Spatial Weights

Spatial Weights 空间矩阵

Spatial Weights, site

1. 概念

空间权重矩阵（Spatial Weight）: \([w_{ij}]\) 用于描述（地理）空间中位置之间的的邻近关系或相似性（similarity）

二元矩阵：\(w_{ij}=1\) 表示位置 \(i\) 和位置 \(j\) 相邻，或者之间存在影响
实数矩阵：数值越大表示位置 \(i\) 和位置 \(j\) 越相似（或者之间影响越大）

2. Spatial Weight Matrix in `PySAL.libpysal.weights`

2.1 `libpysal.weights` module in `PySAL` library

PySAL library spatial weight module : libpysal.weights

import weight module from PySAL

from pysal.lib import weights

2.2 The common Class Object: `libpysal.weights.W()`

libpysal.weights.W() 类为所有 libpysal.weights 模块下构建 spatial weight 的返回类型：

均用于构建 contiguity weight, i.e., binary weight matrix

Observations are usually either considered "linked" (non-zero or one) or "not linked" (zero).

(1) 初始化方法

直接初始化

weights.W(neighbors, weights=None, id_order=None, silence_warnings=False, ids=None)

通过 geopandas.GeoDataFrame() 初始化
.from_dataframe(df, geom_col=None, idVariable=None)
- df (DataFrame): 也可为 GeoDataFrame 类型
- geom_col (str, default=Nonw):
- idVariable (str, default=Nonw), if None, df.index is used

(2) 属性和方法

Attributes
- w.neighbors: return a dictionary recording the neighbor relationships (by id)
- w.s0: return the number of join, represented by a nonzero weight in a weight matrix. See this equation
- w.n return the number of units (nodes, polygon, or some other geometry object)
- w.nonzero: return the number of nonzero weights.
- w.pct_nonzero: return the percentage of nonzero weights, i.e., the density (compliment of sparsity) of the spatial weights matrix, equal to 100 * w.s0 / (w.n * w.n)
- w.cardinalities: return a dictionary reporting the number of neighbors for each observation
- w.histogram: return a tuple list reporting the distribution of the cardinalities
- w.transform: same to w.set_transform()
Functions:
- w.set_transform(value='B'):
  - B : Binary
  - R : Row-standardization (global sum = \(n\), for each row sum = \(1\))
  - D : Double-standardization (global sum = \(1\))
  - V : Variance stabilizing
  - O : Restore original transformation (from instantiation)
- w.full(): generate a full numpy array. Return a tuple:
  - first element being the full numpy array
  - second element keys being the ids associated with each row in the array.
实例代码

# 获得权重矩阵，转换为 pandas.DataFrame 类型
pd.DataFrame(w.full()[0], index=w.full()[1], columns=w.full()[1])

# Matrix normalization
w.transform = "R"	# Row normalization
# or
w.set_transform('R')

2.3 Contiguity weights

w = weights.contiguity.Rook.from_dataframe()
w = weights.contiguity.Queen.from_dataframe()

Rook: Construct a weights object from a collection of pysal polygons that share at least one edge.
Queen: Construct a weights object from a collection of pysal polygons that share at least one vertex.

2.4 Distance based weights

A matrix expressing the distances between all pairs of observations

2.4.1 K-nearest neighbor weights

The distance based weights defines the neighbor set of a particular observation as containing its nearest observations
Return a binary matrix

weights.distance.KNN(data, k=2, p=2, 
                     ids=None, radius=None, 
                     distance_metric='euclidean')

parameters
- k (int): number of nearest neighbors
- p (float): Minkowski p-norm distance metric parameter

2.4.2 Kernel weights

weights.distance.Kernel(bandwidth=None, fixed=True, k=2, 
                        function='triangular', diagonal=False,
                        distance_metric='euclidean', **kwargs)

Parameters:
- bandwidth (float or or array-like), array-like indicates a specific bandwidth \(h_i\) for each \(i\) geometry
- fixed (bool): True for fixed bandwidth, False for adaptive bandwidth
- k (int): the number of nearest neighbors to use for determining bandwidth.
  - For fixed bandwidth, \({h_i = \max_k (d_{KNN}) \forall i}\), where \({d_{KNN}}\) is a vector of k-nearest neighbor distances, i.e., \(h_i\) is the distance to the \(k\)th nearest neighbor for each observation
  - For adaptive bandwidths, \(h_i=d_{KNN,i}\)
- diagonal (bool, default=False): True set diagonal weights = 1.0, False diagonals weights are set to value according to kernel function.
- function (str in {'triangular','uniform', 'quadratic', 'quartic', 'gaussian'}): kernel function

2.4.3 Distance bands and hybrid Weights

libpysal.weights.DistanceBand(data, threshold, p=2, alpha=-1.0, binary=True, ids=None, distance_metric='euclidean', radius=None)

weights.DistanceBand(data, threshold, p=2, alpha=-1.0,
                     binary=True, ids=None,
                     distance_metric='euclidean', radius=None)

\[w_{ij} = d_{ij}^{\alpha}, \quad \text{where} \quad d_{ij} = \left(\sum _{i=1}^{2}|x_{i}-y_{i}|^{p}\right)^{\frac {1}{p}} \]

Parameter:
- threshold (float): distance band
- p (float): Minkowski p-norm distance metric parameter
- binary (bool):
  - If true \(w_{ij}=1\) if \(d_{ij}<=\text{threshold}\), otherwise \(w_{ij}=0\)
  - If false \(w_{ij}=d_{ij}^{\alpha}\)
- alpha (float, default -1.0):
  - distance decay parameter for weight
  - if alpha is positive the weights will not decline with distance
  - If binary=True, alpha is ignored

3.3 Great circle distances

4 Block weights

weights.block_weights(regimes, ids=None, **kwargs)

Construct spatial weights for regime neighbors.

Block contiguity structures are relevant when defining neighbor relations based on membership in a regime. For example, all counties belonging to the same state could be defined as neighbors, in an analysis of all counties in the US.

Parameters:
- regimes (list or array-like), ids of which regime an observation belongs to
- ids (list, array-like), ordered sequence of IDs for the observations

5 Set operations on weights

Reference

Sergio Rey, Dani Arribas-Bel, Levi John Wolf, 2020, "Spatial Weights" in Geographic Data Science with Python, site

posted @ 2023-05-13 20:04 veager 阅读(218) 评论(0) 收藏举报

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