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stblfit.m
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stblfit.m
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function params = stblfit(X,varargin)
%PARAMS = STBLFIT(X) returns an estimate of the four parameters in a
% fit of the alpha-stable distribution to the data X. The output
% PARAMS is a 4 by 1 vector which contains the estimates of the
% characteristic exponent ALPHA, the skewness BETA, the scale GAMMA and
% location DELTA.
%
%PARAMS = STBLFIT(X,METHOD) Specifies the algorithm used to
% estimate the parameters. The choices for METHOD are
% 'ecf' - Fits the four parameters to the empirical characteristic
% function estimated from the data. This is the default.
% Based on Koutrouvelis (1980,1981), see [1],[2] below.
% 'percentile' - Fits the four parameters using various
% percentiles of the data X. This is faster than 'ecf',
% however studies have shown it to be slightly less
% accurate in general.
% Based on McCulloch (1986), see [2] below.
%
%PARAMS = STBLFIT(...,OPTIONS) specifies options used in STBLFIT. OPTIONS
% must be an options stucture created with the STATSET function. Possible
% options are
% 'Display' - When set to 'iter', will display the values of
% alpha,beta,gamma and delta in each
% iteration. Default is 'off'.
% 'MaxIter' - Specifies the maximum number of iterations allowed in
% estimation. Default is 5.
% 'TolX' - Specifies threshold to stop iterations. Default is
% 0.01.
%
% See also: STBLRND, STBLPDF, STBLCDF, STBLINV
%
% References:
% [1] I. A. Koutrouvelis (1980)
% "Regression-Type Estimation of the Paramters of Stable Laws.
% JASA, Vol 75, No. 372
%
% [2] I. A. Koutrouvelis (1981)
% "An Iterative Procedure for the estimation of the Parameters of
% Stable Laws"
% Commun. Stat. - Simul. Comput. 10(1), pages 17-28
%
% [3] J. H. McCulloch (1986)
% "Simple Consistent Estimators of Stable Distribution Parameters"
% Cummun. Stat. Simul. Comput. 15(4)
%
% [4] A. H. Welsh (1986)
% "Implementing Empirical Characteristic Function Procedures"
% Statistics & Probability Letters Vol 4, pages 65-67
% ==== Gather additional options
dispit = false;
maxiter = 5;
tol = .01;
if ~isempty(varargin)
if isstruct(varargin{end})
opt = varargin{end};
try
dispit = opt.Display;
catch ME
error('OPTIONS must be a structure created with STATSET');
end
if ~isempty(opt.MaxIter)
maxiter = opt.MaxIter;
end
if ~isempty(opt.TolX)
tol = opt.TolX;
end
end
end
if strcmp(dispit,'iter')
dispit = true;
fprintf(' iteration\t alpha\t beta\t gamma\t\t delta\n');
dispfmt = '%8d\t%14g\t%8g\t%8g\t%8g\n';
end
% === Find which method.
if any(strcmp(varargin,'percentile'))
maxiter = 0; % This is McCulloch's percentile method
end
% ==== Begin estimation =====
N = numel(X); % data size
% function handle to compute empirical char. functions
I = sqrt(-1);
phi = @(theta,data) 1/numel(data) * sum( exp( I * ...
reshape(theta,numel(theta),1) *...
