def coreTable3_1(c, exp_sig2, exp_beta, var_beta):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 4))
f.suptitle('Bayesian Core Table 3.1', fontsize=14)
log.info("Creating Bayesian Core Table 3.2")
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_data = np.array([c, exp_sig2, exp_beta, var_beta]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.4f}')
col_labels = [
r'c', r'$E^{\pi}(\sigma^{2}|Y,X)$', r'$E^{\pi}(\beta_{0}|Y,X)$',
r'$V^{\pi}(\beta_{0}|Y,X)$'
]
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
cellLoc='center',
loc='center')
tab.set_fontsize(16)
tab.scale(1, 2)
log.log("Saving Figure...")
plt.savefig('Table3_1.png')
log.sucess("Figure created and sucessfully saved")
def mleRegression(x_data, t_data, std):
'''
Completes a MLE regression for the given data
Args:
x_data (np.array): array of input data
y_data (np.array): array of training data
'''
log.log("Calculating MLE regression")
(N,K) = x_data.shape
x = np.zeros((N,K+1)) + 1
x[:,1:] = x_data #Expanding with a colomn of ones for bias terms
#(X'*X)^(-1)*(X')*Y
xtx_i = np.linalg.inv(x.T.dot(x))
beta_hat = xtx_i.dot(x.T).dot(t_data)
#Standard variance
y_star = (t_data - x.dot(beta_hat))
s2 = y_star.dot(y_star)
sig_hat = 1.0/(N-K-1)*s2
std_err = (np.sqrt(np.eye(K+1)*sig_hat*xtx_i)).dot(np.zeros(K+1)+1)
t = (beta_hat/std_err)
p_right = 1.0-sc.stats.t.cdf(np.abs(t),N-K-1)
p_left = sc.stats.t.cdf(-np.abs(t),N-K-1)
p = p_right+p_left
log.sucess("MLE Regression complete")
return beta_hat, std_err, t, p
def readFileData(fileName):
'''
Reads in catapillar data from files
@params:
fileName(string): File name
Returns:
data (np.array): array of data read from file
'''
log.info("Reading in data file")
#Attempt to read text file and extact data into a list
try:
file_object = open(str(fileName), "r").read().splitlines()
data_list = [[float(x.strip()) for x in my_string.split()]
for my_string in file_object]
except OSError as err:
log.error("OS error: {0}".format(err))
return
except IOError as err:
log.error("File read error: {0}".format(err))
return
except:
log.error("Unexpected error:" + sys.exc_info()[0])
return
data = np.asarray(data_list)
log.sucess("Catapillar data successfully read from file")
return data
def coreTable3_11(gibs_info_beta, gibs_noninfo_beta, c):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 5))
f.suptitle('Bayesian Core Table 3.11, C=%d' % (c), fontsize=14)
ax.set_title('Left: Informative, Right: Non-Informative')
log.info("Creating Bayesian Core Table 3.11")
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_data = np.array([gibs_info_beta, gibs_noninfo_beta]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.5f}')
col_labels = [
r'$\gamma_{i}$', r'$\hat{P}^{\pi}(\gamma_{i}=1|Y,X)$',
r'$\hat{P}^{\pi}(\gamma_{i}=1|Y,X)$'
]
gamma_labels = []
for i, val in enumerate(cell_text):
gamma_labels.append(r'$\gamma_{' + str(i) + '}$')
cell_text = util.appendListColumn(cell_text, gamma_labels, 0)
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
colWidths=[0.2, 0.3, 0.3],
cellLoc='center',
loc='center')
tab.set_fontsize(12)
tab.scale(1, 1.75)
log.log("Saving Figure...")
plt.savefig('Table3_11.png')
log.sucess("Figure created and sucessfully saved")
def coreFigure3_2(beta_hat, std_err, t, p):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 6))
f.suptitle('Bayesian Core Figure 3.2', fontsize=14)
log.info("Creating Bayesian Core Figure 3.2")
# hide axes
f.patch.set_visible(False)
ax.axis('off')
ax.axis('tight')
cell_data = np.array([beta_hat, std_err, t, p]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.6f}')
col_labels = ['Estimate', 'Std. Error', 't-value', r'$Pr(>|t|)$']
row_labels = ['intercept']
for i, val in enumerate(beta_hat[1:]):
row_labels.append('XV' + str(i))
tab = ax.table(cellText=cell_text,
rowLabels=row_labels,
colLabels=col_labels,
cellLoc='center',
loc='center',
bbox=[0.15, 0.2, 0.9, 0.7])
tab.set_fontsize(16)
tab.scale(1, 2)
log.log("Saving Figure...")
