,J],'dec','cent',N,epsilon,'hkern','maxdev','gau','suppdf');
// plot the Continuous Large Deviation spectrum
plot2d(a,f,[6]);
xtitle(["Continuous Large Deviation spectrum";" "],"alpha","fgc(alpha)");
References
To be published..SH See Also mch1d, fch1d, fcfg1d, cfg1d (C_LAB routines).
MFAG_continuous, MFAG_epsilon, MFAG_eta, MFAG_epsilon_eta (Matlab and/or
Scilab functions).
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