def PV(data, B, E, f, jacb, jacd, fix):
N = 64 # Number of pts. for best fit curve.
x = np.linspace(B[-1], data.V.min()-1, N) # B[-1] is the last element of B, V0
# Plot the data and the best fit line
figfit = plt.figure(figsize=(9,9))
fig = plt.gcf()
titlestring = 'Fitting Results for '+ data.EOS_type + '-' + str(data.EOS_order)
fig.canvas.set_window_title(titlestring)
plt.subplot(221)
plt.errorbar(x=data.V, y=data.P, xerr=data.Verr, yerr=data.Perr, fmt='ko', label="Data")
plt.plot(x, f(B,x), 'r', linewidth=3, label="Fit")
plt.xlabel(r"Volume, $\AA^{3}$", fontsize=14)
plt.ylabel(r"Pressure, GPa", fontsize=14)
plt.legend(numpoints=1, loc="upper right")
plt.tight_layout()
plt.subplot(222)
K = GEOST_thermo.BM3_V_K(B,x)
plt.plot(x, K, 'r', linewidth=3)
plt.xlabel(r"Volume, $\AA^{3}$", fontsize=14)
plt.ylabel(r"Bulk Modulus, GPa", fontsize=14)
plt.tight_layout()
plt.subplot(223)
fe = GEOST_thermo.EULERIAN_STRAIN(B[-1],x)
dKdP = GEOST_thermo.birch_murnaghan(B,fe).dKdP
plt.plot(x, dKdP, 'r', linewidth=3)
plt.xlabel(r"Volume, $\AA^{3}$", fontsize=14)
plt.ylabel(r"$\partial K / \partial P$", fontsize=14)
plt.tight_layout()
plt.subplot(224)
fe = GEOST_thermo.EULERIAN_STRAIN(B[-1],x)
d2KdP2 = GEOST_thermo.birch_murnaghan(B,fe).d2KdP2
plt.plot(x, d2KdP2, 'r', linewidth=3)
plt.xlabel(r"Volume, $\AA^{3}$", fontsize=14)
plt.ylabel(r"$\partial^{2} K / \partial P^{2}$", fontsize=14)
plt.tight_layout()
return 0
def ON_REFINE_PRESS():
global data
name = thermo.types().names[0] # Which EOS
fix = np.zeros(4, dtype=int) # Fix flags for refinement
beta = np.zeros(4, dtype=float) # Refineable param 1st guesses
fix, beta = GEOST_gui.GUI().FIX_FLAG() # Assign fix flags, and parameter guesses
# Sort out which EOS to work with, derivatives, etc.
if name == app.EOS_names[0]: # Birch-Murnaghan
if app.EOS_order.get() == 2:
f = thermo.BM2_V
jacb = thermo.BM2_V_JACB
jacd = thermo.BM2_V_JACD
elif app.EOS_order.get() == 3:
f = thermo.BM3_V
jacb = thermo.BM3_V_JACB
jacd = thermo.BM3_V_JACD
elif app.EOS_order.get() == 4:
f = thermo.BM4_V
jacb = thermo.BM4_V_JACB
jacd = thermo.BM4_V_JACD
elif name == app.EOS_names[1]: # Natural Strain
if app.EOS_order.get() == 2:
f = thermo.NS2_V
jacb = thermo.NS2_V_JACB
jacd = thermo.NS2_V_JACD
elif app.EOS_order.get() == 3:
f = thermo.NS3_V
jacb = thermo.NS3_V_JACB
jacd = thermo.NS3_V_JACD
elif app.EOS_order.get() == 4:
f = thermo.NS4_V
jacb = thermo.NS4_V_JACB
jacd = thermo.NS4_V_JACD
elif name == app.EOS_names[2]: # Vinet
f = thermo.VINET_V
jacb = thermo.VINET_V_JACB
jacd = thermo.VINET_V_JACD
# ODR SETUP
# model contains information about the ODR functions.
# fcn = model function to refine parameters from
# fjacb = derivatives of fcn w.r.t. parameters
# fjacd = derivatives of fcn w.r.t. independent var.
odr_model = odrpack.Model(fcn=f, fjacb=jacb, fjacd=jacd)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=beta, ifixb=fix, maxit=100)
odr.set_job(fit_type=0, deriv=2, var_calc=0) # Use user-supplied derivatives, but CHECK THEM!!!
