def apply_validation_set(feature_combin, feature_mapping, weights):
feature_combin.append(11)
val_set, field_names = import_data("testing_data_split.csv",
";",
True,
columns=feature_combin)
inputs = val_set[:, 0:val_set.shape[1] - 1]
targets = val_set[:, val_set.shape[1] - 1:]
pred_func = construct_feature_mapping_approx(feature_mapping, weights)
pred_ys = pred_func(inputs)
pred_targets = []
for i in range(pred_ys.size):
pred_targets.append(round(pred_ys[i], 0))
targets = np.rot90(targets, 3)
counter = 0
for i in range(len(pred_targets)):
if pred_targets[i] == targets[:, i]:
counter += 1
print("CORRECT: ", counter)
print("WRONG: ", len(pred_targets) - counter)
def main(inputs, targets, scale, best_no_centres, test_fraction=0.20):
# setting a seed to get the same pseudo-random results every time
np.random.seed(30)
print("\n")
std_inputs = standardise(inputs)
train_part, test_part = train_and_test_split(std_inputs.shape[0],
test_fraction)
train_inputs, train_targets, test_inputs, test_targets = train_and_test_partition(
std_inputs, targets, train_part, test_part)
# specifying the centres of the rbf basis functions
# choosing 10% of the data for the centres of the basis functions or the optimal proportion from earlier analyses
centres = train_inputs[
np.random.choice([False, True],
size=train_inputs.shape[0],
p=[1 - best_no_centres, best_no_centres]), :]
print("centres shape = %r" % (centres.shape, ))
# the width (analogous to standard deviation) of the basis functions
# scale of the basis functions from analysis in external_data file
# We consider the basis function widths to be fixed for simplicity
print("scale = %r" % scale)
# creating the feature mapping
feature_mapping = construct_rbf_feature_mapping(centres, scale)
# plotting the basis functions themselves for reference
display_basis_functions(feature_mapping, train_inputs.shape[1])
# alpha and beta define the shape of our curve when we start
# beta is defining the noise precision of our data, as the reciprocal of the target variance
# it is the spread from the highest point (top) of the curve
# it corresponds to additive Gaussian noise of variance, which is beta to the power of -1
beta = np.reciprocal(0.40365849982557295)
# beta = np.reciprocal(np.var(train_targets))
# beta = 100
# higher beta is going to give us higher precision, so less overlap
# as a side note, could also do beta = 1 / np.var(train_targets)
# location of the highest point of the initial curve / prior distribution
# because targets represent quality ranging from 0 to 10
alpha = mode(targets)[0][0]
# alpha = 100
# now applying our feature mapping to the train inputs and constructing the design matrix
design_matrix = feature_mapping(train_inputs)
# the number of features (phis) is the width of this matrix
# it is equal to the number of centres drawn from the train inputs
# the shape[0] is the number of data points I use for training
M = design_matrix.shape[1]
# defining a prior mean and covariance matrix
# they represent our prior belief over the distribution
# our initial estimate of the range of probabilities
m0 = np.zeros(M)
for m in range(len(m0)):
m0[m] = mode(targets)[0][0] # setting to be the mode of targets
# m0[m] = 0
S0 = alpha * np.identity(M)
