In [ ]:
import numpy
import scipy.stats
from scipy.integrate import quad, dblquad
In [ ]:
from timeit import default_timer as timer
In [ ]:
CHEB_NODES_NB = 15
LOW_BOUND = -10.
UPP_BOUND = 10.
In [ ]:
N = 30
M = 1000000
# Gaussian expectation vector
Mu = numpy.zeros(N)
# Gaussian variance-covariance matrix
Sigma = numpy.array([[ 4.09, -0.89, -1.12, -0.03, 0.21, -0.88, 0.15, 0.04, -1.01,
0.88, 0.98, 0.26, -0.31, 0.94, 0.28, -0.05, -0.89, -1.09,
0.14, -1.83, -0.1 , -0.76, 0.36, 0.24, -0.04, -0.18, 0.94,
0.95, 1.01, 0.45],
[-0.89, 3.17, -0.78, -0.32, 0.6 , 0.34, 0.1 , -0.08, 0.6 ,
-0.1 , 0.59, 0.24, -0.22, 0.55, -0.29, 1.04, 0.03, 0.34,
0.55, 0.9 , -0.39, -0.64, -1. , -0.02, -0.64, -0.01, -0.11,
-0.89, 1.11, -0.67],
[-1.12, -0.78, 4.73, 0.01, 0.2 , 1.36, -0.19, -0.05, -0.2 ,
-1.22, -0.82, -0.01, 0.08, -0.55, 0.6 , 0.14, -0.51, -1.24,
1.05, 1.28, 0.95, -0.23, 2.24, -0.61, 0.64, -0. , -0.52,
-1.92, -1.38, 0.45],
[-0.03, -0.32, 0.01, 3.83, 0.02, -1.1 , -0.22, 0.07, -0.59,
-0.2 , -0.77, 0.1 , 0.26, -0.25, 0.34, 1.38, -0.62, -0.29,
0.33, -1.24, 0.42, 1.08, 1.14, 0.22, -1.03, -0.23, 0.2 ,
1.36, -0.23, -0.91],
[ 0.21, 0.6 , 0.2 , 0.02, 1.15, -0.1 , 0.05, 0.05, -0.08,
-0.6 , -0.09, 0.22, -0.07, 0.15, -0.06, -0.07, -0.46, -0.18,
0.29, 0.12, 0.21, 0.1 , -0.21, -0.26, -0.05, 0.08, -0.26,
-0.99, -0.23, 0.04],
[-0.88, 0.34, 1.36, -1.1 , -0.1 , 5.78, 0.2 , -0.04, -0.64,
-0.31, 1.06, 0.32, -0.16, -0.71, 0.96, 0.88, -1.24, -0.18,
-1.17, -1.07, -0.82, -0.51, 1.69, -1.13, 0.82, 0.49, -0.43,
-0.5 , -0.04, -1.21],
[ 0.15, 0.1 , -0.19, -0.22, 0.05, 0.2 , 0.2 , 0.02, 0.03,
-0. , 0.02, 0.05, -0.03, 0.14, 0.01, -0.03, -0.06, -0.04,
-0.12, -0.17, -0.04, -0.14, -0.01, -0.08, 0.07, 0.06, -0.01,
-0.03, 0.14, 0.09],
[ 0.04, -0.08, -0.05, 0.07, 0.05, -0.04, 0.02, 0.17, 0.04,
0.04, -0. , -0.05, -0. , -0.01, -0.01, -0.14, 0.04, -0.08,
-0.1 , -0.04, 0.03, 0.01, 0.06, -0.06, 0.01, 0.1 , 0.01,
0.12, -0.03, 0.05],
[-1.01, 0.6 , -0.2 , -0.59, -0.08, -0.64, 0.03, 0.04, 2.68,
0.36, -0.12, -0.44, -0.24, -0.57, -0.41, -0.69, -0.33, 0.46,
-0.38, 0.72, 0.15, -0.38, -1.56, 0.09, 0.53, -0.17, 0.52,
0.31, -0.77, 0.71],
[ 0.88, -0.1 , -1.22, -0.2 , -0.6 , -0.31, -0. , 0.04, 0.36,
