光谱波长筛选算法
系列文章目录
近红外光谱分析技术属于交叉领域，需要化学、计算机科学、生物科学等多领域的合作。为此，在（北京邮电大学杨辉华老师团队）指导下，近期准备开源传统的PLS，SVM，ANN，RF等经典算和SG，MSC，一阶导，二阶导等预处理以及GA等波长选择算法以及CNN、AE等最新深度学习算法，以帮助其他专业的更容易建立具有良好预测能力和鲁棒性的近红外光谱模型。
文章目录
- 系列文章目录
- 前言
- 一、SPA算法
- 二、UVE算法
- 三、LAR算法
- 四、CARS算法
- 总结
前言
NIRS是介于可见光和中红外光之间的电磁波，其波长范围为（1100∼2526 nm。 由于近红外光谱区与有机分子中含氢基团（OH、NH、CH、SH）振动的合频和 各级倍频的吸收区一致，通过扫描样品的近红外光谱，可以得到样品中有机分子含氢 基团的特征信息，常被作为获取样本信息的一种有效的载体。 基于NIRS的检测方法具有方便、高效、准确、成本低、可现场检测、不 破坏样品等优势，被广泛应用于各类检测领域。但 近红外光谱存在谱带宽、重叠较严重、吸收信号弱、信息解析复杂等问题，与常用的 化学分析方法不同，仅能作为一种间接测量方法，无法直接分析出被测样本的含量或 类别，它依赖于化学计量学方法，在样品待测属性值与近红外光谱数据之间建立一个 关联模型(或称校正模型，Calibration Model) ，再通过模型对未知样品的近红外光谱 进行预测来得到各性质成分的预测值。现有近红外建模方法主要为经典建模 （预处理+波长筛选进行特征降维和突出，再通过pls、svm算法进行建模）以及深度学习方法（端到端的建模，对预处理、波长选择等依赖性很低）本篇主要讲述常见的波长选择算法(目前是matlab版本的，python版本有时间重写一下)
一、SPA算法
连续投影算法（successive projections algorithm， SPA） 是前向特征变量选择方法。SPA利用向量的投影分析，通过将波长投影到其他波长上，比较投影向量大小，以投影向量最大的波长为待选波长，然后基于矫正模型选择最终的特征波长。SPA选择的是含有最少冗余信息及最小共线性的变量组合。
function chain = projections_qr(X,k,M)
% Projections routine for the Successive Projections Algorithm using the
% built-in QR function of Matlab
%
% chain = projections(X,k,M)
%
% X --> Matrix of predictor variables (# objects N x # variables K)
% k --> Index of the initial column for the projection operations
% M --> Number of variables to include in the chain
%
% chain --> Index set of the variables resulting from the projection operations
X_projected = X;
norms = sum(X_projected.^2); % Square norm of each column vector
norm_max = max(norms); % Norm of the "largest" column vector
X_projected(:,k) = X_projected(:,k)*2*norm_max/norms(k); % Scales the kth column so that it becomes the "largest" column
[dummy1,dummy2,order] = qr(X_projected,0);
chain = order(1:M)';
二、UVE算法
无信息变量去除算法(uninformative variable elimination,UVE)能够去除对建模共效率较小的波长变量，选出特征波长变量，被去除的波长变量我们称之为无信息变量。无信息变量去除算法的建立是基于偏最小二乘(partial least squares,PLS)算法。去除无信息变量，减少了建模所用的变量个数，降低了模型复杂性。为了选择无信息变量，UVE算法通过对PLS模型中添加一组与原始变量数量相同的白噪声变量，然后基于PLS模型的交叉留一法得到每个变量对应的回归系数，包括噪声变量。用每个变量系数的稳定值除以标准差，将他们的商与随机变量矩阵得到的稳定值做比较，删除那些与随机变量一样对建模无效的波长变量。
function [B,C,P,T,U,R,R2X,R2Y]=plssim(X,Y,A,S,XtX);
[n,px] = size(X); [n,m] = size(Y); % size of the input data matrices
if nargin<5, S = []; end, if isempty(S), S=(Y'*X)'; end % if XtX not inputted, S=[]; always when S=[] then S=(Y'*X)'
if nargin<4, XtX=[]; end % if S is not inputted, XtX=[];
if isempty(XtX) & n>3*px, XtX = X'*X; end % when XtX=[] and X is very "tall", the booster XtX is calculated
if nargin<3, A=10; end, A = min([A px n-1]); % if A is not inputted, then the defaul A is min[10 px n-1]
T = zeros(n ,A); U = T; % initialization of variables
R = zeros(px,A); P = R; V = R;
C = zeros(m ,A);
R2Y = zeros(1,A);
z = zeros(m,1); v = zeros(px,1);
if n>px, S0 = S; end, StS = S'*S; % SIMPLS algorithm
nm1 = n-1;
tol = 0;
for a = 1:A
StS = StS-z*z';
[Q,LAMBDA] = eig(StS);
[lambda,j] = max(diag(LAMBDA));
q = Q(:,j(1));
r = S*q;
t = X*r;
if isempty(XtX), p = (t'*X)'; else p = XtX*r; end
if n>px, d = sqrt(r'*p/nm1); else d = sqrt(t'*t/nm1); end
if d<tol,
disp(' ')
disp('WARNING: the required number of factors (A) is too high !')
