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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Spatial Weights
Spatial weights are mathematical structures used to represent spatial relationships. They characterize the relationship of each observation to every other observation using some concept of proximity or closeness that depends on the weight type.
They can be build in PySAL from shapefiles, as well as some types of files.
Step1:  There are functions to construct weights directly from a file path.
Step2:  Weight Types
Contiguity
Step3:  All weights objects have a few traits that you can use to work with the weights object, as well as to get information about the weights object. 
To get the neighbors & weights around an observation, use the observation's index on the weights object, like a dictionary
Step4:  By default, the weights and the pandas dataframe will use the same index. So, we can view the observation and its neighbors in the dataframe by putting the observation's index and its neighbors' indexes together in one list
Step5:  and grabbing those elements from the dataframe
Step6:  A full, dense matrix describing all of the pairwise relationships is constructed using the .full method, or when pysal.full is called on a weights object
Step7:  Note that this matrix is binary, in that its elements are either zero or one, since an observation is either a neighbor or it is not a neighbor. 
However, many common use cases of spatial weights require that the matrix is row-standardized. This is done simply in PySAL using the .transform attribute
Step8:  Now, if we build a new full matrix, its rows should sum to one
Step9:  Since weight matrices are typically very sparse, there is also a sparse weights matrix constructor
Step10:  By default, PySAL assigns each observation an index according to the order in which the observation was read in. This means that, by default, all of the observations in the weights object are indexed by table order. If you have an alternative ID variable, you can pass that into the weights constructor. 
For example, the NAT.shp dataset has a possible alternative ID Variable, a FIPS code.
Step11:  The observation we were discussing above is in the fifth row
Step12:  Now, Pend Oreille county has a different index
Step13:  Note that a KeyError in Python usually means that some index, here 4, was not found in the collection being searched, the IDs in the queen weights object. This makes sense, since we explicitly passed an idVariable argument, and nothing has a FIPS code of 4.
Instead, if we use the observation's FIPS code
Step14:  We get what we need.
In addition, we have to now query the dataframe using the FIPS code to find our neighbors. But, this is relatively easy to do, since pandas will parse the query by looking into python objects, if told to. 
First, let us store the neighbors of our target county
Step15:  Then, we can use this list in .query
Step16:  Note that we have to use @ before the name in order to show that we're referring to a python object and not a column in the dataframe.
Step17:  Of course, we could also reindex the dataframe to use the same index as our weights
Step18:  Now that both are using the same weights, we can use the .loc indexer again
Step19:  Rook Weights
Rook weights are another type of contiguity weight, but consider observations as neighboring only when they share an edge. The rook neighbors of an observation may be different than its queen neighbors, depending on how the observation and its nearby polygons are configured. 
We can construct this in the same way as the queen weights, using the special rook_from_shapefile function
Step20:  These weights function exactly like the Queen weights, and are only distinguished by what they consider "neighbors."
Bishop Weights
In theory, a "Bishop" weighting scheme is one that arises when only polygons that share vertexes are considered to be neighboring. But, since Queen contiguigy requires either an edge or a vertex and Rook contiguity requires only shared edges, the following relationship is true
Step21:  Thus, the vast majority of counties have no bishop neighbors. But, a few do. A simple way to see these observations in the dataframe is to find all elements of the dataframe that are not "islands," the term for an observation with no neighbors
Step22:  Distance
There are many other kinds of weighting functions in PySAL. Another separate type use a continuous measure of distance to define neighborhoods. To use these measures, we first must extract the polygons' centroids.
For each polygon poly in dataframe.geometry, we want poly.centroid. So, one way to do this is to make a list of all of the centroids
Step23:  If we were working with point data, this step would be unncessary. 
KnnW
If we wanted to consider only the k-nearest neighbors to an observation's centroid, we could use the knnW function in PySAL.
This specific type of distance weights requires that we first build a KDTree, a special representation for spatial point data. Fortunately, this is built in to PySAL
Step24:  Then, we can use this to build a spatial weights object where only the closest k observations are considered "neighbors." In this example, let's do the closest 5
Step25:  So, all observations have exactly 5 neighbors. Sometimes, these neighbors are actually different observations than the ones identified by contiguity neighbors. 
For example, Pend Oreille gets a new neighbor, Kootenai county
Step26:  Kernel W
Kernel Weights are continuous distance-based weights that use kernel densities to provide an indication of neighborliness. 
Typically, they estimate a bandwidth, which is a parameter governing how far out observations should be considered neighboring. Then, using this bandwidth, they evaluate a continuous kernel function to provide a weight between 0 and 1.
