def modified_knox(s_coords, t_coords, delta, tau, permutations=99):
"""
Baker's modified Knox test for spatio-temporal interaction. [4]_
Parameters
----------
s_coords : array
nx2 spatial coordinates
t_coords : array
nx1 temporal coordinates
delta : float
threshold for proximity in space
tau : float
threshold for proximity in time
permutations : int
the number of permutations used to establish pseudo-
significance (default is 99)
Returns
-------
modknox_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue)
stat : float
value of the modified knox test for the dataset
pvalue : float
pseudo p-value associated with the statistic
References
----------
.. [4] R.D. Baker. Identifying space-time disease clusters. Acta Tropica,
91(3):291-299, 2004
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt")
>>> events = SpaceTimeEvents(path, 'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
Run the modified Knox test with distance and time thresholds of 20 and 5,
respectively. This counts the events that are closer than 20 units in
space, and 5 units in time.
>>> result = modified_knox(events.space, events.t, delta=20, tau=5, permutations=99)
Next, we examine the results. First, we call the statistic from the
results dictionary. This reports the difference between the observed
and expected Knox statistic.
>>> print("%2.8f" % result['stat'])
2.81016043
Next, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistics. In this case,
the results indicate there is likely no space-time interaction.
>>> print("%2.2f" % result['pvalue'])
0.11
"""
s = s_coords
t = t_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
spacbin = sdistmat <= delta
spacmat = spacmat * spacbin
timemat = np.ones((n, n))
timebin = tdistmat <= tau
timemat = timemat * timebin
# calculate the observed (original) statistic
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
stat = (obsstat - (expstat / (n - 1.0))) / 2.0
# return results (if no inference)
if not permutations:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
timebin = rtdistmat <= tau
timemat = timemat * timebin
# calculate the observed knox again
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value again
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
tempstat = (obsstat - (expstat / (n - 1.0))) / 2.0
distribution.append(tempstat)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
modknox_result = {'stat': stat, 'pvalue': pvalue}
return modknox_result
def modified_knox(s_coords, t_coords, delta, tau, permutations=99):
"""
Baker's modified Knox test for spatio-temporal interaction. [Baker2004]_
Parameters
----------
s_coords : array
(n, 2), spatial coordinates.
t_coords : array
(n, 1), temporal coordinates.
delta : float
threshold for proximity in space.
tau : float
threshold for proximity in time.
permutations : int, optional
the number of permutations used to establish pseudo-
significance (the default is 99).
Returns
-------
modknox_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue).
stat : float
value of the modified knox test for the dataset.
pvalue : float
pseudo p-value associated with the statistic.
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt.shp")
>>> events = SpaceTimeEvents(path, 'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
Run the modified Knox test with distance and time thresholds of 20 and 5,
respectively. This counts the events that are closer than 20 units in
space, and 5 units in time.
>>> result = modified_knox(events.space, events.t, delta=20, tau=5, permutations=99)
Next, we examine the results. First, we call the statistic from the
results dictionary. This reports the difference between the observed
and expected Knox statistic.
>>> print("%2.8f" % result['stat'])
2.81016043
Next, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistics. In this case,
the results indicate there is likely no space-time interaction.
