线性回归-OLS法

%%OLS法下的线性回归
function prodict = Linear_Regression(X,Y)
x = sym(‘x‘);
n = max(size(X));
%%定义画图窗格属性
h = figure;
set(h,‘color‘,‘w‘);
%%回归相关值
XX_s_m = (X-Expection(X,1))*(X-Expection(X,1))‘;
XY_s_m = (X-Expection(X,1))*(Y-Expection(Y,1))‘;
YY_s_m = (Y-Expection(Y,1))*(Y-Expection(Y,1))‘;
%%回归系数
beta = XY_s_m/XX_s_m;
alpha = Expection(Y,1)-beta*Expection(X,1);
%%画出实际值与回归值的差
yyaxis left
for k = 1:n
    delta_Y(k) = Y(k) - (alpha+beta*X(k));
end
width = (max(X)-min(X))/(2*n);
b = bar(X,delta_Y,width,‘Facecolor‘,[0.9 0.9 0.9]);
set(b,‘Edgecolor‘,‘none‘);
%标出差值
disp(‘是否标出差值？‘);
str = input(‘yes or no?‘,‘s‘);
Y_dis = (max(Y) - min(Y))/40;
if strcmp(str,‘yes‘) == 1
    if n<=10
        for k = 1:length(delta_Y)
            if delta_Y(k)>=0
                text(X(k),delta_Y(k)+Y_dis,strcat(num2str(delta_Y(k),‘%.3f‘)),...
                    ‘VerticalAlignment‘,‘middle‘,‘HorizontalAlignment‘,‘center‘);
            else
                text(X(k),delta_Y(k)-Y_dis,strcat(num2str(delta_Y(k),‘%.3f‘)),...
                    ‘VerticalAlignment‘,‘middle‘,‘HorizontalAlignment‘,‘center‘);
            end
        end
    end
end
xlabel(‘X‘);ylabel(‘差值‘);
hold on
%%画出数据点
yyaxis right
plot(X,Y,‘g--*‘)
if n<=20
    grid on;
end
hold on
ylabel(‘Y‘);
%%相关系数
R2 = (XY_s_m)^2/(XX_s_m*YY_s_m);
%%随机项方差无偏估计
sigma_2 = (Y*Y‘-n*Expection(Y,1)^2-beta*(X*Y‘-n*Expection(X,1)*Expection(Y,1)))/(n-2);
%%beta和alpha的方差
Var_beta = sigma_2/(X*X‘-n*Expection(X,1)^2);
Var_alpha = sigma_2*(X*X‘)/(n*(X*X‘-n*Expection(X,1)^2));
%%beta和alpha的协方差
cov_alpha_beta = -Expection(X,1)*sigma_2/(X*X‘-n*Expection(X,1)^2);
%%画出回归直线
x_v = min(X)-Expection(X,1)/sqrt(n):(max(X)-min(X))/100:max(X)+Expection(X,1)/sqrt(n);
y_v = beta*x_v+alpha;
y_ave = ones(1,max(size(x_v)))*Expection(Y,1);
plot(x_v,y_v,‘r‘,x_v,y_ave,‘k‘,‘LineWidth‘,0.2,‘LineStyle‘,‘-‘)
%%标出数据点
disp(‘是否标出数据点？‘);
str = input(‘yes or no?‘,‘s‘);
if strcmp(str,‘yes‘) == 1
    if n <=10
        for k = 1:max(size(X))
            text(X(k),Y(k),[‘(‘,num2str(X(k)),‘,‘,num2str(Y(k)),‘)‘],...
                ‘color‘,‘b‘,‘VerticalAlignment‘,‘middle‘,‘HorizontalAlignment‘,‘center‘);hold on;
        end
    end
end
disp(‘是否输出回归方程的置信区间？‘);
str = input(‘yes or no?‘,‘s‘);
if strcmp(str,‘yes‘) == 1
    str =  input(‘单侧置信系数 或者 双侧置信系数？‘,‘s‘);
    if strcmp(str(1),‘单‘) == 1
        ALPHA = input(‘输入单侧置信系数：‘);
        a = 1-ALPHA*2;
    elseif strcmp(str(1),‘双‘) == 1
        ALPHA = input(‘输入双侧置信系数：‘);
        a = 1-ALPHA;
    end
    disp(‘t(n-2)为：‘);
    t_value = nctinv(a,n-2,0)
    addvalue = t_value*sqrt(sigma_2)*sqrt(1+1/n+(x-Expection(X,1))^2/(X*X‘-n*Expection(X,1)^2));
    disp(‘置信区间上限的方程为：‘);
    vpa(alpha+beta*x+addvalue,4)
    disp(‘置信区间下限的方程为：‘);
    vpa(alpha+beta*x-addvalue,4)
    y_ub = subs(alpha+beta*x+addvalue,x_v);
    y_lb = subs(alpha+beta*x-addvalue,x_v);
    plot(x_v,y_ub,‘--‘,x_v,y_lb,‘-.‘,‘color‘,[0.2 0.5 0.8]);
    title(‘OLS法对数据的线性回归‘);
    legend(‘差值‘,‘Y:实际值‘,‘y:拟合直线‘,‘平均值‘,‘置信上限‘,‘置信下限‘);
    hold off
    yyaxis left
    text(min(X)-Expection(X,1)/sqrt(n),max(delta_Y)-Expection(delta_Y,1)/sqrt(n),[‘置信程度‘,num2str(100*a),‘%‘],‘color‘,[0.3 0.3 0.3]);
    hold off
else
    legend(‘差值‘,‘Y:实际值‘,‘y:拟合直线‘,‘平均值‘);
    title(‘OLS法对数据的线性回归‘);
