spatial_dynamics.rank – Rank and spatial rank mobility measures

New in version 1.0.

Rank and spatial rank mobility measures

class pysal.spatial_dynamics.rank.Theta(y, regime, permutations=999)

Regime mobility measure

For sequence of time periods Theta measures the extent to which rank changes for a variable measured over n locations are in the same direction within mutually exclusive and exhaustive partitions (regimes) of the n locations.

Theta is defined as the sum of the absolute sum of rank changes within the regimes over the sum of all absolute rank changes. [R4]

Parameters:

y : array (n,k) with k>=2

successive columns of y are later moments in time (years, months,etc)

regime : array (n,)

values corresponding to which regime each observation belongs to

permutations : int

number of random spatial permutations to generate for computationally based inference

References

[R4]	(1, 2) Rey, S.J. (2004) “Spatial dependence in the evolution of regional income distributions,” in A. Getis, J. Mur and H.Zoeller (eds). Spatial Econometrics and Spatial Statistics. Palgrave, London, pp. 194-213.

Examples

>>> import pysal
>>> f=pysal.open("../examples/mexico.csv")
>>> vnames=["pcgdp%d"%dec for dec in range(1940,2010,10)]
>>> y=np.transpose(np.array([f.by_col[v] for v in vnames]))
>>> regime=np.array(f.by_col['esquivel99'])
>>> np.random.seed(10)
>>> t=Theta(y,regime,999)
>>> t.theta
array([[ 0.41538462,  0.28070175,  0.61363636,  0.62222222,  0.33333333,
         0.47222222]])
>>> t.pvalue_left
array([ 0.307,  0.077,  0.823,  0.552,  0.045,  0.735])
>>> t.total
array([ 130.,  114.,   88.,   90.,   90.,   72.])
>>> t.max_total
512
>>> 

Attributes

ranks	array	ranks of the original y array (by columns)
regimes	array	the original regimes array
total	array (k-1,)	the total number of rank changes for each of the k periods
max_total	int	the theoretical maximum number of rank changes for n observations
theta	array (k-1,)	the theta statistic for each of the k-1 intervals
permutations	int	the number of permutations
pvalue_left	float	p-value for test that observed theta is significantly lower than its expectation under complete spatial randomness
pvalue_right	float	p-value for test that observed theta is significantly greater than its expectation under complete spatial randomness

class pysal.spatial_dynamics.rank.SpatialTau(x, y, w, permutations=0)

Spatial version of Kendall’s rank correlation statistic

Kendall’s Tau is based on a comparison of the number of pairs of n observations that have concordant ranks between two variables. The spatial Tau decomposes these pairs into those that are spatial neighbors and those that are not, and examines whether the rank correlation is different between the two sets. [R5]

Parameters:

x : array (n,)

first variable

y : array (n,)

second variable

w : W

spatial weights object

permutations : int

number of random spatial permutations for computationally based inference

References

[R5]	(1, 2) Rey, S.J. (2004) “Spatial dependence in the evolution of regional income distributions,” in A. Getis, J. Mur and H.Zoeller (eds). Spatial Econometrics and Spatial Statistics. Palgrave, London, pp. 194-213.

Examples

>>> import pysal
>>> f=pysal.open("../examples/mexico.csv")
>>> vnames=["pcgdp%d"%dec for dec in range(1940,2010,10)]
>>> y=np.transpose(np.array([f.by_col[v] for v in vnames]))
>>> regime=np.array(f.by_col['esquivel99'])
>>> w=pysal.weights.regime_weights(regime)
>>> np.random.seed(10)
>>> res=[SpatialTau(y[:,i],y[:,i+1],w,99) for i in range(6)]
>>> for r in res:
...     "%8.3f %8.3f %8.3f"%(r.wnc,r.ev_wnc,r.p_rand_wnc)
...     
'  44.000   52.354    0.000'
'  47.000   53.576    0.006'
'  52.000   55.747    0.031'
'  54.000   55.556    0.212'
'  53.000   53.384    0.436'
'  57.000   57.566    0.390'
>>> 

Attributes

tau	float	The classic Tau statistic
wn	int	The number of neighboring pairs
tau_w	float	Spatial Tau statistic
tau_nw	float	Tau for non-neighboring pairs
p_tau_diff	float	p-value for test of difference between tau_w and tau_nw based on asymptotic distribution (Use with caution in small samples).
wnc	int	number of concordant neighbor pairs
wdc	int	number of discordant neighbor pairs
ev_wnc	int	average value of concordant neighbor pairs under random spatial permuations
s_wnc	float	standard deviation of the number of concordant neighbor pairs under random spatial permutations.
p_rand_wnc	float	p-value for test of difference between wnc and its expected value under spatial random permutations
z_rand_wnc	z-value for test of difference between wnc and its expected	value under spatial random permutations