%下面程序是灰色模型GM(1,1)程序二次拟合和等维新陈代谢改进预测程序,matlab6.5 ,使用本程序请注明，程序存储为gm1.m
%x = [5999,5903,5848,5700,7884];gm1(x);  测试数据
%二次拟合预测GM(1,1)模型
function  gmcal=gm1(x)
sizexd2 = size(x,2);
%求数组长度
k=0;
for y1=x
    k=k+1;
    if k>1
        x1(k)=x1(k-1)+x(k);
        %累加生成
        z1(k-1)=-0.5*(x1(k)+x1(k-1));
        %z1维数减1，用于计算B
        yn1(k-1)=x(k);
    else
        x1(k)=x(k);
    end
end

%x1,z1,k,yn1
sizez1=size(z1,2);
%size(yn1);
z2 = z1';
z3 = ones(1,sizez1)';
YN = yn1';   %转置
%YN
B=[z2 z3];
au0=inv(B'*B)*B'*YN;
au = au0';
%B,au0,au
afor = au(1);
ufor = au(2);
ua = au(2)./au(1);
ufor,ufor,ua
%输出预测的  a u 和 u/a的值

constant1 = x(1)-ua;
afor1 = -afor;
x1t1 = 'x1(t+1)';
estr = 'exp';
tstr = 't';
leftbra = '(';
rightbra = ')';
%constant1,afor1,x1t1,estr,tstr,leftbra,rightbra

strcat(x1t1,'=',num2str(constant1),estr,leftbra,num2str

(afor1),tstr,rightbra,'+',leftbra,num2str(ua),rightbra)
%输出时间响应方程
%******************************************************

%二次拟合
k2 = 0;
for y2 = x1
    k2 = k2 + 1;
    if k2 > k
    else
        ze1(k2) = exp(-(k2-1)*afor);
    end
end
%ze1

sizeze1 = size(ze1,2);
z4 = ones(1,sizeze1)';
G=[ze1' z4];
X1 = x1';
au20=inv(G'*G)*G'*X1;
au2 = au20';
%z4,X1,G,au20

Aval = au2(1);
Bval = au2(2);
%Aval,Bval

%输出预测的  A,B的值

strcat(x1t1,'=',num2str(Aval),estr,leftbra,num2str
(afor1),tstr,rightbra,'+',leftbra,num2str(Bval),rightbra)

%输出时间响应方程

nfinal = sizexd2-1 + 1;

%决定预测的步骤数5  这个步骤可以通过函数传入
%nfinal = sizexd2 - 1 + 1;

%预测的步骤数 1
for  k3=1:nfinal
    x3fcast(k3) = constant1*exp(afor1*k3)+ua;
end
%x3fcast

%一次拟合累加值
for  k31=nfinal:-1:0
    if k31>1
        x31fcast(k31+1) = x3fcast(k31)-x3fcast(k31-1);
    else
        if k31>0
            x31fcast(k31+1) = x3fcast(k31)-x(1);
        else
            x31fcast(k31+1) = x(1);
        end
    end
end
x31fcast

%一次拟合预测值
for  k4=1:nfinal
    x4fcast(k4) = Aval*exp(afor1*k4)+Bval;
end
%x4fcast

for  k41=nfinal:-1:0
    if k41>1
        x41fcast(k41+1) = x4fcast(k41)-x4fcast(k41-1);
    else
        if k41>0
            x41fcast(k41+1) = x4fcast(k41)-x(1);
        else
            x41fcast(k41+1) = x(1);
        end
    end
end

x41fcast,x

%二次拟合预测值

%***精度检验p C************//////////////////////////////////

k5 = 0;
for y5 = x
    k5 = k5 + 1;
    if k5 > sizexd2
    else
        err1(k5) = x(k5) - x41fcast(k5);
    end
end
%err1

%绝对误差
xavg = mean(x);
%xavg
%x平均值
err1avg = mean(err1);
%err1avg
%err1平均值
k5 = 0;
s1total = 0 ;
for y5 = x
    k5 = k5 + 1;
    if k5 > sizexd2
    else
        s1total = s1total + (x(k5) - xavg)^2;
    end
end
s1suqare = s1total ./ sizexd2;
s1sqrt = sqrt(s1suqare);
%s1suqare,s1sqrt
%s1suqare  残差数列x的方差  s1sqrt 为x方差的平方根S1

k5 = 0;
s2total = 0 ;
for y5 = x
    k5 = k5 + 1;
    if k5 > sizexd2
    else
        s2total = s2total + (err1(k5) - err1avg)^2;
    end
end

s2suqare = s2total ./ sizexd2;
%s2suqare   残差数列err1的方差S2

Cval = sqrt(s2suqare ./ s1suqare);
Cval
%nnn = 0.6745 * s1sqrt

%Cval  C检验值

k5 = 0;
pnum = 0 ;
for y5 = x
    k5 = k5 + 1;
    if abs( err1(k5) - err1avg ) < 0.6745 * s1sqrt
        pnum = pnum + 1;
        %ppp = abs( err1(k5) - err1avg )
    else
    end
end

pval = pnum ./ sizexd2;
pval
%p检验值
%arr1 = x41fcast(1:6)
%预测结果为区间范围  预测步长和数据长度可调整程序参数进行改进
