In [1]:
import numpy as np
#from scipy.optimize import newton as nt
from scipy import optimize as op
import matplotlib.pyplot as plt
from george.kernels import ExpSquaredKernel #For Gaussian Processes
from george.kernels import ExpSine2Kernel #For Gaussian Processes
import george
%matplotlib inline
import emcee
import corner
In [36]:
#First data
#data = astro.io.ascii.read('Inputs/data/rvs_0001.txt', names=['time','vel','err'])
data = np.loadtxt('Inputs/data/rvs_0001.txt')
obser = [data[:,0],data[:,1],data[:,2]]
In [3]:
x = data[:,0]
y = data[:,1]
yerr = data[:,2]
# First illustrate basic pyplot interface, using defaults where possible.
plt.figure(figsize=(15,8))
#plt.xlabel('teste')
plt.xlabel('Time (s)', fontsize=20)
plt.ylabel('Star Radial velocity (m/s)', fontsize=20)
plt.errorbar(x, y, yerr=yerr,fmt='o')
plt.tick_params(labelsize=20)
plt.title("Data0001",fontsize=30)
#plt.plot(x,y,'o')
plt.show()
In [102]:
#From https://en.wikipedia.org/wiki/True_anomaly and
#http://exoplanets.astro.yale.edu/workshop/EPRV/Bibliography_files/Radial_Velocity.pdf
#Returns the mean anomaly in the 2 pi interval
def mean_anomaly(M0,t,t0,tau):
return (2 * np.pi * (t-t0) / tau + M0 ) % (2*np.pi)
#Returns the [cos(f),sin(f)], where f is the true anomaly.
def real_anomaly(M,e):
# if M-e<0:
# ec_anol = M+op.newton(lambda x: e * np.sqrt(1-(np.cos(M+x))**2)-x,e*np.sin(M),maxiter=100)
# elif M+e>2*np.pi:
# ec_anol = M+op.newton(lambda x: -e * np.sqrt(1-(np.cos(M+x))**2)-x,e*np.sin(M),maxiter=100)
# else:
# ec_anol = M+op.newton(lambda x: e * np.sin(M+x)-x,e*np.sin(M),lambda y: e * np.cos(M+y)-1)
ec_anol = M+op.newton(lambda x: e * np.sin(M+x)-x,e*np.sin(M),lambda y: e * np.cos(M+y)-1,\
fprime2=lambda z: -e * np.sin(M+z))
return [(np.cos(ec_anol)-e)/(1-e * np.cos(ec_anol)),(np.sqrt(1-e**2) * np.sin(ec_anol))/(1-e*np.cos(ec_anol))]
def RV(t, K, p, e, w, M0, t0=0):
p = np.abs(p)
K = np.abs(K)
w = np.abs(w)%(2*np.pi)
e=np.abs(e)
if e>1: e=0.9
M0 = np.abs(M0)%(2*np.pi)
cosf, sinf = real_anomaly(mean_anomaly(M0,t,t0,p),e)
return K * (np.cos(w)*cosf-np.sin(w)*sinf+e*np.cos(w))
#Radial velocity for one planet
#Model for velocity function.
#Input: Planets parameters: Last 2 ate C and J. Five extra parameters for each planet
#Special thanks to Ben Nelson
def model(params, times, nPlanets):
planets = [params[i*5:i*5+5] for i in range(nPlanets)]
try:
p, K, e, w, M = np.transpose(planets)
except ValueError:
p, K, e, w, M = 99999999., 0., 0., 0., 0.
offset = params[5*nPlanets]
jitter = params[-1]
mod = np.zeros(len(times))
for j in range(len(times)):
mod[j] += sum([RV(times[j], K[i], p[i], e[i], w[i], M[i])\
for i in range(nPlanets)]) + offset
return mod
#Gaussian Processes kernel
var, length_e, length_s, period = 3.0, 50., 0.5, 20.
gamma = 1/(2.*length_s*length_s)
kernel = var * ExpSquaredKernel(length_e*length_e) * ExpSine2Kernel(gamma=gamma, log_period=np.log(period))
### Define the likelihood function ###
#Thanks to Ben.
