比较ls和pca，实际上，当点比较少的时候，最小二乘法是个不错的选择，但要观察数据的大体方向，pca是个更好的选择。下面说明。
生成在一定范围内的矩形数据，下图中的蓝色的点(我暂时只生成5000个点。)，将这些点进行旋转(这时是-30度),得到图中的红色点。分别用ls和pca分析得到红色点的主要方向。
clear all
clc
clf
%% compare lsfit and pca
% outline:
% 01 gen random points in rectangle
% 02 rotate and shift rectangle
% 03 ls fit
% 04 pca
% 05 plot result
%% main
% parameters ---------------------------------------------
rng default
n=5000; % points number
the=-pi/6; % rotate angle
%% 01 gen random points in rectangle ----------------------
x=rand(n,1)*4-2;
y=rand(n,1)*3-1.5;
%% 02 rotate and shift rectangle --------------------------
x_new= cos(the)*x+sin(the)*y+3;
y_new=-sin(the)*x+cos(the)*y+5;
%% 03 ls fit ----------------------------------------------
p=polyfit(x_new,y_new,1);
y_p=p(1)*x_new+p(2);
%% 04 pca -------------------------------------------------
dat=[x_new';y_new']';
[coeff,score,latent] = princomp(dat); % new version of matlab, use pca
% rotate first PC back
eye_mat=eye(2);
eye_new=eye_mat*(coeff)^(-1);
% gen pca line
x_cen=mean(x_new);
y_cen=mean(y_new);
y_p_pca=eye_new(1,2)/eye_new(1,1)*(x_new-x_cen)+y_cen;
% first PC angle
atan(eye_new(1,2)/eye_new(1,1))*180/pi
% second PC angle
atan(eye_new(2,2)/eye_new(2,1))*180/pi
%% 05 plot result------------------------------------------
hold on
plot(x,y,'b.') % plot ori points
plot(x_new,y_new,'r.') % plot rotate points
plot(x_new,y_p,'k-','linewidth',2) % ls fit line
plot(x_new,y_p_pca,'g-','linewidth',2) % pca line(first PC)
grid on
axis equal
legend('ori point','rot point','ls fit','PCA','Location','southeast')
% option: output fig
% h=gcf;
% fig_na=['../imgs/fig_01_comp_lsfit_pca'];
% fun_work_li_035_myfig_out(h,fig_na,3);
%% logs
% mod :
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