# diagonal regularisation matrix A to punish over-fitting
# A = alpha * np.identity(M)
# E = 0.5 * m0.T * A * m0
# Zp = regularisation constant
# prior_m0 = np.exp(-E)/Zp
# finding the posterior over weights
# if we have enough data, the posteriors will be the same, no matter the initial parameters
# because they will have been updated according to Bayes' rule
mN, SN = calculate_weights_posterior(design_matrix, train_targets, beta,
m0, S0)
# print("mN = %r" % (mN,))
# the posterior mean (also the MAP) gives the central prediction
mean_approx = construct_feature_mapping_approx(feature_mapping, mN)
# getting MAP and calculating root mean squared errors
train_output = mean_approx(train_inputs)
test_output = mean_approx(test_inputs)
bayesian_mean_train_error = root_mean_squared_error(
train_targets, train_output)
bayesian_mean_test_error = root_mean_squared_error(test_targets,
test_output)
print("Root mean squared errors:")
print("Train error of posterior mean (applying Bayesian inference): %r" %
bayesian_mean_train_error)
print("Test error of posterior mean (applying Bayesian inference): %r" %
bayesian_mean_test_error)
# plotting one input variable on the x axis as an example
fig, ax, lines = plot_function_and_data(std_inputs[:, 10], targets)
# creating data to use for plotting
xs = np.ndarray((101, train_inputs.shape[1]))
for column in range(train_inputs.shape[1]):
column_sample = np.linspace(-5, 5, 101)
column_sample = column_sample.reshape((column_sample.shape[0], ))
xs[:, column] = column_sample
ys = mean_approx(xs)
line, = ax.plot(xs[:, 10], ys, 'r-')
lines.append(line)
ax.set_ylim([0, 10])
# now plotting a number of samples from the posterior
for i in range(20):
weights_sample = np.random.multivariate_normal(mN, SN)
sample_approx = construct_feature_mapping_approx(
feature_mapping, weights_sample)
sample_ys = sample_approx(xs)
line, = ax.plot(xs[:, 10], sample_ys, 'm', linewidth=0.5)
lines.append(line)
ax.legend(lines, ['data', 'mean approx', 'samples'])
# now for the predictive distribution
new_designmtx = feature_mapping(xs)
ys, sigma2Ns = predictive_distribution(new_designmtx, beta, mN, SN)
print("(sigma2Ns**0.5).shape = %r" % ((sigma2Ns**0.5).shape, ))
print("np.sqrt(sigma2Ns).shape = %r" % (np.sqrt(sigma2Ns).shape, ))
print("ys.shape = %r" % (ys.shape, ))
ax.plot(xs[:, 10], ys, 'r', linewidth=3)
lower = ys - np.sqrt(sigma2Ns)
upper = ys + np.sqrt(sigma2Ns)
print("lower.shape = %r" % (lower.shape, ))
print("upper.shape = %r" % (upper.shape, ))
ax.fill_between(xs[:, 10], lower, upper, alpha=0.2, color='r')
ax.set_title('Posterior Mean, Samples, and Predictive Distribution')
ax.set_xlabel('standardised alcohol content')
ax.set_ylabel('p(t|x)')
fig.tight_layout()
fig.savefig("../plots/bayesian/bayesian_rbf.png", fmt="png")
plt.show()
# the predictive distribution
test_design_matrix = feature_mapping(test_inputs)
predictions, prediction_sigma2 = predictive_distribution(
test_design_matrix, beta, mN, SN)
sum_joint_log_probabilities = 0
for n in range(len(predictions)):
sum_joint_log_probabilities += math.log(predictions[n])
sum_joint_log_probabilities *= -1
# joint_log_probabilities = (np.array(test_targets).flatten() - np.array(predictions).flatten())
# print(np.mean(joint_log_probabilities))
print("Error as negative joint log probability: %r" %
sum_joint_log_probabilities)
def main():
"""
This function contains example code that demonstrates how to use the
functions defined in poly_fit_base for fitting polynomial curves to data.
