def exNew(exclude=sc.array([1,2,3,4]),
plotfilename='exNew.png',nburn=20000,nsamples=200000,
parsigma=[5,.075,.01,1,.1],dsigma=1.):
"""exMix1: solve the new exercise using MCMC sampling
Input:
exclude - ID numbers to exclude from the analysis (can be None)
plotfilename - filename for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
dsigma - divide uncertainties by this amount
Output:
plot
History:
2010-04-28 - Written - Bovy (NYU)
"""
sc.random.seed(1) #In the interest of reproducibility (if that's a word)
#Read the data
data= read_data('data_yerr.dat')
ndata= len(data)
if not exclude == None:
nsample= ndata- len(exclude)
else:
nsample= ndata
#First find the chi-squared solution, which we will use as an
#initial guess
#Put the data in the appropriate arrays and matrices
Y= sc.zeros(nsample)
X= sc.zeros(nsample)
A= sc.ones((nsample,2))
C= sc.zeros((nsample,nsample))
yerr= sc.zeros(nsample)
jj= 0
for ii in range(ndata):
if not exclude == None and sc.any(exclude == data[ii][0]):
pass
else:
Y[jj]= data[ii][1][1]
X[jj]= data[ii][1][0]
A[jj,1]= data[ii][1][0]
C[jj,jj]= data[ii][2]**2./dsigma**2.
yerr[jj]= data[ii][2]/dsigma
jj= jj+1
#Now compute the best fit and the uncertainties
bestfit= sc.dot(linalg.inv(C),Y.T)
bestfit= sc.dot(A.T,bestfit)
bestfitvar= sc.dot(linalg.inv(C),A)
bestfitvar= sc.dot(A.T,bestfitvar)
bestfitvar= linalg.inv(bestfitvar)
bestfit= sc.dot(bestfitvar,bestfit)
initialguess= sc.array([bestfit[0],bestfit[1],0.,sc.mean(Y),m.log(sc.var(Y))])#(m,b,Pb,Yb,Vb)
#With this initial guess start off the sampling procedure
initialX= objective(initialguess,X,Y,yerr)
currentX= initialX
bestX= initialX
bestfit= initialguess
currentguess= initialguess
naccept= 0
samples= []
samples.append(currentguess)
for jj in range(nburn+nsamples):
#Draw a sample from the proposal distribution
newsample= sc.zeros(5)
newsample[0]= currentguess[0]+stats.norm.rvs()*parsigma[0]
newsample[1]= currentguess[1]+stats.norm.rvs()*parsigma[1]
#newsample[2]= stats.uniform.rvs()
newsample[2]= currentguess[2]+stats.norm.rvs()*parsigma[2]
newsample[3]= currentguess[3]+stats.norm.rvs()*parsigma[3]
newsample[4]= currentguess[4]+stats.norm.rvs()*parsigma[4]
#Calculate the objective function for the newsample
newX= objective(newsample,X,Y,yerr)
#Accept or reject
#Reject with the appropriate probability
u= stats.uniform.rvs()
if u < m.exp(newX-currentX):
#Accept
currentX= newX
currentguess= newsample
naccept= naccept+1
if currentX > bestX:
bestfit= currentguess
bestX= currentX
samples.append(currentguess)
if double(naccept)/(nburn+nsamples) < .2 or double(naccept)/(nburn+nsamples) > .6:
print "Acceptance ratio was "+str(double(naccept)/(nburn+nsamples))
samples= sc.array(samples).T[:,nburn:-1]
print "Best-fit, overall"
print bestfit, sc.mean(samples[2,:]), sc.median(samples[2,:])
histmb,edges= sc.histogramdd(samples.T[:,0:2],bins=round(sc.sqrt(nsamples)/5.))
