Exemple #1
0
def PESIPS(ojf, lb, ub, ki, b, fname):
    sp.random.seed(int(os.urandom(4).encode('hex'), 16))
    para = dict()
    para['kindex'] = ki[0]
    para['mprior'] = ki[1]
    para['sprior'] = ki[2]
    para['d'] = lb.size + 1
    para['ninit'] = 10
    para['volper'] = 1e-7
    para['DH_SAMPLES'] = 12
    para['DM_SAMPLES'] = 12
    para['DM_SUPPORT'] = 1200
    para['DM_SLICELCBPARA'] = 1.
    para['SUPPORT_MODE'] = [
        ESutils.SUPPORT_LAPAPR
    ]  #[ESutils.SUPPORT_SLICELCB,ESutils.SUPPORT_SLICEEI]
    para['sl'] = 0.
    para['su'] = 1.
    para['s'] = 0.
    para['sfn'] = None
    para['axis'] = 0
    para['value'] = para['sl']
    if os.path.exists(fname):
        print "starting from " + str(fname)
        OE = OPTutils.PESIPS(ojf,
                             lb,
                             ub,
                             para,
                             initstate=pickle.load(open(fname, 'rb')))
    else:
        print "fresh start"
        OE = OPTutils.PESIPS(ojf, lb, ub, para)

    if sum(OE.C) >= b:
        print "no further steps needed"
        k = 0
        while sum(OE.C[:k]) < b:

            k += 1
        state = [
            OE.X[:k, :], OE.Y[:k, :], OE.S[:k, :], OE.D[:k], OE.R[:k, :],
            OE.C[:k], OE.T[:k], OE.Tr[:k], OE.Ymin[:k], OE.Xmin[:k, :],
            OE.Yreg[:k, :], OE.Rreg[:k, :]
        ]
    else:
        pbar = tqdm(total=(b - sum(OE.C)))
        while sum(OE.C) < b:
            print "used {0} of {1} eval budget".format(sum(OE.C), b)

            pbar.update(OE.C[-1])
            OE.step()
        state = [
            OE.X, OE.Y, OE.S, OE.D, OE.R, OE.C, OE.T, OE.Tr, OE.Ymin, OE.Xmin,
            OE.Yreg, OE.Rreg
        ]
        pickle.dump(state, open(fname, 'wb'))
    return state
Exemple #2
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def PESVS(ojf, lb, ub, ki, s, b, cfn, lsl, lsu, fname):
    overridemaxsteps = 200
    sp.random.seed(int(os.urandom(4).encode('hex'), 16))
    para = dict()
    para['kindex'] = ki[0]
    para['mprior'] = ki[1]
    para['sprior'] = ki[2]
    para['s'] = s
    para['ninit'] = 10
    para['volper'] = 1e-6
    para['DH_SAMPLES'] = 12
    para['DM_SAMPLES'] = 12
    para['DM_SUPPORT'] = 1200
    para['DM_SLICELCBPARA'] = 1.
    para['SUPPORT_MODE'] = [ESutils.SUPPORT_LAPAPR]
    para['cfn'] = cfn
    para['logsl'] = lsl
    para['logsu'] = lsu
    para['s'] = 10**lsu
    if os.path.exists(fname):
        print "starting from " + str(fname)
        OE = OPTutils.PESVS(ojf,
                            lb,
                            ub,
                            para,
                            initstate=pickle.load(open(fname, 'rb')))
    else:
        print "fresh start"
        OE = OPTutils.PESVS(ojf, lb, ub, para)

    if sum(OE.C) >= b:
        print "no further steps needed"
        k = 0
        while sum(OE.C[:k]) < b:

