加性偏移外推提升参数识别可靠性

python# 复制代码

## 1. 核心功能模块
### 1.1 ODE模型定义
- **Case A**: 结构性不可识别 ODE
  ```python
  def ode_caseA(t, x0, theta):
      theta1, theta2 = theta
      lam = theta1 + theta2
      return x0 * np.exp(-lam * t)

Case B : 先验截断 ODE

python 复制代码

def ode_caseB(t, x0, theta):
    theta1 = theta[0]
    return x0 * np.exp(-theta1 * t)

Case C : 病态尺度参数 ODE

python 复制代码

def ode_caseC(t, x0, theta):
    theta1, theta2 = theta
    lam = theta1 * theta2
    return x0 * np.exp(-lam * t)

Case D : 反向山脊标本 ODE

python 复制代码

def ode_caseD(t, x0, theta):
    theta1, theta2 = theta
    lam = theta1 - theta2
    return x0 * np.exp(-lam * t)

1.2 损失函数构建

python 复制代码

def build_dels_loss(t_obs, y_obs, ode_model, x0, prior_mean, prior_cov):
    inv_prior_cov = np.linalg.inv(prior_cov)
    def loss(theta):
        x_hat = ode_model(t_obs, x0, theta)
        resid = y_obs - x_hat
        ll_resid = -0.5 * np.sum(resid ** 2)
        ll_prior = -0.5 * (theta - prior_mean).T @ inv_prior_cov @ (theta - prior_mean)
        return -(ll_resid + ll_prior)
    return loss

1.3 剖面似然扫描

python 复制代码

def profile_likelihood_scan(
    param_idx: int,
    param_name: str,
    theta_prior_low: float,
    theta_prior_high: float,
    mle_point: np.ndarray,
    mcmc_post_std: float,
    hessian_full: np.ndarray,
    dels_loss_fn,
    base_bounds: List[Tuple[float, float]],
    specimen_id: str,
    extrap_fraction: float = EXTRAP_FRACTION,
    use_spline_ci: bool = True,
    multi_restart: bool = True
) -> ProfileResult:
    # 实现细节略

1.4 山脊方向检测

python 复制代码

def ridge_direction(profile_list: List[ProfileResult]) -> np.ndarray:
    dir_signs = []
    for pr in profile_list:
        grad = np.gradient(pr.profile_ll, pr.scan_grid)
        flat_idx = np.where(pr.profile_ll >= pr.profile_ll.max() - DELTA_LOG_L)[0]
        if len(flat_idx) < 2:
            continue
        mean_grad = np.mean(grad[flat_idx])
        s = np.sign(mean_grad)
        dir_signs.append(s)
    return np.array(dir_signs)

1.5 跨标本聚合

python 复制代码

def cross_specimen_aggregate(profile_results: List[ProfileResult], enable_ridge_check: bool = True) -> ProfileResult:
    n_escaping = sum(1 for r in profile_results if r.boundary_extrap_test == "escaping")
    agg = profile_results[0]

    if n_escaping >= 2:
        if enable_ridge_check:
            signs = ridge_direction(profile_results)
            if len(signs) < 2:
                agg.cross_specimen_verdict = "inconsistent_numerical"
                agg.verdict = "NUMERICAL_COUPLING"
            else:
                if np.all(signs == signs[0]):
                    agg.cross_specimen_verdict = "consistent_structural"
                    agg.verdict = "STRUCTURAL_UNIDENTIFIABLE"
                else:
                    agg.cross_specimen_verdict = "opposite_ridge"
                    agg.verdict = "NUMERICAL_COUPLING"
        else:
            agg.cross_specimen_verdict = "consistent_structural"
            agg.verdict = "STRUCTURAL_UNIDENTIFIABLE"
    else:
        agg.cross_specimen_verdict = "inconsistent_numerical"
        agg.verdict = "NUMERICAL_COUPLING"
    return agg

2. 测试用例分析

2.1 Case A: 结构性不可识别

ODE : dx/dt = -(θ1 + θ2) x
结果 : STRUCTURAL_UNIDENTIFIABLE

断言 :

python 复制代码

assert prA.verdict == "STRUCTURAL_UNIDENTIFIABLE"
assert prA.boundary_extrap_test == "escaping"

2.2 Case B: 先验截断

ODE : dx/dt = -θ x
结果 : PRIOR_TRUNCATED

断言 :

python 复制代码

assert prB.verdict == "PRIOR_TRUNCATED"
assert prB.boundary_extrap_test == "stable"

2.3 Case C: 优化器失效

ODE : dx/dt = -θ1 * θ2 * x
结果 : OPT_FAILURE
断言 :
python 复制代码
```
assert prC.verdict == "OPT_FAILURE"
```

2.4 Case D: 反向山脊双标本

ODE : dx/dt = -(θ1 − θ2) x
结果 : NUMERICAL_COUPLING（生产模式）或 STRUCTURAL_UNIDENTIFIABLE（反事实模式）

断言 :

python 复制代码

assert aggD_prod.verdict == "NUMERICAL_COUPLING"
assert aggD_counterfact.verdict == "STRUCTURAL_UNIDENTIFIABLE"

3. 关键技术点

技术点	描述
增量更新	使用 `--update` 和 `--cluster-only` 实现知识图谱的最小代价维护
山脊方向校验	通过梯度符号判断参数空间中的逃逸方向，避免误判
跨标本聚合	通过 `cross_specimen_aggregate` 实现多标本的一致性验证
剖面似然扫描	通过自适应网格生成和多初值重启提高优化稳定性

4. 总结

该代码实现了对不同ODE模型的参数可识别性分析，通过剖面似然扫描、山脊方向检测和跨标本聚合等方法，能够准确判断模型的可识别性。在实际应用中，需注意先验设置、优化器配置和网格生成策略，以确保结果的可靠性。

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