965 lines
40 KiB
Python
965 lines
40 KiB
Python
import numpy as np
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from scipy import stats
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class RJMCMC_Sampler:
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def __init__(self, y, u, k_min=1, k_max=10, initial_k=None,
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initial_a=None, initial_b=None):
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"""
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初始化采样器
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参数:
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y (array): 观测到的输出序列
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u (array): 已知的输入序列
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k_min (int): 探索的最小模型阶数
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k_max (int): 探索的最大模型阶数
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initial_k (int, optional): 初始模型阶数
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initial_a (dict, optional): 初始多项式系数
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initial_b (array, optional): 初始多项式系数
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"""
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# 定义观测数据
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self.y = y
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self.u = u
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self.N = len(y)
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# 定义模型阶数
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self.k_min = k_min
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self.k_max = k_max
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gamma_sample = np.random.gamma(shape=2, scale=1)
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self.current_k = initial_k if initial_k is not None else min(max(int(gamma_sample), k_min), k_max)
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# 初始化多项式系数
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if initial_a is not None:
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self.current_a = initial_a
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else:
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self.current_a = self._generate_stable_a(self.current_k)
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# 定义标准型的b系数
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if initial_b is not None:
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self.current_b = initial_b
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else:
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self.current_b = self._generate_observable_b(self.current_k)
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# 初始化噪声先验 (假设为 Gamma 先验)
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self.lambda_a_w = 1e-3 # 过程噪声精度的先验
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self.lambda_b_w = 1e-3
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self.lambda_a_z = 1e-3 # 观测噪声精度的先验
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self.lambda_b_z = 1e-3
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# 初始化噪声方差 (从先验采样)
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# 为避免除零错误，在分母上增加一个极小值
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epsilon = 1e-9
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self.current_sigma2_w = 1.0 / (np.random.gamma(self.lambda_a_w, 1.0/self.lambda_b_w) + epsilon)
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self.current_sigma2_z = 1.0 / (np.random.gamma(self.lambda_a_z, 1.0/self.lambda_b_z) + epsilon)
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def _update_current_eigenvalues(self):
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"""一个辅助函数，根据 current_a 更新 current_eigenvalues。"""
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if self.current_k > 0:
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poly_coeffs = np.concatenate(([1], -self.current_a[::-1]))
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self.current_eigenvalues = np.roots(poly_coeffs)
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else:
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self.current_eigenvalues = np.array([])
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def _update_current_a_coeffs(self):
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"""一个辅助函数，根据 current_eigenvalues 更新 current_a。"""
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k = len(self.current_eigenvalues)
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if k > 0:
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# np.poly 返回 [1, a_{k-1}, ..., a_0]，去掉首项“1”，反转剩下的
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poly_coeffs = np.poly(self.current_eigenvalues)
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self.current_a = np.real(poly_coeffs[1:][::-1])
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else:
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self.current_a = np.array([])
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def _cal_current_eigenvalues(self, a_coeffs_temp):
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"""一个辅助函数，根据给定的 a_coeffs 计算对应的特征值。"""
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k = len(a_coeffs_temp)
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if k > 0:
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poly_coeffs = np.concatenate(([1], -a_coeffs_temp[::-1]))
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return np.roots(poly_coeffs)
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else:
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return np.array([])
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def _cal_current_a_coeffs(self, eigenvalues_temp):
