esda.Moran_Rate¶

class esda.Moran_Rate(e, b, w, adjusted=True, transformation='r', permutations=999, two_tailed=True)[source]¶

Adjusted Moran’s I Global Autocorrelation Statistic for Rate Variables [Assuncao and Reis, 1999]

Parameters:¶

e : array¶: an event variable measured across n spatial units
b : array¶: a population-at-risk variable measured across n spatial units
w : W | Graph¶: spatial weights instance as W or Graph aligned with e and b
adjusted : boolean¶: whether or not Moran’s I needs to be adjusted for rate variable
transformation : {'R', 'B', 'D', 'U', 'V'}¶: weights transformation, default is row-standardized “r”. Other options include “B”: binary, “D”: doubly-standardized, “O”: restore original transformation (applicable only if w is passed as W), “V”: variance-stabilizing.
two_tailed : boolean¶: If True (default), analytical p-values for Moran’s I are two-tailed, otherwise they are one tailed.
permutations : int¶: number of random permutations for calculation of pseudo p_values

y[source]¶

rate variable computed from parameters e and b if adjusted is True, y is standardized rates otherwise, y is raw rates

Type:¶: array

z[source]¶

zero-mean, unit standard deviation normalized y

Type:¶: array

w[source]¶

original w object

Type:¶: W | Graph

permutations[source]¶

number of permutations

Type:¶: int

I[source]¶

value of Moran’s I

Type:¶: float

EI[source]¶

expected value under normality assumption

Type:¶: float

VI_norm[source]¶

variance of I under normality assumption

Type:¶: float

seI_norm[source]¶

standard deviation of I under normality assumption

Type:¶: float

z_norm[source]¶

z-value of I under normality assumption

Type:¶: float

p_norm[source]¶

p-value of I under normality assumption

Type:¶: float

VI_rand[source]¶

variance of I under randomization assumption

Type:¶: float

seI_rand[source]¶

standard deviation of I under randomization assumption

Type:¶: float

z_rand[source]¶

z-value of I under randomization assumption

Type:¶: float

p_rand[source]¶

p-value of I under randomization assumption

Type:¶: float

two_tailed[source]¶

If True, p_norm and p_rand are two-tailed p-values, otherwise they are one-tailed.

Type:¶: boolean

sim[source]¶

(if permutations>0) vector of I values for permuted samples

Type:¶: array

p_sim[source]¶

(if permutations>0) p-value based on permutations (one-sided) null: spatial randomness alternative: the observed I is extreme if it is either extremely greater or extremely lower than the values obtained from permutaitons

Type:¶: array

EI_sim[source]¶

(if permutations>0) average value of I from permutations

Type:¶: float

VI_sim[source]¶

(if permutations>0) variance of I from permutations

Type:¶: float

seI_sim[source]¶

(if permutations>0) standard deviation of I under permutations.

Type:¶: float

z_sim[source]¶

(if permutations>0) standardized I based on permutations

Type:¶: float

p_z_sim[source]¶

(if permutations>0) p-value based on standard normal approximation from

Type:¶: float

Examples

>>> import libpysal, numpy
>>> w = libpysal.io.open(libpysal.examples.get_path("sids2.gal")).read()
>>> f = libpysal.io.open(libpysal.examples.get_path("sids2.dbf"))
>>> e = numpy.array(f.by_col('SID79'))
>>> b = numpy.array(f.by_col('BIR79'))
>>> from esda import Moran_Rate
>>> mi = Moran_Rate(e, b,  w, two_tailed=False)
>>> "%6.4f" % mi.I
'0.1662'
>>> "%6.4f" % mi.p_norm
'0.0042'

Methods

`by_col`(df, events, populations[, w, ...])	Function to compute a Moran_Rate statistic on a dataframe
`plot_scatter`([ax, scatter_kwds, ...])	Plot a Moran scatterplot with optional coloring for significant points.
`plot_simulation`([ax, legend, fitline_kwds])	Global Moran's I simulated reference distribution.