plt.savefig('Figure3_2.png')
log.sucess("Figure created and sucessfully saved")
def coreTable3_5(hpd):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 5))
f.suptitle('Bayesian Core Table 3.5', fontsize=14)
log.info("Creating Bayesian Core Table 3.5")
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_text0 = util.npArrayToStrList(hpd, '{0:.4f}')
cell_text1 = [
str(row).replace('\'', '') for i, row in enumerate(cell_text0)
]
cell_text = map(list, zip(cell_text1))
col_labels = [r'$\beta_{i}$', 'HPD Interval']
beta_labels = []
for i, val in enumerate(cell_text):
beta_labels.append(r'$\beta_{' + str(i) + '}$')
cell_text = util.appendListColumn(cell_text, beta_labels, 0)
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
colWidths=[0.2, 0.3],
cellLoc='center',
loc='center')
tab.set_fontsize(16)
tab.scale(1, 2)
log.log("Saving Figure...")
plt.savefig('Table3_5.png')
log.sucess("Figure created and sucessfully saved")
def coreTable_B10(exp_beta, var_beta, log_b10, c):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 5))
f.suptitle('Bayes\' Factor, C=' + str(c), fontsize=14)
log.info("Creating Bayes Factor Table")
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_data = np.array([exp_beta, var_beta, log_b10]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.4f}')
col_labels = [
r'$\beta_{i}$', r'$E^{\pi}(\beta_{i}|Y,X)$',
r'$V^{\pi}(\beta_{i}|Y,X)$', r'$log_{10}(BF)$'
]
beta_labels = []
for i, val in enumerate(exp_beta):
beta_labels.append(r'$\beta_{' + str(i) + '}$')
cell_text = util.appendListColumn(cell_text, beta_labels, 0)
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
colWidths=[0.2, 0.3, 0.3, 0.3],
cellLoc='center',
loc='center')
tab.set_fontsize(16)
tab.scale(1, 2)
log.log("Saving Figure...")
plt.savefig('Table_BayesFactor.png')
log.sucess("Figure created and sucessfully saved")
def coreGibbsModelEvidenceTable(model_evid, gibbs_evid, fignum_str, K, c=None):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 7))
if (c != None):
f.suptitle('Bayesian Core Table ' + fignum_str + ', C = %d' % (c),
fontsize=14)
else:
f.suptitle('Bayesian Core Table ' + fignum_str, fontsize=14)
log.info("Creating Bayesian Core Table " + fignum_str)
t_gamma = []
for e0 in gibbs_evid:
#e0 contains [model id, evidence]
t_gamma_np, q = util.getGammaIndexes(K, int(e0[0]))
model_label = str(t_gamma_np.tolist())
t_gamma.append(re.sub('[^0-9 ,]+', '', model_label))
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_data = np.array([model_evid[:, 1], gibbs_evid[:, 1]]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.5f}')
col_labels = [
r'$t_{1}(\gamma)$', r'$\pi(\gamma|Y,X)$', r'$\hat{\pi}(\gamma|Y,X)$'
]
cell_text = util.appendListColumn(cell_text, t_gamma, 0)
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
colWidths=[0.4, 0.3, 0.3],
cellLoc='center',
loc='center')
tab.set_fontsize(12)
tab.scale(1, 1.75)
log.log("Saving Figure...")
plt.savefig('Table' + fignum_str.replace('.', '_') + '.png')
log.sucess("Figure created and sucessfully saved")
def coreFigure3_1(data):
#Set up subplots
f, ax = plt.subplots(2, 5, figsize=(6, 6))
f.suptitle('Bayesian Core Figure 3.1', fontsize=14)
log.info("Creating Bayesian Core Figure 3.1")
n = 0
for (i, j), ax0 in np.ndenumerate(ax):
ax0.plot(data[:, n], data[:, -1], 'o', markersize=3.5)
ax0.set_yscale('log')
# Get rid of the ticks
ax0.set_xticks([])
ax0.set_yticks([])
ax0.minorticks_off()
#Axis label
ax0.set_xlabel(r'$x_' + str(n + 1) + '$')
n += 1
plt.tight_layout(rect=[0, 0, 1.0, 0.93])
log.log("Saving Figure...")
