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678 lines
21 KiB
Markdown
678 lines
21 KiB
Markdown
# 01.3 概率与不确定性
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## 核心问题
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> 空间分析中的不确定性从何而来？
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> AI系统如何表示和处理不确定性？
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> 如何在不确定性下做出稳健的决策？
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---
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## 概念讲解
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### 不确定性的来源
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在空间分析和AI系统中，不确定性无处不在：
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```
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空间分析中的不确定性来源
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┌─────────────────────────────────────────────────────────────┐
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│ │
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│ 1. 数据不确定性 │
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│ - 测量误差 │
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│ - 空间采样不完整 │
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│ - 分类错误 │
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│ - 时间延迟 │
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│ │
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│ 2. 参数不确定性 │
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│ - 阻力权重不确定 │
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│ - 阈值选择主观 │
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│ - 模型参数拟合误差 │
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│ │
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│ 3. 结构不确定性 │
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│ - 模型选择 │
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│ - 变量关系假设 │
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│ - 尺度效应 │
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│ │
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│ 4. 语义不确定性 │
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│ - 概念模糊（"生态质量"是什么？） │
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│ - 分类边界不清 │
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│ - 专家意见分歧 │
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│ │
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└─────────────────────────────────────────────────────────────┘
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```
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### 不确定性的类型
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| 类型 | 说明 | 例子 |
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|-----|------|------|
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| **偶然不确定性** (Aleatoric) | 系统固有的随机性，无法通过更多数据消除 | 降雨量的随机波动 |
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| **认知不确定性** (Epistemic) | 知识不足导致的不确定性，可通过更多数据减少 | 未调查区域的物种分布 |
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| **模糊性** (Ambiguity) | 概念或分类的不明确 | "高生态价值"的定义 |
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| **冲突** (Conflict) | 不同信息源的不一致 | 两个专家给出相反意见 |
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### AI如何处理不确定性
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**传统GIS vs 概率AI**：
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```
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传统GIS: 确定性输出
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输入 → [处理] → 单一结果
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例如：这个区域是/不是生态源地
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概率AI: 概率输出
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输入 → [处理] → (结果, 置信度)
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例如：这个区域是生态源地的概率是 0.78 ± 0.12
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```
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**置信度的表示**：
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```python
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# 方式1: 点估计 + 置信区间
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estimate = 0.75
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confidence_interval = (0.65, 0.85)
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# 方式2: 概率分布
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from scipy.stats import beta
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distribution = beta(a=8, b=3) # 基于共8次成功，3次失败
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# 方式3: 分类概率
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class_probabilities = {
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"high_suitability": 0.65,
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"medium_suitability": 0.25,
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"low_suitability": 0.10
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}
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# 方式4: 模糊隶属度
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fuzzy_membership = {
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"is_source": 0.72,
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"is_not_source": 0.28
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}
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```
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---
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## 设计原理
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### 不确定性传播
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当多个步骤串联时，不确定性会累积：
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```python
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"""
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不确定性传播示例
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"""
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import numpy as np
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from scipy.stats import norm
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class UncertainValue:
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"""带不确定性的值"""
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def __init__(self, mean, std):
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self.mean = mean
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self.std = std
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def __add__(self, other):
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"""加法：方差相加"""
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return UncertainValue(
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self.mean + other.mean,
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np.sqrt(self.std**2 + other.std**2)
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)