disp('Less PLS factors were extracted from the data in the PLSSIM program !')
disp(' ')
break,
, else tol=max(tol,d/1e5);
end
v = p-V(:,1:max(1,a-1))*(p'*V(:,1:max(1,a-1)))'; v = v/sqrt(v'*v);
z = (v'*S)';
S = S-v*z';
% save results
V(:,a) = v;
R(:,a) = r/d; % X weights
P(:,a) = p/(d*nm1); % X loadings
T(:,a) = t/d; % X scores
U(:,a) = Y*q; % Y scores
C(:,a) = q*(lambda(1)/(nm1*d)); % Y loadings
R2Y(1,a) = lambda(1)/d; % Y-variance accounted for
end
clear StS V LAMBDA Q p q r t v z;
if d<tol,
A=a-1; a=A; T=T(:,1:A); U=U(:,1:A); R=R(:,1:A); P=P(:,1:A); C=C(:,1:A);
end
while a>1
U(:,a) = U(:,a)-T(:,1:a-1)*(U(:,a)'*T(:,1:a-1)/nm1)';
a=a-1;
end
B = R*C'; % B-coefficients of the regression Y on X
if isempty(XtX), sumX2=sum(X.^2); else sumX2 = sum(diag(XtX)); end
R2X = 100*nm1/sum(sumX2)*cumsum(sum(P.^2));
R2Y = 100/nm1/sum(sum(Y.^2))*cumsum(R2Y(1:A).^2);
三、LAR算法
LAR(Least Angel Regression)，Efron于2004年提出的一种变量选择的方法，类似于向前逐步回归(ForwardStepwise)的形式，是lasso regression的一种高效解法。向前逐步回归(Forward Stepwise)不同点在于，Forward Stepwise每次都是根据选择的变量子集，完全拟合出线性模型，计算出RSS，再设计统计量（如AIC）对较高的模型复杂度作出惩罚，而LAR是每次先找出和因变量相关度最高的那个变量, 再沿着LSE的方向一点点调整这个predictor的系数，在这个过程中，这个变量和残差的相关系数会逐渐减小，等到这个相关性没那么显著的时候，就要选进新的相关性最高的变量，然后重新沿着LSE的方向进行变动。而到最后，所有变量都被选中，就和LSE相同了。
function [b info] = lar(X, y, stop, storepath, verbose)
%% Input checking
% Set default values.
if nargin < 5
verbose = false;
end
if nargin < 4
storepath = true;
end
if nargin < 3
stop = 0;
end
if nargin < 2
error('SpaSM:lar', 'Input arguments X and y must be specified.');
end
%% LARS variable setup
[n p] = size(X);
maxVariables = min(n-1,p); % Maximum number of active variables
useGram = false;
% if n is approximately a factor 10 bigger than p it is faster to use a
% precomputed Gram matrix rather than Cholesky factorization when solving
% the partial OLS偏最小二乘法 problem. Make sure the resulting Gram matrix is not
% prohibitively large.