plt.savefig('Figure3_1.png')
log.sucess("Figure created and sucessfully saved")
def priorExpectations(x_data, t_data, beta_hat, beta_tilde, a, b, c):
'''
Computes the expecation and variance of the beta_hats for conjugate priors
@params:
x_data (np.array): array of input data
t_data (np.array): array of training data
beta_hat (np.array): MLE linear regression weights
beta_tilde (np.array): prior hyperparameter
a (float): hyper-parameter
b (float): hyper-parameter
c (float): hyper-parameter
'''
log.log("Computing exp and var for conjugate priors")
log.log("Hyper-parameters a:%.2f b:%.2f c:%.2f"%(a, b, c))
(N,K) = x_data.shape
x = np.zeros((N,K+1)) + 1
x[:,1:] = x_data #Expanding with a colomn of ones for bias terms
y_star = (t_data - x.dot(beta_hat))
s2 = y_star.dot(y_star)
#(X'*X)^(-1)*(X')*Y
xtx_i = np.linalg.inv(x.T.dot(x))
M = np.eye(K+1)/c
c1 = np.linalg.inv(np.linalg.inv(M)+np.linalg.inv(x.T.dot(x)))
c1 = (beta_tilde - beta_hat).T.dot(c1).dot(beta_tilde - beta_hat)
c2 = np.linalg.inv(M+x.T.dot(x))
exp_sig2 = (2*b + s2 + c1)/(N + 2*a - 2)
exp_beta = c2.dot(x.T.dot(x).dot(beta_hat) + M.dot(beta_tilde))
covar_beta = (exp_sig2)*c2
var_beta = (np.eye(K+1)*(covar_beta)).dot(np.zeros(K+1)+1) #Get the variance (diagonal) elements
log.sucess("MLE Regression complete")
return exp_sig2, exp_beta, var_beta
def coreTableExpVar(exp_beta, var_beta, fignum_str, c=None):
#Set up subplots
f, ax = plt.subplots(1, 1, figsize=(6, 5))
if (c != None):
f.suptitle('Bayesian Core Table ' + fignum_str + ', C = %d' % (c),
fontsize=14)
else:
f.suptitle('Bayesian Core Table ' + fignum_str, fontsize=14)
log.info("Creating Bayesian Core Table " + fignum_str)
# hide axes
f.patch.set_visible(False)
ax.axis('off')
cell_data = np.array([exp_beta, var_beta]).T
cell_text = util.npArrayToStrList(cell_data, '{0:.4f}')
col_labels = [
r'$\beta_{i}$', r'$E^{\pi}(\beta_{i}|Y,X)$',
r'$V^{\pi}(\beta_{i}|Y,X)$'
]
beta_labels = []
for i, val in enumerate(exp_beta):
beta_labels.append(r'$\beta_{' + str(i) + '}$')
cell_text = util.appendListColumn(cell_text, beta_labels, 0)
tab = ax.table(cellText=cell_text,
colLabels=col_labels,
colWidths=[0.2, 0.3, 0.3],
cellLoc='center',
loc='center')
tab.set_fontsize(16)
tab.scale(1, 2)
log.log("Saving Figure...")
plt.savefig('Table' + fignum_str.replace('.', '_') + '.png')
log.sucess("Figure created and sucessfully saved")
def gibbsSamplingInformative(x0, y0, beta_tilde, c, T, T0):
'''
Performs Gibb's sampling for Zelner's Informative prior
@params:
x0 (np.array): array of input data
y0 (np.array): array of training data
beta_tilde (np.array): prior hyperparameter
T (int): numder of samples to take
T0 (int): "burn in" samples to ignore
'''
T = int(T)
T0 = int(T0)
(N, K) = x0.shape
x = np.zeros((N, K + 1)) + 1
x[:, 1:] = x0 #Expanding with a colomn of ones for bias terms
zeller_evid = np.zeros(2)
gammas = np.zeros(T)
post_informative = np.zeros((2**K, 2))
beta_evidence = np.zeros((K))
post_informative[:, 0] = range(2**K)
gam_index = random.randint(0, 2**K)
(t_gamma0, q) = util.getGammaIndexes(K, int(gam_index))
log.log("Staring Zeller Informative Prior Gibb's sampling")
log.warning("This will take a while...")