            k += 1
        state = [
            OE.X[:k, :], OE.Y[:k, :], OE.S[:k, :], OE.D[:k], OE.R[:k, :],
            OE.C[:k], OE.T[:k], OE.Tr[:k], OE.Ymin[:k], OE.Xmin[:k, :],
            OE.Yreg[:k, :], OE.Rreg[:k, :]
        ]
    else:
        pbar = tqdm(total=(b - sum(OE.C)))
        while sum(OE.C) < b and len(OE.C) < overridemaxsteps:
            print "used {0} of {1} eval budget".format(sum(OE.C), b)
            pbar.update(OE.C[-1])
            OE.step()
        state = [
            OE.X, OE.Y, OE.S, OE.D, OE.R, OE.C, OE.T, OE.Tr, OE.Ymin, OE.Xmin,
            OE.Yreg, OE.Rreg
        ]
        pickle.dump(state, open(fname, 'wb'))
    return state
Exemple #3
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def MLEFS(ojf, lb, ub, ki, s, b, fname):
    #use kernel ki and evaluate ojf with variance s at step for a budget b
    sp.random.seed(int(os.urandom(4).encode('hex'), 16))
    d = lb.size
    volper = 1e-8
    ninit = 10
    para = [ki[0], ki[1], ki[2], volper, s, ninit]
    if os.path.exists(fname):
        print "starting from " + str(fname)
        OE = OPTutils.EIMLE(ojf,
                            lb,
                            ub,
                            para,
                            initstate=pickle.load(open(fname, 'rb')))
    else:
        print "fresh start"
        OE = OPTutils.EIMLE(ojf, lb, ub, para)
    if sum(OE.C) >= b:
        print "no further steps needed"
        k = 0
        while sum(OE.C[:k]) < b:
            k += 1
        state = [
            OE.X[:k, :], OE.Y[:k, :], OE.S[:k, :], OE.D[:k], OE.R[:k, :],
            OE.C[:k], OE.T[:k], OE.Tr[:k], OE.Ymin[:k], OE.Xmin[:k, :],
            OE.Yreg[:k, :], OE.Rreg[:k, :]
        ]
    else:
        while sum(OE.C) < b:
            print "used {0} of {1} eval budget".format(sum(OE.C), b)
            OE.step()
        state = [
            OE.X, OE.Y, OE.S, OE.D, OE.R, OE.C, OE.T, OE.Tr, OE.Ymin, OE.Xmin,
            OE.Yreg, OE.Rreg
        ]
        print "saving as " + str(fname)
        pickle.dump(state, open(fname, 'wb'))
    return state
Exemple #4
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#!/usr/bin/env python2
#encoding: UTF-8

# To change this license header, choose License Headers in Project Properties.
# To change this template file, choose Tools | Templates
# and open the template in the editor.

import scipy as sp
from matplotlib import pyplot as plt

n=12
def f(x):
    return 0.1*x**2-0.5*x+1.+0.1*(0.2*abs(x)+1)*sp.random.normal()
r=[]
y=[]
for i in xrange(n):
    r.append(sorted(sp.random.uniform(0,10,40)))
    y.append(map(f,r[-1]))
    plt.plot(r[-1],y[-1],'c')

import OPTutils

X,Y,lb,ub = OPTutils.mergelines(r,y)
    
plt.fill_between(X,lb,ub,facecolor='lightgreen')
plt.plot(X,Y)
plt.show()
Exemple #5
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import sys

sys.path.append("./..")
import OPTutils
import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import ESutils
from tqdm import tqdm, tqdm_gui

runn = 60
d = 2
lb = sp.array([[-1.0] * d])
ub = sp.array([[1.0] * d])

[ojf, truexmin, trueymin] = OPTutils.gensquexpdraw(d, sp.array([-1.0] * d), sp.array([1.0] * d))


kindex = GPdc.SQUEXP

mprior = sp.array([0.0] + [-1.0] * d)
sprior = sp.array([1.0] * (d + 1))
maxf = 4000
volper = 1e-7
s = 1e-5
ninit = 10

