def __homogeneous_poisson_sampling(T, S, maximum):
"""
To generate a homogeneous Poisson point pattern in space S X T, it basically
takes two steps:
1. Simulate the number of events n = N(S) occurring in S according to a
Poisson distribution with mean lam * |S X T|.
2. Sample each of the n location according to a uniform distribution on S
respectively.
Args:
lam: intensity (or maximum intensity when used by thining algorithm)
S: [(min_t, max_t), (min_x, max_x), (min_y, max_y), ...] indicates the
range of coordinates regarding a square (or cubic ...) region.
Returns:
samples: point process samples:
[(t1, x1, y1), (t2, x2, y2), ..., (tn, xn, yn)]
"""
_S = [T] + S
# sample the number of events from S
n = utils.lebesgue_measure(_S)
N = tf.random.poisson(lam=maximum * n, shape=[1], dtype=tf.int32)
# simulate spatial sequence and temporal sequence separately.
points = [
tf.random.uniform(shape=N, minval=_S[i][0], maxval=_S[i][1])
for i in range(len(_S))
]
# sort the temporal sequence ascendingly.
points[0] = tf.contrib.framework.sort(points[0], direction="ASCENDING")
points = tf.transpose(tf.stack(points))
return points
def _homogeneous_poisson_sampling(self, T=[0, 1], S=[[0, 1], [0, 1]]):
"""
To generate a homogeneous Poisson point pattern in space S X T, it basically
takes two steps:
1. Simulate the number of events n = N(S) occurring in S according to a
Poisson distribution with mean lam * |S X T|.
2. Sample each of the n location according to a uniform distribution on S
respectively.
Args:
lam: intensity (or maximum intensity when used by thining algorithm)
S: [(min_t, max_t), (min_x, max_x), (min_y, max_y), ...] indicates the
range of coordinates regarding a square (or cubic ...) region.
Returns:
samples: point process samples:
[(t1, x1, y1), (t2, x2, y2), ..., (tn, xn, yn)]
"""
_S = [T] + S
# sample the number of events from S
n = utils.lebesgue_measure(_S)
N = np.random.poisson(size=1, lam=self.lam.upper_bound() * n)
# simulate spatial sequence and temporal sequence separately.
points = [ np.random.uniform(_S[i][0], _S[i][1], N) for i in range(len(_S)) ]
points = np.array(points).transpose()
# sort the sequence regarding the ascending order of the temporal sample.
points = points[points[:, 0].argsort()]
return points
def pdf_with_history():
# triggering probability
log_trig_prob = tf.log(self._lambda(x, y, t, x_his, y_his, t_his))
# variables for calculating tail probability
tn, ti = points[-2, 0], points[:-1, 0]
t_ti, tn_ti = t - ti, tn - ti
# tail probability
log_tail_prob = - \
self.mu * (t - t_his[-1]) * utils.lebesgue_measure(S) - \
tf.reduce_sum(tf.scan(
lambda a, i: self.C * (tf.exp(- self.beta * tn_ti[i]) - tf.exp(- self.beta * t_ti[i])) / self.beta,
tf.range(tf.shape(t_ti)[0]),
initializer=np.array(0., dtype=np.float32)))
return log_trig_prob + log_tail_prob
def pdf_with_history():
# triggering probability
log_trig_prob = tf.log(
tf.clip_by_value(self._lambda(t, s, his_t, his_s), 1e-8,
1e+10))
# variables for calculating tail probability
tn, ti = points[-2, 0], points[:-1, 0]
t_ti, tn_ti = t - ti, tn - ti
# tail probability
# TODO: change to gaussian mixture (add phi)
log_tail_prob = - \
self.mu * (t - his_t[-1]) * utils.lebesgue_measure(self.S) - \
tf.reduce_sum(tf.scan(
lambda a, i: self.C * (tf.exp(- self.beta * tn_ti[i]) - tf.exp(- self.beta * t_ti[i])) / \
tf.clip_by_value(self.beta, 1e-8, 1e+10),
tf.range(tf.shape(t_ti)[0]),
initializer=np.array(0., dtype=np.float32)))
return log_trig_prob + log_tail_prob