3.09, 1.42, -0.15, -0.14, 0.29, 0.39, -0.54, 0.92, 0.22,
-0.41, 0.32, -0.61, -0.3 , -0.14, 0.72, -0.13, -0.57, 0.01,
2.19, 1.69, -0.38],
[ 0.98, 0.59, -0.82, -0.77, -0.09, 1.06, 0.02, -0. , -0.12,
1.42, 2.2 , 0.27, -0.42, -0.12, 0.09, -0.41, 0.15, -0.31,
-0.3 , 0.09, -0.71, -0.43, 0.33, 0.3 , -0.07, -0.14, 0.23,
1.4 , 1.33, -0. ],
[ 0.26, 0.24, -0.01, 0.1 , 0.22, 0.32, 0.05, -0.05, -0.44,
-0.15, 0.27, 0.55, 0.01, 0.29, 0.1 , 0.12, 0.01, -0.21,
-0.09, -0.13, 0.1 , 0.12, 0.46, -0.16, -0.03, -0.11, -0.11,
-0.11, 0.49, 0.08],
[-0.31, -0.22, 0.08, 0.26, -0.07, -0.16, -0.03, -0. , -0.24,
-0.14, -0.42, 0.01, 0.55, 0.02, 0.18, 0.18, 0.25, 0.09,
-0.16, 0.1 , 0.06, 0.1 , -0.01, -0.03, -0.07, -0.11, -0.2 ,
-0.14, 0.07, -0.02],
[ 0.94, 0.55, -0.55, -0.25, 0.15, -0.71, 0.14, -0.01, -0.57,
0.29, -0.12, 0.29, 0.02, 1.97, 0.1 , 0.23, 0.81, -0.21,
-0.1 , -0.1 , 0.5 , -0.25, 0.02, -0.33, 0.23, -0.04, 0.27,
-0.7 , 1.22, -0.26],
[ 0.28, -0.29, 0.6 , 0.34, -0.06, 0.96, 0.01, -0.01, -0.41,
0.39, 0.09, 0.1 , 0.18, 0.1 , 0.96, 0.42, -0.15, -0.69,
-0.19, -0.62, -0.12, -0.22, 0.61, -0.03, 0.13, -0.02, 0.04,
0.69, -0.02, -0.55],
[-0.05, 1.04, 0.14, 1.38, -0.07, 0.88, -0.03, -0.14, -0.69,
-0.54, -0.41, 0.12, 0.18, 0.23, 0.42, 2.59, -1.24, -0.2 ,
-0.09, -0.53, -0.52, -0.14, 0.6 , 0.03, -0.73, -0.14, 0.08,
-0.74, 0.88, -1. ],
[-0.89, 0.03, -0.51, -0.62, -0.46, -1.24, -0.06, 0.04, -0.33,
0.92, 0.15, 0.01, 0.25, 0.81, -0.15, -1.24, 2.97, 0.23,
0.28, 1.14, 0.26, 0.27, -0.62, 0.1 , 0.06, -0.01, -0.42,
0.95, 0.27, -0.37],
[-1.09, 0.34, -1.24, -0.29, -0.18, -0.18, -0.04, -0.08, 0.46,
0.22, -0.31, -0.21, 0.09, -0.21, -0.69, -0.2 , 0.23, 2.77,
0.5 , 0.19, 0.15, 0.6 , -1.25, 0.01, -0.1 , -0.12, -0.5 ,
-1.12, -0.11, -0.67],
[ 0.14, 0.55, 1.05, 0.33, 0.29, -1.17, -0.12, -0.1 , -0.38,
-0.41, -0.3 , -0.09, -0.16, -0.1 , -0.19, -0.09, 0.28, 0.5 ,
3.37, -0.11, 0.71, -0.4 , -0.32, 0.4 , -0.67, -0.01, -0.24,
-0.48, -1.39, -0.66],
[-1.83, 0.9 , 1.28, -1.24, 0.12, -1.07, -0.17, -0.04, 0.72,
0.32, 0.09, -0.13, 0.1 , -0.1 , -0.62, -0.53, 1.14, 0.19,
-0.11, 4.13, -0.05, 0.13, 0.23, 0.44, -0.2 , -0.31, -0.97,
-1.92, 1.56, 1.45],