# para = [kindex,hlb,hub,maxf,s,ninit]
para = [kindex, mprior, sprior, volper, s, ninit]

Exemple #6
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import GPdc
import OPTutils
import search
import os
import pickle
print 'start'
print 'start'
d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
pwr = 0.1
def cfn(s):
    ##print s
    #print 'cfn'+str(((1e-6)/s)**pwr)
    return ((1e-6)/s)**pwr
ojf = OPTutils.genbranin(cfn=cfn)
braninmin = 0.39788735772973816
kindex = GPdc.MAT52
prior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
kernel = [kindex,prior,sprior]
nreps = 4
bd = 15.
slist = [1e3,1e-3]
print 'start'
f,a_ = plt.subplots(2)
fnull,anull=plt.subplots(1)
a = sp.hstack([anull,a_])
import os
tmp=[]
fc = ['lightblue','lightgreen']
Exemple #7
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import GPdc
import OPTutils
import search
import os
import pickle
print 'start'
print 'start'
d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
pwr = 0.3
def cfn(s):
    #print s
    #print 'cfn'+str(((1e-6)/s)**pwr)
    return ((1e-6)/s)**pwr
ojf = OPTutils.genbranin(cfn=cfn)
braninmin = 0.39788735772973816
kindex = GPdc.MAT52
prior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
kernel = [kindex,prior,sprior]


nreps = 2
bd = 10
slist = [1e-4, 1e-5,1e-6,1e-7]
plist = [0.1,0.2,0.3,0.4,0.5,0.6]
print 'start'
lines = [[[],[],[],[]] for s in slist]
pline = [[],[],[],[]]
for l,pwr in enumerate(plist):
Exemple #8
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# To change this license header, choose License Headers in Project Properties.
# To change this template file, choose Tools | Templates
# and open the template in the editor.

import OPTutils
import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import ESutils

d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
[ojf,truexmin] = OPTutils.gensquexpdraw(d,sp.array([-1.]*d),sp.array([1.]*d),ignores=1e-3)

O = OPTutils.opt(ojf,lb,ub)
for i in xrange(20):
    O.step()


para = dict()
para['kindex'] = GPdc.SQUEXPCS
para['mprior'] = sp.array([0.]+[-1.]*d+[-2.])
para['sprior'] = sp.array([1.]*(d+1)+[2.])
#para['s'] = 1e-6
para['ninit'] = 10
#para['maxf'] = 2500
para['volper'] = 1e-7
para['DH_SAMPLES'] = 8
para['DM_SAMPLES'] = 8
para['DM_SUPPORT'] = 400
Exemple #9
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import DIRECT


seed=123
sp.random.seed(seed)
d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
sl=0.
su=1.
sfn = lambda x:1e-8
fls = 1.0
sls = (su-sl)*fls
dcc=1.0
cfn = lambda x: sp.exp(-dcc*x.flatten()[0])
[ojf,truexmin,ymin] = OPTutils.gensquexpIPdraw(d,lb,ub,sl,su,sfn,sls,cfn)

#what are the best mins in planes away from true?
def planemin(xp):
    def dirwrap(x,y):
        z,c = ojf(sp.hstack([xp,x]),-1,[sp.NaN],override=True)
        return (z,0)
    [xm,ym,ierror] = DIRECT.solve(dirwrap,lb,ub,user_data=[], algmethod=1, maxf=80000, logfilename='/dev/null')
    ye,c = ojf(sp.hstack([0.,xm]),-1,[sp.NaN],override=True)
    r=ye-ymin
    return [xm,ym,ye,r]

plane00 = planemin(0.)
plane025 = planemin(0.25)
plane05 = planemin(0.5)
plane001 = planemin(0.01)
Exemple #10
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import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import OPTutils
import search
import os
import pickle


d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
pwr = 0.2
cfn = lambda s:((1e-6)/s)**pwr
ojf = OPTutils.gencamel(cfn=cfn)
camelmin = -1.031628453489877
kindex = GPdc.MAT52
prior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
kernel = [kindex,prior,sprior]
nreps = 8
bd = 15
slist = [1e-4,1e-6,1e-8]
f,a = plt.subplots(3)

for s in slist:
    
    names = ["../cache/camelnoise/EIMLE_"+str(int(100*sp.log10(s)))+"_"+str(pwr)+"_"+str(i)+".p" for i in xrange(nreps)]
    results = search.multiMLEFS(ojf,lb,ub,kernel,s,bd,names)
    yr = [r[11].flatten() for r in results]
Exemple #11
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import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import OPTutils
import search
import os
import pickle