>>> print("%2.2f" % result['pvalue'])
0.11
"""
s = s_coords
t = t_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
spacbin = sdistmat <= delta
spacmat = spacmat * spacbin
timemat = np.ones((n, n))
timebin = tdistmat <= tau
timemat = timemat * timebin
# calculate the observed (original) statistic
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
stat = (obsstat - (expstat / (n - 1.0))) / 2.0
# return results (if no inference)
if not permutations:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
timebin = rtdistmat <= tau
timemat = timemat * timebin
# calculate the observed knox again
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value again
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
tempstat = (obsstat - (expstat / (n - 1.0))) / 2.0
distribution.append(tempstat)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
modknox_result = {'stat': stat, 'pvalue': pvalue}
return modknox_result
def mantel(s_coords,
t_coords,
permutations=99,
scon=1.0,
spow=-1.0,
tcon=1.0,
tpow=-1.0):
"""
Standardized Mantel test for spatio-temporal interaction. [2]_
Parameters
----------
s_coords : array
nx2 spatial coordinates
t_coords : array
nx1 temporal coordinates
permutations : int
the number of permutations used to establish pseudo-
significance (default is 99)
scon : float
constant added to spatial distances
spow : float
value for power transformation for spatial distances
tcon : float
constant added to temporal distances
tpow : float
value for power transformation for temporal distances
Returns
-------
mantel_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue)
stat : float
value of the knox test for the dataset
pvalue : float
pseudo p-value associated with the statistic
References
----------
.. [2] N. Mantel. 1967. The detection of disease clustering and a
generalized regression approach. Cancer Research, 27(2):209-220.
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt")
>>> events = SpaceTimeEvents(path,'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
The standardized Mantel test is a measure of matrix correlation between
the spatial and temporal distance matrices of the event dataset. The
following example runs the standardized Mantel test without a constant
or transformation; however, as recommended by Mantel (1967) [2]_, these
should be added by the user. This can be done by adjusting the constant
and power parameters.
>>> result = mantel(events.space, events.t, 99, scon=1.0, spow=-1.0, tcon=1.0, tpow=-1.0)
Next, we examine the result of the test.
>>> print("%6.6f"%result['stat'])
0.048368
Finally, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistic for each of the 99
permutations. According to these parameters, the results indicate
space-time interaction between the events.
>>> print("%2.2f"%result['pvalue'])
0.01
"""
t = t_coords
s = s_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
distmat = cg.distance_matrix(s)
timemat = cg.distance_matrix(t)
# calculate the transformed standardized statistic
timevec = (util.get_lower(timemat) + tcon)**tpow
distvec = (util.get_lower(distmat) + scon)**spow
stat = stats.pearsonr(timevec, distvec)[0].sum()
# return the results (if no inference)
if not permutations:
return stat
# loop for generating a random distribution to assess significance
dist = []
for i in range(permutations):
trand = util.shuffle_matrix(timemat, range(n))
timevec = (util.get_lower(trand) + tcon)**tpow
m = stats.pearsonr(timevec, distvec)[0].sum()
dist.append(m)
## establish the pseudo significance of the observed statistic
distribution = np.array(dist)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# report the results
mantel_result = {'stat': stat, 'pvalue': pvalue}
return mantel_result
def mantel(s_coords, t_coords, permutations=99, scon=1.0, spow=-1.0, tcon=1.0, tpow=-1.0):
"""
Standardized Mantel test for spatio-temporal interaction. [Mantel1967]_
Parameters
----------
s_coords : array
(n, 2), spatial coordinates.
t_coords : array
(n, 1), temporal coordinates.
permutations : int, optional
the number of permutations used to establish pseudo-
significance (the default is 99).
scon : float, optional
constant added to spatial distances (the default is 1.0).
spow : float, optional
value for power transformation for spatial distances
(the default is -1.0).
tcon : float, optional
constant added to temporal distances (the default is 1.0).
tpow : float, optional
value for power transformation for temporal distances
(the default is -1.0).
Returns
-------
mantel_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue).
stat : float
value of the knox test for the dataset.
pvalue : float
pseudo p-value associated with the statistic.
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt.shp")
>>> events = SpaceTimeEvents(path,'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
The standardized Mantel test is a measure of matrix correlation between
the spatial and temporal distance matrices of the event dataset. The
following example runs the standardized Mantel test without a constant
or transformation; however, as recommended by Mantel (1967) [2]_, these
should be added by the user. This can be done by adjusting the constant
and power parameters.
>>> result = mantel(events.space, events.t, 99, scon=1.0, spow=-1.0, tcon=1.0, tpow=-1.0)
Next, we examine the result of the test.
>>> print("%6.6f"%result['stat'])
0.048368
Finally, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistic for each of the 99
permutations. According to these parameters, the results indicate
space-time interaction between the events.