# ======== ODR RESULTS =========
output = odr.run() # Output of ODR run
ref_B = output.beta # LSQ best-fit parameters
err = output.sd_beta # Parameter errors (1-sigma)
cov_B = output.cov_beta # Covariance matrix
# The following was implemented for testing purposes to understand
# how ODR was computing the covariance matrix, and how to generate
# correlated random data using the Cholesky decomposition.
# Most of this is now out of date. -JPT feb 2015
'''
# ODR covariance matrix
print "ODR covariance:"
print cov_B
# Homemade covariance matrix from C = inv(JtJ), J = jacobian at optimum params.
# NB: ODR wants the TRANSPOSE of J, so we need to transpose this guy
J = jacb(ref_B,V).T
for i in range(J.shape[0]):
J[i,:] = (-1./Perr[i])*J[i,:]
Jt = np.transpose(J)
cov = np.linalg.inv(np.dot(Jt,J)) # Defn. of the covariance
print 'Homemade cov (see Craigs PDF):'
print cov
cor = np.empty(cov.shape, dtype=float) # Correlation matrix. Need for correlated draws
for i in range(cov.shape[0]):
for j in range(cov.shape[1]):
cor[i,j] = cov[i,j]/np.sqrt(np.dot(cov[i,i],cov[j,j]))
# Spit results to stdout, will remove once it's finalized.
print "cor (See Hughes & Hase pp. 94 eqn. 7.30):"
print cor
print ""
print "Why does ODR cov_beta and homemade cov differ?"
print "Because Hughes & Hase compute cov = inv(A)"
print "Where Aij = 1/2 * d2 (chi_sq)/dBi dBj"
print "Therefore one gets cor = 1/4 * inv(A)"
print "As a result, multiply sqrt of diagonal elements"
print "by 2 to compare with ODR std_beta."
print ""
print "spectrum of cor" # Check that cor is positive definite
u,w = np.linalg.eig(cor)
print u
# Compute the Cholesky of the correlation matrix,
# NB: Scipy returns a lower triangular Cholesky, we want upper triangular.
U = np.linalg.cholesky(cor).T
print "Cholesky(cor)"
print U
print "Best fit params:"
print ref_B
print "Standard error:"
print err
'''
# Print the results and plot if ODR was successful
if output.stopreason[0] == 'Iteration limit reached':
app.LOG_PRINT('ITERATION LIMIT REACHED, LSQ NOT CONVERGED!')
else:
RESULTS(ref_B, err, cov_B, output, f)
PLOTS(ref_B, err, f, jacb, jacd, cov_B)
return 0
def f34test(data, B, fix):
F = np.zeros(2, dtype=float) # pdf
C = np.zeros(2, dtype=float) # p-values
B3 = np.zeros(3, dtype=float) # for the 2nd order EOS, assign K0 and V0 from user guesses
B3[0] = B[0]
B3[1] = B[1]
B3[2] = B[3]
if data.EOS_type == GEOST_thermo.types().names[0]: # Figure out what EOS's to use
# If birch murnaghan
odr_model = odrpack.Model(fcn=GEOST_thermo.BM3_V, fjacb=GEOST_thermo.BM3_V_JACB,
fjacd=GEOST_thermo.BM3_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B3, ifixb=[fix[0],fix[1],fix[3]])
odr.set_job(deriv=3) # Use user-supplied derivatives
output = odr.run() # Output of ODR run
ref_B3 = output.beta # LSQ best-fit parameters
err_B3 = output.sd_beta # Parameter errors (1-sigma)
f3 = GEOST_thermo.BM3_V(ref_B3,data.V)
df3 = data.V.shape[0] - 3
odr_model = odrpack.Model(fcn=GEOST_thermo.BM4_V, fjacb=GEOST_thermo.BM4_V_JACB,
fjacd=GEOST_thermo.BM4_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B, ifixb=fix)
odr.set_job(deriv=3) # Use user-supplied derivatives, but CHECK THEM!!!