def lnlike(params, obs, nPlanets):
times, rvs, sigs = obs
jit = params[-1]
sigs_plus_jitter = np.sqrt(sigs*sigs + jit*jit*np.ones(len(times)))
gp = george.GP(kernel)
gp.compute(times, sigs_plus_jitter)
# return gp.lnlikelihood(rvs - model_1(params, times, nPlanets))
return gp.lnlikelihood(rvs - model(params, times, nPlanets))
In [10]:
plt.plot(np.arange(100),model([1.2, 1, 0.1, 3., 4., 1, 2],np.arange(100),1));
plt.title('Star velocity model (1 planet)')
plt.xlabel('Time')
plt.ylabel('Redshift velocity');
In [11]:
IC_0pl = [-0.72851, 2.08718] #Test parameters
lnlike(IC_0pl, [data[:,0],data[:,1],data[:,2]], 0) #Should be around -477
Out[11]:
-477.19968698126718
In [132]:
#Must define priors!
def log_prior( params, nPlanets):
value=0
planets = [params[i*5:i*5+5] for i in range(nPlanets)]
try:
p, K, e, w, M = np.transpose(planets)
except ValueError:
p, K, e, w, M = 99999999., 0., 0., 0., 0.
offset = params[5*nPlanets]
jitter = params[-1]
if (offset<-1000) or (offset>1000) or (jitter<0) or (jitter>99):
return -np.infty
value+=-np.log(2000) #offset priors
value+=-np.log(1+jitter)-np.log(np.log(100)) #Jitter prior
if nPlanets>0:
if ((K<=0) + (K>999) + (e<0) + (e>=1) + (w<0) + (w>2*np.pi) + (M<0) + (M>2*np.pi)).all():
return -np.infty
value+=-np.log(p)-np.log(np.log(10000/1.25)) #Period prior
value+=-np.log(1+K)-np.log(np.log(1000)) #K prior
value+=-np.log(2*np.pi**2) #w, M priors
# if (1-np.exp(-e**2/0.08)==0):
# value+=np.log(e/0.04)-e**2/0.08-np.log((e**2/0.08)) #e prior
# else:
value+=np.log(e/0.04)-e**2/0.08-np.log(-np.expm1(-e**2/0.08)) #e prior
return value.sum()
def log_posterior(params, obs, nPlanets):
return log_prior(params, nPlanets) + lnlike(params, obs, nPlanets)
In [54]:
log_posterior([102,30,0.5,1.1,np.pi,-1,-2],obser,1)
Out[54]:
-inf
In [105]:
def mini(a,b,c):
value = -log_posterior(a,b,c)
if np.isfinite(value):
return value
else:
return 10**10
sol0 = op.minimize(mini,IC_0pl,args=(obser,0))
print(sol0)
fun: 487.454531660062
hess_inv: array([[ 0.35446769, -0.0022139 ],
[-0.0022139 , 0.01958826]])
jac: array([ 0.00000000e+00, 3.81469727e-06])
message: 'Optimization terminated successfully.'
nfev: 28
nit: 4
njev: 7
status: 0
success: True
x: array([-0.72023413, 2.08317069])
In [106]:
IC_1pl = [500, 4, 1e-20, 0, 0,0,1] #Test parameters
mini(IC_1pl,obser,1)
Out[106]:
528.70663220787628
In [107]:
sol1 = op.minimize(mini,IC_1pl,args=(obser,1),method='Powell')
print(sol1)
direc: array([[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.00000000e+00],
[ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
0.00000000e+00],
[ 5.80322207e+00, 6.84341413e-03, 0.00000000e+00,
2.84644162e-02, 3.90629533e-12, -2.63665910e-02,
-1.44862947e-03]])
fun: 452.88576071295586
message: 'Optimization terminated successfully.'