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def __mul__(self, scalar):
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"""乘以标量：标准差也乘"""
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return UncertainValue(
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self.mean * scalar,
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self.std * abs(scalar)
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)
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def __repr__(self):
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return f"{self.mean:.2f} ± {self.std:.2f}"
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# 示例：源地适宜性评估中的不确定性传播
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def assess_suitability_with_uncertainty():
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"""
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每个指标都有测量不确定性，
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最终的适宜性评分会累积这些不确定性
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"""
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# 各项指标（均值 ± 标准差）
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vegetation_quality = UncertainValue(0.75, 0.10)
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connectivity = UncertainValue(0.60, 0.15)
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distance_to_threat = UncertainValue(0.80, 0.08)
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# 加权组合（权重也有不确定性）
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weights = {
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"vegetation": 0.4,
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"connectivity": 0.3,
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"distance": 0.3
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}
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# 计算总分（简化传播）
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total = (vegetation_quality * weights["vegetation"] +
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connectivity * weights["connectivity"] +
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distance_to_threat * weights["distance"])
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print("各指标不确定性:")
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print(f" 植被质量: {vegetation_quality}")
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print(f" 连通性: {connectivity}")
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print(f" 威胁距离: {distance_to_threat}")
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print(f"\n总分: {total}")
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print(f" 置信区间95%: [{total.mean - 1.96*total.std:.2f}, "
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f"{total.mean + 1.96*total.std:.2f}]")
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return total
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if __name__ == "__main__":
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assess_suitability_with_uncertainty()
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```
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### 敏感性分析
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了解哪些参数对结果影响最大：
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```python
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"""
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敏感性分析：识别关键参数
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"""
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import numpy as np
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from typing import Dict, List, Tuple
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def sensitivity_analysis(model_fn, param_ranges: Dict[str, Tuple[float, float]],
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n_samples=1000) -> Dict[str, float]:
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"""
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使用蒙特卡洛方法进行敏感性分析
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Args:
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model_fn: 模型函数，接受参数字典，返回结果
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param_ranges: 参数范围 {param_name: (min, max)}
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n_samples: 采样次数
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Returns:
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各参数的敏感性系数
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"""
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results = {param: [] for param in param_ranges}
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model_outputs = []
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# 蒙特卡洛采样
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for _ in range(n_samples):
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# 随机采样参数
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sample = {k: np.random.uniform(v[0], v[1])
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for k, v in param_ranges.items()}
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# 记录参数值
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for param, value in sample.items():
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results[param].append(value)
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# 计算模型输出
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output = model_fn(sample)
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model_outputs.append(output)
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# 计算相关性作为敏感性指标
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sensitivities = {}
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for param in param_ranges:
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correlation = np.corrcoef(results[param], model_outputs)[0, 1]
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sensitivities[param] = abs(correlation)
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return sensitivities
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# 示例：生态阻力面构建的敏感性分析
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def resistance_model(params):
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"""简化的阻力面模型"""
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# 参数：各土地类型的阻力权重
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forest_weight = params["forest"]
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grass_weight = params["grass"]
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urban_weight = params["urban"]
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# 简化：计算平均阻力
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# 实际应用中会是空间计算
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landscape_composition = {
