Grey forecast and itsMATLAB实现(一）

Grey forecasting is a conventional forecasting method,具有操作简便,Advantages such as less amount of data required,Generally just have4data can be predicted.

Grey forecasting is a forecasting method based on grey system theory.The grey system is developed by Professor Deng Julong, a famous scholar in my country1982年提出,是相对于“白色模型”——Fully informative model,和“black model”——A model concept for models that know nothing about information.The problems solved by using the gray system are mainly uncertain problems：

1. Information is ambiguous,It cannot be precisely characterized by mathematical equations.
2. The mechanism is uncertain.
3. Information-poor uncertainty.

Grey relational matrix

The grey correlation degree analyzes the correlation degree between vectors and between matrices and between matrices,The degree of correlation between vectors can be regarded as a special form of the degree of correlation between matrices.

计算关联度,It must be to calculate a sequence of numbers to be compared with a reference（That is, the reference sequence）之间的相关程度.

Suppose there is a set of reference sequences as follows：
$$x_j=(x_j(1),x_j(2),x_j(3),\cdots ,x_j(k),\cdots ,x_j(n))j=1,2,3,\cdots ,s$$
Suppose there is a set of arrays to be compared as follows：
$$x_i=(x_i(1),x_i(2),x_i(3),\cdots ,x_i(k),\cdots ,x_i(n))i=1,2,3,\cdots ,t$$
The correlation coefficient is defined as follows：
$${\xi }_{ji}(k)=\frac{\underset{i}{\min }\underset{k}{\min }|x_j(k)-x_i(k)|+\rho \cdot \underset{i}{\max }\underset{k}{\max }|x_j(k)-x_i(k)|}{|x_j(k)-x_i(k)|+\rho \cdot \underset{i}{\max }\underset{k}{\max }|x_j(k)-x_i(k)|}$$
The description of this formula is as follows：

1. 变量${\xi }_{ji}(k)$表示的是第$i$number sequence and number$j$个参考数列第$k$个样本之间的关联系数.
2. $\underset{i}{\min }\underset{k}{\min }|x_j(k)-x_i(k)|$ 和$\underset{i}{\max }\underset{k}{\max }|x_j(k)-x_i(k)|$Indicates the minimum and maximum values ​​of the difference between the reference sequence matrix and the comparison sequence matrix,目的是为了保证${\xi }_{ji}(k)$的值在[0,1]区间内,At the same time, the upper and lower symmetrical structure can eliminate the problems of different dimensions and orders of magnitude.
3. $|x_j(k)-x_i(k)|$即为汉明距离（“Hamming distance”）,The reciprocal of the Hamming distance is called the inverse reciprocal distance,The essence of gray correlation degree is to determine the degree of correlation by the size of the inverse reciprocal distance,倒数越大,Indicates that the distance between the two curves is smaller,The more similar the curves are.
4. $\rho$The value convention becomes[0,1],但实际上$\rho$的取值范围为$(0,+\infty )$.但不管$\rho$如何取值,It only changes${\xi }_{ji}(k)$的绝对大小,without changing the relative strength of the correlation.

由于${\xi }_{ji}(k)$It can only reflect the correlation between points and points,相关性信息分散,不方便刻画数列之间的相关性,需要把${\xi }_{ji}(k)$整合起来,定义：
$$r_{ji}=\frac{\stackrel{n}{\underset{k=1}{\Sigma }}{\xi }_{ji}(k)}{n}$$
变量$r_{ji}$is the correlation,Incorporate actual context,A positive effect is a positive correlation,反之则为负相关;$r_{ji}$大于0.7called strong correlation,小于0.3called weak correlation.

将$x_i$与$x_j$The correlations between are written in matrix form：
$$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1t}\\ r_{21}& r_{22}& \cdots & r_{2t}\\ ⋮& ⋮& \ddots & ⋮\\ r_{s1}& r_{s2}& \cdots & r_{st}\end{array}\right]$$
By observing that the value of one column is significantly larger than the value of other columns,This is called the dominant sub-factor;If the value of one row is significantly greater than the value of other rows,This behavior is called the dominant parent factor,The dominant parent factor is easily influenced by the driving influence of the child factor.

Programming of grey relational degree matrices

function [R]=GrayConnect(X,Y)
[xa,xb]=size(X);
[ya,yb]=size(Y);
if (xb==yb)
else
    return ;
end
R=zeros(ya,xa);
q=0.5;
for i = 1:ya
    k=zeros(xa,xb);
    for j=1:xa
        k(j,:)=abs(X(j,:)-Y(i,:));
    end
    temp1=min(min(k));
    temp2=q*max(max(k));
    for j=1:xa
        sum=0;
        for t=1:xb
        sum=sum+(temp1+temp2)/(abs(X(j,t)-Y(i,t))+temp2);
        end
        R(i,j)=sum/xb;
    end
end
end

将数据导入MATLAB,

A=xlsread('Grey.xlsx','B1:F11');

将数据标准化,

for i=1:11
    A(i,:)=A(i,:)/A(i,1);
end

Divide the parent and child factors into X和Y,并使用函数GrayConnect,

X=A(1:5,:);
Y=A(6:11,:);
[R]=GrayConnect(X,Y)

得到结果如下：
$$R=\left[\begin{array}{ccccc}0.8016& 0.7611& 0.5570& 0.8102& 0.9355\\ 0.6887& 0.6658& 0.5287& 0.8854& 0.8004\\ 0.8910& 0.8581& 0.5786& 0.5773& 0.6749\\ 0.6776& 0.6634& 0.5675& 0.7800& 0.7307\\ 0.8113& 0.7742& 0.5648& 0.8038& 0.9205\\ 0.7432& 0.7663& 0.5616& 0.6065& 0.6319\end{array}\right]$$