d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
pwr = 0.2
cfn = lambda s:((1e-6)/s)**pwr
ojf = OPTutils.genbanana(cfn=cfn)
kindex = GPdc.MAT52
prior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
kernel = [kindex,prior,sprior]
nreps = 5
bd = 15
slist = [1e-4,1e-6,1e-8]
f,a = plt.subplots(2)

for s in slist:
    
    names = ["../cache/rosennoise/EIMLE_"+str(int(100*sp.log10(s)))+"_"+str(pwr)+"_"+str(i)+".p" for i in xrange(nreps)]
    results = search.multiMLEFS(ojf,lb,ub,kernel,s,bd,names)
    yr = [r[11].flatten() for r in results]
    C = results[0][5]
Exemple #12
0
lb = sp.array([[-1.] * d])
ub = sp.array([[1.] * d])
[ojf, truexmin] = [lambda x, s, d: [-1., -1.], None]

para = dict()
para['kindex'] = GPdc.SQUEXP
para['mprior'] = sp.array([0.] + [-1.] * d)
para['sprior'] = sp.array([1.] * (d + 1))
para['s'] = 1e-6
para['ninit'] = 10
#para['maxf'] = 2500
para['volper'] = 1e-6
para['DH_SAMPLES'] = 8
para['DM_SAMPLES'] = 8
para['DM_SUPPORT'] = 400
para['DM_SLICELCBPARA'] = 0.025
para['SUPPORT_MODE'] = ESutils.SUPPORT_SLICEPM
OP = OPTutils.PESFS(ojf, lb, ub, para)

import pickle
[OP.X, OP.Y, OP.S, OP.D, OP.R, OP.C, OP.T, OP.Tr,
 OP.Ymin] = pickle.load(open('state.p', 'rb'))

OP.pcs()
#OP.step()
plt.figure(1)
#plt.plot(OP.R[-1,0],OP.R[-1,1],'r.')
plt.plot(OP.X[:, 0], OP.X[:, 1], 'b.')

plt.show()
Exemple #13
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# To change this license header, choose License Headers in Project Properties.
# To change this template file, choose Tools | Templates
# and open the template in the editor.

import OPTutils
import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import ESutils
from tqdm import tqdm, tqdm_gui
import copy
runn = 10
d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
[ojf,truexmin,trueymin] = OPTutils.gensquexpdraw(d,sp.array([-1.]*d),sp.array([1.]*d),ignores=1e-9)
print "truexmin "+ str(truexmin)
f,a = plt.subplots(8)
#ojf = OPTutils.banana
#truexmin=OPTutils.bananamin
#trueymin=0.
O = OPTutils.opt(ojf,lb,ub)
for i in tqdm(xrange(10)):
    O.step()
initstate = copy.deepcopy([O.X,O.Y,O.S,O.D,O.R,O.C,O.T,O.Tr,O.Ymin,O.Xmin,O.Yreg,O.Rreg])

    
kindex = GPdc.MAT52
mprior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
volper=1e-8
Exemple #14
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# To change this template file, choose Tools | Templates
# and open the template in the editor.
import sys
sys.path.append("./..")
import OPTutils
import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import ESutils
from tqdm import tqdm, tqdm_gui
runn = 60
d = 2
lb = sp.array([[-1.] * d])
ub = sp.array([[1.] * d])

[ojf, truexmin, trueymin] = OPTutils.gensquexpdraw(d, sp.array([-1.] * d),
                                                   sp.array([1.] * d))

kindex = GPdc.SQUEXP

mprior = sp.array([0.] + [-1.] * d)
sprior = sp.array([1.] * (d + 1))
maxf = 4000
volper = 1e-7
s = 1e-5
ninit = 10

#para = [kindex,hlb,hub,maxf,s,ninit]
para = [kindex, mprior, sprior, volper, s, ninit]