"""
# specify the centres of the rbf basis functions
centres = np.linspace(0, 1, 9)
# the width (analogous to standard deviation) of the basis functions
scale = 0.1
print("centres = %r" % (centres, ))
print("scale = %r" % (scale, ))
# create the feature mapping
feature_mapping = construct_rbf_feature_mapping(centres, scale)
# plot the basis functions themselves for reference
display_basis_functions(feature_mapping)
# sample number of data-points: inputs and targets
N = 9
# define the noise precision of our data
beta = (1. / 0.1)**2
inputs, targets = sample_data(N,
arbitrary_function_1,
noise=np.sqrt(1. / beta),
seed=37)
# now construct the design matrix for the inputs
designmtx = feature_mapping(inputs)
# the number of features is the widht of this matrix
M = designmtx.shape[1]
# define a prior mean and covaraince matrix
m0 = np.zeros(M)
alpha = 100
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(designmtx, targets, beta, m0, S0)
# the posterior mean (also the MAP) gives the central prediction
mean_approx = construct_feature_mapping_approx(feature_mapping, mN)
fig, ax, lines = plot_function_data_and_approximation(
mean_approx, inputs, targets, arbitrary_function_1)
# now plot a number of samples from the posterior
xs = np.linspace(0, 1, 101)
print("mN = %r" % (mN, ))
for i in range(20):
weights_sample = np.random.multivariate_normal(mN, SN)
sample_approx = construct_feature_mapping_approx(
feature_mapping, weights_sample)
sample_ys = sample_approx(xs)
line, = ax.plot(xs, sample_ys, 'm', linewidth=0.5)
lines.append(line)
ax.legend(lines, ['true function', 'data', 'mean approx', 'samples'])
ax.set_xticks([])
ax.set_yticks([])
fig.tight_layout()
fig.savefig("regression_bayesian_rbf.pdf", fmt="pdf")
# now for the predictive distribuiton
new_inputs = np.linspace(0, 1, 51)
new_designmtx = feature_mapping(new_inputs)
ys, sigma2Ns = predictive_distribution(new_designmtx, beta, mN, SN)
print("(sigma2Ns**0.5).shape = %r" % ((sigma2Ns**0.5).shape, ))
print("np.sqrt(sigma2Ns).shape = %r" % (np.sqrt(sigma2Ns).shape, ))
print("ys.shape = %r" % (ys.shape, ))
fig, ax, lines = plot_function_and_data(inputs, targets,
arbitrary_function_1)
ax.plot(new_inputs, ys, 'r', linewidth=3)
lower = ys - np.sqrt(sigma2Ns)
upper = ys + np.sqrt(sigma2Ns)
print("lower.shape = %r" % (lower.shape, ))
print("upper.shape = %r" % (upper.shape, ))
ax.fill_between(new_inputs, lower, upper, alpha=0.2, color='r')
plt.show()
def main():
"""
This function contains example code that demonstrates how to use the
functions defined in poly_fit_base for fitting polynomial curves to data.
"""
# specify the centres of the rbf basis functions
centres = np.linspace(0,1,7)
# the width (analogous to standard deviation) of the basis functions
scale = 0.15
print("centres = %r" % (centres,))
print("scale = %r" % (scale,))
feature_mapping = construct_rbf_feature_mapping(centres,scale)
datamtx = np.linspace(0,1, 51)
designmtx = feature_mapping(datamtx)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
for colid in range(designmtx.shape[1]):
ax.plot(datamtx, designmtx[:,colid])
ax.set_xlim([0,1])
ax.set_xticks([0,1])
ax.set_yticks([0,1])
# choose number of data-points and sample a pair of vectors: the input
# values and the corresponding target values
N = 20
inputs, targets = sample_data(N, arbitrary_function_1, seed=37)
# define the feature mapping for the data
feature_mapping = construct_rbf_feature_mapping(centres,scale)
# now construct the design matrix
designmtx = feature_mapping(inputs)
#
# find the weights that fit the data in a least squares way
weights = ml_weights(designmtx, targets)
# use weights to create a function that takes inputs and returns predictions
# in python, functions can be passed just like any other object
# those who know MATLAB might call this a function handle
rbf_approx = construct_feature_mapping_approx(feature_mapping, weights)
fig, ax, lines = plot_function_data_and_approximation(
rbf_approx, inputs, targets, arbitrary_function_1)
ax.legend(lines, ['true function', 'data', 'linear approx'])
ax.set_xticks([])
ax.set_yticks([])
fig.tight_layout()
fig.savefig("regression_rbf.pdf", fmt="pdf")
# for a single choice of regularisation strength we can plot the
# approximating function
reg_param = 10**-3
reg_weights = regularised_ml_weights(
designmtx, targets, reg_param)
rbf_reg_approx = construct_feature_mapping_approx(feature_mapping, reg_weights)
fig, ax, lines = plot_function_data_and_approximation(
rbf_reg_approx, inputs, targets, arbitrary_function_1)
ax.set_xticks([])
ax.set_yticks([])
fig.tight_layout()
fig.savefig("regression_rbf_basis_functions_reg.pdf", fmt="pdf")
