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## 請問 ARIMA 模型中的 φ θ 係數是怎麼運用 MLE 決定出來的？

https://stackoverflow.com/questions/44301263/python-arma-mle-implementing-algorithms-from-literature

``````import numpy as np
import statsmodels.api as sm
import statsmodels.tsa.arima_model
import scipy.optimize

class ARMA(object):
def __init__(self, endo, nar, nma):
np.random.seed(0)
# endogenous variables
self.endo = endo
# Number of AR terms
self.nar = nar
# Number of MA terms
self.nma = nma
# "Dimension" of the ARMA fit
self.dim = max(nar, nma+1)
# Current ARMA parameters
self.params = np.zeros(self.nar+self.nma, dtype='float')

def __g(self, ma_params):
'''
Build MA parameter vector
'''
g = np.zeros(self.dim, dtype='float')
g[0] = 1.0
if self.nma > 0:
g[1:self.nma+1] = ma_params
return g

def __F(self, ar_params):
'''
Build AR parameter matrix
'''
F = np.zeros((self.dim, self.dim), dtype='float')
F[:self.nar, 0] =  ar_params
for i in xrange(1, self.dim):
F[i-1, i] = 1.0
return F

def __initial_P(self, R, T):
'''
Solve for initial P matrix

Solves P = TPT' + RR'
'''
v = np.zeros(self.dim*self.dim, dtype='float')
for i in xrange(self.dim):
for j in xrange(self.dim):
v[i+j*self.dim] = R[i]*R[j]
R = np.array([R])
S = np.identity(self.dim**2, dtype='float')-np.kron(T, T)
V = np.outer(R, R).ravel('F')
Pmat = np.linalg.solve(S,V).reshape(self.dim, self.dim, order='F')
return Pmat

def __likelihood(self, params):
'''
Compute log likehood for a parameter vector

Implements the Pearlman 1980 algorithm
'''
# these checks are pilfered from statsmodels
if self.nar > 0 and not np.all(np.abs(np.roots(np.r_[1, -params[:self.nar]]))<1):
print 'AR coefficients are not stationary'
if self.nma > 0 and not np.all(np.abs(np.roots(np.r_[1, -params[-self.nma:]]))<1):
print 'MA coefficients are not stationary'
ar_params = params[:self.nar]
ma_params = params[-self.nma:]
g = self.__g(ma_params)
F = self.__F(ar_params)
w = self.endo
P = self.__initial_P(g, F)
n = len(w)
z = np.zeros(self.dim, dtype='float')
R = np.zeros(n, dtype='float')
a = np.zeros(n, dtype='float')
K = np.dot(F, P[:, 0])
L = K.copy()
R[0] = P[0, 0]
for i in xrange(1, n):
a[i-1] = w[i-1] - z[0]
z = np.dot(F, z) + K*(a[i-1]/R[i-1])
Kupdate = -(L[0]/R[i-1])*np.dot(F, L)
Rupdate = -L[0]*L[0]/R[i-1]
P -= np.outer(L, L)/R[i-1]
L = np.dot(F, L) - (L[0]/R[i-1])*K
K += Kupdate
R[i] = R[i-1] + Rupdate
if np.abs(R[i] - 1.0) < 1e-9:
R[i:] = 1.0
break
for j in xrange(i, n):
a[j] = w[j] - z[0]
z = np.dot(F, z) + K*(a[i-1]/R[i-1])
likelihood = 0.0
for i in xrange(n):
likelihood += np.log(R[i])
likelihood *= -0.5
ssum = 0.0
for i in xrange(n):
ssum += a[i]*a[i]/R[i]
likelihood += -0.5*n*np.log(ssum)
return likelihood

def fit(self):
'''
Fit the ARMA model by minimizing the loglikehood

Uses scipy.optimize.minimize
'''
sm_arma = statsmodels.tsa.arima_model.ARMA(endog=self.endo, order=(self.nar, self.nma, 0))
params = statsmodels.tsa.arima_model.ARMA._fit_start_params_hr(sm_arma, order=(self.nar, self.nma, 0))
opt = scipy.optimize.minimize(fun=self.__likelihood, x0=params, method='L-BFGS-B')
print opt

# Test the code on statsmodels sunspots data
nar = 2
nma = 1