时间序列

数学建模时间序列

原文：http://blog.youkuaiyun.com/qq_34861102/article/details/77075424

---

• 趋势移动平均法：
  
  • 最简单的，使用线性的预测

• 指数平滑预测模型：

  • 二次指数平滑

    

  • 三次指数平滑

    

曲线预测模型：

• Compertz曲线
• 生长曲线
• 修正指数曲线

ARMA模型：

• armax(p,q)函数
  
• 定阶
  我想的话，直接用aic评估一下，画个图，看哪个点的指标最小

 aic(m)

• 预测

predict和forecast

---

正文：

• 1

yt = [80.8 94.0 88.4 101.5 110.3 121.5 134.7 142.7];
m = length(yt);
n = 3;
%未来一步的预测
for i = n+1:m+1
    ythat(i) = sum(yt(i-n:i-1))/n;
end
ythat
%未来多步的预测
for i = m+1:m+3
    yt(i) = ythat(i);
    ythat(i+1) = sum(yt(i-n+1:i))/n;
end
yhat = ythat(end-3:end)
s1 = sqrt(mean(yt(n+1:m)-ythat(n+1:m)).^2)

• 2

  • alpha = 0.6;
     yt = [80.8 94.0 88.4 101.5 110.3 121.5 134.7 142.7];
     n = length(yt);
     st1(1) = mean(yt(1:3));
     st2(1) = st1(1);
     for i = 2:n
        st1(i) = alpha*yt(i) + (1-alpha)*st1(i-1);
        st2(i) = alpha*st1(i) + (1-alpha)*st2(i-1);
     end
     at = 2*st1 - st2;
     bt = alpha/(1-alpha)*(st1-st2);
     yhat = at + bt;
     sigma = sqrt(mean((yt(2:end)-yhat(1:end-1)).^2));
     m = 1:4;
     yhat2 = at(end) + bt(end)*m;

  • alpha = 0.3;
     yt = [80.8 94.0 88.4 101.5 110.3 121.5 134.7 142.7];
     n = length(yt);
     st1(1) = mean(yt(1:3));
     st2(1) = st1(1);
     for i = 2:n
        st1(i) = alpha*yt(i) + (1-alpha)*st1(i-1);
        st2(i) = alpha*st1(i) + (1-alpha)*st2(i-1);
     end
     at = 2*st1 - st2;
     bt = alpha/(1-alpha)*(st1-st2);
     yhat = at + bt;
     sigma = sqrt(mean((yt(2:end)-yhat(1:end-1)).^2));
     m = 1:4;
     yhat2 = at(end) + bt(end)*m;

• 同（1）中的指数模型编写方法，参考三次指数：

  

• 修正指数曲线

  • t = 1969:1983;
    yt = [3.78 4.19 4.83 5.46 6.71 7.99 8.60 9.24 9.67 9.87 10.49 10.92 10.93 12.39 12.59];
    plot(t,yt,'* -');
    n = length(yt);
    
    
    m = n/3;
    dyt = diff(yt);
    for i = 1:n-2
        jy(i) = dyt(i+1)/dyt(i);
    end
    fw = minmax(jy);
    s1 = sum(yt(1:m));
    s2 = sum(yt(m+1:2*m));
    s3 = sum(yt(2*m+1:end));
    b = ((s3-s2)/(s2-s1))^(1/m);
    a = (s2 - s1)*(b - 1)/(b * (b^m - 1)^2);
    k = (s1 - a*b*(b^m-1)/(b-1))/m;
    yuce = @(t) k + a*b.^t;
    bzcha = sqrt(mean((yt - yuce(1:n)).^2))
    ythat = yuce([n+2,n+7])

    

    

• Compertz曲线

  解答：

  

  y = [3.78 4.19 4.83 5.46 6.71 7.99 8.60 9.24 9.67 9.87 10.49 10.92 10.93 12.39 12.59];
  yt = log(y);
  n = length(yt);
  m = n/3;
  s1 = sum(yt(1:m));
  s2 = sum(yt(m+1:2*m));
  s3 = sum(yt(2*m+1:end));
  b = ((s3 - s2)/(s2 - s1))^(1/m);
  a2 = (s2 - s1)*(b - 1)/(b*(b^m - 1)^2);
  k2 = (s1 - a2*b*(b^m - 1)/(b - 1))/m;
  a = exp(a2);
  k = exp(k2);
  yuce = @(t) k*a.^(b.^t);
  yhat = yuce(1:n);
  bzcha = sqrt(mean((y-yhat).^2))
  ythat = yuce([n+2,n+7])

• 生长曲线

  
  

  y = [3.78 4.19 4.83 5.46 6.71 7.99 8.60 9.24 9.67 9.87 10.49 10.92 10.93 12.39 12.59];
  yt = 1./y;
  n = length(yt);
  m = n/3;
  s1 = sum(yt(1:m));
  s2 = sum(yt(m+1:2*m));
  s3 = sum(yt(2*m+1:end));
  b = ((s3-s2)/(s2-s1))^(1/m);
  a = (s2 - s1)*(b - 1)/(b * (b^m - 1)^2);
  k = (s1 - a*b*(b^m-1)/(b-1))/m;
  yuce = @(t)1./(k + a*b.^t);
  bzcha = sqrt(mean((yt - yuce(1:n)).^2))
  ythat = yuce([n+2,n+7])

ARMA模型拟合实例：

elps = randn(100,1);
x = ones(100,1);
for i = 3:100
    x(i,1) = elps(i) - 0.6*elps(i - 1) - 0.2*elps(i - 2);
end
%模拟ARMA(0,2)模型
%x必须为列向量
m = armax(x,[0,2]);
%预测
dx_Forecast = forecast(m,x,10);
x_Forecast = x(end) + cumsum(dx_Forecast)

---

• 具体建模的话，先观察时列图

  • 选择差分

    

  • 确定与非确定结合

    

• 然后进行拟合

• 拟合后预测