reshape(data,1,numel(data)) ) , 2);
% Step 1 - Obtain initial estimates of parameters using McCulloch's method
% then standardize data
[alpha beta] = intAlpBet(X);
[gam delta ] = intGamDel(X,alpha,beta);
if gam==0
% Use standard deviation as initial guess
gam = std(X);
end
s = (X - delta)/gam;
if dispit
fprintf(dispfmt,0,alpha,beta,gam,delta);
end
% Step 2 - Iterate until convergence
alphaold = alpha;
deltaold = delta;
diffbest = inf;
for iter = 1:maxiter
% Step 2.1 - Regress against ecf to refine estimates of alpha & gam
% After iteration 1, use generalized least squares
if iter <= 2
K = chooseK(alpha,N);
t = (1:K)*pi/25;
w = log(abs(t));
end
y = log( - log( abs(phi(t,s)).^2 ) );
if iter == 1 % use ordinary least squares regression
ell = regress(y,[w' ones(size(y))]);
alpha = ell(1);
gamhat = (exp(ell(2))/2)^(1/alpha);
gam = gam * gamhat;
else % use weighted least squares regression
sig = charCov1(t ,N, alpha , beta, 1);
try
ell = lscov([w' ones(size(y))],y,sig);
catch % In case of badly conditioned covariance matrix, just use diagonal entries
try
ell = lscov([w' ones(size(y))],y,eye(K).*(sig+eps));
catch
break
end
end
alpha = ell(1);
gamhat = (exp(ell(2))/2)^(1/alpha);
gam = gam * gamhat;
end
% Step 2.2 - Rescale data by estimated scale, truncate
s = s/gamhat;
alpha = max(alpha,0);
alpha = min(alpha,2);
beta = min(beta,1);
beta = max(beta,-1);
gam = max(gam,0);
% Step 2.3 - Regress against ecf to refine estimates of beta, delta
% After iteration 1, use generalized least squares
if iter <= 2
L = chooseL(alpha,N);
% To ensure g is continuous, find first zero in real part of ecf
A = efcRoot(s);
u = (1:L)*min(pi/50,A/L);
end
ecf = phi(u,s);
U = real(ecf);
V = imag(ecf);
g = atan2(V,U);
if iter == 1 % use ordinary least squares
ell = regress(g, [u', sign(u').*abs(u').^alpha]);
beta = ell(2)/tan(alpha*pi/2) ;
delta = delta + gam* ell(1) ;
else % use weighted least squares regression
sig = charCov2(u ,N, alpha , beta, 1);
try
ell = lscov([u', sign(u').*abs(u').^alpha],g,sig);
catch % In case of badly conditioned covariance matrix, use diagonal entries
try
ell = lscov([u', sign(u').*abs(u').^alpha],g,eye(L).*(sig+eps));
catch
break
end
end
beta = ell(2)/tan(alpha*pi/2) ;
delta = delta + gam* ell(1) ;
end
% Step 2.4 Remove estimated shift
s = s - ell(1);
% display
if dispit
fprintf(dispfmt,iter,alpha,beta,gam,delta);
end
% Check for blow-up
if any(isnan([alpha, beta, gam, delta]) | isinf([alpha, beta, gam, delta]))
break
end
% Step 2.5 Check for convergence, keep track of parameters with
% smallest 'diff'
diff = (alpha - alphaold)^2 + (delta - deltaold)^2;
if abs(diff) < diffbest
bestparams = [alpha; beta; gam; delta];
diffbest = diff;
if diff < tol
break;
end
end
alphaold = alpha;
deltaold = delta;
end
% Pick best
if maxiter > 0 && iter >= 1
alpha = bestparams(1);
beta = bestparams(2);
gam = bestparams(3);
delta = bestparams(4);
end
% Step 3 - Truncate if necessary
alpha = max(alpha,0);
alpha = min(alpha,2);
beta = min(beta,1);
beta = max(beta,-1);
gam = max(gam,0);
params = [alpha; beta; gam; delta];
end % End stblfit
%===============================================================
%===============================================================
function [alpha beta] = intAlpBet(X)
% Interpolates Tables found in MuCulloch (1986) to obtain a starting
% estimate of alpha and beta based on percentiles of data X
% Input tables
nuA = [2.439 2.5 2.6 2.7 2.8 3.0 3.2 3.5 4.0 5.0 6.0 8.0 10 15 25];
nuB = [0 .1 .2 .3 .5 .7 1];
[a b] = meshgrid( nuA , nuB );
alphaTab= [2.000 2.000 2.000 2.000 2.000 2.000 2.000;...
1.916 1.924 1.924 1.924 1.924 1.924 1.924;...
1.808 1.813 1.829 1.829 1.829 1.829 1.829;...
1.729 1.730 1.737 1.745 1.745 1.745 1.745;...
1.664 1.663 1.663 1.668 1.676 1.676 1.676;...
1.563 1.560 1.553 1.548 1.547 1.547 1.547;...
1.484 1.480 1.471 1.460 1.448 1.438 1.438;...
1.391 1.386 1.378 1.364 1.337 1.318 1.318;...