output = odr.run() # Output of ODR run
ref_B4 = output.beta # LSQ best-fit parameters
err_B4 = output.sd_beta # Parameter errors (1-sigma)
f4 = GEOST_thermo.BM3_V(ref_B3,data.V)
df4 = data.V.shape[0] - 4
elif data.EOS_type == GEOST_thermo.types().names[1]:
# If Natural strain
odr_model = odrpack.Model(fcn=GEOST_thermo.NS3_V, fjacb=GEOST_thermo.NS3_V_JACB, fjacd=GEOST_thermo.NS3_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B3, ifixb=[fix[0], fix[1], fix[3]])
odr.set_job(deriv=3) # Use user-supplied derivatives, but CHECK THEM!!!
output = odr.run() # Output of ODR run
ref_B3 = output.beta # LSQ best-fit parameters
err_B3 = output.sd_beta # Parameter errors (1-sigma)
f3 = GEOST_thermo.NS3_V(ref_B3,data.V)
df3 = data.V.shape[0] - 3
odr_model = odrpack.Model(fcn=GEOST_thermo.NS4_V, fjacb=GEOST_thermo.NS4_V_JACB,
fjacd=GEOST_thermo.NS4_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B, ifixb=fix)
odr.set_job(deriv=3) # Use user-supplied derivatives, but CHECK THEM!!!
output = odr.run() # Output of ODR run
ref_B4 = output.beta # LSQ best-fit parameters
err_B4 = output.sd_beta # Parameter errors (1-sigma)
f4 = GEOST_thermo.NS4_V(ref_B4,data.V)
df4 = data.V.shape[0] - 4
elif data.EOS_type == GEOST_thermo.types().names[2]:
self.LOG_PRINT("ERROR in PLOTS: Cannot do F-test using Vinet EOS.")
else:
self.LOG_PRINT("ERROR in PLOTS: Unrecognized value of EOS_SELECT")
chisq_3 = float(0)
chisq_4 = float(0)
for i in range(data.P.shape[0]): # Compute the chi-squared for each EOS
chisq_3 += (data.P[i] - f3[i])**2
chisq_4 += (data.P[i] - f4[i])**2
# Compute the F-statistic
Fx34 = (chisq_3 - chisq_4)/(chisq_4/df4)
x1 = np.linspace(0.01, 2*Fx34, 128)
pdf34 = f_dist.pdf(x1, 1, df4)
cdf34 = f_dist.cdf(x1, 1, df4)
# Use Scipy's built-in F-distribution methods
F = f_dist.pdf(Fx34, 1, df4)
# Compute the P-value
C = 1 - f_dist.cdf(Fx34, 1, df4)
# Finally, Make the plot. SHould be a 1 row 2 column plot showing
# f-test for 2nd to 3rd order EOS and 3rd to 4th order EOS.
plt.figure()
fig = plt.gcf()
fig.canvas.set_window_title("F-Test Results")
plt.plot(x1, pdf34, 'r', linewidth=3, alpha=0.8)
plt.plot(x1, cdf34, 'b', linewidth=3, alpha=0.8)
plt.plot(Fx34, f_dist.pdf(Fx34, 1, df4), 'ko')
plt.fill_between(x1, 0, pdf34, where=f_dist.cdf(Fx34, 1, df4)<cdf34, facecolor='red', alpha=0.2)
plt.xticks(fontsize=14)
plt.ylim([-0.01,1.01])
plt.title("Comparing 3rd vs. 4th order EOS", fontsize=14)
plt.xlabel(r"$\left( \chi^{2}_{3} - \chi^{2}_{4} \right) / \left(\chi^{2}_{4} / \nu_{4} \right)$",
fontsize=12)
plt.legend(['PDF', 'CDF', r"$F_{X}$"], loc='upper right', numpoints=1)
plt.text(Fx34, f_dist.pdf(Fx34, 1, df4)+0.05, "p-value= {:8.4f}".format(1-f_dist.cdf(Fx34, 1, df4)), fontsize=14)
plt.tight_layout()
return [chisq_3/df3, chisq_4/df4, Fx34, f_dist.cdf(Fx34, 1, df4)]
def f23test(data, B, fix):
F = np.zeros(2, dtype=float) # pdf
C = np.zeros(2, dtype=float) # p-values
B2 = np.zeros(2, dtype=float) # for the 2nd order EOS, assign K0 and V0 from user guesses
B2[0] = B[0]
B2[1] = B[-1]
if data.EOS_type == GEOST_thermo.types().names[0]: # Figure out what EOS's to use
# If birch murnaghan
odr_model = odrpack.Model(fcn=GEOST_thermo.BM2_V, fjacb=GEOST_thermo.BM2_V_JACB,
fjacd=GEOST_thermo.BM2_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B2, ifixb=[fix[0],fix[-1]])
odr.set_job(deriv=3) # Use user-supplied derivatives
output = odr.run() # Output of ODR run
ref_B2 = output.beta # LSQ best-fit parameters
err_B2 = output.sd_beta # Parameter errors (1-sigma)
f2 = GEOST_thermo.BM2_V(ref_B2,data.V)
df2 = data.V.shape[0] - 2
odr_model = odrpack.Model(fcn=GEOST_thermo.BM3_V, fjacb=GEOST_thermo.BM3_V_JACB,
fjacd=GEOST_thermo.BM3_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B, ifixb=fix)
odr.set_job(deriv=3) # Use user-supplied derivatives, but CHECK THEM!!!