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"""一个辅助函数，根据给定的特征值计算对应的 a_coeffs。"""
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k = len(eigenvalues_temp)
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if k > 0:
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# np.poly 返回 [1, a_{k-1}, ..., a_0]，去掉首项“1”，反转剩下的
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poly_coeffs = np.poly(eigenvalues_temp)
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return np.real(poly_coeffs[1:][::-1])
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else:
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return np.array([])
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def _log_prior_eigenvalues(self, eigenvalues):
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"""
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计算 k 个特征值 (Lambda) 的对数先验概率 log p(Lambda)。
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该先验基于论文 4.1.1 节 讨论的先验：
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1. 稳定性：如果任何 |lambda| >= 1，先验概率为 0 (log(0) = -inf)。
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2. 实数根 (p_real)：在 (-1, 1) 上均匀分布, p(lambda_r) = 1/2。
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log p(lambda_r) = -log(2)。
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3. 复数根 (p_complex)：在单位圆盘上均匀分布, p(lambda_c) = 1/pi。
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log p(lambda_c) = -log(pi)。
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注意：复共轭对 (lambda, lambda_conj) 只计算一次 (例如上半平面)。
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我们通过只计算 imag(lambda) >= 0 的复数根来实现。
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"""
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log_prior = 0.0
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# 跟踪已处理的复数根 (避免重复计算共轭对)
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processed_complex = set()
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for eig in eigenvalues:
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if np.abs(eig) >= 1.0:
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# 违反稳定性约束
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return -np.inf
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if np.isclose(eig.imag, 0):
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# 是实数
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# 这是 "Uniform real-eigenvalue prior"
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log_prior += -np.log(2.0)
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else:
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# 是复数
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# 检查是否已处理过
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if eig in processed_complex or np.conjugate(eig) in processed_complex:
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continue
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# 我们只计算 imag >= 0 的根 (代表一对)
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if eig.imag >= 0:
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# 这是 "Polar-coordinate prior"
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log_prior += -np.log(np.pi)
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processed_complex.add(eig)
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return log_prior
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def _log_prior_eigenvalues_to_a(self,eigenvalues):
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"""
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计算从特征值 (eigenvalues) 到多项式系数 a = [a_0, ..., a_{k-1}] 的对数先验概率。
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该方法基于 Proposition 4.2 (Change of variables)
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log p(a) = log p(Lambda) - log(|det J|)
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其中 log p(Lambda) 是特征值的先验
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而 log(|det J|) 是维塔变换的对数雅可比行列式。
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"""
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# 计算 log p(Lambda)
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log_prior_lambda = self._log_prior_eigenvalues(eigenvalues)
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# 如果特征值不稳定 (log p(Lambda) = -inf)，则 p(a) 也不可能
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if log_prior_lambda == -np.inf:
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return -np.inf
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# 计算对数雅可比行列式 log(|det J|)
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log_det_jacobian = 0.0
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k = len(eigenvalues)
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if k < 2:
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return log_prior_lambda
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for i in range(k):
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for j in range(i + 1, k):
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diff = eigenvalues[i] - eigenvalues[j]
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abs_diff = np.abs(diff)
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if abs_diff == 0:
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return -np.inf # 重根导致雅可比行列式为零
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log_det_jacobian += np.log(abs_diff)
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if log_det_jacobian == -np.inf:
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return -np.inf
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# 应用变量替换公式
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log_prior_a = log_prior_lambda - log_det_jacobian
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return log_prior_a