classmethod by_col(df, events, populations, w=None, inplace=False, pvalue='sim', outvals=None, swapname='', **stat_kws)[source]¶

Function to compute a Moran_Rate statistic on a dataframe

Parameters:¶

df : pandas.DataFrame¶: a pandas dataframe with a geometry column
events : string or list of strings¶: one or more names where events are stored
populations : string or list of strings¶: one or more names where the populations corresponding to the events are stored. If one population column is provided, it is used for all event columns. If more than one population column is provided but there is not a population for every event column, an exception will be raised.
w : W | Graph¶: spatial weights instance as W or Graph aligned with the dataframe. If not provided, this is searched for in the dataframe’s metadata
inplace : bool¶: a boolean denoting whether to operate on the dataframe inplace or to return a series contaning the results of the computation. If operating inplace, the derived columns will be named ‘column_moran_rate’
pvalue : string¶: a string denoting which pvalue should be returned. Refer to the the Moran_Rate statistic’s documentation for available p-values
outvals : list of strings¶: list of arbitrary attributes to return as columns from the Moran_Rate statistic
**stat_kws : keyword arguments¶: options to pass to the underlying statistic. For this, see the documentation for the Moran_Rate statistic.

Returns:¶

If inplace, None, and operation is conducted on dataframe
in memory. Otherwise, returns a copy of the dataframe with
the relevant columns attached.

plot_scatter(ax=None, scatter_kwds=None, fitline_kwds=None, losh_scaling_factor=False, losh_inference=None, a=2)[source]¶

Plot a Moran scatterplot with optional coloring for significant points.

Parameters:¶

ax : matplotlib.axes.Axes, optional¶: Pre-existing axes for the plot, by default None.
scatter_kwds : dict, optional¶: Additional keyword arguments for scatter plot, by default None.
fitline_kwds : dict, optional¶: Additional keyword arguments for fit line, by default None.
losh_scaling_factor : bool | int | float, by default False¶: Scale the observations by LOSH. When set to a number, it is treated as the multiplicative factor applied to exp(LOSH.Hi) when converting LOSH values into marker areas.
losh_inference : str, optional¶: Inference method for LOSH. See LOSH for supported options. Applies only if losh_scaling_factor is not False.
a : int or float, default=2¶: Residual exponent passed to esda.losh.LOSH.fit(). The default corresponds to a variance-based LOSH measure. Applies only if losh_scaling_factor is not False.

Returns:¶

Axes object with the Moran scatterplot.

Return type:¶

matplotlib.axes.Axes

plot_simulation(ax=None, legend=False, fitline_kwds=None, **kwargs)[source]¶

Global Moran’s I simulated reference distribution.

Parameters:¶

ax : matplotlib.axes.Axes, optional¶: Pre-existing axes for the plot, by default None.
legend : bool, optional¶: Plot a legend, by default False
fitline_kwds : dict, optional¶: Additional keyword arguments for vertical Moran fit line, by default None.
**kwargs : keyword arguments, optional¶: Additional keyword arguments for KDE plot passed to seaborn.kdeplot, by default None.

Returns:¶

Axes object with the Moran scatterplot.

Return type:¶

matplotlib.axes.Axes

Notes

This requires optional dependencies matplotlib and seaborn.

Examples

>>> import libpysal, numpy
>>> w = libpysal.io.open(libpysal.examples.get_path("stl.gal")).read()
>>> f = libpysal.io.open(libpysal.examples.get_path("stl_hom.txt"))
>>> y = numpy.array(f.by_col['HR8893'])
>>> from esda import Moran
>>> mi = Moran(y,  w)

Default plot:

>>> mi.plot_simulation()

Customized styling that turns the distribution into a pink line and line indicating I to a black line:

>>> mi.plot_simulation(fitline_kwds={"color": "k"}, color="pink", shade=False)