建模之数据处理常用方法

灰色系统模型

灰色系统模型在数据处理和预测中经常使用。
灰色系统理论建模特点：原始数据必须等时间间距。
处理思路：首先对原始数据进行累加，弱化原始时间序列数据的随
机因素.然后建立生成数的微分方程。
GM(1,1)模型是灰色系统理论中的单序列一阶灰色微分方程

 Matlab实现程序

%1997--2003年数据
A=[83.0, 79.8,78.1,85.1,86.6,88.2,90.3,86.7,93.3,92.5,90.9,96.9;
101.7,85.1,87.8,91.6,93.4,94.5,97.4,99.5,104.2,102.3,101.0,123.5;
92.2,114.0,93.3,101.0,103.5,105.2,109.5,109.2,109.6,111.2,121.7,131.3;
105.0,125.7,106.6,116.0,117.6,118.0,121.7,118.7,120.2,127.8,121.8,121.9;
139.3,129.5,122.5,124.5,135.7,130.8,138.7,133.7,136.8,138.9,129.6,133.7;
137.5,135.3,133.0,133.4,142.8,141.6,142.9,147.3,159.6,162.1,153.5,155.9;
163.2,159.7,158.4,145.2,124,144.1,157.0,162.6,171.8,180.7,173.5,176.5];
T=A(1:6,1:12);
x0=mean(T');%对前6年求平均
x1=zeros(size(x0));
n=length(x0);
x1(1)=x0(1);
for i=2:n
x1(i)=x1(i-1)+x0(i); %累积求和
end
z=zeros(size(x0));
af=0.4; %参数
for i=2:n
z(i)=af*x1(i)+(1-af)*x1(i-1);
end
Y=zeros(n-1,1); B=zeros(n-1,2);
for i=2:n
Y(i-1,1)=x0(i);
B(i-1,1)=-z(i);
B(i-1,2)=1;
end
Para=inv(B'*B)*B'*Y; %计算参数
a=Para(1);
b=Para(2);
Pred=(x0(1)-b/a)*exp(-a*n)*(1-exp(a));
%预测第n+1年数值(2003年)
Total=12*Pred; %2003年总平均值
%估计各月所占比重；
r=sum(T)/sum(sum(T));
%预测2003年各月销售量
Px=Total*r;
fprintf('输出2003年预测值与实际值.\n');
for i=1:12
fprintf('%5d ',i);
end
fprintf('\n');
for i=1:12
fprintf('%6.1f ',Px(i)); %输出2003年预测值
end
fprintf('\n');
for i=1:12
fprintf('%6.1f ',A(7,i)); %输出2003年实际值
end
fprintf('\n');
Error=sum(Px(4:6))-sum(A(7,4:6));
fprintf('2003年4,5,6月SARS导致减少销售额%6.2f亿元\n',Error);
%作图
subplot(2,1,1);
PA=[A(1,:),A(2,:),A(3,:),A(4,:),A(5,:),A(6,:),A(7,:)];%变为一行数据
plot(PA); grid on
title('原始数据');
subplot(2,1,2);
plot(1:12,A(7,:),'b*',1:12,Px,'r');
title('2003年对比数据');
grid on

 三、实例计算
根据某地6年每年12个月的交通死亡数据。预测未来一年每个月的交通死亡人数。数据见表一。

 Matlab程序

x=[9007,8106,8928,9137,10017,10826,11317,10744,9713,9938,9161,8927,7750,6981,8038,8422,8714,9512,10120,9823,8743,9129,8710,8680,8162,7306,8124,7870,9387,9556,10093,9620,8285,8433,8160,8034,7717,7461,7776,7925,8634,8945,10078,9179,8037,8488,7874,8647,7792,6957,7726,8106,8890,9299,10625,9302,8314,8850,8265,8796,7836,6892,7791,8129,9115,9434,10484,9827,9110,9070,8633,9240];
D=[9007,8106,8928,9137,10017,10826,11317,10744,9713,9938,9161,8927;
7750,6981,8038,8422,8714,9512,10120,9823,8743,9129,8710,8680;
8162,7306,8124,7870,9387,9556,10093,9620,8285,8433,8160,8034;
7717,7461,7776,7925,8634,8945,10078,9179,8037,8488,7874,8647;
7792,6957,7726,8106,8890,9299,10625,9302,8314,8850,8265,8796;
7836,6892,7791,8129,9115,9434,10484,9827,9110,9070,8633,9240];
aver=mean(D');
st=zeros(6,12);
for i=1:6
for j=1:12
st(i,j)=D(i,j)-aver(i);
end
end
NST=zeros(1,12);
nst=sum(st)/6; %对6年月平均作为st的估计
nx=zeros(72,1);
for i=1:6
for j=1:12
k=(i-1)*12+j; nx(k)=x(k)-nst(j);
end
end
%对消去季节项后数据nx
%进行线性拟合并预测
Y=zeros(72,1);
A=zeros(72,2);
for i=1:72
Y(i)=nx(i);
A(i,1)=1; A(i,2)=i;
end
coef=inv(A'*A)*A'*Y;
py=zeros(1,84);
for i=1:84
py(i)=coef(1)+coef(2)*i;
end
subplot(2,1,1);
plot(1:72,nx,1:72,py(1:72));
xx=zeros(1,84);
for i=1:7
for j=1:12
k=(i-1)*12+j;
xx(k)=py(k)+nst(j); %预测各月数值
end
end
subplot(2,1,2);
plot(1:72,x,'*',1:84,xx);

 神经网络

x=[1.24,1.36,1.38,1.38,1.38,1.40,1.48,1.54,1.56,1.14,1.18,1.20,1.26,1.28,1.30;
1.72,1.74,1.64,1.82,1.90,1.70,1.82,1.82,2.08,1.78,1.96,1.86,2.0, 2.0,1.96];
y=[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,1,1,1,1,1,1];
xmin=min(x'); %求各指标最小值
xmax=max(x'); %求各指标最大值
net.trainParam.epochs=2500; %设定迭代步数
net=newff([xmin',xmax'],[5,2],{'logsig','logsig'}); %初始化网络
net=train(net,x,y); %进行网络训练
x1=[1.24,1.28,1.40;
1.80,1.84,2.04];%待分样本
y1=sim(net,x1); %数据泛化
plot(x(1,1:9),x(2,1:9),'*',x(1,10:15),x(2,10:15),'o',x1(1,:),x1(2,:),'p') %画数据图
grid on