# to find a good regularisation parameter, we can performa a parameter
# search (a naive way to do this is to simply try a sequence of reasonable
# values within a reasonable range.
# sample some training and testing inputs
train_inputs, train_targets = sample_data(N, arbitrary_function_1, seed=37)
# we need to use a different seed for our test data, otherwise some of our
# sampled points will be the same
test_inputs, test_targets = sample_data(100, arbitrary_function_1, seed=82)
# convert the raw inputs into feature vectors (construct design matrices)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# now we're going to evaluate train and test error for a sequence of
# potential regularisation strengths storing the results
reg_params = np.logspace(-5,1)
train_errors = []
test_errors = []
for reg_param in reg_params:
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"$\lambda$", reg_params, train_errors, test_errors)
ax.set_xscale('log')
# we may also be interested in choosing the right number of centres, or
# the right width/scale of the rbf functions.
# Here we vary the width and evaluate the performance
reg_param = 10**-3
scales = np.logspace(-2,0)
train_errors = []
test_errors = []
for scale in scales:
# we must construct the feature mapping anew for each scale
feature_mapping = construct_rbf_feature_mapping(centres,scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"scale", scales, train_errors, test_errors)
ax.set_xscale('log')
# Here we vary the number of centres and evaluate the performance
reg_param = 10**-3
scale = 0.15
n_centres_seq = np.arange(3,20)
train_errors = []
test_errors = []
for n_centres in n_centres_seq:
# we must construct the feature mapping anew for each number of centres
centres = np.linspace(0,1,n_centres)
feature_mapping = construct_rbf_feature_mapping(centres,scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"Num. Centres", n_centres_seq, train_errors, test_errors)
plt.show()
def parameter_search_rbf_cross(inputs, targets, folds,test_error_linear,test_inputs,test_targets,normalize=True):
"""
This function will take as inputs the raw data and targets, the folds for cross validation and the test linear error for plotting
"""
if(normalize):
# normalise inputs (meaning radial basis functions are more helpful)
for i in range(inputs.shape[1]):
inputs[:,i]=(inputs[:,i]-np.mean(inputs[:,i]))/np.std(inputs[:,i])
test_inputs[:,i]=(test_inputs[:,i]-np.mean(test_inputs[:,i]))/np.std(test_inputs[:,i])
N = inputs.shape[0]
# for the centres of the basis functions sample 10% of the data
sample_fractions = np.array([0.05,0.1,0.15,0.2,0.25])
scales = np.logspace(0,4,20 ) # of the basis functions
reg_params = np.logspace(-16,-1, 20) # choices of regularisation strength.
# create empty 3d arrays to store the train and test errors
train_mean_errors = np.empty((sample_fractions.size,scales.size,reg_params.size))
test_mean_errors = np.empty((sample_fractions.size,scales.size,reg_params.size))
best_k=0
best_i=0
best_j=0
test_error_temp=10**100
#loop through the possible centres as a percentage (5%, 10%,15%, 20%, 25%)
for k,sample_fraction in enumerate(sample_fractions):
p = (1-sample_fraction,sample_fraction)
centres = inputs[np.random.choice([False,True], size=N, p=p),:]
# iterate over the scales
for i,scale in enumerate(scales):
# i is the index, scale is the corresponding scale
# we must recreate the feature mapping each time for different scales
feature_mapping = construct_rbf_feature_mapping(centres,scale)
designmtx = feature_mapping(inputs)
# iteratre over the regularisation parameters
for j, reg_param in enumerate(reg_params):
# j is the index, reg_param is the corresponding regularisation
# parameter for train and test the data
train_error, test_error,weights = cv_evaluation_linear_model(designmtx, targets, folds,reg_param=reg_param)
#When we've found a lowest than stores test error value, we store it's indices
if (np.mean(test_error)<test_error_temp):
test_error_temp=np.mean(test_error)
best_k=k
best_i=i
