import numpy as np from scipy import stats class RJMCMC_Sampler: def __init__(self, y, u, k_min=1, k_max=10, initial_k=None, initial_a=None, initial_b=None): """ 初始化采样器 参数: y (array): 观测到的输出序列 u (array): 已知的输入序列 k_min (int): 探索的最小模型阶数 k_max (int): 探索的最大模型阶数 initial_k (int, optional): 初始模型阶数 initial_a (dict, optional): 初始多项式系数 initial_b (array, optional): 初始多项式系数 """ # 定义观测数据 self.y = y self.u = u self.N = len(y) # 定义模型阶数 self.k_min = k_min self.k_max = k_max gamma_sample = np.random.gamma(shape=2, scale=1) self.current_k = initial_k if initial_k is not None else min(max(int(gamma_sample), k_min), k_max) # 初始化多项式系数 if initial_a is not None: self.current_a = initial_a else: self.current_a = self._generate_stable_a(self.current_k) # 定义标准型的b系数 if initial_b is not None: self.current_b = initial_b else: self.current_b = self._generate_observable_b(self.current_k) # 初始化噪声先验 (假设为 Gamma 先验) self.lambda_a_w = 1e-3 # 过程噪声精度的先验 self.lambda_b_w = 1e-3 self.lambda_a_z = 1e-3 # 观测噪声精度的先验 self.lambda_b_z = 1e-3 # 初始化噪声方差 (从先验采样) # 为避免除零错误，在分母上增加一个极小值 epsilon = 1e-9 self.current_sigma2_w = 1.0 / (np.random.gamma(self.lambda_a_w, 1.0/self.lambda_b_w) + epsilon) self.current_sigma2_z = 1.0 / (np.random.gamma(self.lambda_a_z, 1.0/self.lambda_b_z) + epsilon) def _update_current_eigenvalues(self): """一个辅助函数，根据 current_a 更新 current_eigenvalues。""" if self.current_k > 0: poly_coeffs = np.concatenate(([1], -self.current_a[::-1])) self.current_eigenvalues = np.roots(poly_coeffs) else: self.current_eigenvalues = np.array([]) def _update_current_a_coeffs(self): """一个辅助函数，根据 current_eigenvalues 更新 current_a。""" k = len(self.current_eigenvalues) if k > 0: # np.poly 返回 [1, a_{k-1}, ..., a_0]，去掉首项“1”，反转剩下的 poly_coeffs = np.poly(self.current_eigenvalues) self.current_a = np.real(poly_coeffs[1:][::-1]) else: self.current_a = np.array([]) def _cal_current_eigenvalues(self, a_coeffs_temp): """一个辅助函数，根据给定的 a_coeffs 计算对应的特征值。""" k = len(a_coeffs_temp) if k > 0: poly_coeffs = np.concatenate(([1], -a_coeffs_temp[::-1])) return np.roots(poly_coeffs) else: return np.array([]) def _cal_current_a_coeffs(self, eigenvalues_temp): """一个辅助函数，根据给定的特征值计算对应的 a_coeffs。""" k = len(eigenvalues_temp) if k > 0: # np.poly 返回 [1, a_{k-1}, ..., a_0]，去掉首项“1”，反转剩下的 poly_coeffs = np.poly(eigenvalues_temp) return np.real(poly_coeffs[1:][::-1]) else: return np.array([]) def _log_prior_eigenvalues(self, eigenvalues): """ 计算 k 个特征值 (Lambda) 的对数先验概率 log p(Lambda)。 该先验基于论文 4.1.1 节 讨论的先验： 1. 稳定性：如果任何 |lambda| >= 1，先验概率为 0 (log(0) = -inf)。 2. 实数根 (p_real)：在 (-1, 1) 上均匀分布, p(lambda_r) = 1/2。 log p(lambda_r) = -log(2)。 3. 复数根 (p_complex)：在单位圆盘上均匀分布, p(lambda_c) = 1/pi。 log p(lambda_c) = -log(pi)。 注意：复共轭对 (lambda, lambda_conj) 只计算一次 (例如上半平面)。 我们通过只计算 imag(lambda) >= 0 的复数根来实现。 """ log_prior = 0.0 # 跟踪已处理的复数根 (避免重复计算共轭对) processed_complex = set() for eig in eigenvalues: if np.abs(eig) >= 1.0: # 违反稳定性约束 return -np.inf if np.isclose(eig.imag, 0): # 是实数 # 这是 "Uniform real-eigenvalue prior" log_prior += -np.log(2.0) else: # 是复数 # 检查是否已处理过 if eig in processed_complex or np.conjugate(eig) in processed_complex: continue # 我们只计算 imag >= 0 的根 (代表一对) if eig.imag >= 0: # 这是 "Polar-coordinate prior" log_prior += -np.log(np.pi) processed_complex.add(eig) return log_prior def _log_prior_eigenvalues_to_a(self,eigenvalues): """ 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。 