von Mises Distribution (冯·米赛斯分布)的随机模拟与参数估计的笔记(二)
von Mises Distribution (冯·米赛斯分布)的随机模拟与参数估计的笔记(二)
1.参数估计算子分析
在上一节中,我们讨论了von Mises Distribution的概率分布函数PDF和累计分布函数CDF,并给出了von Mises Distribution的随机模拟和参数估计matlab程序,其中在此我们就参数估计的细节进行补充。其基于最大似然参数估计算子,如下表:
来源于《Statistical Distributions》
利用如下改进贝塞尔函数的关系求解参数\(\kappa\),如下表达:
\[R=\frac{1}{n}\left[\left(\sum_{i=1}^{n} \cos x_{i}\right)^{2}+\left(\sum_{i=1}^{n} \sin x_{i}\right)^{2}\right]^{1 / 2}
\]
\[\kappa \approx \begin{cases}2 R+R^{3}+\frac{5}{6} R^{5} \quad & R<0.53 \\ -0.4+1.39 R+\frac{0.43}{1-R} & 0.53 \leq R<0.85 \\ \frac{1}{R^{3}-4 R^{2}+3 R} & \text { other }\end{cases}
\]
1.1 \(\mu\)参数估计分析matlab代码
function mu=circ_mean(alpha, w, dim)
%
% mu = circ_mean(alpha, w)
% Computes the mean direction for circular data.
%
% Input:
% alpha sample of angles in radians
% [w weightings in case of binned angle data]
% [dim compute along this dimension, default is 1]
%
% If dim argument is specified, all other optional arguments can be
% left empty: circ_mean(alpha, [], dim)
%
% Output:
% mu mean direction
%
% PHB 7/6/2008
%
% References:
% Statistical analysis of circular data, N. I. Fisher
% Topics in circular statistics, S. R. Jammalamadaka et al.
% Biostatistical Analysis, J. H. Zar
%
% Circular Statistics Toolbox for Matlab
% By Philipp Berens, 2009
% berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html
if nargin < 3
dim = 1;
end
if nargin < 2 || isempty(w)
% if no specific weighting has been specified
% assume no binning has taken place
w = ones(size(alpha));
else
if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1)
error('Input dimensions do not match');
end
end
% compute weighted sum of cos and sin of angles
r = sum(w.*exp(1i*alpha),dim);
% obtain mean by
mu = angle(r);
1.2 \(\kappa\)参数估计的matlab代码
function kappa = circ_kappa(alpha,w)
%
% kappa = circ_kappa(alpha,[w])
% Computes an approximation to the ML estimate of the concentration
% parameter kappa of the von Mises distribution.
%
% Input:
% alpha angles in radians OR alpha is length resultant
% [w number of incidences in case of binned angle data]
%
% Output:
% kappa estimated value of kappa
%
% References:
% Statistical analysis of circular data, Fisher, equation p. 88
%
% Circular Statistics Toolbox for Matlab
% By Philipp Berens, 2009
% berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html
alpha = alpha(:);
if nargin<2
% if no specific weighting has been specified
% assume no binning has taken place
w = ones(size(alpha));
else
if size(w,2) > size(w,1)
w = w';
end
end
N = length(alpha);
if N>1
R = circ_r(alpha,w);
else
R = alpha;
end
if R < 0.53
kappa = 2*R + R^3 + 5*R^5/6;
elseif R>=0.53 && R<0.85
kappa = -.4 + 1.39*R + 0.43/(1-R);
else
kappa = 1/(R^3 - 4*R^2 + 3*R);
end
if N<15 && N>1
if kappa < 2
kappa = max(kappa-2*(N*kappa)^-1,0);
else
kappa = (N-1)^3*kappa/(N^3+N);
end
end
function r = circ_r(alpha, w, d, dim)
% r = circ_r(alpha, w, d)
% Computes mean resultant vector length for circular data.
%
% Input:
% alpha sample of angles in radians
% [w number of incidences in case of binned angle data]
% [d spacing of bin centers for binned data, if supplied
% correction factor is used to correct for bias in
% estimation of r, in radians (!)]
% [dim compute along this dimension, default is 1]
%
% If dim argument is specified, all other optional arguments can be
% left empty: circ_r(alpha, [], [], dim)
%
% Output:
% r mean resultant length
%
% PHB 7/6/2008
%
% References:
% Statistical analysis of circular data, N.I. Fisher
% Topics in circular statistics, S.R. Jammalamadaka et al.
% Biostatistical Analysis, J. H. Zar
%
% Circular Statistics Toolbox for Matlab
% By Philipp Berens, 2009
% berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html
if nargin < 4
dim = 1;
end
if nargin < 2 || isempty(w)
% if no specific weighting has been specified
% assume no binning has taken place
w = ones(size(alpha));
else
if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1)
error('Input dimensions do not match');
end
end
if nargin < 3 || isempty(d)
% per default do not apply correct for binned data
d = 0;
end
% compute weighted sum of cos and sin of angles
r = sum(w.*exp(1i*alpha),dim);
% obtain length
r = abs(r)./sum(w,dim);
% for data with known spacing, apply correction factor to correct for bias
% in the estimation of r (see Zar, p. 601, equ. 26.16)
if d ~= 0
c = d/2/sin(d/2);
r = c*r;
end
2 代码效果分析
clc
clear all
close all
theta=pi/2; %设置模拟参数
kappa=50;
n=3000;
alpha=circ_vmrnd(theta,kappa,n); %生成制定参数的von-Mises分布的随机数
[thetahat1 kappa1]=circ_vmpar(alpha); %对其进行分布参数进行估计分析
%绘制模拟数据直方图
figure(1)
hist(alpha,100);
xlabel('Angle(弧度)');
ylabel('Frequency');
X = categorical({'Really value','Estimate value'});
%估计参数与模型参数对比
figure(2)
subplot(1,2,1)
bar(X,[theta,thetahat1]);
ylabel('theta');
subplot(1,2,2)
bar(X,[kappa,kappa1]);
ylabel('kappa');