[-0.1 , -0.39, 0.95, 0.42, 0.21, -0.82, -0.04, 0.03, 0.15,
-0.61, -0.71, 0.1 , 0.06, 0.5 , -0.12, -0.52, 0.26, 0.15,
0.71, -0.05, 1.37, 0.26, 0.45, -0.39, 0.42, 0.01, 0.18,
-0.77, -0.7 , 0.33],
[-0.76, -0.64, -0.23, 1.08, 0.1 , -0.51, -0.14, 0.01, -0.38,
-0.3 , -0.43, 0.12, 0.1 , -0.25, -0.22, -0.14, 0.27, 0.6 ,
-0.4 , 0.13, 0.26, 1.49, 0.41, 0.11, -0.12, -0.03, -0.33,
0.03, -0.18, -0.08],
[ 0.36, -1. , 2.24, 1.14, -0.21, 1.69, -0.01, 0.06, -1.56,
-0.14, 0.33, 0.46, -0.01, 0.02, 0.61, 0.6 , -0.62, -1.25,
-0.32, 0.23, 0.45, 0.41, 4.4 , -0.03, -0.13, 0.11, -0.31,
-0.29, 1.69, 0.95],
[ 0.24, -0.02, -0.61, 0.22, -0.26, -1.13, -0.08, -0.06, 0.09,
0.72, 0.3 , -0.16, -0.03, -0.33, -0.03, 0.03, 0.1 , 0.01,
0.4 , 0.44, -0.39, 0.11, -0.03, 1.36, -0.56, -0.23, 0.14,
1.07, 0.65, 0.51],
[-0.04, -0.64, 0.64, -1.03, -0.05, 0.82, 0.07, 0.01, 0.53,
-0.13, -0.07, -0.03, -0.07, 0.23, 0.13, -0.73, 0.06, -0.1 ,
-0.67, -0.2 , 0.42, -0.12, -0.13, -0.56, 1.27, 0.05, 0.34,
-0.34, -0.73, 0.23],
[-0.18, -0.01, -0. , -0.23, 0.08, 0.49, 0.06, 0.1 , -0.17,
-0.57, -0.14, -0.11, -0.11, -0.04, -0.02, -0.14, -0.01, -0.12,
-0.01, -0.31, 0.01, -0.03, 0.11, -0.23, 0.05, 0.69, 0.06,
-0.16, -0.48, -0.09],
[ 0.94, -0.11, -0.52, 0.2 , -0.26, -0.43, -0.01, 0.01, 0.52,
0.01, 0.23, -0.11, -0.2 , 0.27, 0.04, 0.08, -0.42, -0.5 ,
-0.24, -0.97, 0.18, -0.33, -0.31, 0.14, 0.34, 0.06, 1.29,
0.96, -0.28, 0.17],
[ 0.95, -0.89, -1.92, 1.36, -0.99, -0.5 , -0.03, 0.12, 0.31,
2.19, 1.4 , -0.11, -0.14, -0.7 , 0.69, -0.74, 0.95, -1.12,
-0.48, -1.92, -0.77, 0.03, -0.29, 1.07, -0.34, -0.16, 0.96,
5.81, -0.35, -0.59],
[ 1.01, 1.11, -1.38, -0.23, -0.23, -0.04, 0.14, -0.03, -0.77,
1.69, 1.33, 0.49, 0.07, 1.22, -0.02, 0.88, 0.27, -0.11,
-1.39, 1.56, -0.7 , -0.18, 1.69, 0.65, -0.73, -0.48, -0.28,
-0.35, 4.81, 1.05],
[ 0.45, -0.67, 0.45, -0.91, 0.04, -1.21, 0.09, 0.05, 0.71,
-0.38, -0. , 0.08, -0.02, -0.26, -0.55, -1. , -0.37, -0.67,
-0.66, 1.45, 0.33, -0.08, 0.95, 0.51, 0.23, -0.09, 0.17,
-0.59, 1.05, 3.02]])
print "Mu = %s" % Mu
print "Sigma = \n %s" % Sigma
ts_rv = timer()
rv = scipy.stats.multivariate_normal (mean=Mu, cov=Sigma, allow_singular=True)
X = rv.rvs(size=M)