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def _log_likelihood(self, k, a_coeffs, b_coeffs, sigma2_w, sigma2_z):
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"""
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计算给定参数下的对数似然 log p(y | k, a, b, sigma_w, sigma_z)。
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使用标准卡尔曼滤波器 (Kalman Filter) (如论文 Appendix B )
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和预测误差分解 (Prediction Error Decomposition) (如论文 Eq. (D.37) )。
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"""
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# 如果 k=0，模型无法运行
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if k == 0:
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# 返回一个非常小的似然
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return -np.inf
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# 获取当前参数下的状态空间矩阵（以参数为输入，避免依赖实例状态）
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A, B, C, D = self.get_controller_canonical_form(k, a_coeffs, b_coeffs)
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# 过程噪声 Sigma (k x k)
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Sigma = np.zeros((k, k))
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Sigma[k-1, k-1] = sigma2_w
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# 测量噪声 Gamma (1 x 1)
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Gamma = np.array([[sigma2_z]])
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# 初始化卡尔曼滤波器
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# 状态 x_t|t-1
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x_pred = np.zeros((k, 1))
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# 状态协方差 P_t|t-1
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P_pred = np.eye(k) * 1e6
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total_log_likelihood = 0.0
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# 运行卡尔曼滤波器 N 步
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for t in range(self.N):
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y_t = self.y[t]
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u_t = self.u[t]
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# --- 预测步 (Prediction) ---
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# (在 t=0 时，x_pred 和 P_pred 是我们初始化的 x_0|-1, P_0|-1)
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# --- 计算似然 ---
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# 预测误差 nu_t
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y_pred = C @ x_pred + D @ np.array([[u_t]])
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nu_t = y_t - y_pred
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# 预测误差协方差 S_t
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S_t = C @ P_pred @ C.T + Gamma
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S_t_inv = 1.0 / S_t
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# 累加对数似然
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total_log_likelihood += -0.5 * (np.log(2 * np.pi) + np.log(S_t) + (nu_t * S_t_inv * nu_t))
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# --- 更新步 (Update) ---
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# 卡尔曼增益 K_t (k x 1)
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K_t = P_pred @ C.T * S_t_inv
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# 更新状态 x_t|t
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x_update = x_pred + K_t @ nu_t
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# 更新协方差 P_t|t
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I_KC = np.eye(k) - K_t @ C
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P_update = I_KC @ P_pred @ I_KC.T + K_t @ Gamma @ K_t.T
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# --- 为下一次循环准备预测 (t+1) ---
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x_pred = A @ x_update + B @ np.array([[u_t]])
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P_pred = A @ P_update @ A.T + Sigma
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return total_log_likelihood # 返回标量值
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def _propose_birth(self):
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"""
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提议一个 "诞生" 转移 (k -> k+1 或 k -> k+2)。
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返回包含提议状态和计算接受率所需的对数比率的字典。
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"""
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k = self.current_k
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current_eigs = self.current_eigenvalues
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# 获取实数特征值数目
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real_eigs_indices = np.where(np.isreal(current_eigs))
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num_real_eigs = len(real_eigs_indices[0])
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# 获取复数特征值数目，只保留 imag > 0 的部分 (共轭对只算一次)
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complex_eigs_indices = np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0)) #type: ignore
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num_complex_eigs = len(complex_eigs_indices[0])
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# 决定是诞生一个实数根还是一对复共轭根
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can_add_complex = (k + 2) <= self.k_max
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add_real = True
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if can_add_complex:
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# 以0.5的概率选择诞生一对复共轭根
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if np.random.rand() < 0.5:
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add_real = False
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if add_real:
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k_new = k + 1