best_j=j
optimal_weights=weights
optimal_feature_mapping=feature_mapping
# store the train and test errors in our 3d matrix
train_mean_errors[k,i,j] = np.mean(train_error)
test_mean_errors[k,i,j] = np.mean(test_error)
print ("The value with the lowest test error at the training stage is:",test_mean_errors[best_k][best_i][best_j])
print("Best joint choice of parameters: sample fractions %.2g scale %.2g and lambda = %.2g" % (sample_fractions[best_k],scales[best_i],reg_params[best_j]))
# now we can plot the error for different scales using the best regularization choice and centres percentage
fig , ax = plot_train_test_errors("scale", scales, train_mean_errors[best_k,:,best_j], test_mean_errors[best_k,:,best_j])
ax.set_xscale('log')
fig.suptitle('RBF regression for the best reg. parameter & centres using cross-validation', fontsize=10)
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:') #graph the linear regression
# ...and the error for different regularisation choices given the best scale choice and centres percentage
fig , ax = plot_train_test_errors("$\lambda$", reg_params, train_mean_errors[best_k,best_i,:], test_mean_errors[best_k,best_i,:])
ax.set_xscale('log')
fig.suptitle('RBF regression for the best scale parameter & centres using cross-validation', fontsize=10)
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:')
# #ax.set_ylim([0,20])
# ...and the error for different centres given the best reg.parameter and the best scale choice
fig , ax = plot_train_test_errors("sample fractions", sample_fractions, train_mean_errors[:,best_i,best_j], test_mean_errors[:,best_i,best_j])
fig.suptitle('RBF regression for the best scale parameter & reg. parameter using cross-validation', fontsize=10)
ax.set_xlim([0.05, 0.25])
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:')
predictive_func=construct_feature_mapping_approx(optimal_feature_mapping, optimal_weights)
final_error=root_mean_squared_error(test_targets,predictive_func(test_inputs))
print("final test error for RBF model:",final_error)
def bayesian_regression_entry_point(data):
"""
This function contains example code that demonstrates how to use the
functions defined in poly_fit_base for fitting polynomial curves to data.
"""
data_targets = data[:, -1]
data = data[:, 0:11]
print(data)
print(data_targets)
for i in range(data.shape[1]):
data[:, i] = (data[:, i] - np.mean(data[:, i])) / np.std(data[:, i])
print("standard deviation is %s" % str(np.std(data, axis=0)))
inputs = data[0:960, :]
targets = data_targets[0:960]
test_inputs = data[1300:1599, :]
test_targets = data_targets[1300:1599]
# specify the centres of the rbf basis functions
N = inputs.shape[0]
centres1 = inputs[np.random.choice([False, True], size=N, p=[0.9, 0.1]), :]
# centres1 = data[10,:]
# centres1 = np.linspace(4,20,10)
print(centres1)
# the width (analogous to standard deviation) of the basis functions
scale = 47
print("centres = %r" % (centres1, ))
print("scale = %r" % (scale, ))
# create the feature mapping
feature_mapping = construct_rbf_feature_mapping(centres1, scale)
# plot the basis functions themselves for reference
# sample number of data-points: inputs and targets
# define the noise precision of our data
beta = (1 / 0.01)**2
# now construct the design matrix for the inputs
designmtx = feature_mapping(inputs)
test_designmtx = feature_mapping(test_inputs)
print(designmtx.shape)
# the number of features is the width of this matrix
M = designmtx.shape[1]
# define a prior mean and covaraince matrix
# m0 = np.random.randn(M)
m0 = np.zeros(M)
print("m0 equals %r" % (m0))
alpha = 50
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(designmtx, targets, beta, m0, S0)
# for i in range(500):
# mN, SN = calculate_weights_posterior(designmtx, targets, beta, mN, SN)
train_error, test_error = train_and_test(designmtx, targets,
test_designmtx, test_targets, mN)
print(train_error, test_error)