te_rv = timer()
ctp = te_rv - ts_rv
We need to solve the following system:
\begin{equation*} \begin{cases} \lambda \left[ 1 + \mathbb{E} \left( \left(X_k - m_k^* \right)^+ \right) + \sum_{j \neq k} \mathbb{E} \left( \left(X_j - m_j^* \right)^+ 1_{X_k > m_k^*} \right) \right] = 1 \text{ for all $k$} \\ \sum_k \mathbb{E} \left( X_k - m_k^* \right) + \sum_k \frac{1}{2} \mathbb{E} \left( \left[ \left( X_k - m_k^* \right)^+ \right]^2 \right) + \sum_{j < k} \mathbb{E} \left( \left( X_j - m_j^* \right)^+ \left( X_k - m_k^* \right)^+ \right) = c \end{cases} \end{equation*}Definitions from the paper:
\begin{equation*} \begin{cases} \lambda \left[ 1 + f_k \left( m_k^* \right) + \sum_{j \neq k} l_{j, k} \left( m_j^*, m_k^* \right) \right] = 1 \text{ for all $k$} \\ \sum_k \left( e_k \left( m_k^* \right) + \frac{1}{2} g_k \left( m_k^* \right) + \sum_{j < k} h \left( m_j^*, m_k^* \right) \right) = c \end{cases} \end{equation*}
In [ ]:
def e(index, x):
mu = Mu[index]
return mu - x
In [ ]:
def f(k, x):
global X
Xk = X[:, k]
Xkmx = Xk - x
Xkmxp = numpy.maximum(Xkmx, 0.)
return numpy.mean(Xkmxp)
In [ ]:
def g(k, x):
global X
Xk = X[:, k]
Xkmx = Xk - x
Xkmxp = numpy.maximum(Xkmx, 0.)
Xkmxp2 = numpy.square(Xkmxp)
return numpy.mean(Xkmxp2)
In [ ]:
def h(j, k, x, y):
global X
Xjk = X[:, [j, k]]
m = numpy.array([x, y])
zero = numpy.zeros(2)
Xmm = numpy.subtract(Xjk, m)
Xmmp = numpy.maximum(Xmm, zero)
prod = numpy.cumprod(Xmmp, axis=1)[:, -1]
return numpy.mean(prod)
In [ ]:
def l(j, k, x, y):
global X
Xjk = X[:, [j, k]]
m = numpy.array([x, y])
zero = numpy.zeros(2)
Xmm = numpy.subtract(Xjk, m)
Xmmp = numpy.maximum(Xmm, zero)
# We use the fact that sgn(0) = 0
Xmmp[:, 1] = numpy.sign(Xmmp[:, 1])
prod = numpy.cumprod(Xmmp, axis=1)[:, -1]
return numpy.mean(prod)
In [ ]:
class WrapperFunc(object):
def __init__(self, func, indexes):
self._i = indexes
self._f = func
def __call__(self, *x):
tmp_args = self._i + list(x)
return self._f(*tmp_args)
In [ ]:
import sys
sys.path.append("../../lib")
In [ ]:
from tchebychev import TchebychevFun
def init_funcs():
from timeit import default_timer as timer
funcs = dict()
interpol_time = 0.