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# 诞生实根的对数概率之比
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log_birth_real_forward = np.log(0.5)
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log_birth_real_backward = np.log(0.5)
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# 从提议分布 q(u) 中采样辅助变量 u = (u_lambda, u_b)
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u_lambda = np.random.uniform(-1.0, 1.0) # 新特征值
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u_b = np.random.normal(0, 1) # 新 b 系数
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# 计算对数提议密度
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log_q_forward = log_birth_real_forward + stats.norm.logpdf(u_b, 0, 1) - np.log(2.0)
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log_q_backward = log_birth_real_backward - np.log(num_real_eigs + 1)
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# 计算对数雅可比行列式 log|J|
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if k == 0:
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log_det_jacobian = 0.0
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else:
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log_det_jacobian = 1
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# 构造新状态
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new_eigs = np.append(current_eigs, u_lambda)
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new_a = np.real(np.poly(new_eigs)[1:][::-1])
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new_b = np.append(self.current_b, u_b)
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# 接受率中的项为 q_backward(u) * |J| / q_forward(u)
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log_ratio = log_q_backward + log_det_jacobian - log_q_forward
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# 计算未归一化的后验之比
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log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_a, self.current_b,
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self.current_sigma2_w, self.current_sigma2_z)
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log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, new_a, new_b, self.current_sigma2_w, self.current_sigma2_z)
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# 计算接受率
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log_acceptance_ratio = log_post_unnormalized_new - log_post_unnormalized_current + log_ratio
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acceptance_ratio = np.exp(log_acceptance_ratio)
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return {"k_new": k_new, "a_new": new_a, "b_new": new_b, "log_ratio": log_ratio, "type": "birth_real", "new_eigs": new_eigs, "acceptance_ratio": acceptance_ratio}
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else:
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# --- 诞生一对复共轭根 (k -> k+2) ---
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k_new = k + 2
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# 诞生实根的对数概率之比
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log_birth_complex_forward = np.log(0.5)
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log_birth_complex_backward = np.log(0.5)
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# 采样辅助变量 u = (rho, theta, u_b1, u_b2)
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rho = np.sqrt(np.random.uniform(0, 1.0))
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theta = np.random.uniform(0, np.pi)
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u_lambda = rho * (np.cos(theta) + 1j * np.sin(theta))
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u_b1, u_b2 = np.random.normal(0, 1, 2)
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# 计算对数提议密度 log(q(u))
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log_q_forward = log_birth_complex_forward + stats.norm.logpdf(u_b1, 0, 1) + stats.norm.logpdf(u_b2, 0, 1) - np.log(np.pi) - np.log(1.0)
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log_q_backward = log_birth_complex_backward - np.log(num_complex_eigs + 1)
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# 计算对数雅可比行列式 log|J|
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# |J| = 2*ρ
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log_jacobian = np.log(2 * rho)
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# 构造新状态
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new_eigs = np.append(current_eigs, [u_lambda, np.conjugate(u_lambda)])
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new_a = np.real(np.poly(new_eigs)[1:][::-1])
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new_b = np.append(self.current_b, [u_b1, u_b2])
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log_ratio = log_q_backward + log_jacobian - log_q_forward
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# 计算未归一化的后验之比
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log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_a, self.current_b,
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self.current_sigma2_w, self.current_sigma2_z)
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log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, new_a, new_b, self.current_sigma2_w, self.current_sigma2_z)
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# 计算接受率
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log_acceptance_ratio = log_post_unnormalized_new - log_post_unnormalized_current + log_ratio
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acceptance_ratio = np.exp(log_acceptance_ratio)
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return {"k_new": k_new, "a_new": new_a, "b_new": new_b, "log_ratio": log_ratio, "type": "birth_complex", "new_eigs": new_eigs, "acceptance_ratio": acceptance_ratio}