indxi= sc.argmax(sc.amax(histmb,axis=1))
indxj= sc.argmax(sc.amax(histmb,axis=0))
print "Best-fit, marginalized"
print edges[0][indxi-1], edges[1][indxj-1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi+1], edges[1][indxj+1]
#2D histogram
plot.bovy_print()
levels= special.erf(0.5*sc.arange(1,4))
#xrange=[edges[0][0],edges[0][-1]]
#yrange=[edges[1][0],edges[1][-1]]
xrange=[-120,120]
yrange=[1.5,3.2]
histmb,edges= sc.histogramdd(samples.T[:,0:2],
range=[[-120,120],[1.5,3.2]],
bins=(round(sc.sqrt(nsamples)/5.)/(edges[0][-1]-edges[0][0])*(xrange[1]-xrange[0]),
round(sc.sqrt(nsamples)/5.)/(edges[1][-1]-edges[1][0])*(yrange[1]-yrange[0])))
aspect=(xrange[1]-xrange[0])/(yrange[1]-yrange[0])
plot.bovy_dens2d(histmb.T,origin='lower',cmap='gist_yarg',
contours=True,cntrmass=True,
xrange=xrange,yrange=yrange,
levels=levels,
aspect=aspect,
xlabel=r'$b$',ylabel=r'$m$')
if dsigma == 1.:
plot.bovy_text(r'$\mathrm{using\ correct\ data\ uncertainties}$',
top_right=True)
else:
plot.bovy_text(r'$\mathrm{using\ data\ uncertainties\ /\ 2}$',
top_right=True)
if dsigma == 1.:
plot.bovy_end_print('exNew1a.png')
else:
plot.bovy_end_print('exNew2a.png')
#Data with MAP line and sampling
plot.bovy_print()
bestb= edges[0][indxi]
bestm= edges[1][indxj]
xrange=[0,300]
yrange=[0,700]
plot.bovy_plot(xrange,bestm*sc.array(xrange)+bestb,'k-',
xrange=xrange,yrange=yrange,
xlabel=r'$x$',ylabel=r'$y$',zorder=2)
errorbar(X,Y,yerr,color='k',marker='o',color='k',linestyle='None',zorder=1)
for ii in range(10):
#Random sample
ransample= sc.floor((stats.uniform.rvs()*nsamples))
ransample= samples.T[ransample,0:2]
bestb= ransample[0]
bestm= ransample[1]
plot.bovy_plot(xrange,bestm*sc.array(xrange)+bestb,
overplot=True,xrange=xrange,yrange=yrange,
xlabel=r'$x$',ylabel=r'$y$',color='0.75',zorder=1)
if dsigma == 1.:
plot.bovy_text(r'$\mathrm{using\ correct\ data\ uncertainties}$',
top_right=True)
else:
plot.bovy_text(r'$\mathrm{using\ data\ uncertainties\ /\ 2}$',
top_right=True)
if dsigma == 1.:
plot.bovy_end_print('exNew1b.png')
else:
plot.bovy_end_print('exNew2b.png')
#Pb plot
plot.bovy_print()
plot.bovy_hist(samples.T[:,2],color='k',bins=round(sc.sqrt(nsamples)/5.),
xlabel=r'$P_\mathrm{b}$',normed=True,histtype='step',
range=[0,1])
if dsigma == 1.:
plot.bovy_text(r'$\mathrm{using\ correct\ data\ uncertainties}$',
top_right=True)
else:
plot.bovy_text(r'$\mathrm{using\ data\ uncertainties\ /\ 2}$',
top_right=True)
if dsigma == 1.:
plot.bovy_end_print('exNew1c.png')
else:
plot.bovy_end_print('exNew2c.png')
return
def exMix1(
exclude=None,
plotfilenameA="exMix1a.png",
plotfilenameB="exMix1b.png",
plotfilenameC="exMix1c.png",
nburn=20000,
nsamples=1000000,
parsigma=[5, 0.075, 0.2, 1, 0.1],
dsigma=1.0,
bovyprintargs={},
sampledata=None,
):
"""exMix1: solve exercise 5 (mixture model) using MCMC sampling
Input:
exclude - ID numbers to exclude from the analysis (can be None)
plotfilename* - filenames for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
dsigma - divide uncertainties by this amount
Output:
plot
History:
2010-04-28 - Written - Bovy (NYU)
"""
sc.random.seed(-1) # In the interest of reproducibility (if that's a word)
# Read the data
data = read_data("data_yerr.dat")
ndata = len(data)
if not exclude == None:
nsample = ndata - len(exclude)
else:
nsample = ndata
# First find the chi-squared solution, which we will use as an
# initial guess
# Put the data in the appropriate arrays and matrices
Y = sc.zeros(nsample)
X = sc.zeros(nsample)
A = sc.ones((nsample, 2))
C = sc.zeros((nsample, nsample))
yerr = sc.zeros(nsample)
jj = 0
for ii in range(ndata):
if not exclude == None and sc.any(exclude == data[ii][0]):
pass
else:
Y[jj] = data[ii][1][1]
X[jj] = data[ii][1][0]
A[jj, 1] = data[ii][1][0]
C[jj, jj] = data[ii][2] ** 2.0 / dsigma ** 2.0
yerr[jj] = data[ii][2] / dsigma
jj = jj + 1
brange = [-120, 120]
mrange = [1.5, 3.2]
# This matches the order of the parameters in the "samples" vector
mbrange = [brange, mrange]
if sampledata is None:
sampledata = runSampler(X, Y, A, C, yerr, nburn, nsamples, parsigma, mbrange)