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"forest": 0.4,
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"grass": 0.3,
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"urban": 0.3
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}
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total_resistance = (
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forest_weight * landscape_composition["forest"] +
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grass_weight * landscape_composition["grass"] +
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urban_weight * landscape_composition["urban"]
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)
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return total_resistance
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def run_sensitivity_example():
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"""运行敏感性分析示例"""
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print("=== 阻力面参数敏感性分析 ===\n")
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# 定义参数范围
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param_ranges = {
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"forest": (1, 10),
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"grass": (10, 50),
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"urban": (50, 200)
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}
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# 运行敏感性分析
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sensitivities = sensitivity_analysis(
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resistance_model,
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param_ranges,
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n_samples=5000
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)
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# 排序并输出
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sorted_sens = sorted(sensitivities.items(),
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key=lambda x: x[1], reverse=True)
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print("参数敏感性排序:")
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for param, sensitivity in sorted_sens:
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bar = "█" * int(sensitivity * 30)
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print(f" {param}: {sensitivity:.3f} {bar}")
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print("\n解释:")
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print(f" 最敏感的参数是 {sorted_sens[0][0]}")
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print(f" 应当优先精确确定该参数的值")
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if __name__ == "__main__":
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run_sensitivity_example()
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```
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### 鲁棒决策
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当不确定性无法消除时，做鲁棒的决策：
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```python
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"""
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鲁棒决策：在不确定性下的稳健决策
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"""
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from typing import List, Callable
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import numpy as np
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def robust_decision_scenarios():
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"""
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鲁棒决策的几种策略
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"""
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# 策略1: 最大最小 (Maximin) - 最坏情况最优
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def maximin(payoff_matrix):
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"""
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选择在最坏情况下表现最好的选项
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payoff_matrix: 选项 × 场景 的收益矩阵
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"""
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worst_case_outcomes = payoff_matrix.min(axis=1)
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best_option = worst_case_outcomes.argmax()
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return best_option, worst_case_outcomes
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# 策略2: 最大平均 (Maximum Expected Value)
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def max_expected(payoff_matrix, probabilities=None):
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"""选择期望收益最大的选项"""
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if probabilities is None:
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probabilities = np.ones(payoff_matrix.shape[1]) / payoff_matrix.shape[1]
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expected_values = payoff_matrix @ probabilities
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best_option = expected_values.argmax()
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return best_option, expected_values
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# 策略3: 最小后悔 (Minimax Regret)
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def minimax_regret(payoff_matrix):
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"""选择最小化最大后悔的选项"""
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# 每个场景的最佳收益
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best_per_scenario = payoff_matrix.max(axis=0)
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# 后悔矩阵：每个选项在每个场景与最佳的差距
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regret_matrix = best_per_scenario - payoff_matrix
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# 每个选项的最大后悔
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max_regret = regret_matrix.max(axis=1)
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# 选择最大后悔最小的选项
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best_option = max_regret.argmin()
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return best_option, max_regret
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# 示例：生态廊道选址决策
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# 选项：3个候选廊道路线
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# 场景：不同的未来土地变化情景
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payoff_matrix = np.array([
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# 情景1 情景2 情景3 情景4
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[80, 60, 40, 70], # 选项A：穿过森林
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[50, 90, 70, 50], # 选项B：沿河流
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[60, 70, 90, 60], # 选项C：绕行城市边缘