nfev: 461
nit: 3
status: 0
success: True
x: array([ 3.70915909e+02, 1.05666623e+00, 1.00000000e-20,
3.15672189e-01, 1.57427744e-10, -4.82983286e-01,
2.07941042e+00])
In [133]:
IC_2pl = np.concatenate([[500, 4, 1e-20, 0, 0],sol1['x']]) #Test parameters
mini(IC_2pl,obser,2)
Out[133]:
442.80393234894291
In [134]:
sol2 = op.minimize(mini,IC_2pl,args=(obser,2),method='Powell')
print(sol2)
/home/arthur/anaconda2/envs/p36/lib/python3.6/site-packages/ipykernel_launcher.py:20: RuntimeWarning: invalid value encountered in log
/home/arthur/anaconda2/envs/p36/lib/python3.6/site-packages/ipykernel_launcher.py:25: RuntimeWarning: invalid value encountered in log
direc: array([[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.00000000e+00],
[ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
1.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 1.00000000e+00, 0.00000000e+00,
0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
0.00000000e+00, 0.00000000e+00, 1.00000000e+00,
0.00000000e+00],
[ 5.80322207e+00, 6.84341413e-03, 0.00000000e+00,
2.84644162e-02, 3.90629533e-12, -2.63665910e-02,
-1.44862947e-03]])
fun: 452.88576071295586
message: 'Optimization terminated successfully.'
nfev: 461
nit: 3
status: 0
success: True
x: array([ 3.70915909e+02, 1.05666623e+00, 1.00000000e-20,
3.15672189e-01, 1.57427744e-10, -4.82983286e-01,
2.07941042e+00])
In [138]:
IC_3pl = np.concatenate([[500, 4, 1e-20, 0, 0],sol2['x']]) #Test parameters
mini(IC_3pl,obser,3)
Out[138]:
388.59337657744152
In [139]:
sol3 = op.minimize(mini,IC_3pl,args=(obser,3),method='Powell')
print(sol3)
/home/arthur/anaconda2/envs/p36/lib/python3.6/site-packages/ipykernel_launcher.py:20: RuntimeWarning: invalid value encountered in log
/home/arthur/anaconda2/envs/p36/lib/python3.6/site-packages/ipykernel_launcher.py:25: RuntimeWarning: invalid value encountered in log
direc: array([[ 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1.,
0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
1., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 1., 0., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 1., 0.],
[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 1.]])
fun: 350.32224336493988
message: 'Optimization terminated successfully.'
nfev: 3003
nit: 7
status: 0
success: True
x: array([ 5.16370020e+02, -1.00000000e+00, 1.00000000e-20,
-5.91639646e-12, -6.74745985e-13, 3.48154671e+02,
-1.00000000e+00, 1.00000000e-20, 1.83403791e-01,
4.33820758e-04, 2.17200194e+02, 1.01240817e+00,
1.00000000e-20, 2.96029162e-01, 1.21412629e-03,
-8.47669348e-01, 2.05499848e+00])
In [140]:
x = data[:,0]
xm = np.arange(0,data[-1,0],data[-1,0]/1000)
y = data[:,1]
y0 = model(sol0['x'],xm,0)
y1 = model(sol1['x'],xm,1)
y2 = model(sol2['x'],xm,2)
y3 = model(sol3['x'],xm,3)
yerr = data[:,2]