(histmb, edges, mbsamples, pbhist, pbedges) = sampledata
# Hack -- produce fake Pbad samples from Pbad histogram.
pbsamples = hstack([array([x] * N) for x, N in zip((pbedges[:-1] + pbedges[1:]) / 2, pbhist)])
indxi = sc.argmax(sc.amax(histmb, axis=1))
indxj = sc.argmax(sc.amax(histmb, axis=0))
print "Best-fit, marginalized"
print edges[0][indxi - 1], edges[1][indxj - 1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi + 1], edges[1][indxj + 1]
# 2D histogram
plot.bovy_print(**bovyprintargs)
levels = special.erf(0.5 * sc.arange(1, 4))
xe = [edges[0][0], edges[0][-1]]
ye = [edges[1][0], edges[1][-1]]
aspect = (xe[1] - xe[0]) / (ye[1] - ye[0])
plot.bovy_dens2d(
histmb.T,
origin="lower",
cmap=cm.gist_yarg,
interpolation="nearest",
contours=True,
cntrmass=True,
extent=xe + ye,
levels=levels,
aspect=aspect,
xlabel=r"$b$",
ylabel=r"$m$",
)
xlim(brange)
ylim(mrange)
plot.bovy_end_print(plotfilenameA)
# Data with MAP line and sampling
plot.bovy_print(**bovyprintargs)
bestb = edges[0][indxi]
bestm = edges[1][indxj]
xrange = [0, 300]
yrange = [0, 700]
plot.bovy_plot(
xrange,
bestm * sc.array(xrange) + bestb,
"k-",
xrange=xrange,
yrange=yrange,
xlabel=r"$x$",
ylabel=r"$y$",
zorder=2,
)
errorbar(X, Y, yerr, marker="o", color="k", linestyle="None", zorder=1)
for m, b in mbsamples:
plot.bovy_plot(
xrange,
m * sc.array(xrange) + b,
overplot=True,
xrange=xrange,
yrange=yrange,
xlabel=r"$x$",
ylabel=r"$y$",
color="0.75",
zorder=1,
)
plot.bovy_end_print(plotfilenameB)
# Pb plot
if not "text_fontsize" in bovyprintargs:
bovyprintargs["text_fontsize"] = 11
plot.bovy_print(**bovyprintargs)
plot.bovy_hist(
pbsamples,
bins=round(sc.sqrt(nsamples) / 5.0),
xlabel=r"$P_\mathrm{b}$",
normed=True,
histtype="step",
range=[0, 1],
edgecolor="k",
)
ylim(0, 4.0)
if dsigma == 1.0:
plot.bovy_text(r"$\mathrm{using\ correct\ data\ uncertainties}$", top_right=True)
else:
plot.bovy_text(r"$\mathrm{using\ data\ uncertainties\ /\ 2}$", top_left=True)
plot.bovy_end_print(plotfilenameC)
return sampledata
def ex17(exclude=sc.array([3]),plotfilename='ex17.png',
nburn=5000,nsamples=200000,
parsigma=[1,m.pi/200.,.1],
bovyprintargs={}):
"""ex17: solve exercise 17 by MCMC
Input:
exclude - ID numbers to exclude from the analysis
plotfilename - filename for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
Output:
plot
History:
2010-05-07 - Written - Bovy (NYU)
"""
#Read the data
data= read_data('data_allerr.dat',allerr=True)
ndata= len(data)
nsample= ndata- len(exclude)
#First find the chi-squared solution, which we will use as an
#initial gues
#Put the dat in the appropriate arrays and matrices
Y= sc.zeros(nsample)
X= sc.zeros(nsample)
A= sc.ones((nsample,2))
C= sc.zeros((nsample,nsample))
Z= sc.zeros((nsample,2))
yerr= sc.zeros(nsample)
ycovar= sc.zeros((2,nsample,2))#Makes the sc.dot easier
jj= 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
pass
else:
Y[jj]= data[ii][1][1]
X[jj]= data[ii][1][0]
Z[jj,0]= X[jj]
Z[jj,1]= Y[jj]
A[jj,1]= data[ii][1][0]
C[jj,jj]= data[ii][2]**2.