该方法基于 Proposition 4.2 (Change of variables) log p(a) = log p(Lambda) - log(|det J|) 其中 log p(Lambda) 是特征值的先验 而 log(|det J|) 是维塔变换的对数雅可比行列式。 """ # 计算 log p(Lambda) log_prior_lambda = self._log_prior_eigenvalues(eigenvalues) # 如果特征值不稳定 (log p(Lambda) = -inf)，则 p(a) 也不可能 if log_prior_lambda == -np.inf: return -np.inf # 计算对数雅可比行列式 log(|det J|) log_det_jacobian = 0.0 k = len(eigenvalues) if k < 2: return log_prior_lambda for i in range(k): for j in range(i + 1, k): diff = eigenvalues[i] - eigenvalues[j] abs_diff = np.abs(diff) if abs_diff == 0: return -np.inf # 重根导致雅可比行列式为零 log_det_jacobian += np.log(abs_diff) if log_det_jacobian == -np.inf: return -np.inf # 应用变量替换公式 log_prior_a = log_prior_lambda - log_det_jacobian return log_prior_a def _log_likelihood(self, k, a_coeffs, b_coeffs, sigma2_w, sigma2_z): """ 计算给定参数下的对数似然 log p(y | k, a, b, sigma_w, sigma_z)。 使用标准卡尔曼滤波器 (Kalman Filter) (如论文 Appendix B ) 和预测误差分解 (Prediction Error Decomposition) (如论文 Eq. (D.37) )。 """ # 如果 k=0，模型无法运行 if k == 0: # 返回一个非常小的似然 return -np.inf # 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） A, B, C, D = self.get_controller_canonical_form(k, a_coeffs, b_coeffs) # 过程噪声 Sigma (k x k) Sigma = np.zeros((k, k)) Sigma[k-1, k-1] = sigma2_w # 测量噪声 Gamma (1 x 1) Gamma = np.array([[sigma2_z]]) # 初始化卡尔曼滤波器 # 状态 x_t|t-1 x_pred = np.zeros((k, 1)) # 状态协方差 P_t|t-1 P_pred = np.eye(k) * 1e6 total_log_likelihood = 0.0 # 运行卡尔曼滤波器 N 步 for t in range(self.N): y_t = self.y[t] u_t = self.u[t] # --- 预测步 (Prediction) --- # (在 t=0 时，x_pred 和 P_pred 是我们初始化的 x_0|-1, P_0|-1) # --- 计算似然 --- # 预测误差 nu_t y_pred = C @ x_pred + D @ np.array([[u_t]]) nu_t = y_t - y_pred # 预测误差协方差 S_t S_t = C @ P_pred @ C.T + Gamma S_t_inv = 1.0 / S_t # 累加对数似然 total_log_likelihood += -0.5 * (np.log(2 * np.pi) + np.log(S_t) + (nu_t * S_t_inv * nu_t)) # --- 更新步 (Update) --- # 卡尔曼增益 K_t (k x 1) K_t = P_pred @ C.T * S_t_inv # 更新状态 x_t|t x_update = x_pred + K_t @ nu_t # 更新协方差 P_t|t I_KC = np.eye(k) - K_t @ C P_update = I_KC @ P_pred @ I_KC.T + K_t @ Gamma @ K_t.T # --- 为下一次循环准备预测 (t+1) --- x_pred = A @ x_update + B @ np.array([[u_t]]) P_pred = A @ P_update @ A.T + Sigma return total_log_likelihood # 返回标量值 def _propose_birth(self): """ 提议一个 "诞生" 转移 (k -> k+1 或 k -> k+2)。 返回包含提议状态和计算接受率所需的对数比率的字典。 """ k = self.current_k current_eigs = self.current_eigenvalues # 获取实数特征值数目 real_eigs_indices = np.where(np.isreal(current_eigs)) num_real_eigs = len(real_eigs_indices[0]) # 获取复数特征值数目，只保留 imag > 0 的部分 (共轭对只算一次) complex_eigs_indices = np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0)) #type: ignore num_complex_eigs = len(complex_eigs_indices[0]) # 决定是诞生一个实数根还是一对复共轭根 can_add_complex = (k + 2) <= self.k_max add_real = True if can_add_complex: # 以0.5的概率选择诞生一对复共轭根 if np.random.rand() < 0.5: add_real = False if add_real: k_new = k + 1 # 诞生实根的对数概率之比 log_birth_real_forward = np.log(0.5) log_birth_real_backward = np.log(0.5) # 从提议分布 q(u) 中采样辅助变量 u = (u_lambda, u_b) u_lambda = np.random.uniform(-1.0, 