# First illustrate basic pyplot interface, using defaults where possible.
plt.figure(figsize=(15,8))
#plt.xlabel('teste')
plt.xlabel('Time (s)', fontsize=20)
plt.ylabel('Star Radial velocity (m/s)', fontsize=20)
plt.errorbar(x, y, yerr=yerr,fmt='o')
plt.plot(xm,y0)
plt.plot(xm,y1)
plt.plot(xm,y2)
plt.plot(xm,y3)
plt.tick_params(labelsize=20)
plt.title("Data0001",fontsize=30)
#plt.plot(x,y,'o')
plt.show()
In [161]:
ndim = 2 # number of parameters in the model
nwalkers = 50 # number of MCMC walkers
nburn = 1000 # "burn-in" period to let chains stabilize
nsteps = 2000 # number of MCMC steps to take
#starting guesses. Something around the IC_0pl
rand_offset = 3*(np.random.rand(nwalkers)-0.5)
rand_jitter = 2*(np.random.rand(nwalkers))
#rand_jitter = np.expm1(np.random.rand(nwalkers)*np.log(100))
starting_guesses = np.column_stack((rand_offset,rand_jitter))
In [162]:
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=(obser,0))
sampler.run_mcmc(starting_guesses, nsteps);
#sample = sampler.chain # shape = (nwalkers, nsteps, ndim)
#sample_join = [sampler.chain[:, nburn:, i].ravel() for i in range(ndim)] # discard burn-in points
Out[162]:
(array([[-0.77531348, 2.18803002],
[-0.08201122, 1.91908872],
[-1.14226818, 2.08128887],
[-1.6454635 , 2.24530195],
[-0.80543169, 2.12255858],
[-0.11695722, 2.09194853],
[ 0.57137351, 2.07763764],
[-0.4671037 , 2.14921612],
[-1.30902158, 2.15285452],
[-0.44833382, 1.92755456],
[-0.30548696, 1.78575719],
[ 0.3502451 , 2.08997356],
[-1.29737341, 2.13973026],
[-0.5300814 , 2.16429224],
[-0.64333288, 2.02417777],
[-1.44680233, 1.95169867],
[-1.42211586, 2.27801838],
[-0.18661804, 2.18758503],
[-0.00473699, 2.19701521],
[-0.70194903, 2.14291161],
[-1.63122367, 1.81417557],
[-0.44623351, 2.0553303 ],
[-1.15030566, 1.99368526],
[-1.02617781, 2.10369021],
[-0.99431191, 2.01912374],
[-1.52752739, 2.20575567],
[-0.4699519 , 1.97026166],
[-0.33640844, 2.05562957],
[-1.0654371 , 2.36252715],
[-0.07212359, 2.25329243],
[-0.81584392, 1.95634673],
[-1.84351301, 2.00577825],
[-0.921945 , 2.20979475],
[-1.24106327, 2.25388127],
[-1.12241813, 2.15622079],
[-1.50201218, 2.17450463],
[-1.69894972, 1.91790248],
[-0.94331984, 2.42768494],
[-0.96601551, 2.49272978],
[-1.73267524, 2.13695131],
[-1.76300274, 2.26886138],
[-1.56053381, 2.14296524],
[-0.45970583, 2.05872139],
[ 0.22184959, 2.12406876],
[ 0.15497733, 1.95828441],
[-0.44158069, 1.96892848],
[-0.83179267, 2.14365982],
[-0.55232604, 2.23510514],
[-0.37087942, 2.27065577],
[-1.39379592, 2.19797348]]),
array([-487.72193684, -488.76003305, -487.70634338, -489.20869721,
-487.50241572, -487.97138628, -489.80930708, -487.6569288 ,
-488.04635995, -488.22207774, -490.34589936, -489.07430198,
-487.99051935, -487.67058935, -487.55329292, -488.71762616,
-488.96464849, -488.13190347, -488.50402978, -487.54349265,
-490.90205868, -487.57848082, -487.9458036 , -487.59497371,
-487.67557699, -488.6895631 , -487.88169646, -487.67939438,
-489.3034447 , -488.74340387, -487.91276321, -489.43920349,
-487.88507923, -488.48057349, -487.80255509, -488.48842331,
-489.65167257, -490.02208721, -490.95716685, -488.9470379 ,
-489.69346564, -488.51690356, -487.5640265 , -488.75543532,
-488.94249169, -487.91050392, -487.56018112, -488.04391897,
-488.45367433, -488.37839451]),
