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');