    hold off
end
disp(‘是否输出回归系数的置信区间？‘);
str = input(‘yes or no?‘,‘s‘);
if strcmp(str,‘yes‘) == 1
    str =  input(‘单侧置信系数 或者 双侧置信系数？‘,‘s‘);
    if strcmp(str(1),‘单‘) == 1
        ALPHA = input(‘输入单侧置信系数：‘);
        a = 1-ALPHA*2;
    elseif strcmp(str(1),‘双‘) == 1
        ALPHA = input(‘输入双侧置信系数：‘);
        a = 1-ALPHA;
    end
    disp(‘t(n-2)为：‘);
    t_value = nctinv(a,n-2,0)
    disp(‘回归系数beta的置信区间为：‘);
    [beta-t_value*sqrt(Var_beta),beta+t_value*sqrt(Var_beta)]
    disp(‘回归系数alpha的置信区间为：‘);
    [Expection(Y,1)-(beta+t_value*sqrt(Var_beta))*Expection(X,1),...
        Expection(Y,1)-(beta-t_value*sqrt(Var_beta))*Expection(X,1)]
end
%%预测值输出
disp(‘是否需要求预测值？‘);
str = input(‘yes or no?‘,‘s‘);
if strcmp(str,‘yes‘) == 1
    X0 = input(‘输入预测向量：‘);
    str =  input(‘单侧置信系数 或者 双侧置信系数？‘,‘s‘);
    if strcmp(str(1),‘单‘) == 1
        ALPHA = input(‘输入单侧置信系数：‘);
        a = 1-ALPHA*2;
    elseif strcmp(str(1),‘双‘) == 1
        ALPHA = input(‘输入双侧置信系数：‘);
        a = 1-ALPHA;
    end
    disp(‘t(n-2)为：‘);
    t_value = nctinv(a,n-2,0)
    addvalue0 = t_value*sqrt(sigma_2).*sqrt(1+1/n+(X0-Expection(X,1)).^2/(X*X‘-n*Expection(X,1)^2));
    Y0 = alpha+beta*X0;
    prodict{1,1} = {‘预测点x‘};
    prodict{2,1} = {‘预测值y‘};
    prodict{3,1} = {‘置信区间‘};
    for k = 2:max(size(X0))+1
        interval(k-1,:) = [Y0(k-1)-addvalue0(k-1),Y0(k-1)+addvalue0(k-1)];
        prodict{1,k} = X0(k-1);
        prodict{2,k} = Y0(k-1);
        prodict{3,k} = interval(k-1,:);
        Y0_lb(k-1) = Y0(k-1)-addvalue0(k-1);
        Y0_ub(k-1) = Y0(k-1)+addvalue0(k-1);
    end
    h2 = figure(2);
    set(h2,‘color‘,‘w‘);
    plot(X0,Y0,‘r-*‘,X0,Y0_ub,‘g-o‘,X0,Y0_lb,‘b-o‘);hold on;
    errorbar(X0,Y0,addvalue0,‘color‘,[0.8 0.5 0.2]);
    xlabel(‘X‘);ylabel(‘y‘);
    grid on;
    title(‘点预测‘);
    legend(‘预测值‘,‘置信上限‘,‘置信下限‘,‘置信限‘);
    Y0_dis = (max(Y0)-min(Y0))/40;
    Y0_lb_dis = (max(Y0_lb)-min(Y0_lb))/40;
    Y0_ub_dis = (max(Y0_ub)-min(Y0_ub))/40;
    disp(‘是否显示数据？‘);
    str = input(‘yes or no?‘,‘s‘);
    if strcmp(str ,‘yes‘) == 1
        if max(size(X0)) <= 3
            for k = 1:max(size(X0))
                text(X0(k),Y0(k)+Y0_dis,[‘(‘,num2str(X0(k)),‘,‘,num2str(Y0(k)),‘)‘],...
                    ‘color‘,[0.2 0.2 0.2]);hold on;
                text(X0(k),Y0_lb(k)+Y0_lb_dis,[‘(‘,num2str(X0(k)),‘,‘,num2str(Y0_lb(k)),‘)‘],...
                    ‘color‘,[0.2 0.2 0.2],‘VerticalAlignment‘,‘middle‘,‘HorizontalAlignment‘,‘center‘);hold on;
                text(X0(k),Y0_ub(k)+Y0_ub_dis,[‘(‘,num2str(X0(k)),‘,‘,num2str(Y0_ub(k)),‘)‘],...
                    ‘color‘,[0.2 0.2 0.2],‘VerticalAlignment‘,‘middle‘,‘HorizontalAlignment‘,‘center‘);hold on;
            end
        end
    end
    text(Expection(X0,1),(max(Y0)+max(Y0_ub))/2,[‘置信程度‘,num2str(100*a),‘%‘],‘color‘,[0.3 0.3 0.3]);
else
    prodict = [];
end
disp(‘回归方程为：‘);
vpa(alpha+beta*x,4)
disp(‘相关系数平方为：‘);
vpa(R2,6)
disp(‘随机项方差的无偏估计为：‘);
vpa(sigma_2,6)
disp(‘回归系数beta和alpha的方差分别为：‘);
vpa(Var_beta,6)
vpa(Var_alpha,6)
disp(‘回归系数beta和alpha的协方差为：‘);
vpa(cov_alpha_beta,6)
%%SST,SSE和SSR
disp(‘SST：‘);
SST = YY_s_m
Y_OLS = reshape(alpha+beta*X‘,1,n);
disp(‘SSE：‘);
SSE = (Y_OLS-Expection(Y,1))*(Y_OLS-Expection(Y,1))‘
disp(‘SSR：‘);
SSR = (Y-Y_OLS)*(Y-Y_OLS)‘

