🔆 文章首发于我的个人博客：欢迎大佬们来逛逛

数学建模：灰色预测模型

文章目录

数学建模：灰色预测模型
灰色预测
- 算法步骤
- 代码实现

灰色预测

三个基本方法：

累加数列：计算一阶累加生成数列

x ( 1 ) ( k ) = ∑ i = 1 k x ( 0 ) ( i ) , k = 1 , 2 , ⋯ , n , x^{(1)}(k)=\sum_{i=1}^kx^{(0)}(i),k=1,2,\cdots,n, x(1)(k)=i=1∑kx(0)(i),k=1,2,⋯,n,

累减数列：计算一阶累减生成数列

x ( 0 ) ( k ) = x ( 1 ) ( k ) − x ( 1 ) ( k − 1 ) , k = 2 , 3 , ⋯ , n , x^{(0)}(k)=x^{(1)}(k)-x^{(1)}(k-1),k=2,3,\cdots,n, x(0)(k)=x(1)(k)−x(1)(k−1),k=2,3,⋯,n,

加权累加 ：计算一阶等权邻接生成数，图片描述有误，此处计算的是一次累加的加权邻值生成

z ( 0 ) ( k ) = 0.5 x ( 1 ) ( k ) + 0.5 x ( 1 ) ( k − 1 ) , k = 2 , 3 , ⋯ , n , z^{(0)}(k)=0.5x^{(1)}(k)+0.5x^{(1)}(k-1),k=2,3,\cdots,n, z(0)(k)=0.5x(1)(k)+0.5x(1)(k−1),k=2,3,⋯,n,

算法步骤

进行级比检验，检查是否满足建立微分方程的前提条件。

λ ( k ) = x ( 0 ) ( k − 1 ) x ( 0 ) ( k ) \lambda(k)=\frac{x^{(0)}(k-1)}{x^{(0)}(k)} λ(k)=x(0)(k)x(0)(k−1)

对原数据做一次累加，计算加权邻值生成数
构造数据矩阵 B B B ，与数据向量 Y Y Y

B = $- 1 2 ( x ( 1 ) ( 1 ) + x ( 1 ) ( 2 ) - 1 2 ( x ( 1 ) ( 2 ) + x ( 1 ) ( 3 ) ) 1 ⋮ ⋮ - 1 2 ( x ( 1 ) ( n - 1 ) + x ( 1 ) ( n ) )$ , Y = $x ( 0 ) ( 2 ) x ( 0 ) ( 3 ) ⋮ x ( 0 ) ( n )$ B~=\left $\\begin{array}{ccccc}-\\dfrac{1}{2}\\big(x\^{(1)}\\big(1\\big)+x\^{(1)}\\big(\&2\\big)\&\\\\-\\dfrac{1}{2}\\big(x\^{(1)}\\big(2\\big)+x\^{(1)}\\big(\&3\\big)\&\\big)\&1\\\\\&\\vdots\&\&\\vdots\\\\-\\dfrac{1}{2}\\big(x\^{(1)}\\big(n-1\\big)+x\^{(1)}\\big(\&n\\big)\&\\big)\&\\end{array}\\right$ ,Y~=\left $\\begin{array}{ccc}x\^{(0)}\\big(\&2\\big)\\\\x\^{(0)}\\big(\&3\\big)\\\\\\vdots\\\\x\^{(0)}\\big(\&n\\big)\\end{array}\\right$ B = −21(x(1)(1)+x(1)(−21(x(1)(2)+x(1)(−21(x(1)(n−1)+x(1)(2)3)⋮n)))1⋮ ,Y = x(0)(x(0)(⋮x(0)(2)3)n)

计算 a a a 与 b b b 的值

u ^ = ( a ^ , b ^ ) T = ( B T ⋅ B ) − 1 B T Y \hat{u}=(\hat{a},\hat{b})^T=(B^T\cdot B)^{-1}B^TY u^=(a^,b^)T=(BT⋅B)−1BTY

构建模型

x ( 1 ) ( t ) = ( x ( 0 ) ( 1 ) − b a ) e − a ( t − 1 ) + b a . x^{(1)}(t)=(x^{(0)}(1)-\frac ba)e^{-a(t-1)}+\frac ba. x(1)(t)=(x(0)(1)−ab)e−a(t−1)+ab.

计算生成模型值 x ^ ( 1 ) ( k ) \hat{x}^{(1)}(k) x^(1)(k) 和模型还原值 x ^ ( 0 ) ( k ) \hat{x}^{(0)}(k) x^(0)(k) 并且带入预测

x ^ ( 0 ) ( k ) = x ^ ( 1 ) ( k ) − x ^ ( 1 ) ( k − 1 ) \hat{x}^{(0)}(k)=\hat{x}^{(1)}(k)-\hat{x}^{(1)}(k-1) x^(0)(k)=x^(1)(k)−x^(1)(k−1)

检验预测值

代码实现

matlab 复制代码

%95至04年数据
clc;clear;
data = [174 179 183 189 207 234 220.5 256 270 285];
n = length(data);
 
%% 级比检验通过
check = [];
for k = 2:n
    lambda(k) = data(k-1)/data(k);
    if (exp(-2/(n+1))<lambda(k))&&(lambda(k)<exp(2/(n+1)))
        check(end+1) = 1;
    else check(end+1) = 0;
    end
end 
 
%% 计算累加数列
X1 = cumsum(data);

%% 计算加权
for i=2:n
    z(i) = 0.5*(X1(i-1)+X1(i));
end
 
%% 数据矩阵B及数据向量Y
Y = data(2:n)';
B = [-z(2:n)',ones(n-1,1)];
u = (B'*B)\B'*Y;
% u = B\Y; 表示B的逆 乘以 Y
a = u(1,1);
b = u(2,1);
 
%% 构造模型并且带入预测值
% 生成预测一次累加数列
f_X1 = [];
f_X0 = [];
for k=1:n-1
    f_X1(1)=data(1);
    f_X1(k+1) = (data(1)-b/a)*exp(-a*k) + b/a;
end
% 前缀和反推原始数据
for k=2:n
    f_X0(1)=data(1);
    f_X0(k)=f_X1(k)-f_X1(k-1);
end
 
%% 残差检验 与 级比偏差值检验
for k=1:n-1
    sigma(k)=abs((data(k)-f_X0(k))/data(k));
    rho(k+1)=abs(1-((1-0.5*a)*lambda(k+1))/(1+0.5*a));
end

%% 预测下n个值
test = input('nums:');
nums = 5;
n=n+test;
f_f_X1 = [];
f_f_X0 = [];
for k=1:n-1
    f_f_X1(1)=data(1);
    f_f_X1(k+1) = (data(1)-b/a)*exp(-a*k) + b/a;
end
for k=2:n
    f_f_X0(1)=data(1);
    f_f_X0(k)=f_f_X1(k)-f_f_X1(k-1);
end

%% 绘图
xAxis = 1995:2004;
xAxisPredict = 1995:1995+n-1; 
h = plot(xAxis,data,'o',xAxisPredict,f_f_X0,'-');
set(gca, 'XScale', 'log', 'YScale', 'log');
set(h,'LineWidth',1.5);