将定期取样的 Ornstein-Uhlenbeck 过程识别为自回归过程 

Mathematica 10 对过程切片的计算支持进行个改进，允许您可以对多变量切片直接使用矩量法在两个过程间创建等效值.

自回归过（AR）过程和 Ornstein–Uhlenbeck（OU）过程都属于高斯过程，由其均值和协方差函数而定.

In[1]:=

X ar\[ScriptCapitalP] = ARProcess[c, {a1}, si2]; ou\[ScriptCapitalP] = OrnsteinUhlenbeckProcess[mu, si, th];

对切片分布使用矩量法，在正则时间网格将平稳 AR 过程参数与平稳 OU 过程的约束相匹配.

In[2]:=

X arSlice\[ScriptCapitalD] = ar\[ScriptCapitalP][{0, 1, 2, 3, 4}]; ouSlice\[ScriptCapitalD] = ou\[ScriptCapitalP][{dt, 2 dt, 3 dt, 4 dt, 5 dt}];

In[3]:=

X {sol1} = (Solve[{Mean[arSlice\[ScriptCapitalD]] == Mean[ouSlice\[ScriptCapitalD]], Covariance[arSlice\[ScriptCapitalD]] == Covariance[ouSlice\[ScriptCapitalD]]}, {c, a1, si2}] // FullSimplify)

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确认协方差函数与符号延迟一致.

In[4]:=

X Simplify[CovarianceFunction[ar\[ScriptCapitalP] /. sol1, h], dt > 0 && th > 0 && h \[Element] Integers]

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In[5]:=

X CovarianceFunction[ou\[ScriptCapitalP], h dt]

In[6]:=

X (E^(-th Abs[dt h]) si^2)/(2 th)

用初始条件重复 AR 和 OU 过程的练习.

In[7]:=

X ar\[ScriptCapitalP]2 = ARProcess[c, {a1}, si2, {x0ar}]; ou\[ScriptCapitalP]2 = OrnsteinUhlenbeckProcess[mu, si, th, x0];

In[8]:=

X arSlice\[ScriptCapitalD]2 = ar\[ScriptCapitalP]2[{0, 1, 2, 3, 4}]; ouSlice\[ScriptCapitalD]2 = ou\[ScriptCapitalP]2[{dt, 2 dt, 3 dt, 4 dt, 5 dt}];

In[9]:=

X {sol2} = (Solve[{Mean[arSlice\[ScriptCapitalD]2] == Mean[ouSlice\[ScriptCapitalD]2], Covariance[arSlice\[ScriptCapitalD]2] == Covariance[ouSlice\[ScriptCapitalD]2]}, {c, a1, si2, x0ar}] // FullSimplify)

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在等间隔时间网格用自回归表示 Ornstein–Uhlenbeck 过程取样.

In[10]:=

X RandomFunction[ ar\[ScriptCapitalP]2 /. sol2 /. {mu -> 1, si -> 2, th -> 2/3, x0 -> 10, dt -> 0.01}, {0, 200}, 5] // ListPlot[#, ImageSize -> 400] &

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