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数学建模：灰色预测模型
文章目录
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
%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);