for k in range(N):
ek = WrapperFunc(e, [k])
fk = WrapperFunc(f, [k])
gk = WrapperFunc(g, [k])
hk = dict()
lk = dict()
for j in range(N):
if j < k:
hjk = WrapperFunc(h, [j, k])
hk[j] = hjk
if j != k:
ljk = WrapperFunc(l, [j, k])
lk[j] = ljk
funcs[k] = {"e": ek, "f": fk, "g": gk,
"hk": hk, "lk": lk}
global ctp
ctp += interpol_time
return funcs
In [ ]:
def prepare_parameterized_funcs(dict_funcs):
res = dict()
for k, v in dict_funcs.iteritems():
for kk, vv in v.iteritems():
if isinstance(vv, dict):
for j, f in vv.iteritems():
key = "%s_%s_%s" % (kk[:-1], j, k)
res[key] = (f, j, k)
else:
key = "%s_%s" % (kk, k)
res[key] = (vv, k)
return res
In [ ]:
def run_par_func(param):
f = param[0]
vm = []
for i_m in param[1:]:
vm.append(M[i_m])
return f(*vm)
In [ ]:
funcs = init_funcs()
flatten_funcs = prepare_parameterized_funcs(funcs)
In [ ]:
from scipy.optimize import fsolve
def optimize(c, **kwargs):
global N
__results = {key: dict() for key in "efghl"}
def fill_results(lbl, res):
m_lbl = lbl.split("_")
fname = m_lbl[0]
k = int(m_lbl[-1])
if len(m_lbl) == 3:
j = int(m_lbl[1])
if k not in __results[fname]:
__results[fname][k] = dict()
__results[fname][k][j] = res
else:
__results[fname][k] = res
def objective(v_x):
_lambda = v_x[-1]
# Local version
global M
M = numpy.clip(v_x, LOW_BOUND, UPP_BOUND)
f_res = map(run_par_func, flatten_funcs.values())
for lbl, fx in zip(flatten_funcs.keys(), f_res):
fill_results(lbl, fx)
last = 0.
res = numpy.zeros(N + 1)
for k in range(N):
res_k = 1 + __results["f"][k]
for j, v in __results["l"][k].iteritems():
res_k += v
res[k] = (_lambda * res_k - 1.)**2
last += __results["e"][k] + .5 * __results["g"][k]
if k in __results["h"]:
for j, v in __results["h"][k].iteritems():
last += v
res[-1] = (last - c)**2
return res
guess = numpy.zeros(N + 1)
if 'guess' in kwargs and kwargs['guess'] is not None:
guess = kwargs.pop('guess')
return fsolve(objective, guess, **kwargs)
In [ ]:
c = 1.
guess = numpy.zeros(N+1)
In [ ]:
tic = timer()
continue_ = True
__i = 1
l_N = 0
while continue_:
print "iteration nb %i" % __i
optim_result = optimize(c, guess=guess, maxfev=1000, full_output=True)
continue_ = optim_result[2] != 1
if continue_:
guess = optim_result[0]
__i += 1
l_N += optim_result[1]["nfev"]
print
toc_m_tic = timer() - tic
print optim_result[-1]
In [ ]:
m_star = optim_result[0][:-1]
lb_star = optim_result[0][-1]
cto = toc_m_tic
print "Time of optimization: %.2f sec" % (cto)
print "m star = %s" % m_star
print "lambda star = %s" % lb_star
In [ ]:
l_N += optim_result[1]["nfev"]
print "l(%i) = %i" % (N, l_N)
print "function evaluated at the output = %s" % optim_result[1]["fvec"]
In [ ]:
print m_star
print
print "%f \n %i \n %.2f \n %.2f" % (lb_star, l_N, ctp, cto)