yerr[jj]= data[ii][2]
ycovar[0,jj,0]= data[ii][3]**2.
ycovar[1,jj,1]= data[ii][2]**2.
ycovar[0,jj,1]= data[ii][4]*m.sqrt(ycovar[0,jj,0]*ycovar[1,jj,1])
ycovar[1,jj,0]= ycovar[0,jj,1]
jj= jj+1
#Now compute the best fit and the uncertainties
bestfit= sc.dot(linalg.inv(C),Y.T)
bestfit= sc.dot(A.T,bestfit)
bestfitvar= sc.dot(linalg.inv(C),A)
bestfitvar= sc.dot(A.T,bestfitvar)
bestfitvar= linalg.inv(bestfitvar)
bestfit= sc.dot(bestfitvar,bestfit)
#Now sample
inittheta= m.acos(1./m.sqrt(1.+bestfit[1]**2.))
if bestfit[1] < 0.:
inittheta= m.pi- inittheta
initialguess= sc.array([bestfit[0]*m.cos(inittheta),inittheta,sc.log(1.)])#(m,b,logV)
#With this initial guess start off the sampling procedure
initialX= objective(initialguess,Z,ycovar)
currentX= initialX
bestX= initialX
bestfit= initialguess
currentguess= initialguess
naccept= 0
samples= []
samples.append(currentguess)
for jj in range(nburn+nsamples):
#Draw a sample from the proposal distribution
newsample= sc.zeros(3)
newsample[0]= currentguess[0]+stats.norm.rvs()*parsigma[0]
newsample[1]= currentguess[1]+stats.norm.rvs()*parsigma[1]
newsample[2]= currentguess[2]+stats.norm.rvs()*parsigma[2]
#Calculate the objective function for the newsample
newX= objective(newsample,Z,ycovar)
#Accept or reject
#Reject with the appropriate probability
u= stats.uniform.rvs()
try:
test= m.exp(newX-currentX)
except OverflowError:
test= 2.
if u < test:
#Accept
currentX= newX
currentguess= newsample
naccept= naccept+1
if currentX > bestX:
bestfit= currentguess
bestX= currentX
samples.append(currentguess)
if double(naccept)/(nburn+nsamples) < .5 or double(naccept)/(nburn+nsamples) > .8:
print "Acceptance ratio was "+str(double(naccept)/(nburn+nsamples))
samples= sc.array(samples).T[:,nburn:-1]
print "Best-fit, overall"
print bestfit, sc.mean(samples[2,:]), sc.median(samples[2,:])
histmb,edges= sc.histogramdd(samples.T[:,0:2],bins=round(sc.sqrt(nsamples)/2.))
indxi= sc.argmax(sc.amax(histmb,axis=1))
indxj= sc.argmax(sc.amax(histmb,axis=0))
print "Best-fit, marginalized"
print edges[0][indxi-1], edges[1][indxj-1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi+1], edges[1][indxj+1]
t= edges[1][indxj]
bcost= edges[0][indxi]
mf= m.sqrt(1./m.cos(t)**2.-1.)
b= bcost/m.cos(t)
print b, mf
#Plot
plot.bovy_print(**bovyprintargs)
hist, bins, patchess= plot.bovy_hist(sc.exp(samples.T[:,2]/2.),edgecolor='k',
bins=round(sc.sqrt(nsamples)/2.),
xlabel=r'$\sqrt{V}$',normed=True,
histtype='step')
cumhist= sc.cumsum(hist)/sc.sum(hist)/(bins[1]-bins[0])
ninefive= 0.
ninenine= 0.
foundfive= False
foundnine= False
for ii in range(len(cumhist)):
if cumhist[ii]*(bins[1]-bins[0]) > 0.95 and not foundfive:
ninefive= bins[ii]
foundfive= True
if cumhist[ii]*(bins[1]-bins[0]) > 0.99 and not foundnine:
ninenine= bins[ii]
foundnine= True
print ninefive, ninenine
axvline(ninefive,color='0.5',lw=2.)