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def _propose_death(self):
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"""
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提议一个 "消亡" 转移 (k -> k-1 或 k -> k-2)。
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返回包含提议状态和计算接受率所需的对数比率的字典。
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"""
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k = self.current_k
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current_eigs = self.current_eigenvalues
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# 识别出可移除的实数根和复共轭对
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real_eigs_indices = np.where(np.isreal(current_eigs))
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complex_eigs_indices = np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0)) #type: ignore
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can_remove_real = len(real_eigs_indices[0]) > 0
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can_remove_complex = len(complex_eigs_indices[0]) > 0
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if not can_remove_real and not can_remove_complex:
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return None # 无法执行消亡
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# 决定是移除实数根还是复共轭对
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remove_real = True
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if can_remove_real and can_remove_complex:
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if np.random.rand() < 0.5:
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remove_real = False
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elif can_remove_complex:
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remove_real = False
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if remove_real:
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# --- 消亡一个实数根 (k -> k-1) ---
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k_new = k - 1
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# 生成与消亡实根对应的诞生过程的对数概率之比
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log_death_real_forward = np.log(0.5)
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log_death_real_backward = np.log(0.5)
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# 1. 随机选择一个实数根移除
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idx_to_remove = np.random.choice(real_eigs_indices[0])
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lambda_removed = current_eigs[idx_to_remove]
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# 2. 随机选择一个实数 b 系数移除
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idx_b_to_remove = np.random.choice(len(self.current_b))
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b_removed = self.current_b[idx_b_to_remove]
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|
||
# 移除的根和b系数构成了逆向（诞生）提议的辅助变量 u
|
||
u_lambda = lambda_removed
|
||
u_b = b_removed
|
||
|
||
# 2. 计算逆向提议的对数密度 log(q(u))
|
||
log_q_forward = log_death_real_forward - np.log(len(real_eigs_indices[0])) - np.log(len(self.current_b))
|
||
log_q_backward = log_death_real_backward + stats.norm.logpdf(u_b, 0, 1) - np.log(2.0)
|
||
|
||
# 3. 计算对应诞生过程的对数雅可比行列式 log|J|
|
||
remaining_eigs = np.delete(current_eigs, idx_to_remove)
|
||
if k_new == 0:
|
||
log_jacobian_birth = 0.0
|
||
else:
|
||
log_jacobian_birth = 1
|
||
|
||
# 构造新状态
|
||
if k_new == 0:
|
||
new_a = np.array([])
|
||
else:
|
||
new_a = np.real(np.poly(remaining_eigs)[1:][::-1])
|
||
new_b = np.delete(self.current_b, idx_b_to_remove)
|
||
|
||
# 接受率中的项为 q_backward(u) / (|J| * q_forward(u))
|
||
log_ratio = -log_q_forward - log_jacobian_birth + log_q_backward
|
||
|
||
# 计算未归一化的后验之比
|
||
log_post_unnormalized_current = self._log_posterior_unnormalized(k, current_eigs, self.current_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, remaining_eigs, new_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
|
||
# 计算接受率
|
||
log_acceptance_ratio = log_post_unnormalized_new - log_post_unnormalized_current + log_ratio
|
||
acceptance_ratio = np.exp(log_acceptance_ratio)
|
||
|
||
return {"k_new": k_new, "a_new": new_a, "b_new": new_b, "log_ratio": log_ratio, "type": "death_real", "new_eigs": remaining_eigs, "acceptance_ratio": acceptance_ratio}
|
||
|
||
else:
|
||
# --- 消亡一对复共轭根 (k -> k-2) ---
|
||
k_new = k - 2
|
||
rho = np.sqrt(np.random.uniform(0, 1.0))
|
||
|
||
# 生成与消亡复共轭对对应的诞生过程的对数概率之比
|
||
log_death_complex_forward = np.log(0.5)
|
||
log_death_complex_backward = np.log(0.5)
|
||
|
||
# 随机选择一对共轭根移除
|
||
complex_idx_to_remove = np.random.choice(complex_eigs_indices[0])
|
||
lambda_removed = current_eigs[complex_idx_to_remove]
|
||
|
||
# 随机选择一对 b 系数移除，需要确保两次选的不一样
|
||
idx_b1_to_remove, idx_b2_to_remove = np.random.choice(len(self.current_b), size=2, replace=False)
|
||
b1_removed = self.current_b[idx_b1_to_remove]
|
||
b2_removed = self.current_b[idx_b2_to_remove]
|
||
|
||
# 找到其共轭对
|
||
conjugate_idx_to_remove_array = np.where(current_eigs == np.conjugate(lambda_removed))[0]
|
||
if len(conjugate_idx_to_remove_array) == 0:
|
||
return None # 找不到共轭对，无法执行消亡
|
||
conjugate_idx = conjugate_idx_to_remove_array[0]
|
||
|
||
# 逆向提议的辅助变量 u
|
||
u_lambda = lambda_removed
|
||
u_b1, u_b2 = self.current_b[-2:]
|
||
|
||
# 计算逆向提议的对数密度 log(q(u))
|
||
n_b = len(self.current_b)
|
||
pair_count = n_b * (n_b - 1) / 2.0
|
||
log_q_forward = log_death_complex_forward - np.log(len(complex_eigs_indices[0])) - np.log(pair_count)
|
||
log_q_backward = log_death_complex_backward + stats.norm.logpdf(u_b1, 0, 1) + stats.norm.logpdf(u_b2, 0, 1) - np.log(np.pi) - np.log(1.0)
|
||
|
||
# 计算对应诞生过程的对数雅可比行列式
|
||
remaining_eigs = np.delete(current_eigs, [complex_idx_to_remove, conjugate_idx])
|
||
log_jacobian = np.log(2 * rho)
|
||
|
||
# 构造新状态
|
||
if k_new == 0:
|
||
new_a = np.array([])
|
||
else:
|
||