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])
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print("=== 生态廊道选址：鲁棒决策分析 ===\n")
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print("收益矩阵（廊道质量评分）:")
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print(" 情景1 情景2 情景3 情景4")
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for i, row in enumerate(payoff_matrix, ord('A')):
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print(f"选项{i}: {row}")
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print("\n--- 策略1: 最大最小 (最坏情况最优) ---")
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option, worst = maximin(payoff_matrix)
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print(f"推荐: 选项{chr(ord('A') + option)}")
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print(f"各选项最坏情况: {worst}")
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print(" 原理: 选择在最坏情景下表现最好的")
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print("\n--- 策略2: 最大期望 (平均收益最大) ---")
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option, expected = max_expected(payoff_matrix)
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print(f"推荐: 选项{chr(ord('A') + option)}")
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print(f"各选项期望收益: {expected}")
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print(" 原理: 选择平均表现最好的")
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print("\n--- 策略3: 最小最大后悔 ---")
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option, regret = minimax_regret(payoff_matrix)
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print(f"推荐: 选项{chr(ord('A') + option)}")
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print(f"各选项最大后悔: {regret}")
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print(" 原理: 选择让'选错'的后悔最小的")
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return payoff_matrix
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if __name__ == "__main__":
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robust_decision_scenarios()
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```
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---
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## 代码示例
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### 概率源地识别
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```python
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"""
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带不确定性的生态源地识别
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"""
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import numpy as np
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from scipy.stats import beta
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from typing import Dict, List, Tuple
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class ProbabilisticSource:
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"""概率源地：带置信度的源地"""
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def __init__(self, source_id: int, geometry,
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probability: float, confidence: float):
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self.id = source_id
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self.geometry = geometry
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self.probability = probability # 是源地的概率
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self.confidence = confidence # 概率估计的置信度
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def __repr__(self):
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return (f"Source({self.id}, P={self.probability:.2f}±{self.confidence:.2f})")
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class ProbabilisticSourceIdentifier:
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"""概率源地识别器"""
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def __init__(self, threshold=0.5):
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self.threshold = threshold
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def identify(self, landscape_data) -> List[ProbabilisticSource]:
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"""
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识别源地，返回概率源地
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返回的不是"是/否"的判断，而是"是源地的概率"
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"""
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sources = []
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# 模拟：对每个斑块计算是源地的概率
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for i, patch in enumerate(landscape_data):
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# 基于多个指标计算概率
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probability = self._calculate_source_probability(patch)
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# 估计置信度（基于数据质量）
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confidence = self._estimate_confidence(patch)
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if probability >= self.threshold:
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source = ProbabilisticSource(
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source_id=i,
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geometry=patch['geometry'],
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probability=probability,
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confidence=confidence
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)
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sources.append(source)
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return sources
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def _calculate_source_probability(self, patch) -> float:
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"""计算斑块是源地的概率"""
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# 使用贝叶斯推理
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# P(是源地|数据) ∝ P(数据|是源地) × P(是源地)
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# 各指标的似然
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area_likelihood = self._area_likelihood(patch['area'])
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veg_likelihood = self._vegetation_likelihood(patch['vegetation'])
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shape_likelihood = self._shape_likelihood(patch['shape_index'])
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# 先验概率
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prior = 0.3 # 假设30%的斑块可能是源地
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# 后验概率（简化）
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probability = (area_likelihood * veg_likelihood *
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shape_likelihood * prior)
