gray system prediction
In the solution of the problem, the GM (1,1) gray system theory is used to predict, and according to the comprehensive data of many known elements in the system, the time series of the data is fitted according to the differential equation to approximate the time series mentioned above The described dynamic process can then be extrapolated to achieve the purpose of prediction. Generally speaking, the gray system theory has better predictability, effectively overcomes the good characteristics of "less data" modeling to seek realistic laws, and overcomes insufficient data or systematic Paradox of short cycle. The model fitted by the gray system theory is a first-order differential equation of time series, so the GM (1,1) model is abbreviated.
Note: When using gray forecasting to forecast a small amount of data, try not to forecast too long in the future, and the error will be large.
1. Model establishment
GM (1,1 ) model:
The posterior difference test can be used to judge the accuracy of the model by referring to Table 1 :
Table 5-2-1 Posterior difference test discrimination reference table
| C |
model accuracy |
| <0.35 |
excellent |
| <0.5 |
pass |
| <0.65 |
good |
| >0.65 |
unqualified |
2. Examples (such as global temperature prediction)
gray forecasting model
GM(1,1) modeling
As shown in the figure, the red dots are the real data we input, and the blue dots are the data we predicted. By calculating the percentage absolute error is 0.082598%, the error is very small, so the result is convincing.
And based on the data before 2023, the temperature in 2023 and beyond can be predicted:
2023——
The predicted value is:
Table 5-2-2 Temperature forecast from 2023 to 2050
| forecast temperature |
years |
forecast temperature |
years |
forecast temperature |
|
| 2023 |
15.7692 |
2035 |
16.1319 |
2047 |
16.5028 |
| 2024 |
15.7991 |
2036 |
16.1625 |
2048 |
16.5341 |
| 2025 |
15.8291 |
2037 |
16.1931 |
2049 |
16.5655 |
| 2026 |
15.8591 |
2038 |
16.2238 |
2050 |
16.5969 |
| 2027 |
15.8892 |
2039 |
16.2546 |
||
| 2028 |
15.9193 |
2040 |
16.2854 |
||
| 2029 |
15.9495 |
2041 |
16.3163 |
||
| 2030 |
15.9798 |
2042 |
16.3472 |
||
| 2031 |
16.0101 |
2043 |
16.3782 |
||
| 2032 |
16.0404 |
2044 |
16.4093 |
||
| 2033 |
16.0709 |
2045 |
16.4404 |
||
| 2034 |
16.1013 |
2046 |
16.4716 |
3. The code is as follows (y is the predicted value of the input)
% Matlab的灰色预测程序：
%y=input('请输入数据');
clc;clear
y=[15.47141667 15.36183333 15.38175 15.40033333 15.47175 15.6155 15.75225 15.65633333 15.58066667 15.71575 15.74108333 15.58908333 15.6526]
n=length(y);
yy=ones(n,1);
yy(1)=y(1);
for i=2:n
yy(i)=yy(i-1)+y(i)
end
B=ones(n-1,2);
for i=1:(n-1)
B(i,1)=-(yy(i)+yy(i+1))/2;
B(i,2)=1;
end
BT=B';
for j=1:(n-1)
YN(j)=y(j+1);
end
YN=YN';
A=inv(BT*B)*BT*YN;
a=A(1);
u=A(2);
t=u/a;
t_test=input('输入需要预测的个数');
i=1:t_test+n;
yys(i+1)=(y(1)-t).*exp(-a.*i)+t;
yys(1)=y(1);
for j=n+t_test:-1:2
ys(j)=yys(j)-yys(j-1);
end
x=1:n;
xs=2:n+t_test;
yn=ys(2:n+t_test);
plot(x,y,'^r',xs,yn,'*-b');
det=0;
for i=2:n
det=det+abs(yn(i)-y(i));
end
det=det/(n-1);
disp(['百分绝对误差为：',num2str(det),'%']);
disp(['预测值为：',num2str(ys(n+1:n+t_test))]);
%
Of course, there is a simpler code that can still make predictions:
clc,clear
x0=[15.47141667 15.36183333 15.38175 15.40033333 15.47175 15.6155 15.75225 15.65633333 15.58066667 15.71575 15.74108333 15.58908333 15.6526];
n=length(x0);
lamda=x0(1:n-1)./x0(2:n)
range=minmax(lamda)
x1=cumsum(x0)
for i=2:n
z(i)=0.5*(x1(i))+x1(i-1);
end
B=[-z(2:n)',ones(n-1,1)];
Y=x0(2:n)';
u=B\Y
x=dsolve('Dx+a*x=b','x(0)=x0');
x=subs(x,{'a','b','x0'},{u(1),u(2),x1(1)});
yuce1=subs(x,'t',[0:n-1]);
digits(6),y=vpa(x)
yuce=[x0(1),diff(yuce1)]
epsilon=x0-yuce %计算残差
delta=abs(epsilon./x0) %相对误差
rho=1-(1-0.5*u(1))/(1+0.5*u(1))*lamda %计算级比偏差值