new_a = np.real(np.poly(remaining_eigs)[1:][::-1])
|
||
new_b = np.delete(self.current_b, [idx_b1_to_remove, idx_b2_to_remove])
|
||
|
||
log_ratio = -log_q_forward - log_jacobian + log_q_backward
|
||
|
||
# 计算未归一化的后验之比
|
||
log_post_unnormalized_current = self._log_posterior_unnormalized(k, current_eigs, self.current_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, remaining_eigs, new_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
|
||
# 计算接受率
|
||
log_acceptance_ratio = log_post_unnormalized_new - log_post_unnormalized_current + log_ratio
|
||
acceptance_ratio = np.exp(log_acceptance_ratio)
|
||
|
||
return {"k_new": k_new, "a_new": new_a, "b_new": new_b, "log_ratio": log_ratio, "type": "death_complex", "new_eigs": remaining_eigs, "acceptance_ratio": acceptance_ratio}
|
||
|
||
def _propose_within_model(self):
|
||
"""
|
||
随机选择实根变复根，复根变实根，或微调现有根，调用现有函数_merge_eigenvalues，_split_eigenvalues，_disturbe_swimming。
|
||
"""
|
||
k = self.current_k
|
||
current_eigs = self.current_eigenvalues.copy()
|
||
|
||
if k == 0:
|
||
return None # 无法在 k=0 时进行模型内提议
|
||
|
||
proposal_type = np.random.choice(['merge', 'split', 'disturb'], p=[0.3, 0.3, 0.4])
|
||
|
||
if proposal_type == 'merge':
|
||
return self._merge_eigenvalues(current_eigs) # type: ignore
|
||
elif proposal_type == 'split':
|
||
return self._split_eigenvalues(current_eigs) # type: ignore
|
||
else:
|
||
return self._disturbe_swimming(current_eigs) # type: ignore
|
||
|
||
|
||
def _log_prior_b(self, b_coeffs):
|
||
"""
|
||
计算 C_c 系数 (b_0, ..., b_{k-1}) 的对数先验概率。
|
||
根据论文 4.2 节，我们使用独立标准正态先验。
|
||
b_i ~ N(0, 1)
|
||
"""
|
||
#
|
||
return stats.norm.logpdf(b_coeffs, 0, 1).sum()
|
||
|
||
|
||
def _generate_stable_a(self, k):
|
||
"""
|
||
生成 k 个稳定的 a = [a_0, a_1, ..., a_{k-1}] 系数, 并保存对应的特征值。
|
||
|
||
该方法基于论文 4.1.1 节 [cite: 314-320] 和 6.2 节 的讨论，
|
||
生成 k 个稳定的特征值 (根)，然后使用维塔定理 (Proposition 4.1) [cite: 290]
|
||
计算多项式系数。
|
||
|
||
为确保系数a为实数，特征值必须是实数或复共轭对 。
|
||
"""
|
||
|
||
eigenvalues = []
|
||
i = 0
|
||
|
||
while i < k:
|
||
# 如果 k-i > 1，随机决定采样一个实数根还是一个复共轭对
|
||
# 如果 k-i == 1，则只能采样一个实数根
|
||
if i == k - 1 or np.random.rand() < 0.5:
|
||
# --- 1. 采样一个实数根 ---
|
||
# 使用 "Uniform real-eigenvalue prior" (在 (-1, 1) 上均匀分布)
|
||
real_eig = np.random.uniform(-1.0, 1.0)
|
||
eigenvalues.append(real_eig)
|
||
i += 1
|
||
else:
|
||
# --- 2. 采样一个复共轭对 ---
|
||
# 使用 "Polar-coordinate prior" (极坐标先验)
|
||
# 为确保在单位圆盘内面积均匀，rho = sqrt(U(0,1))
|
||
rho = np.sqrt(np.random.uniform(0, 1.0))
|
||
|
||
# theta 在 (0, pi) 均匀分布 (上半平面)
|
||
theta = np.random.uniform(0, np.pi)
|
||
|
||
# 计算复特征值
|
||
complex_eig = rho * (np.cos(theta) + 1j * np.sin(theta))
|
||
|
||
# 添加该值及其共轭
|
||
eigenvalues.append(complex_eig)
|
||
eigenvalues.append(np.conjugate(complex_eig))
|
||
i += 2
|
||
|
||
# --- 3. (维塔定理) 从根计算多项式系数 ---
|
||
# np.poly(roots) 返回 [1, a_{k-1}, a_{k-2}, ..., a_0]
|
||
coefficients_descending = np.poly(eigenvalues)
|
||
|
||
# 返回 [a_0, a_1, ..., a_{k-1}]
|
||
a_coefficients = coefficients_descending[1:][::-1]
|
||
|
||
self.current_eigenvalues = np.array(eigenvalues) # 顺便保存特征值
|
||
return np.real(a_coefficients)
|
||
|
||
|
||
def _generate_observable_b(self, k):
|
||
"""
|
||
生成 b = [b_0, ..., b_{k-1}] 系数 (共 k 个)。
|
||
"""
|
||
b_array = np.random.normal(0, 1, size=k)
|
||
return b_array
|
||
|
||
|
||
def get_controller_canonical_form(self, k=None, a_coeffs=None, b_coeffs=None):
|
||
"""
|
||
返回给定参数的规范型 (Controller Canonical Form) 矩阵 A_c, B_c, C_c, D_c。
|
||
|
||
如果提供了 k、a_coeffs、b_coeffs，则使用这些输入（使函数成为纯函数）；
|
||
否则回退到实例的当前状态（向后兼容）。
|
||
"""
|
||
# 回退到实例状态以保持向后兼容
|
||
if k is None:
|
||
k = self.current_k
|
||
if a_coeffs is None:
|
||
a_coeffs = self.current_a
|
||
if b_coeffs is None:
|
||
b_coeffs = self.current_b
|
||
|
||
# k=0 是无效情况，但 k=1 时 np.diag(np.ones(0), 1) 会创建一个空 1x1 矩阵
|
||
if k == 0:
|
||
return np.array([]), np.array([]), np.array([]), np.array([[0.0]])
|
||
|
||
# 1. 构造 A_c (k x k)
|
||
# 先创建一个 k x k 的零矩阵
|
||
A_c = np.zeros((k, k))
|
||
if k > 1:
|
||
# 创建上对角线为 1
|
||
np.fill_diagonal(A_c[0:-1, 1:], 1)
|
||
|
||
# 填充最后一行
|
||
# 确保 a_coeffs 的形状/长度与 k 匹配
|
||
a_arr = np.asarray(a_coeffs).ravel()
|
||
if a_arr.size != k:
|
||
raise ValueError(f"a_coeffs length {a_arr.size} does not match k={k}")
|
||
A_c[-1, :] = -a_arr
|
||
|
||
# 2. 构造 B_c (k x 1)
|
||
B_c = np.zeros((k, 1))
|
||
B_c[-1] = 1
|
||
|
||
# 3. 构造 C_c (1 x k)
|
||
b_arr = np.asarray(b_coeffs).ravel()
|
||
if b_arr.size != k:
|
||
raise ValueError(f"b_coeffs length {b_arr.size} does not match k={k}")
|
||
C_c = b_arr.reshape(1, k)
|
||
|
||
# 4. 构造 D_c (1 x 1)
|
||
# 论文 Definition 3.1 包含 d_0 ，但实验中设为 0
|
||
D_c = np.array([[0.0]])
|
||
|
||
return A_c, B_c, C_c, D_c
|
||
|
||
def _log_posterior_unnormalized(self, k, eigenvalues, b_coeffs, sigma2_w, sigma2_z):
|
||
"""计算所有参数的非归一化对数后验。"""
|
||
log_prior_k = -np.log(self.k_max - self.k_min + 1)
|
||
|
||
if k > 0:
|
||
log_prior_eigen = self._log_prior_eigenvalues(eigenvalues)
|
||
else:
|
||
log_prior_eigen = 0.0
|
||
|
||
log_prior_b = self._log_prior_b(b_coeffs)
|
||
precision_w = 1.0 / sigma2_w
|
||
precision_z = 1.0 / sigma2_z
|
||
log_prior_w = stats.gamma.logpdf(precision_w, a=self.lambda_a_w, scale=1.0/self.lambda_b_w)
|
||
log_prior_z = stats.gamma.logpdf(precision_z, a=self.lambda_a_z, scale=1.0/self.lambda_b_z)
|
||
log_likelihood = self._log_likelihood(k, self._cal_current_a_coeffs(eigenvalues), b_coeffs, sigma2_w, sigma2_z)
|
||
return log_prior_k + log_prior_eigen + log_prior_b + log_prior_w + log_prior_z + log_likelihood
|
||
|
||
def _purpose_b_swimming(self):
|
||
"""
|
||
对 b 系数进行游动：对每个 b_i 添加一个小的高斯扰动, 返回b游动前后的先验
|
||
"""
|
||
k = self.current_k
|
||
current_b = self.current_b
|
||
proposal_b = np.copy(current_b)
|
||
|
||
# 对每个 b_i 添加高斯扰动
|
||
for i in range(k):
|
||
b_proposal = proposal_b[i] + np.random.normal(0, 0.1) # 小的高斯扰动