# cross-validation
# train_error, test_error = cv_evaluation_linear_model(designmtx, targets, folds, mN)
# print(train_error, test_error, np.mean(train_error), np.mean(test_error))
# the posterior mean (also the MAP) gives the central prediction
mean_approx = construct_feature_mapping_approx(feature_mapping, mN)
fig, ax, lines = plot_function_data_and_approximation(
mean_approx, test_inputs, test_targets)
ax.legend(lines, ['Prediction', 'True value'])
ax.set_xticks([])
ax.set_ylabel("Quality")
fig.suptitle('Prediction vlaue against True value', fontsize=10)
fig.savefig("regression_bayesian_rbf.pdf", fmt="pdf")
# search the optimum alpha for baysian model regression
train_inputs = data[0:960, :]
train_targets = data_targets[0:960]
test_inputs = data[960:1300, :]
test_targets = data_targets[960:1300]
# folds = create_cv_folds(train_inputs.shape[0], num_folds)
alphas = np.logspace(1, 3)
# convert the raw inputs into feature vectors (construct design matrices)
# train_errors = np.empty(alphas.size)
# test_errors = np.empty(alphas.size)
train_errors = []
test_errors = []
for a, alpha in enumerate(alphas):
# we must construct the feature mapping anew for each scale
feature_mapping = construct_rbf_feature_mapping(centres1, scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
beta = (1 / 0.01)**2
M = train_designmtx.shape[1]
# define a prior mean and covaraince matrix
m0 = np.zeros(M)
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(train_designmtx, train_targets,
beta, m0, S0)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(train_designmtx,
train_targets, test_designmtx,
test_targets, mN)
train_errors.append(train_error)
test_errors.append(test_error)
# train_error, test_error = cv_evaluation_linear_model(train_designmtx, train_targets, folds, mN)
# train_errors[a] = np.mean(train_error)
# test_errors[a] = np.mean(test_error)
# plot the results
min_error = np.min(test_errors)
min_error_index = np.argmin(test_errors)
fig, ax = plot_train_test_errors("alpha", alphas, train_errors,
test_errors)
fig.suptitle('Alpha vs Error in Bayesian', fontsize=10)
ax.plot(alphas[min_error_index], min_error, "ro")
# ax.text(scales[min_error_index],min_error,(str(scales[min_error_index]),str(min_error)))
ax.annotate((str(alphas[min_error_index]), str(min_error)),
xy=(alphas[min_error_index], min_error),
xytext=(alphas[min_error_index] + 0.01, min_error + 0.01),
arrowprops=dict(facecolor='green', shrink=0.1))
ax.set_xscale('log')
fig.savefig("alpha.pdf", fmt="pdf")
# search the optimum beta for baysian model regression
train_inputs = data[0:960, :]
train_targets = data_targets[0:960]
test_inputs = data[960:1300, :]
test_targets = data_targets[960:1300]
# folds = create_cv_folds(train_inputs.shape[0], num_folds)
betas = (1. / np.logspace(-3, 1))**2
# convert the raw inputs into feature vectors (construct design matrices)
# train_errors = np.empty(betas.size)
# test_errors = np.empty(betas.size)
train_errors = []
test_errors = []
for b, beta in enumerate(betas):
# we must construct the feature mapping anew for each scale
feature_mapping = construct_rbf_feature_mapping(centres1, scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
M = train_designmtx.shape[1]
# define a prior mean and covaraince matrix
m0 = np.zeros(M)
alpha = 50
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(train_designmtx, train_targets,
beta, m0, S0)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(train_designmtx,
train_targets, test_designmtx,
test_targets, mN)
train_errors.append(train_error)
test_errors.append(test_error)
# train_error, test_error = cv_evaluation_linear_model(train_designmtx, train_targets, folds, mN)
# train_errors[b] = np.mean(train_error)
# test_errors[b] = np.mean(test_error)
# plot the results
min_error = np.min(test_errors)
min_error_index = np.argmin(test_errors)
fig, ax = plot_train_test_errors("beta", betas, train_errors, test_errors)
fig.suptitle('Beta vs Error in Bayesian', fontsize=10)
ax.plot(betas[min_error_index], min_error, "ro")
# ax.text(scales[min_error_index],min_error,(str(scales[min_error_index]),str(min_error)))
ax.annotate((str(betas[min_error_index]), str(min_error)),
