使用PyMC进行时间序列分层建模

在统计建模领域，理解总体趋势的同时解释群体差异的一个强大方法是分层(或多层)建模。这种方法允许参数随组而变化，并捕获组内和组间的变化。在时间序列数据中，这些特定于组的参数可以表示不同组随时间的不同模式。

今天，我们将深入探讨如何使用PyMC(用于概率编程的Python库)构建分层时间序列模型。

让我们从为多个组生成一些人工时间序列数据开始，每个组都有自己的截距和斜率。

 import numpy as np
 import matplotlib.pyplot as plt
 import pymc as pm
 
 # Simulating some data
 np.random.seed(0)
 n_groups = 3  # number of groups
 n_data_points = 100  # number of data points per group
 x = np.tile(np.linspace(0, 10, n_data_points), n_groups)
 group_indicator = np.repeat(np.arange(n_groups), n_data_points)
 slope_true = np.random.normal(0, 1, size=n_groups)
 intercept_true = np.random.normal(2, 1, size=n_groups)
 y = slope_true[group_indicator]*x + intercept_true[group_indicator] + np.random.normal(0, 1, size=n_groups*n_data_points)

我们生成了三个不同组的时间序列数据。每组都有自己的时间趋势，由唯一的截距和斜率定义。

 colors = ['b', 'g', 'r']  # Define different colors for each group
 
 plt.figure(figsize=(10, 5))
 
 # Plot raw data for each group
 for i in range(n_groups):
     plt.plot(x[group_indicator == i], y[group_indicator == i], 'o', color=colors[i], label=f'Group {i+1}')
 
 plt.title('Raw Data with Groups')
 plt.xlabel('Time')
 plt.ylabel('Value')
 plt.legend()
 plt.show()

下一步是构建层次模型。我们的模型将具有组特定的截距(alpha)和斜率(beta)。截距和斜率是从具有超参数mu_alpha、sigma_alpha、mu_beta和sigma_beta的正态分布中绘制的。这些超参数分别表示截距和斜率的组水平均值和标准差。

 with pm.Model() as hierarchical_model:
     # Hyperpriors
     mu_alpha = pm.Normal('mu_alpha', mu=0, sigma=10)
     sigma_alpha = pm.HalfNormal('sigma_alpha', sigma=10)
     mu_beta = pm.Normal('mu_beta', mu=0, sigma=10)
     sigma_beta = pm.HalfNormal('sigma_beta', sigma=10)
   
     # Priors
     alpha = pm.Normal('alpha', mu=mu_alpha, sigma=sigma_alpha, shape=n_groups)  # group-specific intercepts
     beta = pm.Normal('beta', mu=mu_beta, sigma=sigma_beta, shape=n_groups)  # group-specific slopes
     sigma = pm.HalfNormal('sigma', sigma=1)
 
     # Expected value
     mu = alpha[group_indicator] + beta[group_indicator] * x
 
     # Likelihood
     y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y)
 
     # Sampling
     trace = pm.sample(2000, tune=1000)

现在我们已经定义了模型并对其进行了采样。让我们检查不同参数的模型估计:

 # Checking the trace
 pm.plot_trace(trace,var_names=['alpha','beta'])
 plt.show()

最后一步是将原始数据和模型预测可视化:

 # Posterior samples
 alpha_samples = trace.posterior['alpha'].values
 beta_samples = trace.posterior['beta'].values
 
 # New x values for predictions
 x_new = np.linspace(0, 10, 200)
 
 plt.figure(figsize=(10, 5))
 
 # Plot raw data and predictions for each group
 for i in range(n_groups):
     # Plot raw data
     
     plt.plot(x[group_indicator == i], y[group_indicator == i], 'o', color=colors[i], label=f'Group {i+1} observed')
     x_new = x[group_indicator == i]
     # Generate and plot predictions
     alpha = trace.posterior.sel(alpha_dim_0=i,beta_dim_0=i)['alpha'].values
     beta = trace.posterior.sel(alpha_dim_0=i,beta_dim_0=i)['beta'].values
     y_hat = alpha[..., None] + beta[..., None] * x_new[None,:]
     y_hat_mean = y_hat.mean(axis=(0, 1))
     y_hat_std = y_hat.std(axis=(0, 1))
     plt.plot(x_new, y_hat_mean, color=colors[i], label=f'Group {i+1} predicted')
     plt.fill_between(x_new, y_hat_mean - 2*y_hat_std, y_hat_mean + 2*y_hat_std, color=colors[i], alpha=0.3)
 
 plt.title('Raw Data with Posterior Predictions by Group')
 plt.xlabel('Time')
 plt.ylabel('Value')
 plt.legend()
 plt.show()