1.279 1.273 1.266 1.250 1.210 1.184 1.150;...
1.128 1.121 1.114 1.101 1.067 1.027 0.973;...
1.029 1.021 1.014 1.004 0.974 0.935 0.874;...
0.896 0.892 0.887 0.883 0.855 0.823 0.769;...
0.818 0.812 0.806 0.801 0.780 0.756 0.691;...
0.698 0.695 0.692 0.689 0.676 0.656 0.595;...
0.593 0.590 0.588 0.586 0.579 0.563 0.513]';
betaTab= [ 0.000 2.160 1.000 1.000 1.000 1.000 1.000;...
0.000 1.592 3.390 1.000 1.000 1.000 1.000;...
0.000 0.759 1.800 1.000 1.000 1.000 1.000;...
0.000 0.482 1.048 1.694 1.000 1.000 1.000;...
0.000 0.360 0.760 1.232 2.229 1.000 1.000;...
0.000 0.253 0.518 0.823 1.575 1.000 1.000;...
0.000 0.203 0.410 0.632 1.244 1.906 1.000;...
0.000 0.165 0.332 0.499 0.943 1.560 1.000;...
0.000 0.136 0.271 0.404 0.689 1.230 2.195;...
0.000 0.109 0.216 0.323 0.539 0.827 1.917;...
0.000 0.096 0.190 0.284 0.472 0.693 1.759;...
0.000 0.082 0.163 0.243 0.412 0.601 1.596;...
0.000 0.074 0.147 0.220 0.377 0.546 1.482;...
0.000 0.064 0.128 0.191 0.330 0.478 1.362;...
0.000 0.056 0.112 0.167 0.285 0.428 1.274]';
% Calculate percentiles
Xpcts = prctile(X,[95 75 50 25 5]);
nuAlpha = (Xpcts(1) - Xpcts(5))/(Xpcts(2) - Xpcts(4));
nuBeta = (Xpcts(1) + Xpcts(5) - 2*Xpcts(3))/(Xpcts(1) - Xpcts(5));
% Bring into range
if nuAlpha < 2.4390
nuAlpha = 2.439 + 1e-12;
elseif nuAlpha > 25
nuAlpha = 25 - 1e-12;
end
s = sign(nuBeta);
% Get alpha
alpha = interp2(a,b,alphaTab,nuAlpha,abs(nuBeta));
% Get beta
beta = s * interp2(a,b,betaTab,nuAlpha,abs(nuBeta));
% Truncate beta if necessary
if beta>1
beta = 1;
elseif beta < -1
beta =-1;
end
end
function [gam delta] = intGamDel(X,alpha,beta)
% Uses McCulloch's Method to obtain scale and location of data X given
% estimates of alpha and beta.
% Get percentiles of data and true percentiles given alpha and beta;
Xpcts = prctile(X,[75 50 25]);
% If alpha is very close to 1, truncate to avoid numerical instability.
warning('off','stblcdf:ScaryAlpha');
warning('off','stblpdf:ScaryAlpha');
if abs(alpha - 1) < .02
alpha = 1;
end
% With the 'quick' option, these are equivalent to McCulloch's tables
Xquart = stblinv([.75 .25],alpha,beta,1,0,'quick');
Xmed = stblinv(.5,alpha,beta,1,-beta*tan(pi*alpha/2),'quick');
% Obtain gamma as ratio of interquartile ranges
gam = (Xpcts(1) - Xpcts(3))/(Xquart(1) - Xquart(2));
% Obtain delta using median of shifted data and estimate of gamma
zeta = Xpcts(2) - gam * Xmed;
delta = zeta - beta*gam*tan(alpha*pi/2);
end
function K = chooseK(alpha,N)
% Interpolates Table 1 in [1] to calculate optimum K given alpha and N
% begin parameters into correct ranges.
alpha = max(alpha,.3);
alpha = min(alpha,1.9);
N = max(N,200);
N = min(N,1600);
a = [1.9, 1.5: -.2: .3];
n = [200 800 1600];
[X Y] = meshgrid(a,n);
Kmat = [ 9 9 9 ; ...
11 11 11 ; ...
22 16 14 ; ...
24 18 15 ; ...
28 22 18 ; ...
30 24 20 ; ...