xy=(betas[min_error_index], min_error),
xytext=(betas[min_error_index] + 0.05, min_error + 0.05),
arrowprops=dict(facecolor='green', shrink=0.1))
ax.set_xscale('log')
fig.savefig("beta.pdf", fmt="pdf")
# search the optimum scale for baysian model regression
scales = np.logspace(0.5, 3)
train_inputs = data[0:960, :]
train_targets = data_targets[0:960]
test_inputs = data[960:1300, :]
test_targets = data_targets[960:1300]
# folds = create_cv_folds(train_inputs.shape[0], num_folds)
# convert the raw inputs into feature vectors (construct design matrices)
# train_errors = np.empty(scales.size)
# test_errors = np.empty(scales.size)
train_errors = []
test_errors = []
for j, scale in enumerate(scales):
# we must construct the feature mapping anew for each scale
feature_mapping = construct_rbf_feature_mapping(centres1, scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
beta = (1. / 0.01)**2
M = train_designmtx.shape[1]
# define a prior mean and covaraince matrix
m0 = np.zeros(M)
alpha = 50
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(train_designmtx, train_targets,
beta, m0, S0)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(train_designmtx,
train_targets, test_designmtx,
test_targets, mN)
# train_error, test_error = cv_evaluation_linear_model(train_designmtx, train_targets, folds, mN)
# train_errors[j] = np.mean(train_error)
# test_errors[j] = np.mean(test_error)
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
min_error = np.min(test_errors)
min_error_index = np.argmin(test_errors)
fig, ax = plot_train_test_errors("scale", scales, train_errors,
test_errors)
fig.suptitle('Scale vs Error in Bayesian', fontsize=10)
ax.plot(scales[min_error_index], min_error, "ro")
# ax.text(scales[min_error_index],min_error,(str(scales[min_error_index]),str(min_error)))
ax.annotate((str(scales[min_error_index]), str(min_error)),
xy=(scales[min_error_index], min_error),
xytext=(scales[min_error_index] + 0.2, min_error + 0.2),
arrowprops=dict(facecolor='green', shrink=0.1))
ax.set_xscale('log')
fig.savefig("scale.pdf", fmt="pdf")
# Here we vary the number of centres and evaluate the performance
scale = 60
train_inputs = data[0:960, :]
train_targets = data_targets[0:960]
test_inputs = data[960:1300, :]
test_targets = data_targets[960:1300]
# folds = create_cv_folds(train_inputs.shape[0], num_folds)
cent_parts = np.linspace(0.05, 0.8, 16)
# train_errors = np.empty(cent_parts.size)
# test_errors = np.empty(cent_parts.size)
train_errors = []
test_errors = []
N = train_inputs.shape[0]
for n, cent_part in enumerate(cent_parts):
# we must construct the feature mapping anew for each number of centres
centres1 = train_inputs[np.random.choice(
[False, True], size=N, p=[1 - cent_part, cent_part]), :]
feature_mapping = construct_rbf_feature_mapping(centres1, scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# evaluate the test and train error for this regularisation parameter
M = train_designmtx.shape[1]
# define a prior mean and covaraince matrix
m0 = np.zeros(M)
beta = (1. / 0.01)**2
alpha = 50
S0 = alpha * np.identity(M)
# find the posterior over weights
mN, SN = calculate_weights_posterior(train_designmtx, train_targets,
beta, m0, S0)
train_error, test_error = train_and_test(train_designmtx,
train_targets, test_designmtx,
test_targets, mN)
train_errors.append(train_error)
test_errors.append(test_error)
# train_error, test_error = cv_evaluation_linear_model(train_designmtx, train_targets, folds, mN)
# train_errors[n] = np.mean(train_error)
# test_errors[n] = np.mean(test_error)
# plot the results
min_error = np.min(test_errors)
min_error_index = np.argmin(test_errors)
fig, ax = plot_train_test_errors("Num. Centres", cent_parts, train_errors,
test_errors)
fig.suptitle('Num. Centres vs Error in Bayesian', fontsize=10)
ax.plot(cent_parts[min_error_index], min_error, "ro")
ax.text(cent_parts[min_error_index], min_error,
(str(cent_parts[min_error_index]), str(min_error)))
fig.savefig("Num. centres.pdf", fmt="pdf")
plt.show()