axvline(ninenine,color='0.5',lw=2.)
plot.bovy_end_print(plotfilename)
return
#Plot result
plot.bovy_print()
xrange=[0,300]
yrange=[0,700]
plot.bovy_plot(sc.array(xrange),mf*sc.array(xrange)+b,
'k--',xrange=xrange,yrange=yrange,
xlabel=r'$x$',ylabel=r'$y$',zorder=2)
for ii in range(10):
#Random sample
ransample= sc.floor((stats.uniform.rvs()*nsamples))
ransample= samples.T[ransample,0:2]
mf= m.sqrt(1./m.cos(ransample[1])**2.-1.)
b= ransample[0]/m.cos(ransample[1])
bestb= b
bestm= mf
plot.bovy_plot(sc.array(xrange),bestm*sc.array(xrange)+bestb,
overplot=True,color='0.75',zorder=0)
#Add labels
nsamples= samples.shape[1]
for ii in range(nsample):
Pb= 0.
for jj in range(nsamples):
Pb+= Pbad(samples[:,jj],Z[ii,:],ycovar[:,ii,:])
Pb/= nsamples
text(Z[ii,0]+5,Z[ii,1]+5,'%.1f'%Pb,color='0.5',zorder=3)
#Plot the data OMG straight from plot_data.py
data= read_data('data_allerr.dat',True)
ndata= len(data)
#Create the ellipses and the data points
id= sc.zeros(nsample)
x= sc.zeros(nsample)
y= sc.zeros(nsample)
ellipses=[]
ymin, ymax= 0, 0
xmin, xmax= 0,0
jj= 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
continue
id[jj]= data[ii][0]
x[jj]= data[ii][1][0]
y[jj]= data[ii][1][1]
#Calculate the eigenvalues and the rotation angle
ycovar= sc.zeros((2,2))
ycovar[0,0]= data[ii][3]**2.
ycovar[1,1]= data[ii][2]**2.
ycovar[0,1]= data[ii][4]*m.sqrt(ycovar[0,0]*ycovar[1,1])
ycovar[1,0]= ycovar[0,1]
eigs= linalg.eig(ycovar)
angle= m.atan(-eigs[1][0,1]/eigs[1][1,1])/m.pi*180.
thisellipse= Ellipse(sc.array([x[jj],y[jj]]),2*m.sqrt(eigs[0][0]),
2*m.sqrt(eigs[0][1]),angle)
ellipses.append(thisellipse)
if (x[jj]+m.sqrt(ycovar[0,0])) > xmax:
xmax= (x[jj]+m.sqrt(ycovar[0,0]))
if (x[jj]-m.sqrt(ycovar[0,0])) < xmin:
xmin= (x[jj]-m.sqrt(ycovar[0,0]))
if (y[jj]+m.sqrt(ycovar[1,1])) > ymax:
ymax= (y[jj]+m.sqrt(ycovar[1,1]))
if (y[jj]-m.sqrt(ycovar[1,1])) < ymin:
ymin= (y[jj]-m.sqrt(ycovar[1,1]))
jj= jj+1
#Add the error ellipses
ax=gca()
for e in ellipses:
ax.add_artist(e)
e.set_facecolor('none')
ax.plot(x,y,color='k',marker='o',linestyle='None')
plot.bovy_end_print(plotfilename)
def ex17(exclude=sc.array([3]),
plotfilename='ex17.png',
nburn=5000,
nsamples=200000,
parsigma=[1, m.pi / 200., .1],
bovyprintargs={}):
"""ex17: solve exercise 17 by MCMC
Input:
exclude - ID numbers to exclude from the analysis
plotfilename - filename for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
Output:
plot
History:
2010-05-07 - Written - Bovy (NYU)
"""
#Read the data
data = read_data('data_allerr.dat', allerr=True)
ndata = len(data)
nsample = ndata - len(exclude)
#First find the chi-squared solution, which we will use as an
#initial gues
#Put the dat in the appropriate arrays and matrices
Y = sc.zeros(nsample)
X = sc.zeros(nsample)
A = sc.ones((nsample, 2))
C = sc.zeros((nsample, nsample))
Z = sc.zeros((nsample, 2))
yerr = sc.zeros(nsample)
ycovar = sc.zeros((2, nsample, 2)) #Makes the sc.dot easier
jj = 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
pass
else:
Y[jj] = data[ii][1][1]
X[jj] = data[ii][1][0]
Z[jj, 0] = X[jj]
Z[jj, 1] = Y[jj]
A[jj, 1] = data[ii][1][0]
C[jj, jj] = data[ii][2]**2.