    function En = Expection(M,n)                                          %%n阶原点矩%%
        En = sum(M.^n)/size(M,2);
    end
end

clear all
clc
X = [-2.269 -1.248 -0.269 1.248 2.006 4.598 5.264 7.002];
Y = [4582 4027 4958 5078 6072 4156 5948 5027];
S = Linear_Regression(X,Y)

是否标出差值？
yes or no?yes
是否标出数据点？
yes or no?no
是否输出回归方程的置信区间？
yes or no?yes
单侧置信系数 或者 双侧置信系数？双
输入双侧置信系数：0.05
t(n-2)为：

t_value =

    1.9432

置信区间上限的方程为：
 
ans =
 
78.43*x + 1466.0*(0.013*(x - 2.041)^2 + 1.125)^(1/2) + 4821.0
 
置信区间下限的方程为：
 
ans =
 
78.43*x - 1466.0*(0.013*(x - 2.041)^2 + 1.125)^(1/2) + 4821.0
 
是否输出回归系数的置信区间？
yes or no?yes
单侧置信系数 或者 双侧置信系数？双
输入双侧置信系数：0.05
t(n-2)为：

t_value =

    1.9432

回归系数beta的置信区间为：

ans =

  -88.7367  245.5885

回归系数alpha的置信区间为：

ans =

   1.0e+03 *

    4.4796    5.1622

是否需要求预测值？
yes or no?yes
输入预测向量：[-2.156 -1.306 0.248 1.296]
单侧置信系数 或者 双侧置信系数？双
输入双侧置信系数：0.05
t(n-2)为：

t_value =

    1.9432

是否显示数据？
yes or no?no
回归方程为：
 
ans =
 
78.43*x + 4821.0
 
相关系数平方为：
 
ans =
 
0.121668
 
随机项方差的无偏估计为：
 
ans =
 
569067.0
 
回归系数beta和alpha的方差分别为：
 
ans =
 
7400.35
 
 
ans =
 
101976.0
 
回归系数beta和alpha的协方差为：
 
ans =
 
-15107.8
 
SST：

SST =

     3887366

SSE：

SSE =

   4.7297e+05

SSR：

SSR =

   3.4144e+06


S =

  3×5 cell 数组

  列 1 至 4

    {1×1 cell}    {[   -2.1560]}    {[   -1.3060]}    {[    0.2480]}
    {1×1 cell}    {[4.6518e+03]}    {[4.7185e+03]}    {[4.8403e+03]}
    {1×1 cell}    {1×2 double  }    {1×2 double  }    {1×2 double  }

  列 5

    {[    1.2960]}
    {[4.9225e+03]}
    {1×2 double  }