Many different choices of kernel functions are supported, and bandwidth can be estimated at each point or over the entire map.
For example, if we wanted to use a single estimated bandwidth for the entire map and weight according to a gaussian kernel
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Python Code:
import pysal as ps
import numpy as np
Explanation: Spatial Weights
Spatial weights are mathematical structures used to represent spatial relationships. They characterize the relationship of each observation to every other observation using some concept of proximity or closeness that depends on the weight type.
They can be build in PySAL from shapefiles, as well as some types of files.
End of explanation
shp_path = ps.examples.get_path('NAT.shp')
Explanation: There are functions to construct weights directly from a file path.
End of explanation
qW = ps.queen_from_shapefile(shp_path)
dataframe = ps.pdio.read_files(shp_path)
qW
Explanation: Weight Types
Contiguity:
Queen Weights
A commonly-used type of weight is a queen contigutiy weight, which reflects adjacency relationships as a binary indicator variable denoting whether or not a polygon shares an edge or a verted each another polygon. These weights are symmetric, in that when polygon $A$ neighbors polygon $B$, both $w_{AB} = 1$ and $w_{BA} = 1$.
To construct queen weights from a shapefile, use the queen_from_shapefile function:
End of explanation
qW[4] #neighbors & weights of the 5th observation
Explanation: All weights objects have a few traits that you can use to work with the weights object, as well as to get information about the weights object. 
To get the neighbors & weights around an observation, use the observation's index on the weights object, like a dictionary:
End of explanation
self_and_neighbors = [4]
self_and_neighbors.extend(qW.neighbors[4])
print(self_and_neighbors)
Explanation: By default, the weights and the pandas dataframe will use the same index. So, we can view the observation and its neighbors in the dataframe by putting the observation's index and its neighbors' indexes together in one list:
End of explanation
dataframe.loc[self_and_neighbors]
Explanation: and grabbing those elements from the dataframe:
End of explanation
Wmatrix, ids = qW.full()
#Wmatrix, ids = ps.full(qW)
Wmatrix
Explanation: A full, dense matrix describing all of the pairwise relationships is constructed using the .full method, or when pysal.full is called on a weights object:
End of explanation
qW.transform = 'r'
Explanation: Note that this matrix is binary, in that its elements are either zero or one, since an observation is either a neighbor or it is not a neighbor. 
However, many common use cases of spatial weights require that the matrix is row-standardized. This is done simply in PySAL using the .transform attribute
End of explanation
Wmatrix, ids = qW.full()
Wmatrix.sum(axis=1) #numpy axes are 0:column, 1:row, 2:facet, into higher dimensions
Explanation: Now, if we build a new full matrix, its rows should sum to one:
End of explanation
qW.sparse
Explanation: Since weight matrices are typically very sparse, there is also a sparse weights matrix constructor:
End of explanation
dataframe.head()
Explanation: By default, PySAL assigns each observation an index according to the order in which the observation was read in. This means that, by default, all of the observations in the weights object are indexed by table order. If you have an alternative ID variable, you can pass that into the weights constructor. 
For example, the NAT.shp dataset has a possible alternative ID Variable, a FIPS code.
End of explanation
qW = ps.queen_from_shapefile(shp_path, idVariable='FIPS')
Explanation: The observation we were discussing above is in the fifth row: Pend Oreille county, Washington. Note that its FIPS code is 53051.
Then, instead of indexing the weights and the dataframe just based on read-order, use the FIPS code as an index:
End of explanation
qW[4] #fails, since no FIPS is 4. 
Explanation: Now, Pend Oreille county has a different index:
End of explanation
qW['53051']
Explanation: Note that a KeyError in Python usually means that some index, here 4, was not found in the collection being searched, the IDs in the queen weights object. This makes sense, since we explicitly passed an idVariable argument, and nothing has a FIPS code of 4.
Instead, if we use the observation's FIPS code:
End of explanation
self_and_neighbors = ['53051']
self_and_neighbors.extend(qW.neighbors['53051'])
Explanation: We get what we need.
In addition, we have to now query the dataframe using the FIPS code to find our neighbors. But, this is relatively easy to do, since pandas will parse the query by looking into python objects, if told to. 
First, let us store the neighbors of our target county:
End of explanation
dataframe.query('FIPS in @self_and_neighbors')
Explanation: Then, we can use this list in .query:
End of explanation
#dataframe.query('FIPS in neighs') will fail because there is no column called 'neighs'
Explanation: Note that we have to use @ before the name in order to show that we're referring to a python object and not a column in the dataframe.