output = odr.run() # Output of ODR run
ref_B3 = output.beta # LSQ best-fit parameters
err_B3 = output.sd_beta # Parameter errors (1-sigma)
f3 = GEOST_thermo.BM3_V(ref_B3,data.V)
df3 = data.V.shape[0] - 3
elif data.EOS_type == GEOST_thermo.types().names[1]:
# If Natural strain
odr_model = odrpack.Model(fcn=GEOST_thermo.NS2_V, fjacb=GEOST_thermo.NS2_V_JACB, fjacd=GEOST_thermo.NS2_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B2, ifixb=[fix[0],fix[-1]])
odr.set_job(deriv=3) # Use user-supplied derivatives
output = odr.run() # Output of ODR run
ref_B2 = output.beta # LSQ best-fit parameters
err_B2 = output.sd_beta # Parameter errors (1-sigma)
f2 = GEOST_thermo.NS2_V(ref_B2,data.V)
df2 = data.V.shape[0] - 2
odr_model = odrpack.Model(fcn=GEOST_thermo.NS3_V, fjacb=GEOST_thermo.NS3_V_JACB,
fjacd=GEOST_thermo.NS3_V_JACD)
odr_data = odrpack.RealData(x=data.V, y=data.P, sx=data.Verr, sy=data.Perr)
odr = odrpack.ODR(odr_data, odr_model, beta0=B, ifixb=fix)
odr.set_job(deriv=3) # Use user-supplied derivatives, but CHECK THEM!!!
output = odr.run() # Output of ODR run
ref_B3 = output.beta # LSQ best-fit parameters
err_B3 = output.sd_beta # Parameter errors (1-sigma)
f3 = GEOST_thermo.NS3_V(ref_B3,data.V)
df3 = data.V.shape[0] - 3
chisq_2 = float(0)
chisq_3 = float(0)
for i in range(data.P.shape[0]): # Compute the chi-squared for each EOS
chisq_2 += (data.P[i] - f2[i])**2
chisq_3 += (data.P[i] - f3[i])**2
# Compute the F-statistic
Fx23 = (chisq_2 - chisq_3)/(chisq_3/df3)
# Use Scipy's built-in F-distribution methods
F = f_dist.pdf(Fx23, 1, df3)
# Compute the P-value
C = 1 - f_dist.cdf(Fx23, 1, df3)
x1 = np.linspace(0.01, 2*Fx23, 128)
pdf23 = f_dist.pdf(x1, 1, df3)
cdf23 = f_dist.cdf(x1, 1, df3)
# Results
plt.figure()
fig = plt.gcf()
fig.canvas.set_window_title("F-Test Results")
plt.plot(x1, pdf23, 'r', linewidth=3, alpha=0.8)
plt.plot(x1, cdf23, 'b', linewidth=3, alpha=0.8)
plt.plot(Fx23, f_dist.pdf(Fx23, 1, df3), 'ko')
plt.fill_between(x1, 0, pdf23, where=f_dist.cdf(Fx23, 1, df3)<cdf23, facecolor='red', alpha=0.2)
plt.xticks(fontsize=14)
plt.ylim([-0.01,1.01])
plt.title("F test for 2nd vs. 3rd order EOS", fontsize=14)
plt.xlabel(r"$\left( \chi^{2}_{2} - \chi^{2}_{3} \right) / \left(\chi^{2}_{3} / \nu_{3} \right)$",
fontsize=12)
plt.ylabel(r"PDF/CDF")
plt.legend(['PDF', 'CDF', r"$F_{X}$"], loc='upper right', numpoints=1)
plt.text(Fx23, f_dist.pdf(Fx23, 1, df3)+0.05, "p-value= {:8.4f}".format(1-f_dist.cdf(Fx23, 1, df3)), fontsize=14)
plt.tight_layout()
return [chisq_2/df2, chisq_3/df3, Fx23, f_dist.cdf(Fx23, 1, df3)]
#!/usr/bin/env python
import numpy as np
import GEOST_thermo as thermo
import matplotlib.pyplot as plt
kB = 8.6173324E-05 #eV/K
B = [1, 100., 300.]
