def _perimeter_area_ratio_spatial_dissim(data,
group_pop_var,
total_pop_var,
standardize=True):
"""
Calculation of Perimeter/Area Ratio Spatial Dissimilarity index
Parameters
----------
data : a geopandas DataFrame with a geometry column.
group_pop_var : string
The name of variable in data that contains the population size of the group of interest
total_pop_var : string
The name of variable in data that contains the total population of the unit
standardize : boolean
A condition for standardisation of the weights matrices.
If True, the values of cij in the formulas gets standardized and the overall sum is 1.
Attributes
----------
statistic : float
Perimeter/Area Ratio Spatial Dissimilarity Index
core_data : a geopandas DataFrame
A geopandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Wong, David WS. "Spatial indices of segregation." Urban studies 30.3 (1993): 559-572.
"""
if (str(type(data)) != '<class \'geopandas.geodataframe.GeoDataFrame\'>'):
raise TypeError(
'data is not a GeoDataFrame and, therefore, this index cannot be calculated.'
)
if ('geometry' not in data.columns):
data['geometry'] = data[data._geometry_column_name]
data = data.drop([data._geometry_column_name], axis=1)
data = data.set_geometry('geometry')
if (type(standardize) is not bool):
raise TypeError('std is not a boolean object')
D = _dissim(data, group_pop_var, total_pop_var)[0]
data = data.rename(columns={
group_pop_var: 'group_pop_var',
total_pop_var: 'total_pop_var'
})
# If a unit has zero population, the group of interest frequency is zero
data = data.assign(
pi=np.where(data.total_pop_var == 0, 0, data.group_pop_var /
data.total_pop_var))
if not standardize:
cij = _return_length_weighted_w(data).full()[0]
else:
cij = _return_length_weighted_w(data).full()[0]
cij = cij / cij.sum()
peri = data.length
ai = data.area
aux_sum = np.add(
np.array(list((peri / ai))),
np.array(list((peri / ai))).reshape((len(list((peri / ai))), 1)))
max_pa = max(peri / ai)
num = np.multiply(np.multiply(manhattan_distances(data[['pi']]), cij),
aux_sum).sum()
den = 4 * max_pa
PARD = D - (num / den)
PARD
core_data = data[['group_pop_var', 'total_pop_var', 'geometry']]
return PARD, core_data
def _bias_corrected_dissim(data, group_pop_var, total_pop_var, B=500):
"""
Calculation of Bias Corrected Dissimilarity index
Parameters
----------
data : a pandas DataFrame
group_pop_var : string
The name of variable in data that contains the population size of the group of interest
total_pop_var : string
The name of variable in data that contains the total population of the unit
B : int
The number of iterations to calculate Dissimilarity simulating randomness with multinomial distributions. Default value is 500.
Attributes
----------
statistic : float
Dissimilarity with Bias-Correction (bias correction from Allen, Rebecca et al. (2015))
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Allen, Rebecca, et al. "More reliable inference for the dissimilarity index of segregation." The econometrics journal 18.1 (2015): 40-66.
"""
if (type(B) is not int):
raise TypeError('B must be an integer')
if (B < 2):
raise TypeError('B must be greater than 1.')
D = _dissim(data, group_pop_var, total_pop_var)[0]
data = data.rename(columns={
group_pop_var: 'group_pop_var',
total_pop_var: 'total_pop_var'
})
x = np.array(data.group_pop_var)
t = np.array(data.total_pop_var)
other_group_pop = t - x
# Group 0: minority group
p0_i = x / x.sum()
n0 = x.sum()
sim0 = np.random.multinomial(n0, p0_i, size=B)
# Group 1: complement group
p1_i = other_group_pop / other_group_pop.sum()
n1 = other_group_pop.sum()
sim1 = np.random.multinomial(n1, p1_i, size=B)
Dbcs = np.empty(B)
for i in np.array(range(B)):
data_aux = {
'simul_group': sim0[i].tolist(),
'simul_tot': (sim0[i] + sim1[i]).tolist()
}
df_aux = pd.DataFrame.from_dict(data_aux)
Dbcs[i] = _dissim(df_aux, 'simul_group', 'simul_tot')[0]
Db = Dbcs.mean()
Dbc = 2 * D - Db
Dbc # It expected to be lower than D, because D is upwarded biased
core_data = data[['group_pop_var', 'total_pop_var']]
return Dbc, core_data
def _spatial_dissim(data, group_pop_var, total_pop_var, w = None, standardize = False):
"""
Calculation of Spatial Dissimilarity index
Parameters
----------
data : a geopandas DataFrame with a geometry column.
group_pop_var : string
The name of variable in data that contains the population size of the group of interest
total_pop_var : string
The name of variable in data that contains the total population of the unit
w : W
A PySAL weights object. If not provided, Queen contiguity matrix is used.
standardize : boolean
A condition for row standardisation of the weights matrices. If True, the values of cij in the formulas gets row standardized.
For the sake of comparison, the seg R package of Hong, Seong-Yun, David O'Sullivan, and Yukio Sadahiro. "Implementing spatial segregation measures in R." PloS one 9.11 (2014): e113767.
works by default with row standardization.
Attributes
----------
statistic : float
Spatial Dissimilarity Index
core_data : a geopandas DataFrame
A geopandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Morrill, R. L. (1991) "On the Measure of Geographic Segregation". Geography Research Forum.
