在统计建模畛域，了解总体趋势的同时解释群体差别的一个弱小办法是分层(或多层)建模。这种办法容许参数随组而变动，并捕捉组内和组间的变动。在工夫序列数据中，这些特定于组的参数能够示意不同组随工夫的不同模式。
明天，咱们将深入探讨如何应用PyMC(用于概率编程的Python库)构建分层工夫序列模型。
让咱们从为多个组生成一些人工工夫序列数据开始，每个组都有本人的截距和斜率。
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)咱们生成了三个不同组的工夫序列数据。每组都有本人的工夫趋势，由惟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。咱们的模型将具备组特定的截距(alpha)和斜率(beta)。截距和斜率是从具备超参数mu_alpha、sigma_alpha、mu_beta和sigma_beta的正态分布中绘制的。这些超参数别离示意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)当初咱们曾经定义了模型并对其进行了采样。让咱们查看不同参数的模型预计:
# Checking the trace pm.plot_trace(trace,var_names=['alpha','beta']) plt.show()最初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()从图中能够看出，分层工夫序列模型很好地捕捉了每组中的单个趋势,而暗影区域给出了预测的不确定性。
层次模型为捕捉工夫序列数据中的组级变动提供了一个弱小的框架。它们容许咱们在组之间共享统计数据，提供局部信息池和对数据结构的轻微了解。应用像PyMC这样的库，实现这些模型变得相当简略，为强壮且可解释的工夫序列剖析铺平了路线。
https://avoid.overfit.cn/post/56ad545325504850ab2b7b7b9a264a61
作者：Charles Copley