esda.moran — Moran’s I global and local measures of spatial autocorrelation

New in version 1.0.

Moran’s I Spatial Autocorrelation Statistics

class pysal.esda.moran.Moran(y, w, transformation='r', permutations=999)

Moran’s I Global Autocorrelation Statistic

Parameters:

y : array

variable measured across n spatial units

w : W

spatial weights instance

permutations : int

number of random permutations for calculation of pseudo-p_values

Examples

>>> import pysal
>>> w=pysal.open("../examples/stl.gal").read()
>>> f=pysal.open("../examples/stl_hom.txt")
>>> y=np.array(f.by_col['HR8893'])
>>> mi=Moran(y,w)
>>> "%7.5f"%mi.I
'0.24366'
>>> mi.EI
-0.012987012987012988
>>> mi.p_norm
0.00027147862770937614

SIDS example replicating OpenGeoda

>>> w=pysal.open("../examples/sids2.gal").read()
>>> f=pysal.open("../examples/sids2.dbf")
>>> SIDR=np.array(f.by_col("SIDR74"))
>>> mi=pysal.Moran(SIDR,w)
>>> "%6.4f"%mi.I
'0.2477'
>>> mi.p_norm
0.0001158330781489969

Attributes

y	array	original variable
w	W	original w object
permutations	int	number of permutations
I	float	value of Moran’s I
EI	float	expected value under normality assumption
VI_norm	float	variance of I under normality assumption
seI_norm	float	standard deviation of I under normality assumption
z_norm	float	z-value of I under normality assumption
p_norm	float	p-value of I under normality assumption (1-tailed)
VI_rand	float	variance of I under randomization assumption
seI_rand	float	standard deviation of I under randomization assumption
z_rand	float	z-value of I under randomization assumption
p_rand	float	p-value of I under randomization assumption (1-tailed)
sim	array (if permutations>0)	vector of I values for permutated samples
p_sim	array (if permutations>0)	p-value based on permutations
EI_sim	float (if permutations>0)	average value of I from permutations
VI_sim	float (if permutations>0)	variance of I from permutations
seI_sim	float (if permutations>0)	standard deviation of I under permutations.
z_sim	float (if permutations>0)	standardized I based on permutations
p_z_sim	float (if permutations>0)	p-value based on standard normal approximation from

class pysal.esda.moran.Moran_Local(y, w, transformation='r', permutations=999)

Local Moran Statistics

Parameters:

y : n*1 array

w : weight instance assumed to be aligned with y

transformation : string

weights transformation, default is row-standardized “r”. Other options include “B”: binary, “D”: doubly-standardized, “U”: untransformed (general weights), “V”: variance-stabilizing.

permutations : number of random permutations for calculation of pseudo-p_values

Notes

p-values are one sided - where side is based on the original I value for each observation (in self.Is). In other words extreme is considered being further away from the origin and in the same direction than original I statistic for the focal observation.

Examples

>>> import pysal
>>> import numpy as np
>>> np.random.seed(10)
>>> w=pysal.open("../examples/desmith.gal").read()
>>> f=pysal.open("../examples/desmith.txt")
>>> y=np.array(f.by_col['z'])
>>> lm=Moran_Local(y,w,transformation="r",permutations=99)
>>> lm.q
array([4, 4, 4, 2, 3, 3, 1, 4, 3, 3])
>>> lm.p_z_sim[0]
0.99036648060872201

Note random components result is slightly different values across architectures so the results have been removed from doctests and will be moved into unittests that are conditional on architectures

Attributes

y	array	original variable
w	W	original w object
permutations	int	number of random permutations for calculation of pseudo-p_values
I	float	value of Moran’s I
q	array (if permutations>0)	values indicate quadrat location 1 HH, 2 LH, 3 LL, 4 HL
sim	array (if permutations>0)	vector of I values for permutated samples
p_sim	array (if permutations>0)	p-value based on permutations
EI_sim	float (if permutations>0)	average value of I from permutations
VI_sim	float (if permutations>0)	variance of I from permutations
seI_sim	float (if permutations>0)	standard deviation of I under permutations.
z_sim	float (if permutations>0)	standardized I based on permutations
p_z_sim	float (if permutations>0)	p-value based on standard normal approximation from permutations

Methods

calc

class pysal.esda.moran.Moran_BV(x, y, w, transformation='r', permutations=999)

Bivariate Moran’s I

Parameters:

x : array

x-axis variable

y : array

(wy will be on y axis)

w : W

weight instance assumed to be aligned with y

transformation : string

weights transformation, default is row-standardized “r”. Other options include “B”: binary, “D”: doubly-standardized, “U”: untransformed (general weights), “V”: variance-stabilizing.

permutations : int

number of random permutations for calculation of pseudo-p_values

Notes

Inference is only based on permutations as analytical results are none too reliable.

Examples

>>> import pysal
>>> import numpy as np
>>> np.random.seed(10)
>>> f=pysal.open("../examples/sids2.dbf")
>>> SIDR74=np.array(f.by_col['SIDR74'])
>>> SIDR79=np.array(f.by_col['SIDR79'])
>>> w=pysal.open("../examples/sids2.gal").read()
>>> mbi=Moran_BV(SIDR79,SIDR74,w)
>>> print mbi.I
0.156131961696
>>> mbi.p_z_sim
0.0028373234843530604

Attributes

zx	array	original x variable standardized by mean and std
zy	array	original y variable standardized by mean and std
w	W	original w object
permutation	int	number of permutations
I	float	value of bivariate Moran’s I
sim	array (if permutations>0)	vector of I values for permutated samples
p_sim	float (if permutations>0)	p-value based on permutations
EI_sim	array (if permutations>0)	average value of I from permutations
VI_sim	array (if permutations>0)	variance of I from permutations
seI_sim	array (if permutations>0)	standard deviation of I under permutations.
z_sim	array (if permutations>0)	standardized I based on permutations
p_z_sim	float (if permutations>0)	p-value based on standard normal approximation from permutations

pysal.esda.moran.Moran_BV_matrix(variables, w, permutations=0, varnames=None)

Bivariate Moran Matrix

Calculates bivariate Moran between all pairs of a set of variables.

Parameters:

variables : list

sequence of variables

w : W

a spatial weights object

permutations : int

number of permutations

varnames : list

strings for variable names. If specified runtime summary is printed

Returns:

results : dictionary

(i,j) is the key for the pair of variables, values are the Moran_BV object.

Examples

>>> import pysal
>>> f=pysal.open("../examples/sids2.dbf")
>>> varnames=['SIDR74','SIDR79','NWR74','NWR79']
>>> vars=[np.array(f.by_col[var]) for var in varnames]
>>> w=pysal.open("../examples/sids2.gal").read()
>>> res=Moran_BV_matrix(vars,w,varnames=varnames)
           x           wy        I
      SIDR74       SIDR79    0.194
      SIDR79       SIDR74    0.156
      SIDR74        NWR74    0.384
       NWR74       SIDR74    0.375
      SIDR74        NWR79    0.388
       NWR79       SIDR74    0.377
      SIDR79        NWR74    0.116
       NWR74       SIDR79    0.103
      SIDR79        NWR79    0.122
       NWR79       SIDR79    0.110
       NWR74        NWR79    0.735
       NWR79        NWR74    0.737
>>>