|
||
proposal_b[i] = b_proposal
|
||
|
||
# 计算新b的先验
|
||
log_prior_current = self._log_prior_b(current_b)
|
||
log_prior_proposal = self._log_prior_b(proposal_b)
|
||
|
||
return proposal_b, log_prior_current, log_prior_proposal
|
||
|
||
def _purpose_lumbda_swimming(self):
|
||
"""
|
||
对 lumbda 进行游动：对每个 lumbda 添加一个小的高斯扰动，虚部也要有扰动
|
||
"""
|
||
k = self.current_k
|
||
current_eigs = np.copy(self.current_eigenvalues)
|
||
proposal_eigs = np.copy(current_eigs)
|
||
|
||
# 对每个 lumbda 添加高斯扰动
|
||
for i in range(k):
|
||
eig_proposal = proposal_eigs[i] + np.random.normal(0, 0.1) # 小的高斯扰动
|
||
proposal_eigs[i] = eig_proposal
|
||
|
||
# 虚部也要有扰动，共轭的虚部记得相等
|
||
for i in range(k):
|
||
if np.iscomplex(proposal_eigs[i]) and proposal_eigs[i].imag != 0:
|
||
imag_perturbation = np.random.normal(0, 0.1)
|
||
proposal_eigs[i] = proposal_eigs[i].real + 1j * (proposal_eigs[i].imag + imag_perturbation)
|
||
# 找到共轭根并更新
|
||
conjugate_idx = np.where(proposal_eigs == np.conjugate(current_eigs[i]))[0]
|
||
if len(conjugate_idx) > 0:
|
||
proposal_eigs[conjugate_idx[0]] = proposal_eigs[conjugate_idx[0]].real - 1j * (current_eigs[i].imag + imag_perturbation)
|
||
|
||
# 计算新lumbda的先验
|
||
log_prior_current = self._log_prior_eigenvalues(current_eigs)
|
||
log_prior_proposal = self._log_prior_eigenvalues(proposal_eigs)
|
||
|
||
return proposal_eigs, log_prior_current, log_prior_proposal
|
||
|
||
|
||
|
||
def _merge_eigenvalues(self, current_eigs):
|
||
"""合并游动：选择两个实数根，确定性地合并为一个复共轭对。"""
|
||
k = self.current_k
|
||
|
||
# 随机选择两个不同的实数根(可能是虚部为0的复数)，并删除他们
|
||
real_indices = np.where(np.isreal(current_eigs))
|
||
if len(real_indices[0]) < 2:
|
||
return # 不足两个实数根，无法合并
|
||
idx_pair = np.random.choice(real_indices[0], size=2, replace=False)
|
||
idx1, idx2 = idx_pair
|
||
lambda1 = current_eigs[idx1].real # type: ignore
|
||
lambda2 = current_eigs[idx2].real # type: ignore
|
||
|
||
current_eigs = np.delete(current_eigs, [idx1, idx2])
|
||
|
||
# 生成一对共轭复根
|
||
new_theta = np.random.uniform(0, np.pi)
|
||
new_rho = np.random.uniform(0, 1.0)
|
||
new_eigenvalue = new_rho * (np.cos(new_theta) + 1j * np.sin(new_theta))
|
||
current_eigs = np.append(current_eigs, [new_eigenvalue, np.conjugate(new_eigenvalue)])
|
||
|
||
# 对b进行游动
|
||
proposal_b, log_prior_b_current, log_prior_b_proposal = self._purpose_b_swimming()
|
||
|
||
# 计算提议密度
|
||
# 正向过程是任选两个实根，合并为复共轭对
|
||
log_q_merge_forward = -np.log(len(real_indices[0]) * (len(real_indices[0]) - 1) / 2.0) - np.log(np.pi) - np.log(1.0) # 提议密度 q_forward(u) 在 (0, pi) x (0,1)
|
||
# 逆向过程是从复共轭对中任选一个删去，然后抽样两个新实根,提议分布就是均匀分布
|
||
num_complex_eigs = len(np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0))[0]) #type: ignore
|
||
log_q_merge_backward = -np.log(num_complex_eigs + 1) - np.log(2) - np.log(2)
|
||
|
||
# 计算对数提议比
|
||
log_proposal_ratio = log_q_merge_backward - log_q_merge_forward
|
||
|
||
# 计算后验比
|
||
log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_eigenvalues, self.current_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
log_post_unnormalized_proposal = self._log_posterior_unnormalized(k, current_eigs, proposal_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
|
||
# 计算接受率
|
||
log_acceptance_ratio = (log_post_unnormalized_proposal - log_post_unnormalized_current) + log_proposal_ratio
|
||
|
||
acceptance_ratio = np.exp(log_acceptance_ratio)
|
||
|
||
return {"a_new": np.real(np.poly(current_eigs)[1:][::-1]), "b_new": proposal_b, "log_ratio": log_proposal_ratio, "type": "merge", "new_eigs": current_eigs, "acceptance_ratio": acceptance_ratio}
|
||
|
||
|
||
def _split_eigenvalues(self, current_eigs):
|
||
"""
|
||
分裂游动：选择一个复共轭对，随机地分裂为两个实数根。
|
||
"""
|
||
k = self.current_k
|
||
|
||
# 随机选择一个复共轭对，并删除它们
|
||
complex_indices = np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0)) #type: ignore
|
||
if len(complex_indices[0]) < 1:
|
||
return # 没有复共轭对，无法分裂
|
||
idx_to_remove = np.random.choice(complex_indices[0])
|
||
lambda_removed = current_eigs[idx_to_remove]
|
||
conjugate_idx_to_remove = np.where(current_eigs == np.conjugate(lambda_removed))[0]
|
||
|
||
if len(conjugate_idx_to_remove) == 0:
|
||
return # 找不到共轭对，无法分裂
|
||
conjugate_idx = conjugate_idx_to_remove[0]
|
||
|
||
# 删除选中的复共轭对
|
||
current_eigs = np.delete(current_eigs, [idx_to_remove, conjugate_idx])
|
||
# 生成两个实数根
|
||
real_eig1 = np.random.uniform(-1.0, 1.0)
|
||
real_eig2 = np.random.uniform(-1.0, 1.0)
|
||
current_eigs = np.append(current_eigs, [real_eig1, real_eig2])
|
||
|
||
# 对b进行游动
|
||
proposal_b, log_prior_b_current, log_prior_b_proposal = self._purpose_b_swimming()
|
||
|
||
# 计算提议密度
|
||
# 正向过程是任选一个复共轭对，分裂为两个实根
|
||
log_q_split_forward = -np.log(len(complex_indices[0])) - np.log(2) - np.log(2) # 提议密度 q_forward(u) 在 (-1,1) x (-1,1)
|
||
# 逆向过程是从实根中任选两个删去，然后抽样一个复共轭对,提议分布就是均匀分布
|
||
num_real_eigs = len(np.where(np.isreal(current_eigs))[0])
|
||
log_q_split_backward = -np.log(num_real_eigs * (num_real_eigs - 1) / 2.0) - np.log(np.pi) - np.log(1.0)
|
||
|
||
# 计算对数提议比
|
||
log_proposal_ratio = log_q_split_backward - log_q_split_forward
|
||
|
||
# 计算后验比
|
||
log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_eigenvalues, self.current_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
log_post_unnormalized_proposal = self._log_posterior_unnormalized(k, current_eigs, proposal_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
|
||
# 计算接受率
|
||
log_acceptance_ratio = (log_post_unnormalized_proposal - log_post_unnormalized_current) + log_proposal_ratio
|
||
acceptance_ratio = np.exp(log_acceptance_ratio)
|
||
|
||
return {"a_new": np.real(np.poly(current_eigs)[1:][::-1]), "b_new": proposal_b, "log_ratio": log_proposal_ratio, "type": "split", "new_eigs": current_eigs, "acceptance_ratio": acceptance_ratio}
|
||
|
||
def _disturbe_swimming(self, current_eigs):
|
||
"""
|
||
扰动游动：保持维度不变，保持实数根和复共轭对的数量不变，进行扰动。