yerr[jj] = data[ii][2]
ycovar[0, jj, 0] = data[ii][3]**2.
ycovar[1, jj, 1] = data[ii][2]**2.
ycovar[0, jj, 1] = data[ii][4] * m.sqrt(
ycovar[0, jj, 0] * ycovar[1, jj, 1])
ycovar[1, jj, 0] = ycovar[0, jj, 1]
jj = jj + 1
#Now compute the best fit and the uncertainties
bestfit = sc.dot(linalg.inv(C), Y.T)
bestfit = sc.dot(A.T, bestfit)
bestfitvar = sc.dot(linalg.inv(C), A)
bestfitvar = sc.dot(A.T, bestfitvar)
bestfitvar = linalg.inv(bestfitvar)
bestfit = sc.dot(bestfitvar, bestfit)
#Now sample
inittheta = m.acos(1. / m.sqrt(1. + bestfit[1]**2.))
if bestfit[1] < 0.:
inittheta = m.pi - inittheta
initialguess = sc.array(
[bestfit[0] * m.cos(inittheta), inittheta,
sc.log(1.)]) #(m,b,logV)
#With this initial guess start off the sampling procedure
initialX = objective(initialguess, Z, ycovar)
currentX = initialX
bestX = initialX
bestfit = initialguess
currentguess = initialguess
naccept = 0
samples = []
samples.append(currentguess)
for jj in range(nburn + nsamples):
#Draw a sample from the proposal distribution
newsample = sc.zeros(3)
newsample[0] = currentguess[0] + stats.norm.rvs() * parsigma[0]
newsample[1] = currentguess[1] + stats.norm.rvs() * parsigma[1]
newsample[2] = currentguess[2] + stats.norm.rvs() * parsigma[2]
#Calculate the objective function for the newsample
newX = objective(newsample, Z, ycovar)
#Accept or reject
#Reject with the appropriate probability
u = stats.uniform.rvs()
try:
test = m.exp(newX - currentX)
except OverflowError:
test = 2.
if u < test:
#Accept
currentX = newX
currentguess = newsample
naccept = naccept + 1
if currentX > bestX:
bestfit = currentguess
bestX = currentX
samples.append(currentguess)
if double(naccept) / (nburn + nsamples) < .5 or double(naccept) / (
nburn + nsamples) > .8:
print "Acceptance ratio was " + str(
double(naccept) / (nburn + nsamples))
samples = sc.array(samples).T[:, nburn:-1]
print "Best-fit, overall"
print bestfit, sc.mean(samples[2, :]), sc.median(samples[2, :])
histmb, edges = sc.histogramdd(samples.T[:, 0:2],
bins=round(sc.sqrt(nsamples) / 2.))
indxi = sc.argmax(sc.amax(histmb, axis=1))
indxj = sc.argmax(sc.amax(histmb, axis=0))
print "Best-fit, marginalized"
print edges[0][indxi - 1], edges[1][indxj - 1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi + 1], edges[1][indxj + 1]
t = edges[1][indxj]
bcost = edges[0][indxi]
mf = m.sqrt(1. / m.cos(t)**2. - 1.)
b = bcost / m.cos(t)
print b, mf
#Plot
plot.bovy_print(**bovyprintargs)
hist, bins, patchess = plot.bovy_hist(sc.exp(samples.T[:, 2] / 2.),
edgecolor='k',
bins=round(sc.sqrt(nsamples) / 2.),
xlabel=r'$\sqrt{V}$',
normed=True,
histtype='step')
cumhist = sc.cumsum(hist) / sc.sum(hist) / (bins[1] - bins[0])
ninefive = 0.
ninenine = 0.