"""
if (str(type(data)) != '<class \'geopandas.geodataframe.GeoDataFrame\'>'):
raise TypeError('data is not a GeoDataFrame and, therefore, this index cannot be calculated.')
if ('geometry' not in data.columns):
data['geometry'] = data[data._geometry_column_name]
data = data.drop([data._geometry_column_name], axis = 1)
data = data.set_geometry('geometry')
if (type(standardize) is not bool):
raise TypeError('std is not a boolean object')
if w is None:
w_object = Queen.from_dataframe(data)
else:
w_object = w
if (not issubclass(type(w_object), libpysal.weights.W)):
raise TypeError('w is not a PySAL weights object')
D = _dissim(data, group_pop_var, total_pop_var)[0]
data = data.rename(columns={group_pop_var: 'group_pop_var',
total_pop_var: 'total_pop_var'})
x = np.array(data.group_pop_var)
t = np.array(data.total_pop_var)
# If a unit has zero population, the group of interest frequency is zero
pi = np.where(t == 0, 0, x / t)
if not standardize:
cij = w_object.full()[0]
else:
cij = w_object.full()[0]
cij = cij / cij.sum(axis = 1).reshape((cij.shape[0], 1))
# Inspired in (second solution): https://stackoverflow.com/questions/22720864/efficiently-calculating-a-euclidean-distance-matrix-using-numpy
# Distance Matrix
abs_dist = abs(pi[..., np.newaxis] - pi)
# manhattan_distances used to compute absolute distances
num = np.multiply(abs_dist, cij).sum()
den = cij.sum()
SD = D - num / den
SD
core_data = data[['group_pop_var', 'total_pop_var', 'geometry']]
return SD, core_data
def _boundary_spatial_dissim(data,
group_pop_var,
total_pop_var,
standardize=False):
"""
Calculation of Boundary Spatial Dissimilarity index
Parameters
----------
data : a geopandas DataFrame with a geometry column.
group_pop_var : string
The name of variable in data that contains the population size of the group of interest
total_pop_var : string
The name of variable in data that contains the total population of the unit
standardize : boolean
A condition for row standardisation of the weights matrices. If True, the values of cij in the formulas gets row standardized.
For the sake of comparison, the seg R package of Hong, Seong-Yun, David O'Sullivan, and Yukio Sadahiro. "Implementing spatial segregation measures in R." PloS one 9.11 (2014): e113767.
works by default without row standardization. That is, directly with border length.
Attributes
----------
statistic : float
Boundary Spatial Dissimilarity Index
core_data : a geopandas DataFrame
A geopandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
The formula is based on Hong, Seong-Yun, David O'Sullivan, and Yukio Sadahiro. "Implementing spatial segregation measures in R." PloS one 9.11 (2014): e113767.
Original paper by Wong, David WS. "Spatial indices of segregation." Urban studies 30.3 (1993): 559-572.
"""
if (str(type(data)) != '<class \'geopandas.geodataframe.GeoDataFrame\'>'):
raise TypeError(
'data is not a GeoDataFrame and, therefore, this index cannot be calculated.'
)
if ('geometry' not in data.columns):
data['geometry'] = data[data._geometry_column_name]
data = data.drop([data._geometry_column_name], axis=1)
data = data.set_geometry('geometry')
if (type(standardize) is not bool):
raise TypeError('std is not a boolean object')
D = _dissim(data, group_pop_var, total_pop_var)[0]
data = data.rename(columns={
group_pop_var: 'group_pop_var',
total_pop_var: 'total_pop_var'
})
# If a unit has zero population, the group of interest frequency is zero
data = data.assign(
pi=np.where(data.total_pop_var == 0, 0, data.group_pop_var /
data.total_pop_var))
if not standardize:
cij = _return_length_weighted_w(data).full()[0]
else:
cij = _return_length_weighted_w(data).full()[0]
cij = cij / cij.sum(axis=1).reshape((cij.shape[0], 1))
# manhattan_distances used to compute absolute distances
num = np.multiply(manhattan_distances(data[['pi']]), cij).sum()
den = cij.sum()
BSD = D - num / den
BSD
core_data = data[['group_pop_var', 'total_pop_var', 'geometry']]
return BSD, core_data
def _modified_dissim(data, group_pop_var, total_pop_var, iterations = 500):
"""
Calculation of Modified Dissimilarity index
Parameters
----------
data : a pandas DataFrame
group_pop_var : string
The name of variable in data that contains the population size of the group of interest
total_pop_var : string
The name of variable in data that contains the total population of the unit
iterations : int
The number of iterations the evaluate average classic dissimilarity under eveness. Default value is 500.
Attributes
----------
statistic : float
Modified Dissimilarity Index (Dissimilarity from Carrington and Troske (1997))
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Carrington, William J., and Kenneth R. Troske. "On measuring segregation in samples with small units." Journal of Business & Economic Statistics 15.4 (1997): 402-409.
"""
if(type(iterations) is not int):
raise TypeError('iterations must be an integer')
if(iterations < 2):
raise TypeError('iterations must be greater than 1.')
D = _dissim(data, group_pop_var, total_pop_var)[0]
data = data.rename(columns={group_pop_var: 'group_pop_var',
total_pop_var: 'total_pop_var'})
# core_data has to be in the beggining of the call because assign methods will be used later
core_data = data[['group_pop_var', 'total_pop_var']]
x = np.array(data.group_pop_var)
t = np.array(data.total_pop_var)
p_null = x.sum() / t.sum()
Ds = np.empty(iterations)
for i in np.array(range(iterations)):
freq_sim = np.random.binomial(n = np.array([t.tolist()]),
p = np.array([[p_null] * data.shape[0]]),
size = (1, data.shape[0])).tolist()[0]
data = data.assign(group_pop_var = freq_sim)
aux = _dissim(data, 'group_pop_var', 'total_pop_var')[0]
Ds[i] = aux
D_star = Ds.mean()
if (D >= D_star):
Dct = (D - D_star)/(1 - D_star)
else:
Dct = (D - D_star)/D_star
return Dct, core_data