1.0) # 新特征值 u_b = np.random.normal(0, 1) # 新 b 系数 # 计算对数提议密度 log_q_forward = log_birth_real_forward + stats.norm.logpdf(u_b, 0, 1) - np.log(2.0) log_q_backward = log_birth_real_backward - np.log(num_real_eigs + 1) # 计算对数雅可比行列式 log|J| if k == 0: log_det_jacobian = 0.0 else: log_det_jacobian = 1 # 构造新状态 new_eigs = np.append(current_eigs, u_lambda) new_a = np.real(np.poly(new_eigs)[1:][::-1]) new_b = np.append(self.current_b, u_b) # 接受率中的项为 q_backward(u) * |J| / q_forward(u) log_ratio = log_q_backward + log_det_jacobian - log_q_forward # 计算未归一化的后验之比 log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_a, self.current_b, self.current_sigma2_w, self.current_sigma2_z) log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, new_a, 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": "birth_real", "new_eigs": new_eigs, "acceptance_ratio": acceptance_ratio} else: # --- 诞生一对复共轭根 (k -> k+2) --- k_new = k + 2 # 诞生实根的对数概率之比 log_birth_complex_forward = np.log(0.5) log_birth_complex_backward = np.log(0.5) # 采样辅助变量 u = (rho, theta, u_b1, u_b2) rho = np.sqrt(np.random.uniform(0, 1.0)) theta = np.random.uniform(0, np.pi) u_lambda = rho * (np.cos(theta) + 1j * np.sin(theta)) u_b1, u_b2 = np.random.normal(0, 1, 2) # 计算对数提议密度 log(q(u)) 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) log_q_backward = log_birth_complex_backward - np.log(num_complex_eigs + 1) # 计算对数雅可比行列式 log|J| # |J| = 2*ρ log_jacobian = np.log(2 * rho) # 构造新状态 new_eigs = np.append(current_eigs, [u_lambda, np.conjugate(u_lambda)]) new_a = np.real(np.poly(new_eigs)[1:][::-1]) new_b = np.append(self.current_b, [u_b1, u_b2]) log_ratio = log_q_backward + log_jacobian - log_q_forward # 计算未归一化的后验之比 log_post_unnormalized_current = self._log_posterior_unnormalized(k, self.current_a, self.current_b, self.current_sigma2_w, self.current_sigma2_z) log_post_unnormalized_new = self._log_posterior_unnormalized(k_new, new_a, 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": "birth_complex", "new_eigs": new_eigs, "acceptance_ratio": acceptance_ratio} def _propose_death(self): """ 提议一个 "消亡" 转移 (k -> k-1 或 k -> k-2)。 返回包含提议状态和计算接受率所需的对数比率的字典。 """ k = self.current_k current_eigs = self.current_eigenvalues # 识别出可移除的实数根和复共轭对 real_eigs_indices = np.where(np.isreal(current_eigs)) complex_eigs_indices = np.where((np.iscomplex(current_eigs)) & (current_eigs.imag > 0)) #type: ignore can_remove_real = len(real_eigs_indices[0]) > 0 can_remove_complex = len(complex_eigs_indices[0]) > 0 if not can_remove_real and not can_remove_complex: return None # 无法执行消亡 # 决定是移除实数根还是复共轭对 remove_real = True if can_remove_real and can_remove_complex: if np.random.rand() < 0.5: remove_real = False elif can_remove_complex: remove_real = False if remove_real: # --- 消亡一个实数根 (k -> k-1) --- k_new = k - 1 # 生成与消亡实根对应的诞生过程的对数概率之比 log_death_real_forward = np.log(0.5) log_death_real_backward = np.log(0.5) # 1. 随机选择一个实数根移除 idx_to_remove = np.random.choice(real_eigs_indices[0]) lambda_removed = current_eigs[idx_to_remove] # 2. 随机选择一个实数 b 系数移除 idx_b_to_remove = np.random.choice(len(self.current_b)) b_removed = self.current_b[idx_b_to_remove] # 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()