foundfive = False
foundnine = False
for ii in range(len(cumhist)):
if cumhist[ii] * (bins[1] - bins[0]) > 0.95 and not foundfive:
ninefive = bins[ii]
foundfive = True
if cumhist[ii] * (bins[1] - bins[0]) > 0.99 and not foundnine:
ninenine = bins[ii]
foundnine = True
print ninefive, ninenine
axvline(ninefive, color='0.5', lw=2.)
axvline(ninenine, color='0.5', lw=2.)
plot.bovy_end_print(plotfilename)
return
#Plot result
plot.bovy_print()
xrange = [0, 300]
yrange = [0, 700]
plot.bovy_plot(sc.array(xrange),
mf * sc.array(xrange) + b,
'k--',
xrange=xrange,
yrange=yrange,
xlabel=r'$x$',
ylabel=r'$y$',
zorder=2)
for ii in range(10):
#Random sample
ransample = sc.floor((stats.uniform.rvs() * nsamples))
ransample = samples.T[ransample, 0:2]
mf = m.sqrt(1. / m.cos(ransample[1])**2. - 1.)
b = ransample[0] / m.cos(ransample[1])
bestb = b
bestm = mf
plot.bovy_plot(sc.array(xrange),
bestm * sc.array(xrange) + bestb,
overplot=True,
color='0.75',
zorder=0)
#Add labels
nsamples = samples.shape[1]
for ii in range(nsample):
Pb = 0.
for jj in range(nsamples):
Pb += Pbad(samples[:, jj], Z[ii, :], ycovar[:, ii, :])
Pb /= nsamples
text(Z[ii, 0] + 5, Z[ii, 1] + 5, '%.1f' % Pb, color='0.5', zorder=3)
#Plot the data OMG straight from plot_data.py
data = read_data('data_allerr.dat', True)
ndata = len(data)
#Create the ellipses and the data points
id = sc.zeros(nsample)
x = sc.zeros(nsample)
y = sc.zeros(nsample)
ellipses = []
ymin, ymax = 0, 0
xmin, xmax = 0, 0
jj = 0
for ii in range(ndata):
if sc.any(exclude == data[ii][0]):
continue
id[jj] = data[ii][0]
x[jj] = data[ii][1][0]
y[jj] = data[ii][1][1]
#Calculate the eigenvalues and the rotation angle
ycovar = sc.zeros((2, 2))
ycovar[0, 0] = data[ii][3]**2.
ycovar[1, 1] = data[ii][2]**2.
ycovar[0, 1] = data[ii][4] * m.sqrt(ycovar[0, 0] * ycovar[1, 1])
ycovar[1, 0] = ycovar[0, 1]
eigs = linalg.eig(ycovar)
angle = m.atan(-eigs[1][0, 1] / eigs[1][1, 1]) / m.pi * 180.
thisellipse = Ellipse(sc.array([x[jj], y[jj]]), 2 * m.sqrt(eigs[0][0]),
2 * m.sqrt(eigs[0][1]), angle)
ellipses.append(thisellipse)
if (x[jj] + m.sqrt(ycovar[0, 0])) > xmax:
xmax = (x[jj] + m.sqrt(ycovar[0, 0]))
if (x[jj] - m.sqrt(ycovar[0, 0])) < xmin:
xmin = (x[jj] - m.sqrt(ycovar[0, 0]))
if (y[jj] + m.sqrt(ycovar[1, 1])) > ymax:
ymax = (y[jj] + m.sqrt(ycovar[1, 1]))
if (y[jj] - m.sqrt(ycovar[1, 1])) < ymin:
ymin = (y[jj] - m.sqrt(ycovar[1, 1]))
jj = jj + 1
#Add the error ellipses
ax = gca()
for e in ellipses:
ax.add_artist(e)
e.set_facecolor('none')
ax.plot(x, y, color='k', marker='o', linestyle='None')
plot.bovy_end_print(plotfilename)
def exMix1(exclude=None,
plotfilenameA='exMix1a.png',
plotfilenameB='exMix1b.png',
plotfilenameC='exMix1c.png',
nburn=20000,
nsamples=1000000,
parsigma=[5, .075, .2, 1, .1],
dsigma=1.,
bovyprintargs={},
sampledata=None):
"""exMix1: solve exercise 5 (mixture model) using MCMC sampling
Input:
exclude - ID numbers to exclude from the analysis (can be None)
plotfilename* - filenames for the output plot
nburn - number of burn-in samples
nsamples - number of samples to take after burn-in
parsigma - proposal distribution width (Gaussian)
dsigma - divide uncertainties by this amount
Output:
plot
History:
2010-04-28 - Written - Bovy (NYU)
"""
sc.random.seed(-1) #In the interest of reproducibility (if that's a word)
#Read the data
data = read_data('data_yerr.dat')
ndata = len(data)
if not exclude == None:
nsample = ndata - len(exclude)
else:
nsample = ndata
#First find the chi-squared solution, which we will use as an
#initial guess
#Put the data in the appropriate arrays and matrices
Y = sc.zeros(nsample)
X = sc.zeros(nsample)
A = sc.ones((nsample, 2))
C = sc.zeros((nsample, nsample))
yerr = sc.zeros(nsample)
jj = 0
for ii in range(ndata):
if not exclude == None and sc.any(exclude == data[ii][0]):
pass
else:
Y[jj] = data[ii][1][1]
X[jj] = data[ii][1][0]
A[jj, 1] = data[ii][1][0]
C[jj, jj] = data[ii][2]**2. / dsigma**2.