D = thermo.debye(B)
X = np.zeros((2,24), dtype=float)
for i in range(X.shape[1]):
X[0,i] = 100. - float(i)
X[1,i] = 1000.
B = np.zeros(3)
B[0] = 1000.
B[1] = 1.5
B[2] = 1.
TD = np.zeros(X.shape[1])
U = np.zeros(X.shape[1])
Cv = np.zeros(X.shape[1])
for i in range(U.shape[0]):
TD[i] = 1000.*np.exp(B[1] - B[1]*(100./X[0,i])**(-B[2]))
U[i] = D.U(TD[i], X[1,i])
Cv[i] = D.Cv(TD[i], X[1,i])
plt.figure()
plt.subplot(221)
plt.title(r'$\Theta_{D}$')
plt.xlabel("Volume")
VTerr[0,i+1*n] = Verrhit[i+1*n]
VT[0,i+2*n] = Vhit[i+2*n]
VTerr[0,i+2*n] = Verrhit[i+2*n]
VT[1,i+0*n] = 1000. # Hi-T's
VTerr[1,i+0*n] = 10.
VT[1,i+1*n] = 1500.
VTerr[1,i+1*n] = 20.
VT[1,i+2*n] = 2000.
VTerr[1,i+2*n] = 30.
# Generate pressures
A = np.zeros(3, dtype=float)
A[0] = 700. # ThetaD0
A[1] = 1.5 # gD0
A[2] = 1. # q
D = GEOST_thermo.debye([20, ref_Biso[-1], 273.]) # Instantiate debye
Phit = D.P_thermal(A, VT) + np.random.normal(0,0.5,3*n)
print 'Phit', Phit
Perrhit = abs(np.random.normal(0,1,3*n))
Perrhit = np.sort(Perrhit)
print "Thermal EOS refinement using Debye"
print "True values & initial guesses:", A
# ODR SETUP for isothermal EOS
# model contains information about the ODR functions.
# fcn = model function to refine parameters from
# fjacb = derivatives of fcn w.r.t. parameters
# fjacd = derivatives of fcn w.r.t. independent var.
odr_model = odrpack.Model(fcn=D.P_thermal)
odr_data = odrpack.RealData(x=VT, y=Phit, sx=VTerr, sy=Perrhit)
odr = odrpack.ODR(odr_data, odr_model, beta0=A, maxit=100)
A = np.zeros(3) # Debye setup A[0] = 20. # No. atoms/cell A[1] = B[1] # V0 A[2] = 273. # Reference temperature (K) data273 = data(N,B) X273 = np.zeros((2,N), dtype=float) # Volume X273[0,:] = data(N,B).V X273[1,:] = 273. Y273 = data(N,B).P e273 = np.zeros((2,N), dtype=float) # errs e273[0,:] = data(N,B).Verr e273[1,:] = data(N,B).Perr debye273 = thermo.debye(A) C = np.zeros(3) C[0] = 1000. C[1] = 1.5 C[2] = 1. ### # 700 K N = 24 K0 = 100. V0 = 202. B = np.ones(2) # isothermal EOS initial parameters B[0] = K0 # K0 B[1] = V0 # V0