def iid_order_stat(X, n, k, **kwargs):
"""It gives distribution of $k$-th observation in ordered sample size of n."""
pdf = X.get_piecewise_pdf()
cdf = X.get_piecewise_cdf()
ccdf = 1 - X.get_piecewise_cdf()
fun = (k * binomial_coeff(n, k)) * pdf * (pow(cdf, k-1) * pow(ccdf, n-k))
#return PDistr(fun.toInterpolated())
return FunDistr(fun = fun.toInterpolated(), breakPoints = X.get_piecewise_pdf().getBreaks(), **kwargs)
def iid_order_stat(X, n, k, **kwargs):
"""It gives distribution of $k$-th observation in ordered sample size of n."""
pdf = X.get_piecewise_pdf()
cdf = X.get_piecewise_cdf()
ccdf = 1 - X.get_piecewise_cdf()
fun = (k * binomial_coeff(n, k)) * pdf * (pow(cdf, k-1) * pow(ccdf, n-k))
#return PDistr(fun.toInterpolated())
return FunDistr(fun = fun.toInterpolated(), breakPoints = X.get_piecewise_pdf().getBreaks(), **kwargs)
def get_nodes(self, q):
"""Compute nodes for current ni."""
nodes = []
node_inds_map = {}
subgrid_map = {} # maps subgrids to nodes
ps = gen_Q(
q, self.d
) # compute partitions. Each partition corresponds to a subgrid
for p in ps:
# each new node is described with a set of indices. Each
# index is the number of interpolation root for given node
# and dimension
max_q = q - self.d + 1
max_d = self.exps[max_q]
if self.init_ni == 1:
new_nodes = itertools.product(*[
list(range(0, max_d, 1 << (max_q -
e))) if e > 1 else [max_d / 2]
for e in p
])
else:
new_nodes = itertools.product(
*[list(range(0, max_d, 1 << (max_q - e))) for e in p])
# node indices for this subgrid
subgrid_indices = []
for ni in new_nodes:
if ni in node_inds_map:
idx = node_inds_map[ni]
else:
idx = len(nodes)
node_inds_map[ni] = idx
nodes.append(ni)
subgrid_indices.append(idx)
# subgrid's coefficient
si = sum(p)
c = (-1)**(q - si) * binomial_coeff(self.d - 1, q - si)
subgrid_map[p] = (c, array(subgrid_indices))
if params.interpolation_nd.debug_info:
print(len(subgrid_map),
"rect. subgrids,",
sum(len(subgrid_map[k]) for k in subgrid_map),
"ops per X,",
len(nodes),
"nodes,",
end=' ')
print("full grid size", self.exps[q - self.d + 1]**self.d, "dim",
self.d)
return nodes, subgrid_map
def get_nodes(self, q):
"""Compute nodes for current ni."""
nodes = []
node_inds_map = {}
subgrid_map = {} # maps subgrids to nodes
ps = gen_Q(q, self.d) # compute partitions. Each partition corresponds to a subgrid
for p in ps:
# each new node is described with a set of indices. Each
# index is the number of interpolation root for given node
# and dimension
max_q = q-self.d+1
max_d = self.exps[max_q]
if self.init_ni == 1:
new_nodes = itertools.product(*[list(range(0, max_d, 1 << (max_q - e))) if e > 1 else [max_d/2] for e in p])
else:
new_nodes = itertools.product(*[list(range(0, max_d, 1 << (max_q - e))) for e in p])
# node indices for this subgrid
subgrid_indices = []
for ni in new_nodes:
if ni in node_inds_map:
idx = node_inds_map[ni]
else:
idx = len(nodes)
node_inds_map[ni] = idx
nodes.append(ni)
subgrid_indices.append(idx)
# subgrid's coefficient
si = sum(p)
c = (-1)**(q-si) * binomial_coeff(self.d - 1, q-si)
subgrid_map[p] = (c, array(subgrid_indices))
if params.interpolation_nd.debug_info:
print(len(subgrid_map), "rect. subgrids,", sum(len(subgrid_map[k]) for k in subgrid_map), "ops per X,", len(nodes), "nodes,", end=' ')
print("full grid size", self.exps[q-self.d+1]**self.d, "dim", self.d)
return nodes, subgrid_map
D = UniformDistr(0, 1)
for i in range(m - 1):
D += UniformDistr(0, 1)
d = N / D
figure()
demo_distr(d, xmax=10)
a = 0.9
nmfact = 1
for i in range(2, n + m + 1):
nmfact *= i
nrm = a**m * nmfact
s = 0
for i in range(int(numpy.floor(m * a) + 1)):
for j in range(int(numpy.floor((m * a - i) / a) + 1)):
s += (-1)**(i + j) * binomial_coeff(n, i) * binomial_coeff(
m, j) * ((m - j) * a - i)**(n + m)
cdf_formula = s / nrm
cdf_pacal = d.cdf(a)
print("n={0},m={1} cdf({2})={3}, err = {4}".format(
n, m, a, cdf_pacal, abs(cdf_pacal - cdf_formula)))
#! Exercise 4.32
for l1, l2 in [(1, 1), (10, 1), (1, 10)]:
d = ExponentialDistr(l1) / ExponentialDistr(l2)
figure()
alpha = float(l2) / l1
demo_distr(d, xmax=10, theoretical=lambda x: alpha / (x + alpha)**2)
print("time=", time.time() - tic)
show()
for i in xrange(n-1):
N += UniformDistr(0,1)
D = UniformDistr(0,1)
for i in xrange(m-1):
D += UniformDistr(0,1)
d = N/D
figure()
demo_distr(d, xmax = 10)
a = 0.9
nmfact = 1
for i in xrange(2, n+m+1):
nmfact *= i
nrm = a**m * nmfact
s = 0
for i in xrange(int(numpy.floor(m*a)+1)):
for j in xrange(int(numpy.floor((m*a-i)/a)+1)):
s += (-1)**(i+j) * binomial_coeff(n, i) * binomial_coeff(m, j) * ((m-j)*a-i)**(n+m)
cdf_formula = s / nrm
cdf_pacal = d.cdf(a)
print "n={0},m={1} cdf({2})={3}, err = {4}".format(n, m, a, cdf_pacal, abs(cdf_pacal - cdf_formula))
#! Exercise 4.32
for l1, l2 in [(1,1), (10,1), (1,10)]:
d = ExponentialDistr(l1) / ExponentialDistr(l2)
figure()
alpha = float(l2) / l1
demo_distr(d, xmax = 10, theoretical = lambda x: alpha / (x+alpha)**2)
print "time=", time.time() - tic
show()