for i in range(T):
if (i % 100 == 0):
log.print_progress(i, T)
for j in range(K):
gam_i0 = gam_index
gam_i1 = gam_index
if (int(gam_index / 2**j) % 2 == 0):
gam_i1 = gam_index + 2**j #With current model parameter
else:
gam_i0 = gam_index - 2**j #With out current model parameter
for i0, gam0 in enumerate([int(gam_i0), int(gam_i1)]):
(t_gamma, q) = util.getGammaIndexes(K, gam0)
x_g = x[:, t_gamma]
bt_g = beta_tilde[t_gamma]
xtxi = np.linalg.inv(x_g.T.dot(x_g))
c0 = y0.T.dot(x_g.dot(xtxi).dot(x_g.T)).dot(y0)
c1 = y0.T.dot(y0)-c/(c+1.0)*c0+1/(c+1.0)*bt_g.T.dot(x_g.T.dot(x_g)).dot(bt_g) -\
2.0/(c+1.0)*y0.T.dot(x_g).dot(bt_g)
zeller_evid[i0] = (c + 1.0)**(-0.5 * (q + 1)) * c1**(-0.5 * N)
zeller_evid_norm = zeller_evid[0] / np.sum(zeller_evid, 0)
if (random.random() < zeller_evid_norm):
gam_index = gam_i0
else:
gam_index = gam_i1
gammas[i] = gam_index
log.print_progress(T, T)
log.sucess("Gibb's Sampling for informative prior complete!")
log.log("Calculating Model Evidence")
#Count all instances of a certain model to get model evidence
for i in range(2**K):
count = np.count_nonzero(gammas[T0:T] == i)
post_informative[i, 1] = count / (T - T0 + 1.0)
post_informative_sort = post_informative[post_informative[:, 1].argsort()
[::-1]]
#Count all instances of a certain certain to get variable evidence
for i in range(K):
count = np.where((gammas[T0:T] / (2**i)).astype(int) %
2 == 1)[0].shape[0]
beta_evidence[i] = count / (T - T0 + 1.0)
return post_informative_sort, beta_evidence
def variableSelection(x0, y0, beta_tilde, c, clim):
'''
Performs variable (model) selection by computing the evidence of each model
@params:
x0 (np.array): array of input data
y0 (np.array): array of training data
beta_tilde (np.array): prior hyperparameter
c (float): prior hyperparameter
clim (int): upper limit of the c summation for the non-informative prior
'''
(N, K) = x0.shape
x = np.zeros((N, K + 1)) + 1
x[:, 1:] = x0 #Expanding with a colomn of ones for bias terms
zeller_posterior = np.zeros((2**K, 2))
post_informative = np.zeros((2**K, 2))
post_noninformative = np.zeros((2**K, 2))
#Save the index of each model in the array for when we max sort it
post_informative[:, 0] = range(2**K)
post_noninformative[:, 0] = range(2**K)
log.log("Starting model variable selection for both priors")
for i in range(2**K):
(gamma_index, q) = util.getGammaIndexes(K, i)
x_g = x[:, gamma_index]
bt_g = beta_tilde[gamma_index]
xtxi = np.linalg.inv(x_g.T.dot(x_g))
c0 = y0.T.dot(x_g.dot(xtxi).dot(x_g.T)).dot(y0)
c1 = y0.T.dot(y0)-c/(c+1.0)*c0+1/(c+1.0)*bt_g.T.dot(x_g.T.dot(x_g)).dot(bt_g) -\
2.0/(c+1.0)*y0.T.dot(x_g).dot(bt_g)
zeller_posterior[i, 0] = (c + 1.0)**(-0.5 * (q + 1)) * c1**(-0.5 * N)
zeller_noninform = 0
for ci in range(1, int(clim)):
c1 = (1.0 / ci) * (ci + 1)**(-0.5 * (q + 1))
zeller_noninform += c1 * (y0.T.dot(y0) - ci /
(ci + 1.0) * c0)**(-0.5 * N)
zeller_posterior[i, 1] = zeller_noninform
log.sucess("Variable selection posteriors calculated")
log.log("Normalizing and sorting Model Evidence")
#Normalize model evidence
post_informative[:, 1] = zeller_posterior[:, 0] / np.sum(
zeller_posterior[:, 0])
post_noninformative[:, 1] = zeller_posterior[:, 1] / np.sum(
zeller_posterior[:, 1])
#Sort array largest to smallest based on model evidence
#https://stackoverflow.com/questions/2828059/sorting-arrays-in-numpy-by-column
post_informative_sort = post_informative[post_informative[:, 1].argsort()
[::-1]]
post_noninformative_sort = post_noninformative[
post_noninformative[:, 1].argsort()[::-1]]
return post_informative_sort, post_noninformative_sort