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In [164]:
labeles = ['C','J']
_ = corner.corner(sampler.flatchain,truths=sol0['x'],labels=labeles)
In [17]:
ndim = 2 # number of parameters in the model
nwalkers = 50 # number of MCMC walkers
nburn = 1000 # "burn-in" period to let chains stabilize
nsteps = 2000 # number of MCMC steps to take
nburn = 0 #test
nsteps = 20 #test
#starting guesses. Something around the IC_0pl
rand_offset = IC_0pl[0]+2*(np.random.rand(nwalkers)-0.5)
rand_jitter = IC_0pl[1]+4*(np.random.rand(nwalkers)-0.5)
starting_guesses = np.column_stack((rand_offset,rand_jitter))
In [18]:
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[[data[:,0],data[:,1],data[:,2]],0])
sampler.run_mcmc(starting_guesses, nsteps)
sample = sampler.chain # shape = (nwalkers, nsteps, ndim)
sample0 = sampler.chain[:, nburn:, 0].ravel() # discard burn-in points
sample1 = sampler.chain[:, nburn:, 1].ravel() # discard burn-in points
In [20]:
# plot a histogram of the sample
plt.hist(sample0, bins=50, histtype="stepfilled", alpha=0.3, normed=True)
plt.hist(sample1, bins=50, histtype="stepfilled", alpha=0.3, normed=True);
In [22]:
fig, ax = plt.subplots(2, sharex=True)
test = [sample0,sample1]
for i in range(2):
ax[i].plot(test[i].T, '-k', alpha=0.2);
In [23]:
fig, axs = plt.subplots(1, 3, sharex=True, sharey=True)
for i in range(axs.size):
axs[i].errorbar(sampler.chain[:, i, 0], sampler.chain[:, i, 1], fmt="-o", alpha=0.5, c='k');
In [24]:
from corner import hist2d
fig, axs = plt.subplots(1, 2, sharex=True, sharey=True)
axs[0].hist2d(sampler.flatchain[:, 0], sampler.flatchain[:, 1], 30, cmap="ocean_r");
hist2d(sampler.flatchain[:, 0], sampler.flatchain[:, 1], ax=axs[1])
axs[0].set_xlabel('C', fontsize=20); axs[0].set_ylabel('J', fontsize=20)
axs[1].set_xlabel('C', fontsize=20);
plt.ylim([0,5])
fig.tight_layout();
In [25]:
# SOLUTION
thechain = sampler.flatchain
print('Mean values:', np.mean(thechain, axis=0))
print('Standard deviation:', np.std(thechain, axis=0))
Mean values: [-0.65641624 1.72968661]
Standard deviation: [ 0.64206305 1.12894944]
In [26]:
plt.plot(data[:,0],model(np.mean(thechain, axis=0),data[:,0],0))
plt.errorbar(data[:,0], data[:,1], yerr=yerr,fmt='o');
plt.title('Star radial velocity',fontsize=25)
plt.xlabel('Time (days)',fontsize=20)
plt.ylabel('Velocity (m/s)',fontsize=20);
In [28]:
ndim = 7 # number of parameters in the model
nwalkers = 50 # number of MCMC walkers
nburn = 1000 # "burn-in" period to let chains stabilize
nsteps = 2000 # number of MCMC steps to take
nburn = 0 #test
nsteps = 20 #test
#starting guesses. Something around the IC_0pl
rand_offset = IC_0pl[0]+2*(np.random.rand(nwalkers)-0.5)
rand_jitter = IC_0pl[1]+4*(np.random.rand(nwalkers)-0.5)
rand_K = 30 + 10*(np.random.rand(nwalkers)-0.5)
rand_p = 1000 + 100*(np.random.rand(nwalkers)-0.5)
rand_M0 = np.pi+(np.random.rand(nwalkers)-0.5)
rand_w = np.pi+(np.random.rand(nwalkers)-0.5)
rand_e = 0.5+0.3*(np.random.rand(nwalkers)-0.5)
starting_guesses = np.column_stack((rand_p,rand_K,rand_e,rand_w,rand_M0,rand_offset,rand_jitter))
In [42]:
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[[data[:,0],data[:,1],data[:,2]],1])
sampler.run_mcmc(starting_guesses, nsteps)
sample = sampler.chain # shape = (nwalkers, nsteps, ndim)
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