if (n/p) > 10 && p < 1000
useGram = true;
Gram = X'*X;
end
% set up the LAR coefficient vector
if storepath
b = zeros(p, p+1);
else
b = zeros(p, 1);
b_prev = b;
end
mu = zeros(n, 1); % current "position" as LARS travels towards lsq solution
I = 1:p; % inactive set
A = []; % active set
if ~useGram
R = []; % Cholesky factorization R'R = X'X where R is upper triangular
end
stopCond = 0; % Early stopping condition boolean
step = 1; % step count
if verbose
fprintf('Step\tAdded\tActive set size\n');
end
%% LARS main loop
% while not at OLS solution or early stopping criterion is met
while length(A) < maxVariables && ~stopCond
r = y - mu;
% find max correlation
c = X(:,I)'*r; % X的每一维与当前残差的相关系数
[cmax cidx] = max(abs(c));
% add variable
if ~useGram
R = cholinsert(R,X(:,I(cidx)),X(:,A));
end
if verbose
fprintf('%d\t\t%d\t\t%d\n', step, I(cidx), length(A) + 1);
end
A = [A I(cidx)];
I(cidx) = [];
c(cidx) = []; % 删除原来的这一项后其他各项的相关系数（也即删除了cidx这一维的相关系数（因为已经把第cidx维加进了active set）），后面的项补上来。上一行同理
% partial OLS solution and direction from current position to the OLS
% solution of X_A
if useGram
b_OLS = Gram(A,A)\(X(:,A)'*y); % same as X(:,A)\y, but faster
else
% b_OLS = R\(R'\(X(:,A)'*y)); % same as X(:,A)\y, but faster
b_OLS = X(:,A)\y; %\是matlab里面的左除。用来求（以你问题为例）X*a=y这个线性方程组的（最小二乘）解 因为运算的时候有时会出现说 警告: '矩阵为奇异工作精度'，所以把上面那行换成这个形式了，尽管运行会慢一点 参考：http://www.ilovematlab.cn/thread-301697-1-1.html和http://zhidao.baidu.com/link?url=GzrsyLqNSkvuryZvr4UAMpxi4PIxjpUtJ9HmJ2nL8jVadDjz5CRMbpoxfmk-mRJZ6bsvqgsPHbV7pslq248SpURQkYUmSNhYBh2VoPYSUQe
end
d = X(:,A)*b_OLS - mu;
if isempty(I)
% if all variables active, go all the way to the OLS solution
gamma = 1;
else
% compute length of walk along equiangular direction
cd = (X(:,I)'*d);
gamma = [ (c - cmax)./(cd - cmax); (c + cmax)./(cd + cmax) ];
gamma = min(gamma(gamma > 0)); % 取gamma正数部分的最小值
end
% update beta
if storepath
b(A,step + 1) = b(A,step) + gamma*(b_OLS - b(A,step)); % update beta
else
b_prev = b;
b(A) = b(A) + gamma*(b_OLS - b(A)); % update beta
end
% update position
mu = mu + gamma*d;
% increment step counter
step = step + 1;
% Early stopping at specified bound on L1 norm of beta
if stop > 0
if storepath
t2 = sum(abs(b(:,step)));
if t2 >= stop
t1 = sum(abs(b(:,step - 1)));
s = (stop - t1)/(t2 - t1); % interpolation factor 0 < s < 1
b(:,step) = b(:,step - 1) + s*(b(:,step) - b(:,step - 1));
stopCond = 1;
end
else
t2 = sum(abs(b));
if t2 >= stop
t1 = sum(abs(b_prev));
s = (stop - t1)/(t2 - t1); % interpolation factor 0 < s < 1
b = b_prev + s*(b - b_prev);
stopCond = 1;
end
end
end
% Early stopping at specified number of variables
if stop < 0
stopCond = length(A) >= -stop;
end
end
% trim beta
if storepath && size(b,2) > step
b(:,step + 1:end) = [];
end
%% Compute auxilliary measures
if nargout == 2 % only compute if asked for
info.steps = step - 1;
b0 = pinv(X)*y; % regression coefficients of low-bias model
penalty0 = sum(abs(b0)); % L1 constraint size of low-bias model低偏置模型
indices = (1:p)';
if storepath % for entire path
q = info.steps + 1;
info.df = zeros(1,q);
info.Cp = zeros(1,q);
info.AIC = zeros(1,q);
info.BIC = zeros(1,q);
info.s = zeros(1,q);
sigma2e = sum((y - X*b0).^2)/n;
for step = 1:q
A = indices(b(:,step) ~= 0); % active set激活集，有效集，也就是每一列不为0的那几个数的索引
% compute godness of fit measurements Cp, AIC and BIC
r = y - X(:,A)*b(A,step); % residuals残差
rss = sum(r.^2); % residual sum-of-squares残差平方和
info.df(step) = step - 1;
info.Cp(step) = rss/sigma2e - n + 2*info.df(step);
info.AIC(step) = rss + 2*sigma2e*info.df(step);
info.BIC(step) = rss + log(n)*sigma2e*info.df(step);
info.s(step) = sum(abs(b(A,step)))/penalty0;
end
else % for single solution
info.s = sum(abs(b))/penalty0;
info.df = info.steps;
end
四、CARS算法
竞争性自适应重加权采样法（competitive adapative reweighted sampling， CARS）是一种结合蒙特卡洛采样与PLS模型回归系数的特征变量选择方法，模仿达尔文理论中的 ”适者生存“ 的原则（Li et al., 2009）。CARS 算法中，每次通过自适应加权采样（adapative reweighted sampling， ARS）保留PLS模型中 回归系数绝对值权重较大的点作为新的子集，去掉权值较小的点，然后基于新的子集建立PLS模型，经过多次计算，选择PLS模型交互验证均方根误差（RMSECV）最小的子集中的波长作为特征波长
function F=carspls(X,y,A,fold,method,num)
tic;
%+++ Initial settings.
