地理加权回归 | Geographically Weighted Regression

Geographically weighted regression (GWR): at each regression point, a distance-decay kernel (bisquare/gaussian) is used for weighted least squares, producing local coefficients that vary in space; the optimal bandwidth is selected by AICc cross-validation (Hurvich & Tsai 1989); outputs include local coefficient rasters and local R².

核心算法（Fotheringham/Brunsdon/Charlton 范式） / 核心算法（Fotheringham/Brunsdon/Charlton 范式）

• Kernel functions:
  • Bisquare (Tukey biweight): w(d,h) = (1 - (d/h)²)² for d<h, 0 otherwise
  • Gaussian: w(d,h) = exp(-0.5·(d/h)²)
• Pointwise weighted OLS: β_i = (X^T W_i X)^{-1} X^T W_i y (Fotheringham et al. 2002, Chapter 4)
• Bandwidth selection: 6 fixed bandwidth candidates (0.1× / 0.2× / 0.3× / 0.5× / 0.7× / 1.0× of the data extent), with the best chosen by AICc (Hurvich & Tsai 1989 correction)
• AICc formula: AICc = n·log(σ²) + n·log(2π) + n·(n + tr(S))/(n − 2 − tr(S)), where tr(S) = Σ_i x_i^T (X^T W_i X)^{-1} x_i·w_i is the trace of the hat matrix
• Local R²: 1 − resid²/var(y) at each regression point, clipped to [0,1] (per-point residual normalization, used as a proxy for local goodness of fit)

Methodology References / 方法学引用

Dependencies / 依赖

pip install numpy rasterio scipy

Usage / 使用方法

Example 1 (synthetic data, offline)

python geoskill-geographically-weighted-regression.py --bbox 116.0 39.0 117.0 40.0 --synthetic --output-dir ./out

Example 2 (specify area + silence)

python geoskill-geographically-weighted-regression.py --bbox 121.0 31.0 122.0 32.0 --synthetic --output-dir ./out2 --quiet

Example 3 (real CSV input)

python geoskill-geographically-weighted-regression.py --input housing.csv --output-dir ./out3

CSV columns: must contain x, y, dep; optionally multiple independent variables ind1, ind2, ... (no intercept; the script adds it automatically).

Example 4 (tiny-area boundary test)

python geoskill-geographically-weighted-regression.py --bbox 116.39 39.90 116.40 39.91 --synthetic --output-dir ./out4 --quiet

Output / 输出

Limitations / 局限

• Distance metric: Euclidean distance in degree coordinate space. The maximum error within 1°×1° is <1% (cos 39° ≈ 0.777), but the bias grows over large areas / at high latitudes. Fotheringham recommends great-circle distance (haversine), which is not implemented.
• Local R²: per-point residual normalization (pseudo R²), not Fotheringham's standard "weighted RSS/TSS within the bandwidth". Both are widely used but have different meanings.
• Significance tests: local coefficient t-tests / pseudo t-tests (Fotheringham Chapter 6) are not implemented.
• Fixed bandwidth: distance-based only; adaptive (neighbor-count) bandwidths are not implemented.
• CSV NoData/NaN: _load_csv uses float(v); a "NaN" string raises ValueError (exit 2); no NoData filtering.

Data Source / 数据源 / Source

• Synthetic mode: locally generated simulated samples with spatially varying coefficients.
• Real mode: reads a local CSV (x, y, dep, ind...), no network requests.

Privacy / 隐私声明 / Privacy

• Fully offline by default; no network requests are made.
• --synthetic mode reads no external data.
• All computation is performed locally; user data is never uploaded.

License / License

MIT

name: geoskill-geographically-weighted-regression description: 'GWR局部回归+带宽选择+局部系数空间图+局部R2 (Fotheringham/Brunsdon/Charlton 范式, bisquare/gaussian核, AICc带宽选择)'

地理加权回归 | Geographically Weighted Regression

地理加权回归（GWR）：在每个回归点用距离衰减核（双平方/高斯）做加权最小二乘，得到随空间变化的局部系数；用 AICc 交叉验证（Hurvich & Tsai 1989）选最优带宽；输出局部系数栅格与局部 R²。

核心算法（Fotheringham/Brunsdon/Charlton 范式）

• 核函数：
  • 双平方（bisquare / Tukey biweight）：w(d,h) = (1 - (d/h)²)² for d<h, 0 otherwise
  • 高斯（gaussian）：w(d,h) = exp(-0.5·(d/h)²)
• 逐点加权 OLS：β_i = (X^T W_i X)^{-1} X^T W_i y（Fotheringham et al. 2002, Chapter 4）
• 带宽选择：6 个固定带宽候选（数据跨度的 0.1× / 0.2× / 0.3× / 0.5× / 0.7× / 1.0×），用 AICc（Hurvich & Tsai 1989 修正）选最优
• AICc 公式：AICc = n·log(σ²) + n·log(2π) + n·(n + tr(S))/(n − 2 − tr(S))，其中 tr(S) = Σ_i x_i^T (X^T W_i X)^{-1} x_i·w_i 是帽矩阵迹
• 局部 R²：每个回归点的 1 − resid²/var(y) 截到 [0,1]（per-point 残差归一化，作为局部拟合优度的代理指标）

方法学引用

依赖

pip install numpy rasterio scipy

使用方法

示例 1（合成数据，离线）

python geoskill-geographically-weighted-regression.py --bbox 116.0 39.0 117.0 40.0 --synthetic --output-dir ./out

示例 2（指定区域 + 静默）

python geoskill-geographically-weighted-regression.py --bbox 121.0 31.0 122.0 32.0 --synthetic --output-dir ./out2 --quiet

示例 3（真实输入 CSV）

python geoskill-geographically-weighted-regression.py --input housing.csv --output-dir ./out3

CSV 列：必须含 x, y, dep；可选多个 ind1, ind2, ... 自变量（不含截距，脚本自动加）。

示例 4（极小区域边界测试）

python geoskill-geographically-weighted-regression.py --bbox 116.39 39.90 116.40 39.91 --synthetic --output-dir ./out4 --quiet

输出

局限

• 距离度量：欧氏距离在度坐标空间。1°×1° 内最大误差 <1%（cos 39°≈0.777），大区域/高纬度偏差增大。Fotheringham 推荐用大圆距离（haversine），未实现。
• 局部 R²：per-point 残差归一化（pseudo R²），非 Fotheringham 标准的"带宽内加权 RSS/TSS"。两者皆被广泛使用，含义不同。
• 显著性检验：局部系数 t 检验 / 伪 t 检验（Fotheringham Chapter 6）未实现。
• 固定带宽：仅 distance-based，未实现 adaptive（neighbor-count）带宽。
• CSV NoData/NaN：_load_csv 用 float(v)，遇 "NaN" 字符串抛 ValueError（exit 2），未做 NoData 过滤。

数据源 / Source

• 合成模式：本地生成系数随空间变化的模拟样本。
• 真实模式：读取本地 CSV（x, y, dep, ind...），无网络请求。

隐私声明 / Privacy

• 默认完全离线运行，不发起任何网络请求。
• --synthetic 模式不读取任何外部数据。
• 所有计算在本地完成，不上传用户数据。

License

MIT