|
||
"""
|
||
k = self.current_k
|
||
|
||
# 对特征值进行游动
|
||
proposal_eigs, log_prior_eigs_current, log_prior_eigs_proposal = self._purpose_lumbda_swimming()
|
||
|
||
# 对b进行游动
|
||
proposal_b, log_prior_b_current, log_prior_b_proposal = self._purpose_b_swimming()
|
||
|
||
# 计算提议密度比 (对称提议，密度相等)
|
||
log_proposal_ratio = 0.0
|
||
|
||
# 计算后验比
|
||
log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_eigenvalues, self.current_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
log_post_unnormalized_proposal = self._log_posterior_unnormalized(k, proposal_eigs, proposal_b, self.current_sigma2_w, self.current_sigma2_z)
|
||
|
||
# 计算接受率
|
||
log_acceptance_ratio = (log_post_unnormalized_proposal - log_post_unnormalized_current) + log_proposal_ratio
|
||
acceptance_ratio = np.exp(log_acceptance_ratio)
|
||
|
||
return {"a_new": np.real(np.poly(proposal_eigs)[1:][::-1]), "b_new": proposal_b, "log_ratio": log_proposal_ratio, "type": "disturbe", "new_eigs": proposal_eigs, "acceptance_ratio": acceptance_ratio}
|
||
|
||
|
||
def run_MCMC(self, num_iterations):
|
||
"""
|
||
运行 RJMCMC 采样器指定次数的迭代。
|
||
"""
|
||
# 存储采样历史
|
||
k_history = []
|
||
a_history = []
|
||
b_history = []
|
||
sigma2_w_history = []
|
||
sigma2_z_history = []
|
||
eigenvalues_history = []
|
||
|
||
print(f"Starting MCMC with initial k={self.current_k}")
|
||
|
||
for i in range(num_iterations):
|
||
# 随机选择一种转移类型
|
||
# 这里的概率可以根据需要调整
|
||
move_type = np.random.choice(['birth_death', 'within_model'], p=[0.5, 0.5])
|
||
|
||
proposal = None
|
||
accepted = False
|
||
move_name = 'None'
|
||
|
||
if move_type == 'birth_death':
|
||
# 决定是 birth 还是 death
|
||
if self.current_k == self.k_min:
|
||
proposal = self._propose_birth()
|
||
elif self.current_k == self.k_max:
|
||
proposal = self._propose_death()
|
||
else:
|
||
if np.random.rand() < 0.5:
|
||
proposal = self._propose_birth()
|
||
else:
|
||
proposal = self._propose_death()
|
||
|
||
elif move_type == 'within_model':
|
||
proposal = self._propose_within_model()
|
||
|
||
# 处理提议
|
||
if proposal and 'acceptance_ratio' in proposal:
|
||
move_name = proposal.get('type', 'N/A')
|
||
if np.random.rand() < proposal['acceptance_ratio']:
|
||
# 接受提议，更新状态
|
||
if 'k_new' in proposal: self.current_k = proposal['k_new']
|
||
if 'new_eigs' in proposal: self.current_eigenvalues = proposal['new_eigs']
|
||
self.current_a = self._cal_current_a_coeffs(self.current_eigenvalues)
|
||
if 'b_new' in proposal: self.current_b = proposal['b_new']
|
||
# sigma2_w 和 sigma2_z 在这些提议中没有更新，保持不变
|
||
accepted = True
|
||
|
||
# 存储当前状态 (无论是否接受，都存储当前链的状态)
|
||
k_history.append(self.current_k)
|
||
a_history.append(self.current_a)
|
||
b_history.append(self.current_b)
|
||
sigma2_w_history.append(self.current_sigma2_w)
|
||
sigma2_z_history.append(self.current_sigma2_z)
|
||
eigenvalues_history.append(self.current_eigenvalues)
|
||
|
||
if (i + 1) % 100 == 0:
|
||
status = "Accepted" if accepted else "Rejected"
|
||
print(f"Iteration {i+1}/{num_iterations}, k: {self.current_k}, Move: {move_name}, Status: {status}")
|
||
|
||
return {
|
||
"k": np.array(k_history),
|
||
"a": a_history,
|
||
"b": b_history,
|
||
"sigma2_w": np.array(sigma2_w_history),
|
||
"sigma2_z": np.array(sigma2_z_history),
|
||
"eigenvalues": eigenvalues_history
|
||
}
|
||
|
||
|
||
if __name__ == "__main__":
|
||
# 导入生成数据的模块
|
||
from generateSimData import simulate_lti_data
|
||
from generateGroudTruth import generate_ground_truth_system
|
||
import matplotlib.pyplot as plt
|
||
|
||
# 配置 matplotlib 支持中文显示
|
||
plt.rcParams['font.sans-serif'] = ['SimHei', 'Microsoft YaHei', 'Arial Unicode MS'] # 用来正常显示中文标签
|
||
plt.rcParams['axes.unicode_minus'] = False # 用来正常显示负号
|
||
|
||
# 1. 生成 Ground Truth 系统和仿真数据
|
||
print("--- 1. 生成仿真数据 ---")
|
||
# 真实系统阶数为 2
|
||
A_true, B_true, C_true, D_true = generate_ground_truth_system(dx=2, rng_seed=42)
|
||
|
||
T_steps = 400
|
||
sigma_proc = 0.1 # 过程噪声标准差
|
||
sigma_meas = 0.1 # 测量噪声标准差
|
||
|
||
u_data, y_data = simulate_lti_data(
|
||
A_true, B_true, C_true, D_true,
|
||
T=T_steps,
|
||
sigma_process=sigma_proc,
|
||
sigma_measurement=sigma_meas,
|
||
rng_seed=123
|
||
)
|
||
# RJMCMC_Sampler 需要一维的 y 和 u
|
||
y_data_flat = y_data.flatten()
|
||
u_data_flat = u_data.flatten()
|
||
|
||
print(f"仿真数据生成完毕。 y shape: {y_data_flat.shape}, u shape: {u_data_flat.shape}")
|
||
|
||
# 2. 初始化 RJMCMC 采样器
|
||
print("\n--- 2. 初始化 RJMCMC 采样器 ---")
|
||
sampler = RJMCMC_Sampler(
|
||
y=y_data_flat,
|
||
u=u_data_flat,
|
||
k_min=1,
|
||
k_max=4, # 探索的最大阶数
|
||
initial_k=3 # 从一个不等于真实阶数的阶数开始
|
||
)
|
||
|
||
# 3. 运行 MCMC
|
||
print("\n--- 3. 开始运行 MCMC 采样 ---")
|
||
num_iterations = 20000
|
||
results = sampler.run_MCMC(num_iterations)
|
||
print("MCMC 采样完成。")
|
||
|
||
# 4. 分析和可视化结果
|
||
print("\n--- 4. 分析结果 ---")
|
||
|
||
# 丢弃早期样本 (burn-in)
|
||
burn_in = 1000
|
||
k_samples = results['k'][burn_in:]
|
||
|
||
# 计算模型阶数的后验分布
|
||
# 使用 bincount 统计每个 k 出现的次数
|
||
k_posterior_counts = np.bincount(k_samples, minlength=sampler.k_max + 1)
|
||
k_posterior_prob = k_posterior_counts / len(k_samples)
|
||
|
||
# 打印后验概率
|
||
print("模型阶数的后验概率分布:")
|
||
for k_val in range(sampler.k_min, sampler.k_max + 1):
|
||
print(f" P(k={k_val} | y) ≈ {k_posterior_prob[k_val]:.4f}")
|
||
|
||
# 可视化 k 的后验分布
|
||
plt.figure(figsize=(10, 6))
|
||
plt.bar(range(sampler.k_min, sampler.k_max + 1),
|
||
k_posterior_prob[sampler.k_min:sampler.k_max + 1],
|
||
color='skyblue', alpha=0.8, label='Posterior Probability')
|
||
plt.axvline(x=2, color='red', linestyle='--', label='True Model Order (k=2)')
|
||
plt.xlabel("模型阶数 (k)")
|
||
plt.ylabel("后验概率 P(k|y)")
|
||
plt.title("模型阶数的后验分布 (After Burn-in)")
|
||
plt.xticks(range(sampler.k_min, sampler.k_max + 1))
|
||
plt.legend()
|
||
plt.grid(axis='y', linestyle='--', alpha=0.7)
|
||
plt.show()
|
||
|
||
|
||
|
||
|