def _multi_gini_seg(data, groups):
"""
Calculation of Multigroup Gini Segregation index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Gini Segregation Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
K = df.shape[1]
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
Is = (Pk * (1 - Pk)).sum()
elements_sum = np.empty(K)
for k in range(K):
aux = np.multiply(np.outer(ti, ti),
manhattan_distances(pik[:, k].reshape(-1, 1))).sum()
elements_sum[k] = aux
multi_Gini_Seg = elements_sum.sum() / (2 * (T**2) * Is)
return multi_Gini_Seg, core_data, groups
def _multi_diversity(data, groups, normalized=False):
"""
Calculation of Multigroup Diversity index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Diversity Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
normalized : bool. Default is False.
Wheter the resulting index will be divided by its maximum (natural log of the number of groups)
Notes
-----
Based on Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67 and Theil, Henry. "Statistical decomposition analysis; with applications in the social and administrative sciences". No. 04; HA33, T4.. 1972.
This is also know as Theil's Entropy Index (Equation 2 of page 37 of Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67)
High diversity means less segregation.
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
Pk = df.sum(axis=0) / df.sum()
E = -(Pk * np.log(Pk)).sum()
if normalized:
K = df.shape[1]
E = E / np.log(K)
return E, core_data, groups
def _multi_dissim(data, groups):
"""
Calculation of Multigroup Dissimilarity index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Dissimilarity Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Sakoda, James M. "A generalized index of dissimilarity." Demography 18.2 (1981): 245-250.
Reference: :cite:`sakoda1981generalized`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
n = df.shape[0]
K = df.shape[1]
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
Is = (Pk * (1 - Pk)).sum()
multi_D = 1 / (2 * T * Is) * np.multiply(
abs(pik - Pk),
np.repeat(ti, K, axis=0).reshape(n, K)).sum()
return multi_D, core_data, groups
def _multi_information_theory(data, groups):
"""
Calculation of Multigroup Information Theory index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Information Theory Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
# The natural logarithm is used, but this could be used with any base following Footnote 3 of pg. 37
# of Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
E = (Pk * np.log(1 / Pk)).sum()
MIT = np.nansum(ti[:, None] * pik * np.log(pik / Pk)) / (T * E)
return MIT, core_data, groups
def _multi_relative_diversity(data, groups):
"""
Calculation of Multigroup Relative Diversity index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Relative Diversity Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Reardon, Sean F. "Measures of racial diversity and segregation in multigroup and hierarchically structured populations." annual meeting of the Eastern Sociological Society, Philadelphia, PA. 1998.
High diversity means less segregation.
Reference: :cite:`reardon1998measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
Is = (Pk * (1 - Pk)).sum()
MRD = (ti[:, None] * (pik - Pk)**2).sum() / (T * Is)
return MRD, core_data, groups
def _simpsons_concentration(data, groups):
"""
Calculation of Simpson's Concentration index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Simpson's Concentration Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Simpson, Edward H. "Measurement of diversity." nature 163.4148 (1949): 688.
Simpson's concentration index (Lambda) can be simply interpreted as the probability that two individuals chosen at random and independently from the population will be found to belong to the same group.
Higher values means higher segregation.
Simpson's Concentration + Simpson's Interaction = 1
Reference: :cite:`simpson1949measurement`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
Pk = df.sum(axis=0) / df.sum()
Lambda = (Pk * Pk).sum()
return Lambda, core_data, groups
def _multi_squared_coefficient_of_variation(data, groups):
"""
Calculation of Multigroup Squared Coefficient of Variation index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Squared Coefficient of Variation Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
K = df.shape[1]
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
C = ((ti[:, None] * (pik - Pk)**2) / (T * (K - 1) * Pk)).sum()
return C, core_data, groups
def _multi_divergence(data, groups):
"""
Calculation of Multigroup Divergence index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Divergence Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Roberto, Elizabeth. "The Divergence Index: A Decomposable Measure of Segregation and Inequality." arXiv preprint arXiv:1508.01167 (2015).
Reference: :cite:`roberto2015divergence`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
Di = np.nansum(pik * np.log(pik / Pk), axis=1)
Divergence_Index = ((ti / T) * Di).sum()
return Divergence_Index, core_data, groups
def _simpsons_interaction(data, groups):
"""
Calculation of Simpson's Interaction index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Simpson's Interaction Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Equation 1 of page 37 of Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
Simpson's interaction index (I) can be simply interpreted as the probability that two individuals chosen at random and independently from the population will be found to not belong to the same group.
Higher values means lesser segregation.
Simpson's Concentration + Simpson's Interaction = 1
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
Pk = df.sum(axis=0) / df.sum()
I = (Pk * (1 - Pk)).sum()
return I, core_data, groups
def _multi_normalized_exposure(data, groups):
"""
Calculation of Multigroup Normalized Exposure index
Parameters
----------
data : a pandas DataFrame
groups : list of strings.
The variables names in data of the groups of interest of the analysis.
Returns
-------
statistic : float
Multigroup Normalized Exposure Index
core_data : a pandas DataFrame
A pandas DataFrame that contains the columns used to perform the estimate.
Notes
-----
Based on Reardon, Sean F., and Glenn Firebaugh. "Measures of multigroup segregation." Sociological methodology 32.1 (2002): 33-67.
Reference: :cite:`reardon2002measures`.
"""
core_data = data[groups]
data = _nan_handle(core_data)
df = np.array(core_data)
T = df.sum()
ti = df.sum(axis=1)
pik = df / ti[:, None]
Pk = df.sum(axis=0) / df.sum()
MNE = ((ti[:, None] * (pik - Pk)**2) / (1 - Pk)).sum() / T
return MNE, core_data, groups