OE = OPTutils.EIMLE(ojf, lb, ub, para)
Exemple #15
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print 'start'
print 'start'
d = 2
lb = sp.array([[-1.] * d])
ub = sp.array([[1.] * d])
pwr = 0.3


def cfn(s):
    #print s
    #print 'cfn'+str(((1e-6)/s)**pwr)
    return ((1e-6) / s)**pwr


ojf = OPTutils.genbranin(cfn=cfn)
braninmin = 0.39788735772973816
kindex = GPdc.MAT52
prior = sp.array([0.] + [-1.] * d)
sprior = sp.array([1.] * (d + 1))
kernel = [kindex, prior, sprior]

nreps = 2
bd = 10
slist = [1e-4, 1e-5, 1e-6, 1e-7]
plist = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6]
print 'start'
lines = [[[], [], [], []] for s in slist]
pline = [[], [], [], []]
for l, pwr in enumerate(plist):
    f, a = plt.subplots(len(slist) + 1)
Exemple #16
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sprior = sp.array([1.]*(d+2)+[2.])
kernel = [kindex,prior,sprior]

fnames = ["../cache/fith/PI{}.p".format(i) for i in xrange(5)]
statesPI = search.multiPESIPS(ojfa,lb,ub,kernel,10,fnames)


x=[]
y=[]
for stateEI in statesEI:
    a[2].plot(stateEI[11],'r')
    #a[3].plot([sum(stateEI[5][:i]) for i in xrange(len(stateEI[5]))],stateEI[11],'r')
    x.append([sum(stateEI[5][:i]) for i in xrange(len(stateEI[5]))])
    y.append(stateEI[11].flatten())
    print 'reccomended under EI: {} : {}'.format([10**i for i in stateEI[4][-1]],ojf(stateEI[4][-1],None,None)[0])
X_,Y_,lb_,ub_ = OPTutils.mergelines(x,y)
a[3].fill_between(X_,lb_,ub_,facecolor='lightcoral',edgecolor='lightcoral',alpha=0.5)
a[3].plot(X_,Y_,'r')

x=[]
y=[]
for statePE in statesPE:
    a[2].plot(statePE[11],'b')
    #a[3].plot([sum(statePE[5][:i]) for i in xrange(len(statePE[5]))],statePE[11],'b')
    x.append([sum(statePE[5][:i]) for i in xrange(len(statePE[5]))])
    y.append(statePE[11].flatten())
    print 'reccomended under PE: {} : {}'.format([10**i for i in statePE[4][-1]],ojf(statePE[4][-1],None,None)[0])
    
X_,Y_,lb_,ub_ = OPTutils.mergelines(x,y)
a[3].fill_between(X_,lb_,ub_,facecolor='lightblue',edgecolor='lightblue',alpha=0.5)
a[3].plot(X_,Y_,'b')
Exemple #17
0
#encoding: UTF-8

# To change this license header, choose License Headers in Project Properties.
# To change this template file, choose Tools | Templates
# and open the template in the editor.

import scipy as sp
from matplotlib import pyplot as plt

n = 12


def f(x):
    return 0.1 * x**2 - 0.5 * x + 1. + 0.1 * (0.2 * abs(x) +
                                              1) * sp.random.normal()


r = []
y = []
for i in xrange(n):
    r.append(sorted(sp.random.uniform(0, 10, 40)))
    y.append(map(f, r[-1]))
    plt.plot(r[-1], y[-1], 'c')

import OPTutils

X, Y, lb, ub = OPTutils.mergelines(r, y)

plt.fill_between(X, lb, ub, facecolor='lightgreen')
plt.plot(X, Y)
plt.show()
Exemple #18
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kernel = [kindex, prior, sprior]

fnames = ["../cache/fith/PI{}.p".format(i) for i in xrange(5)]
statesPI = search.multiPESIPS(ojfa, lb, ub, kernel, 10, fnames)