if nargin<6;num=50;end;
if nargin<5;method='center';end;
if nargin<4;fold=5;end;
if nargin<3;A=2;end;
%+++ Initial settings.
[Mx,Nx]=size(X);
A=min([Mx Nx A]);
index=1:Nx;
ratio=0.9;
r0=1;
r1=2/Nx;
Vsel=1:Nx;
Q=floor(Mx*ratio);
W=zeros(Nx,num);
Ratio=zeros(1,num);
%+++ Parameter of exponentially decreasing function.
b=log(r0/r1)/(num-1); a=r0*exp(b);
%+++ Main Loop
for iter=1:num
perm=randperm(Mx);
Xcal=X(perm(1:Q),:); ycal=y(perm(1:Q)); %+++ Monte-Carlo Sampling.
PLS=pls(Xcal(:,Vsel),ycal,A,method); %+++ PLS model
w=zeros(Nx,1);coef=PLS.coef_origin(1:end-1,end);
w(Vsel)=coef;W(:,iter)=w;
w=abs(w); %+++ weights
[ws,indexw]=sort(-w); %+++ sort weights
ratio=a*exp(-b*(iter+1)); %+++ Ratio of retained variables.
Ratio(iter)=ratio;
K=round(Nx*ratio);
w(indexw(K+1:end))=0; %+++ Eliminate some variables with small coefficients.
Vsel=randsample(Nx,Nx,true,w); %+++ Reweighted Sampling from the pool of retained variables.
Vsel=unique(Vsel);
fprintf('The %dth variable sampling finished.\n',iter); %+++ Screen output.
end
%+++ Cross-Validation to choose an optimal subset;
RMSEP=zeros(1,num);
Q2_max=zeros(1,num);
Rpc=zeros(1,num);
for i=1:num
vsel=find(W(:,i)~=0);
CV=plscvfold(X(:,vsel),y,A,fold,method,0);
RMSEP(i)=CV.RMSECV;
Q2_max(i)=CV.Q2_max;
Rpc(i)=CV.optPC;
fprintf('The %d/%dth subset finished.\n',i,num);
end
[Rmin,indexOPT]=min(RMSEP);
Q2_max=max(Q2_max);
%+++ save results;
time=toc;
%+++ output
F.W=W;
F.time=time;
F.cv=RMSEP;
F.Q2_max=Q2_max;
F.minRMSECV=Rmin;
F.iterOPT=indexOPT;
F.optPC=Rpc(indexOPT);
Ft.ratio=Ratio;
F.vsel=find(W(:,indexOPT)~=0)';
function sel=weightsampling_in(w)
%Bootstrap sampling
%2007.9.6,H.D. Li.
w=w/sum(w);
N1=length(w);
min_sec(1)=0; max_sec(1)=w(1);
for j=2:N1
max_sec(j)=sum(w(1:j));
min_sec(j)=sum(w(1:j-1));
end
% figure;plot(max_sec,'r');hold on;plot(min_sec);
for i=1:N1
bb=rand(1);
ii=1;
while (min_sec(ii)>=bb | bb>max_sec(ii)) & ii<N1;
ii=ii+1;
end
sel(i)=ii;
end % w is related to the bootstrap chance
%+++ subfunction: booststrap sampling
% function sel=bootstrap_in(w);
% V=find(w>0);
% L=length(V);
% interval=linspace(0,1,L+1);
% for i=1:L;
% rn=rand(1);
% k=find(interval<rn);
% sel(i)=V(k(end));
% end
总结
完整代码可从获得GitHub仓库 如果对您有用，请点赞！
代码仅供学术使用，如有问题，联系方式