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probability = min(probability, 1.0) # 限制在[0,1]
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return probability
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def _area_likelihood(self, area: float) -> float:
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"""面积似然：大面积更像源地"""
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if area > 1000:
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return 1.0
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elif area > 500:
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return 0.7
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else:
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return 0.3
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def _vegetation_likelihood(self, veg_quality: float) -> float:
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"""植被质量似然"""
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return veg_quality # 假设已归一化到[0,1]
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def _shape_likelihood(self, shape_index: float) -> float:
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"""形状指数似然：紧凑形状更好"""
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return 1.0 - min(abs(shape_index - 1.0), 0.5)
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def _estimate_confidence(self, patch) -> float:
|
||
"""估计概率的置信度"""
|
||
# 基于数据质量、分辨率等因素
|
||
data_quality = patch.get('data_quality', 0.8)
|
||
resolution_factor = patch.get('resolution', 30) / 30 # 归一化
|
||
|
||
return data_quality * min(resolution_factor, 1.0)
|
||
|
||
def uncertainty_propagation_example():
|
||
"""不确定性传播示例"""
|
||
print("=== 不确定性传播示例 ===\n")
|
||
|
||
# 创建一些模拟斑块
|
||
patches = [
|
||
{'id': 1, 'area': 1200, 'vegetation': 0.85, 'shape_index': 1.2,
|
||
'geometry': 'POLYGON(...)', 'data_quality': 0.9},
|
||
{'id': 2, 'area': 800, 'vegetation': 0.75, 'shape_index': 1.5,
|
||
'geometry': 'POLYGON(...)', 'data_quality': 0.7},
|
||
{'id': 3, 'area': 400, 'vegetation': 0.65, 'shape_index': 1.8,
|
||
'geometry': 'POLYGON(...)', 'data_quality': 0.6},
|
||
]
|
||
|
||
identifier = ProbabilisticSourceIdentifier(threshold=0.4)
|
||
sources = identifier.identify(patches)
|
||
|
||
print("识别到的概率源地:")
|
||
for source in sources:
|
||
print(f" {source}")
|
||
|
||
# 计算整体不确定性
|
||
if sources:
|
||
avg_prob = np.mean([s.probability for s in sources])
|
||
avg_conf = np.mean([s.confidence for s in sources])
|
||
print(f"\n总体置信度: {avg_conf:.2f}")
|
||
print(f"平均源地概率: {avg_prob:.2f}")
|
||
|
||
# 置信区间
|
||
margin_of_error = (1 - avg_conf) * 0.2 # 简化计算
|
||
print(f"源地数量估计: {len(sources)} ± {margin_of_error * len(sources):.1f}")
|
||
|
||
if __name__ == "__main__":
|
||
uncertainty_propagation_example()
|
||
```
|
||
|
||
---
|
||
|
||
## 案例分析
|
||
|
||
### ENAgent中的不确定性处理
|
||
|
||
在ENAgent项目中，不确定性处理体现在：
|
||
|
||
**1. 源地识别的不确定性**
|
||
|
||
```python
|
||
class ENAgentSourceIdentifier:
|
||
"""ENAgent的源地识别模块"""
|
||
|
||
def identify_with_uncertainty(self, landcover, species_params):
|
||
"""
|
||
识别源地，同时估计不确定性
|
||
|
||
Returns:
|
||
sources: 源地列表
|
||
uncertainty_map: 不确定性空间分布
|
||
"""
|
||
sources = []
|
||
uncertainty_map = np.zeros_like(landcover)
|
||
|
||
# 对每个候选斑块
|
||
for patch in self._candidate_patches(landcover):
|
||
# 计算适宜性（考虑物种参数）
|
||
suitability = self._calculate_suitability(patch, species_params)
|
||
|
||
# 估计不确定性（来自多个来源）
|
||
uncertainty = self._estimate_uncertainty(
|
||
patch,
|
||
data_quality=landcover.metadata['quality'],
|
||
species_uncertainty=species_params['uncertainty']
|
||
)
|
||
|
||
# 记录
|
||
if suitability > self.threshold:
|
||
sources.append({
|
||
'geometry': patch,
|
||
'suitability': suitability,
|
||
'uncertainty': uncertainty
|
||
})
|
||
|
||
# 更新不确定性地图
|
||
self._add_to_uncertainty_map(uncertainty_map, patch, uncertainty)
|
||
|
||
return sources, uncertainty_map
|
||
|
||
def _estimate_uncertainty(self, patch, data_quality, species_uncertainty):
|
||
"""
|
||
估计源地识别的不确定性
|
||
|
||
来源：
|
||
1. 数据质量 (data_quality)
|
||
2. 物种参数的不确定性 (species_uncertainty)
|
||
3. 分类误差 (classification_error)
|
||
4. 边界效应 (edge_effect)
|
||
"""
|
||
# 组合各种不确定性源
|
||
uncertainty = np.sqrt(
|
||
(1 - data_quality)**2 +
|
||
species_uncertainty**2 +
|
||
0.1**2 + # 分类误差
|
||
self._edge_uncertainty(patch)**2
|
||
)
|
||
|
||
return min(uncertainty, 1.0)
|
||
```
|
||
|
||
**2. 阻力面的敏感性分析**
|
||
|
||
```python
|
||
class ResistanceSurfaceSensitivity:
|
||
"""阻力面敏感性分析"""
|
||
|
||
def analyze(self, base_weights, variation_ranges, n_simulations=1000):
|
||
"""
|
||
分析阻力面权重对结果的敏感性
|
||
|
||
Args:
|
||
base_weights: 基础权重 {land_type: weight}
|
||
variation_ranges: 权重变化范围 {land_type: (min, max)}
|
||
n_simulations: 蒙特卡洛模拟次数
|
||
|
||
Returns:
|
||
sensitivity_results: 敏感性分析结果
|
||
"""
|
||
results = []
|
||
|
||
for _ in range(n_simulations):
|
||
# 随机采样权重
|
||
sample_weights = {}
|
||
for land_type, (min_w, max_w) in variation_ranges.items():
|
||
sample_weights[land_type] = np.random.uniform(min_w, max_w)
|
||
|
||
# 计算对应的阻力面
|
||
resistance = self._compute_resistance(sample_weights)
|
||
|
||
# 评估结果（例如：平均连通性）
|
||
connectivity = self._assess_connectivity(resistance)
|
||
|
||
results.append({
|
||
'weights': sample_weights,
|
||
'connectivity': connectivity
|
||
})
|
||
|
||
# 分析敏感性
|
||
sensitivity = self._compute_sensitivity(results, base_weights)
|
||
|
||
return sensitivity
|
||
|
||
def _compute_sensitivity(self, results, base_weights):
|
||
"""计算各土地类型的敏感性"""
|
||
# 计算每个权重变化与连通性变化的相关性
|
||
sensitivities = {}
|
||
|
||
for land_type in base_weights:
|
||
weight_values = [r['weights'][land_type] for r in results]
|
||
connectivity_values = [r['connectivity'] for r in results]
|
||
|
||
correlation = np.corrcoef(weight_values, connectivity_values)[0, 1]
|
||
sensitivities[land_type] = abs(correlation)
|
||
|
||
return sensitivities
|
||
```
|
||
|
||
---
|
||
|
||
## 反思与延伸
|
||
|
||
### 思考问题
|
||
|
||
1. **不确定性识别**：在你的项目中，不确定性来自哪些方面？哪些是可减少的，哪些是固有的？
|
||
|
||
2. **表示选择**：你应该用标准差、置信区间，还是概率分布？各有什么优劣？
|
||
|
||
3. **决策权衡**：当不确定性很高时，你应该继续分析还是寻求更多数据？
|
||
|
||
4. **沟通问题**：如何向非专家解释不确定性？
|
||
|
||
### 实践练习
|
||
|
||
1. **不确定性审计**：对一个分析流程，识别所有不确定性来源并分类
|
||
|
||
2. **敏感性分析**：对你熟悉的空间模型进行敏感性分析
|
||
|
||
3. **鲁棒决策**：为你的项目设计一个鲁棒决策框架
|
||
|
||
### 延伸阅读
|
||
|
||
- **"Uncertainty Quantification in Predictive Modeling"** - 不确定性量化的理论基础
|
||
- **"Flaw of Averages"** (Sam Savage) - 为什么平均值会误导
|
||
- **空间数据质量标准** - 空间不确定性的行业实践
|
||
|
||
---
|
||
|
||
## 关键要点
|
||
|
||
1. **不确定性在空间分析中普遍存在**，有多个来源
|
||
2. **偶然不确定性无法消除**，认知不确定性可以通过更多数据减少
|
||
3. **不确定性会传播**，多步骤分析需要考虑累积效应
|
||
4. **敏感性分析识别关键参数**，优先减少高敏感参数的不确定性
|
||
5. **鲁棒决策在不确定性下做稳健选择**，而非追求最优
|