86 68 56 ; ...
134 124 118 ];
K = round(interp2(X,Y,Kmat',alpha,N,'linear'));
end
function L = chooseL(alpha,N)
% Interpolates Table 2 in [1] to calculate optimum L given alpha and N
alpha = max(alpha,.3);
alpha = min(alpha,1.9);
N = max(N,200);
N = min(N,1600);
a = [1.9, 1.5, 1.1:-.2:.3];
n = [200 800 1600];
[X Y] = meshgrid(a,n);
Lmat = [ 9 10 11 ; ...
12 14 15 ; ...
16 18 17 ; ...
14 14 14 ; ...
24 16 16 ; ...
40 38 36 ; ...
70 68 66 ];
L = round(interp2(X,Y,Lmat',alpha,N,'linear'));
end
function A = efcRoot(X)
% An iterative procedure to find the first positive root of the real part
% of the empirical characteristic function of the data X. Based on [4].
N = numel(X);
U = @(theta) 1/N * sum( cos( ...
reshape(theta,numel(theta),1) *...
reshape(X,1,N) ) , 2 ); % Real part of ecf
m = mean(abs(X));
A = 0;
val = U(A);
iter1 = 0;
while abs(val) > 1e-3 && iter1 < 10^4
A = A + val/m;
val = U(A);
iter1 = iter1 + 1;
end
end
function sig = charCov1(t ,N, alpha , beta,gam)
% Compute covariance matrix of y = log (- log( phi(t) ) ), where phi(t) is
% ecf of alpha-stable random variables. Based on Theorem in [2].
K = length(t);
w = tan(alpha*pi/2);
calpha = gam^alpha;
Tj = repmat( t(:) , 1 , K);
Tk = repmat( t(:)' , K , 1);
Tjalpha = abs(Tj).^alpha;
Tkalpha = abs(Tk).^alpha;
TjxTk = abs(Tj .* Tk);
TjpTk = Tj + Tk ;
TjpTkalpha = abs(TjpTk).^alpha;
TjmTk = Tj - Tk ;
TjmTkalpha = abs(TjmTk).^alpha;
A = calpha*( Tjalpha + Tkalpha - TjmTkalpha);
B = calpha * beta *...
(-Tjalpha .* sign(Tj) * w ...
+ Tkalpha .* sign(Tk) * w ...
+ TjmTkalpha .* sign(TjmTk) * w) ;
D = calpha * (Tjalpha + Tkalpha - TjpTkalpha);
E = calpha * beta *...
( Tjalpha .* sign(Tj) * w ...
+ Tkalpha .* sign(Tk) * w ...
- TjpTkalpha .* sign(TjpTk) * w);
sig = (exp(A) .* cos(B) + exp(D).*cos(E) - 2)./...
( 2 * N * gam^(2*alpha) * TjxTk.^alpha);
end
function sig = charCov2(t ,N, alpha , beta, gam)
% Compute covariance matrix of z = Arctan(imag(phi(t))/real(phi(t)),
% where phi(t) is ecf of alpha-stable random variables.
% Based on Theorem in [2].
K = length(t);
w = tan(alpha*pi/2);
calpha = gam^alpha;
Tj = repmat( t(:) , 1 , K);
Tk = repmat( t(:)' , K , 1);
Tjalpha = abs(Tj).^alpha;
Tkalpha = abs(Tk).^alpha;
TjpTk = Tj + Tk ;
TjpTkalpha = abs(TjpTk).^alpha;
TjmTk = Tj - Tk ;
TjmTkalpha = abs(TjmTk).^alpha;
B = calpha * beta *...
(-Tjalpha .* sign(Tj) * w ...
+ Tkalpha .* sign(Tk) * w ...
+ TjmTkalpha .* sign(TjmTk) * w) ;
E = calpha * beta *...
( Tjalpha .* sign(Tj) * w ...
+ Tkalpha .* sign(Tk) * w ...
- TjpTkalpha .* sign(TjpTk) * w);
F = calpha * (Tjalpha + Tkalpha);
G = -calpha * TjmTkalpha;
H = -calpha * TjpTkalpha;
sig = exp(F) .*(exp(G) .* cos(B) - exp(H) .* cos(E))/(2*N);
end