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 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 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)]