>>> print("%2.2f"%result['pvalue'])
0.01
"""
t = t_coords
s = s_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
distmat = cg.distance_matrix(s)
timemat = cg.distance_matrix(t)
# calculate the transformed standardized statistic
timevec = (util.get_lower(timemat) + tcon) ** tpow
distvec = (util.get_lower(distmat) + scon) ** spow
stat = stats.pearsonr(timevec, distvec)[0].sum()
# return the results (if no inference)
if not permutations:
return stat
# loop for generating a random distribution to assess significance
dist = []
for i in range(permutations):
trand = util.shuffle_matrix(timemat, range(n))
timevec = (util.get_lower(trand) + tcon) ** tpow
m = stats.pearsonr(timevec, distvec)[0].sum()
dist.append(m)
## establish the pseudo significance of the observed statistic
distribution = np.array(dist)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# report the results
mantel_result = {'stat': stat, 'pvalue': pvalue}
return mantel_result
def modified_knox(s_coords, t_coords, delta, tau, permutations=99):
s = s_coords
t = t_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
spacbin = sdistmat <= delta
spacmat = spacmat * spacbin
timemat = np.ones((n, n))
timebin = tdistmat <= tau
timemat = timemat * timebin
# calculate the observed (original) statistic
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
stat = (obsstat - (expstat / (n - 1.0))) / 2.0
# return results (if no inference)
if not permutations:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
timebin = rtdistmat <= tau
timemat = timemat * timebin
# calculate the observed knox again
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value again
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
eknox = expstat / (n - 1.0)
# calculate the modified stat
tempstat = (obsstat - (expstat / (n - 1.0))) / 2.0
distribution.append(tempstat)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
modknox_result = {'stat': stat, 'pvalue': pvalue,'eknox': eknox }
return modknox_result
def knox(events, delta, tau, permutations=99):
"""
Knox test for spatio-temporal interaction. [1]_
Parameters
----------
events : space time events object
an output instance from the class SpaceTimeEvents
delta : float
threshold for proximity in space
tau : float
threshold for proximity in time
permutations : int
the number of permutations used to establish pseudo-
significance (default is 99)
Returns
-------
knox_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue)
stat : float
value of the knox test for the dataset
pvalue : float
pseudo p-value associated with the statistic
References
----------
.. [1] E. Knox. 1964. The detection of space-time
interactions. Journal of the Royal Statistical Society. Series C
(Applied Statistics), 13(1):25-30.
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt")
>>> events = SpaceTimeEvents(path,'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
Run the Knox test with distance and time thresholds of 20 and 5,
respectively. This counts the events that are closer than 20 units in
space, and 5 units in time.
>>> result = knox(events,delta=20,tau=5,permutations=99)
Next, we examine the results. First, we call the statistic from the
results results dictionary. This reports that there are 13 events close
in both space and time, according to our threshold definitions.
>>> print(result['stat'])
13.0
Next, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistics. In this case,
the results indicate there is likely no space-time interaction between
the events.
