Python PyMC3中的再生Hamilton-1989马尔可夫切换模型
我试图理解汉密尔顿1989年的马尔可夫切换自回归模型。我试图用贝叶斯方法重现结果。我使用编写了许多PyMC3模型,最新的迭代可以在下面找到 在没有自回归的情况下,模型收敛到接近汉密尔顿(1.16和-0.36)和实际转移概率的μ值。 然而,当加入自回归时,该模型无法收敛,拟合系数接近汉密尔顿的结果。过渡概率的拟合特别差 我错过了什么Python PyMC3中的再生Hamilton-1989马尔可夫切换模型,python,pymc3,hidden-markov-models,Python,Pymc3,Hidden Markov Models,我试图理解汉密尔顿1989年的马尔可夫切换自回归模型。我试图用贝叶斯方法重现结果。我使用编写了许多PyMC3模型,最新的迭代可以在下面找到 在没有自回归的情况下,模型收敛到接近汉密尔顿(1.16和-0.36)和实际转移概率的μ值。 然而,当加入自回归时,该模型无法收敛,拟合系数接近汉密尔顿的结果。过渡概率的拟合特别差 我错过了什么 # %% import pymc3 as pm import theano.tensor as tt import theano.tensor.slinalg as
# %%
import pymc3 as pm
import theano.tensor as tt
import theano.tensor.slinalg as sla # theano-wrapped scipy linear algebra
import theano.tensor.nlinalg as nla # theano-wrapped numpy linear algebra
import theano
theano.config.gcc.cxxflags = "-Wno-c++11-narrowing"
import pandas as pd
dta = pd.read_stata('https://www.stata-press.com/data/r14/rgnp.dta').iloc[1:]
dta.index = pd.DatetimeIndex(dta.date, freq='QS')
dta_hamilton = dta.rgnp
# Plot the data
dta_hamilton.plot(title='Growth rate of Real GNP', figsize=(12,3))
# %%
fig, ax = plt.subplots(figsize=(12, 4))
#plt.plot(np.round(trace["hmm_states"].mean(axis=0)), label="inferred")
plt.plot(dta_hamilton.values, label="true")
# %%
def solve_equilibrium(n_states, p_transition):
A = tt.dmatrix('A')
A = tt.eye(n_states) - p_transition + tt.ones(shape=(n_states, n_states))
p_equilibrium = pm.Deterministic("p_equilibrium", sla.solve(A.T, tt.ones(shape=(n_states))))
return p_equilibrium
class HMMStates(pm.Categorical):
def __init__(self, p_transition, p_equilibrium, n_states, *args, **kwargs):
"""You can ignore this section for the time being."""
super(pm.Categorical, self).__init__(*args, **kwargs)
self.p_transition = p_transition
self.p_equilibrium = p_equilibrium
# This is needed
self.k = n_states
# This is only needed because discrete distributions must define a mode.
self.mode = tt.cast(0,dtype='int64')
def logp(self, x):
"""Focus your attention here!"""
p_eq = self.p_equilibrium
# Broadcast out the transition probabilities,
# so that we can broadcast the calculation
# of log-likelihoods
p_tr = self.p_transition[x[:-1]]
# the logp of the initial state evaluated against the equilibrium probabilities
initial_state_logp = pm.Categorical.dist(p_eq).logp(x[0])
# the logp of the rest of the states.
x_i = x[1:]
ou_like = pm.Categorical.dist(p_tr).logp(x_i)
transition_logp = tt.sum(ou_like)
return initial_state_logp + transition_logp
# %%
class HamiltonEmissions(pm.Continuous):
def __init__(self, states, phi, sigma, mu, *args, **kwargs):
super().__init__(*args, **kwargs)
self.states = states
self.phi = phi
self.mu = mu
self.sigma = sigma # variance
def logp(self, x):
"""
x: observations
"""
states = self.states
sigma = self.sigma[states]
mu = self.mu[states]
phi = self.phi
z = x - mu # Centered version of x
ar_mean = \
phi[0] * z[0:-4] + \
phi[1] * z[1:-3] + \
phi[2] * z[2:-2] + \
phi[3] * z[3:-1]
ar_like = tt.sum(pm.Normal.dist(mu= ar_mean + mu[4:], sigma=sigma[4:]).logp(x[4:]))
boundary_like = pm.Normal.dist(mu=0, sigma=sigma[:4]).logp(x[:4])
return ar_like + boundary_like
# %%
n_states = 2
with pm.Model() as model:
# Priors for transition matrix
p_transition = pm.Dirichlet("p_transition",
a=tt.ones((n_states, n_states)),
shape=(n_states, n_states))
# Solve for the equilibrium state
p_equilibrium = solve_equilibrium(n_states, p_transition)
# HMM state
hmm_states = HMMStates(
"hmm_states",
p_transition=p_transition,
p_equilibrium=p_equilibrium,
n_states=n_states,
shape=(len(dta_hamilton),)
)
# Prior for mu and sigma
mu = pm.Normal("mu", mu=0, sigma=1, shape=(n_states,))
sigma = pm.Exponential("sigma", lam=2, shape=(n_states,))
phi = pm.Normal("phi", 0, 0.5, shape=(4, ))
# Observed emission likelihood
obs = HamiltonEmissions(
"emission",
states=hmm_states,
mu=mu,
sigma=sigma,
phi=phi,
observed=dta_hamilton
)
# %%
with model:
start = pm.find_MAP()
step1 = pm.Metropolis(vars=[mu, sigma, phi, p_transition, emission])
step2 = pm.BinaryGibbsMetropolis(vars=[hmm_states])
trace = pm.sample(2500, cores=1, chains=2, step=[step1, step2], tune=1500)
# %%
import arviz as az
az.plot_trace(trace, var_names=["p_transition"])
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