如何在 Python 中修復我的 Ornstein-Uhlenbeck 參數 MLE？

我正在嘗試將時間序列數據擬合到 Ornstein-Uhlenbeck 過程中。到目前為止，這是我的程式碼：

# source for computation: https://arxiv.org/pdf/1411.5062.pdf
import math
from math import sqrt, exp, log  # exp(n) == e^n, log(n) == ln(n)
import scipy.optimize as so
import numpy as np


def __compute_log_likelihood(params, *args):
   '''
   Compute the average Log Likelihood, this function will by minimized by scipy.
   Find in (2.2) in linked paper

   returns: the average log likelihood from given parameters
   '''
   # functions passed into scipy's minimize() needs accept one parameter, a tuple of
   #   of values that we adjust to minimize the value we return.
   #   optionally, *args can be passed, which are values we don't change, but still want
   #   to use in our function (e.g. the measured heights in our sample or the value Pi)

   theta, mu, sigma = params
   X, dt = args
   n = len(X)

   sigma_tilde_squared = sigma ** 2 * (1 - exp(-2 * mu * dt)) / 2 * mu

   summation_term = 0

   for i in range(1, len(X)):
       summation_term += (X[i] - X[i - 1] * exp(-mu * dt) - theta * (1 - exp(-mu * dt))) ** 2

   summation_term = -summation_term / (2 * n * sigma_tilde_squared)

   log_likelihood = (-log(2 * math.pi) / 2) + (-log(sqrt(sigma_tilde_squared))) + summation_term

   return -log_likelihood
   # since we want to maximize this total log likelihood, we need to minimize the
   #   negation of the this value (scipy doesn't support maximize)


def estimate_coefficients_MLE(X, dt):
   '''
   Estimates Ornstein-Uhlenbeck coefficients (θ, µ, σ) of the given array
   using the Maximum Likelihood Estimation method

   input: X - array-like data to be fit as an OU process
   returns: θ, µ, σ, Total Log Likelihood
   '''

   bounds = ((0, None), (None, None), (0, None))  # theta > 0, mu ∈ ℝ, sigma > 0
   mu_init = np.mean(X)
   result = so.minimize(__compute_log_likelihood, (1e-6, 1e-6, 1e-6), args=(X, dt), bounds=bounds)
   theta, mu, sigma = result.x
   max_log_likelihood = -result.fun  # undo negation from __compute_log_likelihood
   return theta, mu, sigma, max_log_likelihood

但是，當我使用以下內容模擬 OU 流程時：

# simulate Ornstein-Uhlenbeck Process
import numpy as np
import matplotlib.pyplot as plt
t_0 = 0 # define model parameters
t_end = 2
length = 1000
theta = 1.1
mu = 0
sigma = 0.3
t = np.linspace(t_0,t_end,length) # define time axis
dt = np.mean(np.diff(t))

y = np.zeros(length)
y0 = np.random.normal(loc=0.0,scale=1.0) # initial condition
drift = lambda y,t: theta*(mu-y) # define drift term, google to learn about lambda
diffusion = lambda y,t: sigma # define diffusion term
noise = np.random.normal(loc=0.0,scale=1.0,size=length)*np.sqrt(dt) #define noise process
# solve SDE
for i in range(1,length):
   y[i] = y[i-1] + drift(y[i-1],i*dt)*dt + diffusion(y[i-1],i*dt)*noise[i]

plt.plot(t,y)
plt.show()

然後使用我的函式擬合數據（儲存在 y 中）：

theta, mu, sigma, max_ll = estimate_coefficients_MLE(y, 1/len(y))

我要麼得到“值錯誤：數學域錯誤”，要麼我的係數非常偏離。如果有人能指出我正確的方向，我將不勝感激，網上缺乏關於這個主題的資源。

加上 Japser 的回答，要解決除以 0 的問題，我們可以為 mu 和 sigma 的下限設置一個非常小的值（例如 1x10^-5）。要查看實際的算法，請參閱此

那是因為sigma_tilde_squared == 0，您可以在添加時添加 0.01 以避免它== 0

引用自：https://quant.stackexchange.com/questions/55645