>>> print("%2.2f"%result['pvalue'])
0.18
"""
n = events.n
s = events.space
t = events.t
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
test = sdistmat <= delta
spacmat = spacmat * test
timemat = np.ones((n, n))
test = tdistmat <= tau
timemat = timemat * test
# calculate the statistic
knoxmat = timemat * spacmat
stat = (knoxmat.sum() - n) / 2
# return results (if no inference)
if permutations == 0:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
test = rtdistmat <= tau
timemat = timemat * test
knoxmat = timemat * spacmat
k = (knoxmat.sum() - n) / 2
distribution.append(k)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
knox_result = {'stat': stat, 'pvalue': pvalue}
return knox_result
def modified_knox(s_coords, t_coords, delta, tau, permutations=99):
s = s_coords
t = t_coords
n = len(t)
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
spacbin = sdistmat <= delta
spacmat = spacmat * spacbin
timemat = np.ones((n, n))
timebin = tdistmat <= tau
timemat = timemat * timebin
# calculate the observed (original) statistic
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
# calculate the modified stat
stat = (obsstat - (expstat / (n - 1.0))) / 2.0
# return results (if no inference)
if not permutations:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
timebin = rtdistmat <= tau
timemat = timemat * timebin
# calculate the observed knox again
knoxmat = timemat * spacmat
obsstat = (knoxmat.sum() - n)
# calculate the expectated value again
ssumvec = np.reshape((spacbin.sum(axis=0) - 1), (n, 1))
tsumvec = np.reshape((timebin.sum(axis=0) - 1), (n, 1))
expstat = (ssumvec * tsumvec).sum()
eknox = expstat / (n - 1.0)
# calculate the modified stat
tempstat = (obsstat - (expstat / (n - 1.0))) / 2.0
distribution.append(tempstat)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
modknox_result = {'stat': stat, 'pvalue': pvalue, 'eknox': eknox}
return modknox_result
def knox(events, delta, tau, permutations=99):
"""
Knox test for spatio-temporal interaction. [1]_
Parameters
----------
events : space time events object
an output instance from the class SpaceTimeEvents
delta : float
threshold for proximity in space
tau : float
threshold for proximity in time
permutations : int
the number of permutations used to establish pseudo-
significance (default is 99)
Returns
-------
knox_result : dictionary
contains the statistic (stat) for the test and the
associated p-value (pvalue)
stat : float
value of the knox test for the dataset
pvalue : float
pseudo p-value associated with the statistic
References
----------
.. [1] E. Knox. 1964. The detection of space-time
interactions. Journal of the Royal Statistical Society. Series C
(Applied Statistics), 13(1):25-30.
Examples
--------
>>> import numpy as np
>>> import pysal
Read in the example data and create an instance of SpaceTimeEvents.
>>> path = pysal.examples.get_path("burkitt")
>>> events = SpaceTimeEvents(path,'T')
Set the random seed generator. This is used by the permutation based
inference to replicate the pseudo-significance of our example results -
the end-user will normally omit this step.
>>> np.random.seed(100)
Run the Knox test with distance and time thresholds of 20 and 5,
respectively. This counts the events that are closer than 20 units in
space, and 5 units in time.
>>> result = knox(events,delta=20,tau=5,permutations=99)
Next, we examine the results. First, we call the statistic from the
results results dictionary. This reports that there are 13 events close
in both space and time, according to our threshold definitions.
>>> print(result['stat'])
13.0
Next, we look at the pseudo-significance of this value, calculated by
permuting the timestamps and rerunning the statistics. In this case,
the results indicate there is likely no space-time interaction between
the events.
>>> print("%2.2f"%result['pvalue'])
0.18
"""
n = events.n
s = events.space
t = events.t
# calculate the spatial and temporal distance matrices for the events
sdistmat = cg.distance_matrix(s)
tdistmat = cg.distance_matrix(t)
# identify events within thresholds
spacmat = np.ones((n, n))
test = sdistmat <= delta
spacmat = spacmat * test
timemat = np.ones((n, n))
test = tdistmat <= tau
timemat = timemat * test
# calculate the statistic
knoxmat = timemat * spacmat
stat = (knoxmat.sum() - n) / 2
# return results (if no inference)
if permutations == 0:
return stat
distribution = []
# loop for generating a random distribution to assess significance
for p in range(permutations):
rtdistmat = util.shuffle_matrix(tdistmat, range(n))
timemat = np.ones((n, n))
test = rtdistmat <= tau
timemat = timemat * test
knoxmat = timemat * spacmat
k = (knoxmat.sum() - n) / 2
distribution.append(k)
# establish the pseudo significance of the observed statistic
distribution = np.array(distribution)
greater = np.ma.masked_greater_equal(distribution, stat)
count = np.ma.count_masked(greater)
pvalue = (count + 1.0) / (permutations + 1.0)
# return results
knox_result = {'stat': stat, 'pvalue': pvalue}
return knox_result