x = []
y = []
for stateEI in statesEI:
    a[2].plot(stateEI[11], 'r')
    #a[3].plot([sum(stateEI[5][:i]) for i in xrange(len(stateEI[5]))],stateEI[11],'r')
    x.append([sum(stateEI[5][:i]) for i in xrange(len(stateEI[5]))])
    y.append(stateEI[11].flatten())
    print 'reccomended under EI: {} : {}'.format(
        [10**i for i in stateEI[4][-1]],
        ojf(stateEI[4][-1], None, None)[0])
X_, Y_, lb_, ub_ = OPTutils.mergelines(x, y)
a[3].fill_between(X_,
                  lb_,
                  ub_,
                  facecolor='lightcoral',
                  edgecolor='lightcoral',
                  alpha=0.5)
a[3].plot(X_, Y_, 'r')

x = []
y = []
for statePE in statesPE:
    a[2].plot(statePE[11], 'b')
    #a[3].plot([sum(statePE[5][:i]) for i in xrange(len(statePE[5]))],statePE[11],'b')
    x.append([sum(statePE[5][:i]) for i in xrange(len(statePE[5]))])
    y.append(statePE[11].flatten())
Exemple #19
0
print 'start'
print 'start'
d = 2
lb = sp.array([[-1.] * d])
ub = sp.array([[1.] * d])
pwr = 0.1


def cfn(s):
    ##print s
    #print 'cfn'+str(((1e-6)/s)**pwr)
    return ((1e-6) / s)**pwr


ojf = OPTutils.genbranin(cfn=cfn)
braninmin = 0.39788735772973816
kindex = GPdc.MAT52
prior = sp.array([0.] + [-1.] * d)
sprior = sp.array([1.] * (d + 1))
kernel = [kindex, prior, sprior]
nreps = 4
bd = 15.
slist = [1e3, 1e-3]
print 'start'
f, a_ = plt.subplots(2)
fnull, anull = plt.subplots(1)
a = sp.hstack([anull, a_])
import os

tmp = []
Exemple #20
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import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import OPTutils
import search
import os
import pickle


d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
pwr = 0.2
cfn = lambda s:((1e-6)/s)**pwr
ojf = OPTutils.gencamel(cfn=cfn)
camelmin = -1.031628453489877
kindex = GPdc.MAT52
prior = sp.array([0.]+[-1.]*d)
sprior = sp.array([1.]*(d+1))
kernel = [kindex,prior,sprior]
nreps = 12
bd = 100
s = 1e-6
f,a = plt.subplots(2)

names = ["../cache/camelrecc/EIMLE_"+str(int(100*sp.log10(s)))+"_"+str(pwr)+"_"+str(i)+".p" for i in xrange(nreps)]
results = search.multiMLEFS(ojf,lb,ub,kernel,s,bd,names)

C = results[0][5]
Exemple #21
0
sys.path.append("./..")
import OPTutils
import scipy as sp
from matplotlib import pyplot as plt
import GPdc
import ESutils
from tqdm import tqdm, tqdm_gui
d=2
lb = sp.array([[-1.]*d])
ub = sp.array([[1.]*d])
sl=0.
su=1.
sfn = lambda x:1e-8
sls = (su-sl)*0.75
cfn = lambda x: sp.exp(-0.5*x.flatten()[0])
[ojf,truexmin,ymin] = OPTutils.gensquexpIPdraw(d,lb,ub,sl,su,sfn,sls,cfn)

print truexmin
print ojf(sp.array([sl,truexmin[0],truexmin[1]]),1e-8,[sp.NaN])
u = sp.empty(100)
v = sp.empty(100)
w = sp.empty(100)
sup = sp.linspace(-1,1,100)
spp = sp.linspace(sl,su,100)
f,a = plt.subplots(3)
for i in xrange(100):
    u[i] = ojf(sp.array([sl,sup[i],truexmin[1]]),1e-8,[sp.NaN])[0]
    v[i] = ojf(sp.array([sl,truexmin[0],sup[i]]),1e-8,[sp.NaN])[0]
    w[i] = ojf(sp.array([spp[i],truexmin[0],truexmin[1]]),1e-8,[sp.NaN])[0]
a[0].plot(sup,u)
a[1].plot(sup,v)