End of explanation
fips_frame = dataframe.set_index(dataframe.FIPS)
fips_frame.head()
Explanation: Of course, we could also reindex the dataframe to use the same index as our weights:
End of explanation
fips_frame.loc[self_and_neighbors]
Explanation: Now that both are using the same weights, we can use the .loc indexer again:
End of explanation
rW = ps.rook_from_shapefile(shp_path, idVariable='FIPS')
rW['53051']
Explanation: Rook Weights
Rook weights are another type of contiguity weight, but consider observations as neighboring only when they share an edge. The rook neighbors of an observation may be different than its queen neighbors, depending on how the observation and its nearby polygons are configured. 
We can construct this in the same way as the queen weights, using the special rook_from_shapefile function
End of explanation
bW = ps.w_difference(qW, rW, constrained=False, silent_island_warning=True) #silence because there will be a lot of warnings
bW.histogram
Explanation: These weights function exactly like the Queen weights, and are only distinguished by what they consider "neighbors."
Bishop Weights
In theory, a "Bishop" weighting scheme is one that arises when only polygons that share vertexes are considered to be neighboring. But, since Queen contiguigy requires either an edge or a vertex and Rook contiguity requires only shared edges, the following relationship is true:
$$ \mathcal{Q} = \mathcal{R} \cup \mathcal{B} $$
where $\mathcal{Q}$ is the set of neighbor pairs via queen contiguity, $\mathcal{R}$ is the set of neighbor pairs via Rook contiguity, and $\mathcal{B}$ via Bishop contiguity. Thus:
$$ \mathcal{Q} \setminus \mathcal{R} = \mathcal{B}$$
Bishop weights entail all Queen neighbor pairs that are not also Rook neighbors.
PySAL does not have a dedicated bishop weights constructor, but you can construct very easily using the w_difference function. This function is one of a family of tools to work with weights, all defined in ps.weights, that conduct these types of set operations between weight objects.
End of explanation
islands = bW.islands
dataframe.query('FIPS not in @islands')
Explanation: Thus, the vast majority of counties have no bishop neighbors. But, a few do. A simple way to see these observations in the dataframe is to find all elements of the dataframe that are not "islands," the term for an observation with no neighbors:
End of explanation
centroids = [list(poly.centroid) for poly in dataframe.geometry]
centroids[0:5] #let's look at the first five
Explanation: Distance
There are many other kinds of weighting functions in PySAL. Another separate type use a continuous measure of distance to define neighborhoods. To use these measures, we first must extract the polygons' centroids.
For each polygon poly in dataframe.geometry, we want poly.centroid. So, one way to do this is to make a list of all of the centroids:
End of explanation
kdtree = ps.cg.KDTree(centroids)
Explanation: If we were working with point data, this step would be unncessary. 
KnnW
If we wanted to consider only the k-nearest neighbors to an observation's centroid, we could use the knnW function in PySAL.
This specific type of distance weights requires that we first build a KDTree, a special representation for spatial point data. Fortunately, this is built in to PySAL:
End of explanation
nn5 = ps.knnW(kdtree, k=5)
nn5.histogram
Explanation: Then, we can use this to build a spatial weights object where only the closest k observations are considered "neighbors." In this example, let's do the closest 5:
End of explanation
nn5[4]
dataframe.loc[nn5.neighbors[4] + [4]]
fips_frame.loc[qW.neighbors['53051'] + ['53051']]
Explanation: So, all observations have exactly 5 neighbors. Sometimes, these neighbors are actually different observations than the ones identified by contiguity neighbors. 
For example, Pend Oreille gets a new neighbor, Kootenai county:
End of explanation
kernelW = ps.Kernel(centroids, fixed=True, function='gaussian')
#ps.Kernel(centroids, fixed=False, function='gaussian') #same kernel, but bandwidth changes at each observation
dataframe.loc[kernelW.neighbors[4] + [4]]
Explanation: Kernel W
Kernel Weights are continuous distance-based weights that use kernel densities to provide an indication of neighborliness. 
Typically, they estimate a bandwidth, which is a parameter governing how far out observations should be considered neighboring. Then, using this bandwidth, they evaluate a continuous kernel function to provide a weight between 0 and 1.
Many different choices of kernel functions are supported, and bandwidth can be estimated at each point or over the entire map.
For example, if we wanted to use a single estimated bandwidth for the entire map and weight according to a gaussian kernel:
End of explanation
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