yerr[jj] = data[ii][2] / dsigma
jj = jj + 1
brange = [-120, 120]
mrange = [1.5, 3.2]
# This matches the order of the parameters in the "samples" vector
mbrange = [brange, mrange]
if sampledata is None:
sampledata = runSampler(X, Y, A, C, yerr, nburn, nsamples, parsigma,
mbrange)
(histmb, edges, mbsamples, pbhist, pbedges) = sampledata
# Hack -- produce fake Pbad samples from Pbad histogram.
pbsamples = hstack([
array([x] * N)
for x, N in zip((pbedges[:-1] + pbedges[1:]) / 2, pbhist)
])
indxi = sc.argmax(sc.amax(histmb, axis=1))
indxj = sc.argmax(sc.amax(histmb, axis=0))
print "Best-fit, marginalized"
print edges[0][indxi - 1], edges[1][indxj - 1]
print edges[0][indxi], edges[1][indxj]
print edges[0][indxi + 1], edges[1][indxj + 1]
#2D histogram
plot.bovy_print(**bovyprintargs)
levels = special.erf(0.5 * sc.arange(1, 4))
xe = [edges[0][0], edges[0][-1]]
ye = [edges[1][0], edges[1][-1]]
aspect = (xe[1] - xe[0]) / (ye[1] - ye[0])
plot.bovy_dens2d(histmb.T,
origin='lower',
cmap=cm.gist_yarg,
interpolation='nearest',
contours=True,
cntrmass=True,
extent=xe + ye,
levels=levels,
aspect=aspect,
xlabel=r'$b$',
ylabel=r'$m$')
xlim(brange)
ylim(mrange)
plot.bovy_end_print(plotfilenameA)
#Data with MAP line and sampling
plot.bovy_print(**bovyprintargs)
bestb = edges[0][indxi]
bestm = edges[1][indxj]
xrange = [0, 300]
yrange = [0, 700]
plot.bovy_plot(xrange,
bestm * sc.array(xrange) + bestb,
'k-',
xrange=xrange,
yrange=yrange,
xlabel=r'$x$',
ylabel=r'$y$',
zorder=2)
errorbar(X, Y, yerr, marker='o', color='k', linestyle='None', zorder=1)
for m, b in mbsamples:
plot.bovy_plot(xrange,
m * sc.array(xrange) + b,
overplot=True,
xrange=xrange,
yrange=yrange,
xlabel=r'$x$',
ylabel=r'$y$',
color='0.75',
zorder=1)
plot.bovy_end_print(plotfilenameB)
#Pb plot
if not 'text_fontsize' in bovyprintargs:
bovyprintargs['text_fontsize'] = 11
plot.bovy_print(**bovyprintargs)
plot.bovy_hist(pbsamples,
bins=round(sc.sqrt(nsamples) / 5.),
xlabel=r'$P_\mathrm{b}$',
normed=True,
histtype='step',
range=[0, 1],
edgecolor='k')
ylim(0, 4.)
if dsigma == 1.:
plot.bovy_text(r'$\mathrm{using\ correct\ data\ uncertainties}$',
top_right=True)
else:
plot.bovy_text(r'$\mathrm{using\ data\ uncertainties\ /\ 2}$',
top_left=True)
plot.bovy_end_print(plotfilenameC)
return sampledata
